modeling bio electrical

Modeling and Imagingof Bioelectrical ActivityPrinciples and Applications

BIOELECTRIC ENGINEERING

Series Editor: Bin HeUniversity of MinnesotaMinneapoH~ Minnesota

MODELING AND IMAGING OF BIOELECTRICAL ACTIVITYPrinciples and ApplicationsEdited by Bin He

Modeling and Imagingof Bioelectrical ActivityPrinciples and Applications

Edited by

Bin HeUniversity of MinnesotaMinneapolis, Minnesota

Kluwer Academic/ Plenum PublishersNew York, Boston, Dordrecht, London, Moscow

Library of Congress Cataloging-in-Publicat ion Data

Modeling and imaging of bioelectrical activity: principles and applications/edited by Bin He.p. ; cm. - (Bioelectric engineering)

Includes bibliographical references and index.ISBN 0-306-48112-X1. Heart-Electric properties-Mathematical models. 2. Heart-Electric

properties-Computer simulation. 3. Brain-Electric properties-Mathematical models. 4.Brain-Electric properties-Computer simulation. I. He, Bin, 1957- II. Series.

QP112.5.E46M634 2004612'.0142T 011- dc22

2003061963

ISBN 0-306-48112-X

©2004 Kluwer Academic /Plenum Publishers, New York233 Spring Street, New York, New York 10013

http://www.wkap.nl/

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PREFACE

Bioelectrical activity is associated with living excitable tissue. It has been known, owing toefforts of numerous investigators, that bioelectrical activity is closely related to the mechanisms and functions of excitable membranes in living organs such as the heart and thebrain. A better understanding of bioelectrical activity, therefore, will lead to a better understanding of the functions of the heart and the brain as well as the mechanisms underlyingthe bioelectric phenomena.

Bioelectrical activity can be better understood through two common approaches. Thefirst approach is to directly measure bioelectrical activity within the living tissue. A representative example is the direct measurement using microelectrodes or a microelectrodearray. In this direct measurement approach, important characteristics of bioelectrical activity, such as transmembrane potentials and ionic currents, have been recorded to study thebioelectricity of living tissue. Recently, direct measurement of bioelectrical activity has alsobeen made using optical techniques. These electrical and optical techniques have playedan important role in our investigations of the mechanisms of cellular dynamics in the heartand the brain.

The second approach is to noninvasively study bioelectrical activity by means of modeling and imaging. Mathematical and computer models have offered a unique capability ofcorrelating vast experimental observations and exploring the mechanisms underlying experimental data. Modeling also provides a virtual experimental setting, which enables wellcontrolled testing of hypothesis and theory. Based on the modeling of bioelectrical activity,noninvasive imaging approaches have been developed to detect, localize, and image bioelectrical sources that generate clinical measurements such as electrocardiogram (ECG) andelectroencephalogram (EEG). Information obtained from imaging allows for elaborationof the mechanisms and functions of organ systems such as the heart and the brain.

During the past few decades, significant progress has been made in modeling andimaging of bioelectrical activity in the heart and the brain. Most literature, however, hastreated these research efforts in parallel. The similarity arises from the biophysical point ofview that membrane excitation in both cardiac cells and neurons can be treated as volumecurrent sources. The clinical observations of ECG and EEG are the results of volume conduction of currents within a body volume conductor. The difference among bioelectricalactivity originating from different organ systems is primarily due to the different physiological mechanisms underlying the phenomena. From the methodological point of view,

v

vi Preface

therefore, modeling and imaging of bioelectrical activity can be treated within one theoretical framework. Although this book focuses on bioelectric activity of the heart and thebrain, the theory, methodology, and state-of-the-art research that are presented in this bookshould also be applicable to a variety of applications.

The purpose of this book is to provide a state-of-the-art coverage of basic principles,theories , and methods of modeling and imaging of bioelectrical activity with applications tocardiac and neural electrical activity. It is aimed at serving as a reference book for researchersworking in the field of modeling and imaging of bioelectrical activity, as an introductionto investigators who are interested in entering the field or acquiring knowledge about thecurrent state of the field, and as a textbook for graduate students and seniors in a biomedicalengineering, bioengineering, or medical physics curriculum.

The first three chapters deal with the modeling of cellular activity, cell networks,and whole organ for bioelectrical activity in the heart. Chapter I provides a systematicreview of one-cell models and cell network models as applied to cardiac electrophysiology.It illustrates how modeling can help elucidate the mechanisms of cardiac cells and cellnetworks, and increase our understanding of cardiac pathology in three-dimension andwhole heart models . Chapter 2 provides a thorough theoretical treatment of the forwardproblem of bioelectricity, and in particular electrocardiography. Following a review of thetheoretical basis ofequivalent dipole source models and state-of-the-art numerical methodsof computing the electrical potential fields, Chapter 2 discusses the applications of forwardtheory to whole heart modeling and defibrillation. Chapter 3 reviews important issues inwhole heart modeling and its implementation as well as various applications of wholeheart modeling and simulations of cardiac pathologies. Chapter 3 also illustrates importantclinical applications the modeling approach can offer.

The following two chapters review the theory and methods of inverse imaging withapplications to the heart . Chapter 4 provides a systematic treatment of the methods andapplications of heart surface inverse solutions . Many investigations have been made inorder to inversely estimate and reconstruct potential distribution over the epicardium, oractivation sequence, over the heart surface from body surface electrocardiograms. Progresshas also been made to estimate endocardial surface potentials and activation sequence fromcatheter recordings. These approaches and activities are well reviewed in Chapter 4. Chapter5 reviews the recent development in three dimensional electrocardiography tomographicimaging . Recent research shows that, by incorporating a priori information into the inversesolutions, it is possible to estimate three-dimensional distributions of electrophysiologicalcharacteristics such as activation time and transmembrane potentials, or equivalent currentdipole distribution. Inparticular, a whole-heart-model based tomographic imaging approachis introduced, which illustrates the close relationship between modeling and imaging andthe merits of model-based imaging .

Chapter 6 deals with a noninvasive body surface mapping technology - surface Laplacian mapping. Compared with well-established body surface potential mapping , body surface Laplacian mapping has received relatively recent attention in its enhanced capability ofidentifying and mapping spatially separated multiple activities . This chapter also illustratesthat a noninvasive mapping technique can be applied to imaging of bioelectrical activityoriginated from different organ systems, such as the heart and the brain.

The subsequent two chapters treat inverse imaging of the brain from neuromagneticand neuroelectric measurements, as well as functional magnetic resonance imaging (fMRI).

Preface vii

Chapter 7 reviews the forward modeling of magnetoencephalogram (MEG), and neuromagnetic source imaging with a focus on spatial filtering approach. Chapter 8 provides a generalreview of tMR!, linear inverse solutions for EEG and MEG, and multimodal imaging integrating EEG, MEG and tMR!. Along with Chapters 4 and 5, these four chapters are intendedto provide a solid foundation in inverse imaging methods as applied to imaging bioelectricalactivity.

Chapter 9 deals with tissue conductivity, an important parameter that is required inbioelectric inverse solutions. The conductivity parameter is needed in establishing accurateforward models of the body volume conductor and obtaining accurate inverse solutionsusing model-based inverse imaging. As most inverse solutions are derived from noninvasivemeasurements with the assumption of known tissue conductivity distribution, the accuracyof tissue conductivity is crucial in ensuring accurate and robust imaging of bioelectricalactivity. Chapter 9 systematically addresses this issue for various living tissues.

This book is a collective effort by researchers who specialize in the field of modelingand imaging of bioelectrical activity. I am very grateful to them for their contributionsduring their very busy schedules and their patience during this process. I am indebted toAaron Johnson Brian Halm, Shoshana Sternlicht, and Kevin Sequeira of Kluwer AcademicPublisher for their great support during this project. Financial support from the NationalScience Foundation, through grants of NSF CAREER Award BES-9875344, NSF BES0218736 and NSF BES-020l939, is also greatly appreciated.

We hope this book will provide an intellectual resource for your research and/or educational purpose in the fascinating field of modeling and imaging of bioelectrical activity.

Bin HeMinneapolis

CONTENTS

1 FROM CELLULAR ELECTROPHYSIOLOGY TOELECTROCARDIOGRAPHy......................................................... 1Nitish V. Thakor, Vivek Iyer; and Mahesh B. Shenai

1 Introduction 11.1 The One-cell Model 31.1.1 Voltage Gating Ion Channel Kinetics (Hodgkin-HuxleyFormalism) .. 31.1.2 Modeling the Cardiac Action Potential.......... .. 71.1.3 Modeling Pathologic Action Potentials 101.2 Network Models 171.2.1 Cell-cell Coupling and Linear Cable Theory 171.2.2 Multidimensional Networks .. 181.2.3 Reconstruction of the Local Extracellular Electrogram (Forward Problem) 201.2.4 Modeling Pathology in Cellular Networks 231.3 Modeling Pathology in Three-dimensional and Whole Heart Models 291.3.1 Myocardial Ischemia 311.3.2 Preexcitation Studies 311.3.3 Hypertrophic Cardiomyopathy 341.3.4 Drug Integration in Three-dimensional Whole Heart Models 351.3.5 Genetic Integration in Three-dimensional Whole Heart Models. 351.4 Discussion 36

References 38

2 THE FORWARD PROBLEM OF ELECTROCARDIOGRAPHY:THEORETICAL UNDERPINNINGS AND APPLICATIONS................ 43Ramesh M. Gulrajani

2.1 Introduction.................................................................................. 432.2 Dipole Source Representations 442.2.1 Fundamental Equations 442.2.2 The Bidomain Myocardium.............................................................. 462.3 Torso Geometry Representations 532.4 Solution Methodologies for the Forward problem 532.4.1 Surface Methods............................................................................ 54

ix

x Contents

2.4.2 VolumeMethods............................................................................ 582.4.3 Combination Methods 612.5 Applications of the Forward Problem................................................... 612.5.1 Computer Heart Models 622.5.2 Effects of Torso Conductivity Inhomogeneities 702.5.3 Defibrillation................................................................................ 722.6 Future Trends 75

References 75

3 WHOLE HEART MODELING AND COMPUTER SIMULATION 81Darning Wei

3.1 Introduction 813.2 Methodology in 3D Whole Heart Modeling........................................... 823.2.1 Heart-torso Geometry Modeling......................................................... 823.2.2 Inclusion of Specialized Conduction System 833.2.3 Incorporating Rotating Fiber Directions 853.2.4 Action Potentials and Electrophysiologic Properties 893.2.5 Propagation Models........................................................................ 943.2.6 Cardiac Electric Sources and Surface ECG Potentials 1003.3 Computer Simulations and Applications 1033.3.1 Simulation of the Normal Electrocardiogram 1033.3.2 Simulation of ST-T Wavesin Pathologic Conditions 1073.3.3 Simulation of Myocardial Infarction 1083.3.4 Simulation of Pace Mapping 1103.3.5 Spiral Waves-A New Hypothesis of VentricularFibrillation 1103.3.6 Simulation of Antiarrhythmic Drug Effect 1103.4 Discussion 111

References 114

4 HEART SURFACEELECTROCARDIOGRAPHICINVERSE SOLUTIONS 119FredGreensite

4.1 Introduction 1194.1.1 The Rationale for Imaging Cardiac Electrical Function 1204.1.2 A Historical Perspective 1204.1.3 Notation and Conventions 1234.2 The Basic Model and Source Formulations 1234.3 Heart Surface Inverse Problems Methodology 1284.3.1 Solution Nonuniqueness and Instability 1294.3.2 Linear Estimation and Regularization 1324.3.3 Stochastic Processes and Time Series ofInverse Problems 1354.4 Epicardial Potential Imaging 1384.4.1 Statistical Regularization 1384.4.2 Tikhonov Regularization and Its Modifications 1394.4.3 Truncation Schemes 141

Contents xi

4.4.4 Specific Constraints in Regularization 1424.4.5 Nonlinear Regularization Methodology 1434.4.6 An Augmented Source Formulation 1434.4.7 Different Methods for Regularization Parameter Selection 1434.4.8 The Body Surface Laplacian Approach 1444.4.9 Spatiotemporal Regularization 1454.4.10 Recent in Vitro and in Vivo Work 1464.5 Endocardial Potential Imaging 1474.6 Imaging Features of the Action Potential 1494.6.1 Myocardial Activation Imaging 1494.6.2 Imaging Other Features of the Action Potential 1544.7 Discussion 155

References 156

5 THREE-DIMENSIONAL ELECTROCARDIOGRAPHICTOMOGRAPHIC IMAGING 161Bin He

5.1 Introduction ' " 1615.2 Three-Dimensional Myocardial Dipole Source Imaging 1635.2.1 Equivalent Moving Dipole Model 1635.2.2 Equivalent Dipole Distribution Model 1635.2.3 Inverse Estimation of 3D Dipole Distribution 1645.2.4 Numerical Example of 3D Myocardial Dipole Source Imaging 1655.3 Three-Dimensional Myocardial Activation Imaging 1675.3.1 Outline of the Heart-Model based 3D Activation Time Imaging Approach 1675.3.2 Computer Heart Excitation Model 1685.3.3 Preliminary Classification System 1695.3.4 Nonlinear Optimization System 1705.3.5 Computer Simulation 1715.3.6 Discussion 1745.4 Three-Dimensional Myocardial Transmembrane Potential Imaging 1755.5 Discussion 178

References 180

6 BODY SURFACE LAPLACIAN MAPPING OFBIOELECTRIC SOURCES 183Bin He and lie Lian

6.1 Introduction 1836.1.1 High-resolution ECG and EEG 1836.1.2 Biophysical Background of the Surface Laplacian 1846.2 Surface Laplacian Estimation Techniques 1866.2.1 Local Laplacian Estimates 1866.2.2 Global Laplacian Estimates 1886.2.3 Surface Laplacian Based Inverse Problem 1906.3 Surface Laplacian Imaging of Heart Electrical Activity 192

xii Contents

6.3.1 High-resolution Laplacian ECG Mapping 1926.3.2 Performance Evaluation of the Spline Laplacian ECG 1936.3.3 Surface Laplacian Based Epicardial Inverse Problem 1996.4 Surface Laplacian Imaging of Brain Electrical Activity 2006.4.1 High-resolution Laplacian EEG Mapping 2006.4.2 Performance Evaluation of the Spline Laplacian EEG 2006.4.3 Surface Laplacian Based Cortical Imaging 2066.5 Discussion 208

References 209

7 NEUROMAGNETIC SOURCE RECONSTRUCTION ANDINVERSE MODELING 213Kensuke Sekihara and Srikantan S. Nagarajan

7.1 Introduction 2137.2 Brief Summary of Neuromagnetometer Hardware 2147.3 Forward Modeling 2157.3.1 Definitions 2157.3.2 Estimation of the Sensor Lead Field 2167.3.3 Low-rank Signals and Their Properties 2197.4 Spatial Filter Formulation and Non-adaptive Spatial

Filter Techniques 2217.4.1 Spatial Filter Formulation 2217.4.2 Resolution Kernel 2227.4.3 Non-adaptive Spatial Filter 2227.4.4 Noise Gain and Weight Normalization 2257.5 Adaptive Spatial Filter Techniques 2267.5.1 Scalar Minimum-variance-based Beamformer Techniques 2267.5.2 Extension to Eigenspace-projection Beamformer 2277.5.3 Comparison between Minimum-variance and Eigenspace

Beamformer Techniques 2287.5.4 Vector-type Adaptive Spatial Filter 2307.6 Numerical Experiments: Resolution Kernel Comparison between

Adaptive and Non-adaptive Spatial Filters 2327.6.1 Resolution Kernel for the Minimum-norm Spatial Filter 2327.6.2 Resolution Kernel for the Minimum-variance Adaptive

Spatial Filter 2347.7 Numerical Experiments: Evaluation of Adaptive

Beamformer Performance 2357.7.1 Data Generation and Reconstruction Condition 2357.7.2 Results from Minimum-variance Vector Beamformer 2387.7.3 Results from the Vector-extended Borgiotti-Kaplan Beamformer 2387.7.4 Results from the Eigenspace Projected Vector-extendedBorgiotti-Kaplan

Beamformer 2387.8 Application of Adaptive Spatial Filter Technique to MEG Data 2437.8.1 Application to Auditory-somatosensory Combined Response 243

Contents xiii

7.8.2 Application to Somatosensory Response: High-resolutionImaging Experiments 245References 247

8 MULTIMODAL IMAGING FROM NEUROELECTROMAGNETICAND FUNCTIONAL MAGNETIC RESONANCE RECORDINGS 251Fabio Babiloniand Febo Cincotti

8.1 Introduction 2518.2 Generalities on Functional Magnetic Resonance Imaging 2528.2.1 Block-design and Event-Related tMRI 2548.3 Inverse Techniques 2548.3.1 Acquisition of Volume Conductor Geometry 2558.3.2 Dipole Localization Techniques 2568.3.3 Cortical Imaging 2578.3.4 Distributed Linear Inverse Estimation 2598.4 Multimodal Integration of EEG, MEG and tMRI Data 2618.4.1 Visible and Invisible Sources 2618.4.2 Experimental Design and Co-registration Issues 2628.4.3 Integration of EEG and MEG Data 2638.4.4 Functional Hemodynamic Coupling and Inverse Estimation of

Source Activity 2678.5 Discussion 275

References 276

9 THE ELECTRICAL CONDUCTIVITY OF LIVING TISSUE: APARAMETER IN THE BIOELECTRICAL INVERSE PROBLEM 281Maria J. Peters, Jeroen G. Stinstra, and Ibolya Leveles

9.1 Introduction 2819.1.1 Scope of this Chapter 2829.1.2 Ambiguity of the Effective Conductivity 2839.1.3 Measuring the Effective Conductivity 2849.1.4 Temperature Dependence 2879.1.5 Frequency Dependence 2879.2 Models of Human Tissue 2899.2.1 Composites of Human Tissue 2899.2.2 Conductivities of Composites of Human Tissue 2929.2.3 Maxwell's Mixture Equation 2969.2.4 Archie's Law 3009.3 Layered Structures 3079.3.1 The Scalp 3079.3.2 The Skull 3089.3.3 A Layer of Skeletal Muscle 3109.4 Compartments 3119.4.1 Using Implanted Electrodes 3119.4.2 Combining Measurements of the Potential and the Magnetic Field 312

xiv Contents

9.4.3 Estimation of the Equivalent Conductivity using Impedance Tomography 3129.5 Upper and Lower Bounds 3139.5.1 White Matter 3149.5.2 The Fetus 3149.6 Discussion 316

References 316

INDEX 321

1

FROM CELLULARELECTROPHYSIOLOGY TOELECTROCARDIOGRAPHY

by Nitish V. Thakor, Vivek Iyer, and Mahesh B. Shenait Department of Biomedical Engineering, The Johns Hopkins University, 720 Rutland Ave.,

Baltimore MD 21205

INTRODUCTION

Since many cardiac pathologies manifest themselves at the cellular and molecular levels,extrapolation to clinical variables, such as the electrocardiogram (ECG), would prove invaluable to diagnosis and treatment. One ultimate goal of the cardiac modeler is to integratecellular level detail with quantitative properties of the ECG (a property of the whole heart).This magnificent task is not unlike a forest ranger attempting to document each leaf in amassive forest. Both the modeler and ranger need to place fundamental elements in thecontext of a broader landscape. But now, with the recent genome explosion, the modelerneeds to examine the "leaves" at even much greater molecular detail. Fortunately, the rapidexplosion in computational power allows the modeler to span the details of each molecular"leaf" to the "forest" of the whole heart. Thus, cardiac modeling is beginning to span thespectrum from DNA to the ECG, from nucleotide to bedside.

Extending cellular detail to whole-heart electrocardiography requires spanning severallevels of analysis (Figure 1.1). The one-cell model describes an action potential recording from a single cardiac myocyte. By connecting an array of these individual myocytes(via gap junctions), a linear network (cable), two-dimensional (20) network or threedimensional (3D) network (slab) model of action potential propagation can be constructed.The bulk electrophysiological signal recorded from these networks is called the local extracellular electrogram. Subsequently, networks representing tissue diversity and realisticheart geometries can be molded into a whole heart model, and finally, the whole heartmodel can be placed in a torso model replicating lung, cartilage, bone and dermis. At eachlevel, one can reconstruct the salient electric signal (action potential, electrogram, ECG)from the cardiac sources by solving the forward problem of electrophysiology (Chapter 2).

Simply put, cardiac modeling is equivalent to solving a system of non-linear differential(or partial differential) equations, though vigorous reference must be made to numerous

2 N.V. Thakor, V. Iyer, and M. B.Shenai

Cell Network (lD, 2D) Whole

~Action Potential

·SG

·100 -f--~--r------,

~Electrogram

.aECG

FIGURE 1.1. Levels of Analysis. One-cell models include the study of compartments and ion channels andtheir interactions. The basic electrophysiological recording is the action potential. Network models investigate theconnectivity of one-cell units organized in arrays. An electrical measure of bulk network activity is the extracellularelectrogram. Finally, many patches molded into the shape of a whole heart (in addition to torso variables) givesrise to the ECG. See the attached CD for color figure.

laboratory experiments which aim to determine the nature and coefficients of each equation.These equations provide a quantitative measure of each channel, each cell, and networks ofcells. As more experiments are done and data obtained, the model can be made more complexby adding appropriate differential equations to the system. Thus, as more informationabout the cellular networks, tissue structure, heart and torso anatomy are obtained, a betterreconstruction of the ECG becomes possible. Until recently, however, modeling efforts haveprimarily focused on accurately reconstructing normal behavior. But with the accumulatingexperimental history of cardiac disease (such as myocardial ischemia, long-QT syndromeand heart failure), modelers have also begun to revise and extend the quantitative descriptionof these models to include important abnormal behaviors.

This chapter will first focus on the theoretical one-cell equations, which are only solvedin the time domain. Subsequently, the one-cell model will be expanded to represent multipledimensions with the incorporation of partial differential equations in space. At each levelof analysis, the appropriate electrical reconstruction is discussed in the context of relevantpathology to emphasize the usefulness of cardiac modeling.

From Cellular Electrophysiology to Electrocardiography

1.1 THE ONE-CELL MODEL

3

The origins of the one-cell model actually take root from classical neuroscience workconducted by A.L. Hodgkin and A.F. Huxley in 1952 (Hodgkin and Huxley 1952). In famousexperiments conducted on the giant axon of the squid, they were able to derive a quantitativedescription for current flow across the cell membrane, and the resulting action potential (AP).This model mathematically formulated the voltage-dependent "gating" characteristics ofsodium and potassium ion channels in the nerve membrane. Since similar ion channels existin cardiac cells, this Hodgkin-Huxley formalism was applied to model the Purkinje fiberaction potential by McCallister, Noble and Tsien (McAllister et al. 1975).

However, it was determined that the cardiac action potential is considerably morecomplex than the neuronal action potential, presumably due to a larger diversity of ion channels present in the cardiac myocyte, the intercellular connections, and its coupling to muscular contraction. With the addition of the "slow-inward" calcium current in 1976, Beelerand Reuter (Beeler and Reuter 1976) were able to successfully describe the ventricular actionpotential with the characteristic "plateau phase" necessary for proper cardiac contraction.Since then, numerous ion channels and intracellular calcium compartment dynamics havebeen added (DiFrancesco and Noble 1985; Luo and Rudy 1991; Luo and Rudy 1994),making the current AP model considerably more complex and robust. Nevertheless, manyof these membrane channels still follow the same Hodgkin-Huxley formalism, reviewedbelow for the cardiac myocyte. In addition, the cardiac myocyte contains a prominentintracellular calcium compartment-the sarcoplasmic reticulum.

1.1.1 VOLTAGE GATING ION CHANNEL KINETICS (HODGKIN-HUXLEYFORMAUSM)

At the most fundamental level of electrophysiology, an ion (K+, Na", Ca2+) mustcross the membrane via the transmembrane ion channel. Typically, the ion channel is amultidomain transmembrane protein with "gates" that open and close at certain transmembrane voltages, Vm(= Vin - Vout ). The problem, however, is to characterize the opening andclosing of these gates, a process symbolically represented by the following equation:

(1.1)

where k1 and L 1 are the forward and reverse rates of the process, respectively, and n open

and nclosed are the percentage of open or closed channels (which is proportional to channel"concentration"). Thus, by simple rate theory, one would expect the rate ofchannel opening(dn/dr) to equal (note that nclosed = 1 - n open):

(1.2)

The voltage dependence of these ion channels can be understood if these gates aretreated as an "energy-barrier" model, described with Eyring Rate Theory (Eyring et al.1949; Moore and Pearson 1981). Given the concentration of the charged particle on theinside and outside ([Cil, [Co]), an energy barrier (LlGo) located at a relative barrier position

4

V1l1..

N. V. Thakor, V. Iyer, and M. B. Shenai

Extracel lular

K+ Na+

FIGURE 1.2. A Battery-Resistor-Capacitor model of a generic excitable membrane. Ions flow (current) to andfrom the extra- and intracellular domains. across a resistor (or conductance). The membrane has an inherentcapacitance, due to its charge-separating function. The current relates to a transmembrane voltage, Vrn-

(8) along the transmembrane route , and a transmembrane voltage (Vm), Eyring Rate Theorypredicts the forward and reverse rates for ion transfer as:

(

_~ ) ( -( )-. )t FVm )k I = K · e RT • e RT

( "GO) ( ~)k_1 = K· e-7/T . e RT (1.3)

where K is a constant, R is the gas constant, T is the absolute temperature, and z is thevalence of the ion. While, the solution in Eq. (1.3) is an extremely simplified version ofreality, it readily suggests that the forward and reverse rates are voltage-dependent (thusthese rates can be represented as k, (V) and L) (V».

While the "energy-barrier" model predicts voltage-dependence, it does not account forthe time-varying features in opening and closing channels. A model that takes time-varianceinto account was developed by Hodgkin and Huxley in 1952 (Hodgkin and Huxley 1952).The Hodgkin and Huxley model likens the biological membrane to a Battery-ResistorCapacitor (BRC model, Figure 1.2) circuit. The resistor (1/conductance) represents the ionchannel, through which ions pass to create an ionic current (lion). Since the membraneconfines a large amount of negatively-charged protein within the cell, it separates positivelyand negatively charged compartments, thus acting as a capacitor (Cm ). Finally, as ions crossthe membrane and enter (or leave) the intracellular compartment, electrical repellant chargebegins to build that counteracts Vm . The Vm at which a certain ion is at equilibrium (lion = 0)is termed the Nemst potential (Eion), the "battery" which depends on valence , intracellular[C] i and extracellular [C]o ion concentrations:

_ - RT ([Clo)Eion - --In - -zF [Cli

(1.4)

Thus , from simple circuit analysi s of Figure 1.2, the ionic current for a certain ion can be


written as:

5

(1.5)

Where g(V, t) is the voltage-dependent, time-varying ion channel conductance. To determine the dynamics of an individual ion channel, Hodgkin and Huxley assumed that thechannel was a "gate" as described in Eq. (1.2), which can be rewritten solely in terms ofopen probability nopen or simply, n (the forward and reverse rates, k1(V) and k-1(V) arereplaced with a(V) and {3(V), respectively) :

dn(t, V)-d-t- = {3(V)[l - n] - a(V)[n] (1.6)

Eq. (1.6) is a first-order differential equation, which has a particular solution under severalboundary conditions. Following a voltage step LlV(Vm = Vrest + LlV) from the restingmembrane potential, n(t) follows an inverted exponential time course with the followingcharacteristics:

1r(v' )----

n m - a(Vm) + {3(Vm)(1.7)

The quantity of noo(Vm) represents the steady-state proportion of open channels after astep voltage has been applied for a near-infinite amount of time. The variable roo(Vm)characterizes the time the system takes to reach this noo(Vm). Rewriting Eq. (1.6) in termsof the quantities derived in Eq. (1.7), gives a differential equation that describes the timecourse of the open probability for a channel:

dn noo(Vm) - n

dt r(Vm)(1.8)

Using an elegant experimental set-up that applied a voltage-clamp to a giant-squid axon(Cole 1949; Marmont 1949), Hodgkin and Huxley were able to define regression equationsfor noo(V) and rm(V), which represent the gating variables for the potassium channel. Toobtain a suitable fit to experimental data, they arrived at the open channel probability ofn(V, t)4 .Thus, by substituting the open probability into Eq. (1.5), the outward potassiumcurrent can be represented as:

dV 4l« = C- = gK . n(V, t) (V - E K )

dt(1.9)

An analogous equation can be written for the inward sodium current with the addition ofan inactivation mechanism (Figure 1.3). Following the data fitting, the experimental sodiumchannel was represented by Hodgkin and Huxley as three voltage-activated gates similarto the potassium activation gates described by Eq. (1.8). As with the potassium channel,increased membrane voltages stochastically increase the probability that these three gatesopen. Inactivation follows the same kinetics as Eq. (1.8), except that the inactivation gatecloses with increased voltages (Figure lAc). Thus, the sodium response to an appliedvoltage stimulation is biphasic. First, the faster activation gates rapidly open, allowing

K+ Open PrOb3bility Na+ Open Prob 3bility

n 111

n 111

n 111

n

h

Probability aJJ[

1141 IIII" Igates are open

FIGURE 1.3. Idealized ion channels. The potassium channel is generally modeled with four voltage-activationgates. The sodium channel is represented by three rapidly-activating voltage-sensitive gates, with an additionalslowly acting voltage-senstive inactivation gate. The lumped probability that all potassium gates will be open isn", while the probability that the activation and inactivation gates of the sodium channel is m'h.

A) I1aJ(V)

09

-sc

Volts (mV)

B) Illoo(V)

0 9.c;..06....

~

~0 7·

~ 06

~ 0$

I::l.,04

~! 03

§l02

01

.?oo -ee

Volts (mV)

C)~(V)

09 ·

~. .... 0&~~ 07 ,

~~ 06 'Cl... 0$ ·

I::l., 0 4 '

~ 031

C l 02 '

0 \ '

so .?oo -eo se

Volts (mV)

FIGURE 1.4. Activation curves for (A) potassium channels, n; (B) activation curve for sodium channel, m; and(C) inactivation curve for sodium channel, h.

From Cellular Electrophysiology to Electrocardiography 7

inward current to develop. However, with increased voltage, the slower inactivation gateswill close, forcing a decrease in the inward current. There is no conceptual change in thenature of the current equation-the activation gate n is simply replaced with m and h (thoughthese gates all differ quantitatively, m and n both increase with more positive Vm» while thevalue of h decreases with more positive Vm). The sodium current can be represented as:

(1.10)

The biphasic nature of the inward sodium current is crucial to the rapid elicitation of anaction potential and the characteristic biphasic shape of the action potential.

This simplified approach assumes that the cell membrane contains two distinct typesof voltage-gated channels (Na+ and K+) that conducting currents in the opposite direction.With the addition of other inward and outward channels (see later sections), a generalizeddifferential equation can be written:

dV 1dt = C

MUK + INa + Iotherchannels + I stim) (1.11)

where I stim represents a stimulation current (provided from a stimulating lead or adjacentcells), and Iotherchannels is provided via many other channels that vary among cell-types(atrial vs. ventricular cells) and various excitable tissues (heart vs. nervous system). Notethat l«, INa, and other channels are represented by non-linear terms (i.e. n4 and m3h), andare both voltage and time-dependent. Thus, Eq. (1.11) coupled with gating equations foreach channel (Eq. (1.8)), represents a system of non-linear differential equations that mustbe solved using techniques of numerical integration.

1.1.2 MODELING THE CARDIAC ACTION POTENTIAL

While the model of an action potential was originally described for a neuron, themethods were quickly adapted to represent the cardiac action potential. Although thereare slight differences in the quantitative description of the sodium and potassium channelsdescribed above, the cardiac myocyte also exhibits a considerable inward calcium currentthat is responsible for the distinguishable "plateau" phase-which coincides with the muscular contraction in the ventricular myocyte. Additionally, the cardiac myocyte uniquelyexpresses a diverse set of ion channels-which give unique electrophysiological propertiesto different types of heart tissue, in normal and diseased heart function.

Within the heart, there exist a variety of cell types that require different considerationswhen developing a model. Pacemaker cells in the sino-atrial node express channels thatallow an autonomous train of action potentials, while Purkinje fibers represents an efficientconducting system specialized for the fast, uniform excitation of the ventricular myocytes.Ventricular myocytes express the proper proteome to parlay the electrical excitation intoforce generating elements that ultimately produce the cardiac output and blood delivery tothe rest of the body. Even within the ventricle, different models exist for transmural orientation (endocardial cells, middle-myocardial cells (M-cells), and epicardial cells). Modelsfor each type of these cells have been extensively developed and are described in Table 1.1,and the history of these modeling developments is described below.

8 N. V.Thakor, V. Iyer, and M. B. Shena!

TABLE1.1. Classical and Modern Models of Various Cardiac Cell Types

Classical ModelsHodgkin-Huxley (1952)McCallister, Noble, Tsien (1974)Beeler-Reuter (1977)

Modem ModelsDiFrancesco-Noble (1985)Luo-Rudy Phase I (1991)Luo-Rudy Phase II (1994)

Priebe-Beuckelmann (1998)Zhang et al. (2000)

Type

Squid AxonPurkinje CellVentricular Cell

Purkinje CellVentricular CellVentricular Cell

Human Ventricular CellSinoatrial Nodal cells

Novelty

INa,IK

Ix!,IK2

lSi (slow-inward Ica)

INaCa, INaK, ICa-L, Ica-T

Updated INa, IK

Updated INaCa, INaK, Ica-L, Ica-T;Ca-buffering

Updated with human dataUpdated Ca handling

1.1.2.1 Classical modelsofthe cardiac actionpotential

In 1975, McCallister, Noble and Tsien introduced a prototype numeric model for therhythmic "pacemaker activity" of cardiac Purkinje cells by using the voltage-clamp methodto study an outward potassium current, IK2 (McAllister et al. 1975). After repolarizationof the action potential, the deactivation of outward IK2 current allows a net inward currentto produce a diastolic slow wave of depolarization in between action potentials (Figure1.5). As this slow wave of depolarization brings membrane potential towards threshold, IK2

is a prominent current in producing the automaticity of pacemaker cells. Additionally, theMcCallister, Noble and Tsien (M-N-T) model reconstructed the entire action potential, usinga modified Hodgkin-Huxley sodium conductance for the rapid upstroke phases, while usingvoltage-clamp methods to describe an lXI, a generalized plateau and repolarization current.Thus, this landmark model was able to simultaneously describe characteristic pacemakeractivity and rapid conduction velocities associated with Purkinje cells.

However, given the vast diversity of cardiac cell types, the M-N-T model could notdescribe the characteristics of ventricular action potentials-namely, the prominent plateauphase that is crucial for forceful contraction. To this end, Beeler and Reuter developed anumerical model (the B-R model) for the ventricular myocyte in 1977 (Beeler and Reuter1976). This model incorporates an Is component, a slow inward calcium current that isresponsible for the slow depolarization and the prominent plateau phase. This Is currentfollows Hodgkin-Huxley formalism, in that state variables d (activation) and f (inactivation) describe time-varying conductances of the slow inward current. However, unlike otherHodgkin-Huxley ions, the initial low level of intracellular calcium, [Ca2+]j does not remainconstant with the arrival of the transmembrane Is current. In fact, the range of [Ca2+]i canrange from 1 to 10- 7 M, widely altering the Nemst potential, Es. Thus, Beeler and Reutermodeled the intracellular handling of calcium by assuming it flows into the cell and accumulates while being exponentially reduced by an uptake mechanism (in the sarcoplasmicreticulum). At any given state, the flux of [Ca2+]j can be described by:

d[Ca]·T = _10-7 . Is + .07(10- 7 - [Cali) (1.12)

En toto, the model incorporated four major components: the familiar INa current, theIs calcium current, the time-activated outward IXl current and IK1, a time-independent

McCallister, Noble, Ts ien(Purkinje Fiber)

Beeler and Reuter(Ventricular Fiber)

IJOO

(OM<')

'v- --.......( 0 __ • __ ........, . __, _ _ --,-_ --,

- 9)

- x

~ .. . 8

t

-.•.. . - .

t.CJ~ ~.- ,t. ou r ". . -':;I~I ! <,:' , I ~ -- = ._.':f'i!..'l /':~

- {., ; . ' / __' I .,

. 1 .c . ~ - _-:-- .--• • •• .. au II:': ' '-'

,.... . <-

Luo -Rudy(membrane)

Luo-Rudy(sarcoplasmic reticulum)

f I~ I~LCa.t\ J~-<::

...... I • .,. _ ,,_

I.A..dL le.1JSR

~"p.d"""Ia JS:!l

a 1'41 (CSQ. I

"c ~~"~TlJlNII'Ll ~-...

" lII)'opIMm

E;]~CMON).., J~----

F ik=§;I.' ot-ote I. .. I CI "'"

FIGURE 1.5. A comparison of classical (top) and modern models (bottom).

outward potassium current. With this model, Beeler and Reuter began to predict pathological phenomena, including determinants of action potential duration, and oscilliatorybehavior in ventricular cells.

1.1.2.2 Modern models ofcardiac action potentials

While modem models utilize many of the concepts introduced in the classical models described above, current models now incorporate a larger repertoire of ion channels,

10 N. V.Thakor, V. Iyer, and M. B. Shenai

a richer history of experimentation, and complex intracellular and sarcoplasmic calciumhandling. In addition, improved computational power and numerical techniques can solvehefty systems ofdifferential equations, allowing a more precise description of cellular electrophysiology (one-cell) and the interaction of many cells (network models). As a result, thefocus of modeling has shifted from describing normal behavior of myocytes to describingpathological phenomena.

In 1985, DiFrancesco and Noble described an improved model of the Purkinje actionpotential (D-N model) (DiFrancesco and Noble 1985), that included the traditional ionchannel formulation, along with improved assumptions on calcium channels (L-type andT-Type) and intracellular calcium handling. Nevertheless, the experimental recording technique at the time was rather limited, and could not account for important arrhythmogenicphenomena. In 1991, Luo and Rudy published an updated version of the D-N model thatincluded more recent experimental data for the sodium and potassium currents, but omittedthe B-R formation of the inward calcium current (lsi), citing a lack of single-channel andone-cell experimental history (Luo and Rudy 1991). But in 1994, Luo and Rudy publishedan updated model which comprehensively updated the D-N description of the sarcolemmaL-type Calcium channel (lea.d, the sarcolemma Na+ ICaH exchanger, the sarcolemmaNa/K pump, the sarcoplasmic Ca-ATPase, and CaH -induced CaH release. Processes notdescribed in the D-N model were also added, such as the buffering of CaH in the myoplasm, and a non-specific Calcium current (Luo and Rudy 1994). The model consists ofthree compartments-the myoplasm, network sarcoplasmic reticulum, and the junctionalsarcoplasmic reticulum. This enhanced model has provided a breakthrough in simulationsof excitation-contraction (E-C) coupling and reentrant mechanisms of arrhythmogenesis.In 1998, the Luo-Rudy model was updated by substituting animal data in favor of recenthuman data (Priebe and Beuckelmann 1998).

While the Luo-Rudy model describes ventricular action potentials, several other models exist for other cardiac tissues. Recently, Zhang et at. have incorporated recent sinoatrial data to formulate a modem model of various sinoatrial nodal cells (central nodal andperipheral nodal cells) (Zhang et at. 2000). Lindblad et at. have used existing biophysical data to simulate a family of action potentials recorded in rabbit atria (Lindblad et at.1996).

1.1.3 MODELING PATHOLOGIC ACTION POTENTIALS

Currently, there is a comprehensive understanding of basic ionic mechanisms andtheir behavior in normal cardiac cells. The various cardiac models listed in Table 1.1 havewidely contributed to this theoretical understanding. However, less is accepted about howimpairments of these ionic mechanisms ultimately predict or provoke gross events, such asinfarction and/or arrhythmogenesis. Among many others, two areas of cardiac pathology,myocardial ischemia and long-Q'I' syndromes (LQTS), are now the focus of intense modeling research. These studies have contributed not only to a theoretical understanding of thediseases, but also to electrocardiographic detection and appropriate pharmaceutical intervention. Though both myocardial ischemia and long-Q'I' syndromes can lead to fatal arrhythmias (Wit and Janse 1993; El-Sherifet al. 1996), myocardial ischemia does so by shorteningthe action potential duration (APD) while LQTS induces arrhythmias by lengthening theAPD.


1. Acidosis2. Plateau EADs

1.Phase-3 EADs2. DADs

1.Acidosis2. LQT33. Rotors

1. Phase-3 EADs2. DADs

FIGURE 1.6. Cellular phenomena associated with myocyte ion channel currents. Various ion channels have beenimplicated in pathologic phenomena.

To study impaired cells, one must modify existing models of normal behavior. Thesemodifications may be achieved by: (1) adding novel channels to the existing repertoire ofknown membrane channels; (2) altering the quantitative dynamics of known channels-forexample by altering ionic concentrations or pH; or (3) a combination of new channels andaltered channel dynamics. Figure 1.6 summarizes the various cellular phenomena associatedwith myocyte ion channels.

1.1.3.1 Myocardial ischemia

Myocardial ischemia results from a withdrawal of oxygen from myocardial tissue (dueto inefficient or absent perfusion), resulting in disturbances to aerobic respiration and ATPproduction. Alterations in intracellular ATP ([ATP]i), can alter the activity of membranepumps, and thus the distribution of critical ions (Na+ and K+) that are largely responsiblefor the electrophysiological characteristics of myocardium and proper action potential propagation. Thus, ischemia develops at the cellular level, when the amount of oxygen (Poz) in

12 N. V.Thakor, V.Iyer, and M. B. Shena!

the vicinity of the mitochondria fails to meet the demand of rephosphorylation in the Kreb'scycle (Factor and Bache 1998). Myocardial ischemia has at least four cellular sequellae: (1)hyperkalemia, or an increase in extracellular potassium [K+]o; (2) acidosis , or a decrease incell-medium pH (intracellular) or interstitial space pH (extracellular); (3) anoxia, or oxygenwithdrawal that results in a decrease in [ATP]i; and (4) decoupling of cells . The effects ofthese individual manifestations on excitability have been widely reported, experimentally(Kagiyama et al. 1982; Kodama et al. 1984; Kleber et al. 1986; Weiss et al. 1992; Yan et al.1993) and theoretically (Ferrero et al. 1996; Shaw and Rudy 1997; Shaw and Rudy 1997).

Hyperkalemia

As the intracellular stores of ATP diminish due to reduced aerobic respiration, Na+/K+pumps responsible for ion distribution also demonstrate reduced activity. Though normallythis pump acts to relocate sodium out of the cell and potassium into the cell, a lethargicpump performs this process inefficiently. Thus, there is an extracellular accumulation ofpotassium, referred to as "hyperkalemia". The electrophysiological consequences of hyperkalemia are two-fold. First, the upstroke velocity (dV/dtmax) of the action potential can bediminished. With the increased extracellular potassium, the resting membrane potential(RMP) becomes more positive, increasing sodium channel inactivation and reducing theinward sodium current (Weidmann 1955; Morena et al. 1980). This dominating effect issomewhat mitigated by the increased RMP being closer to the action potential threshold.Thus, moderate increases (5.4 mmol to 7.5 mmol) in potassium (c-7.5 mmol) may actuallyincrease upstroke velocity (this is termed "superconduction"), while large increases in extracellular potassium begin to inactivate the sodium current and decrease upstroke velocity.Even larger increases can prevent the upstroke entirely and produce conduction block (Witand lanse 1993; Cascio et al. 1995). Hyperkalemia can also significantly decrease the APD.This effect is due to exaggerated outward potassium current late in the action potential thatis able to overcome the inward calcium current relatively earlier, reducing the APD (Figure1.7). Both effects of hyperkalemia, APD shortening and conduction depression, have beensuccessfully modeled by Shaw and Rudy (Shaw and Rudy 1997; Shaw and Rudy 1997).

41.7

o

-65

(mV)

-100o(msec) 120

FIGURE 1.7. Action potential simulations with varying degrees of [K+]o. Increasing extracellular potassiu m(hyperkalemia) results in decreasing APD. (From Shaw and Rudy 1997; used by permission)


Acidosis

13

In the absence of aerobic respiration, alternate pathways that attempt to maintain energy production result in the formation of acidic species , thus initially creating intracellularacidosis. An increase in the intracellular proton concentration leads to proton extrusioninto the extracellular space-resulting in extacellular acidosis. Changes in acidity can subtly change three-dimensional protein structures, including ion channels embedded in thesarcolemma. Most notably, the sodium channel experiences a decrease in maximum conductance (gNu)with extracellular acidosis . Intracellular acidosis reduces the availability ofthe L-type calcium channel (described below) . These changes considerably affect upstrokevelocity (Shaw and Rudy 1997).

Hypoxia

The accumulation of intracellular ADP (at the expense of intracellular ATP) activatesa special K-ATP channel in the sarcolemma, described by the following equation:

where !ATP is represented by:

! ATP = ( [AT PJ . )H1+ I

35.8 + l7.9[ADPJ/ 56

(1.13)

(1.14)

where H is the Hill coefficient that decreases exponentially with [ADPli- From Eq. (14) ,a decrease in the [ATPli/[ADPJi leads to an increase in the ! ATP coefficient and the outward lx_ATP. This outward potassium current supplements the normal potassium current,enhancing the total outward current and drastically reducing the APD (Ferrero et al. 1996)(Figure 1.8).

Incidentally, the power of computer modeling was used to settle the controversy surrounding APD shortening and the role of the K-ATP channel. Because experiments showed

50

lOOms

FIGURE 1.8. AP simulations with varying degrees of fATP. An increase in the fraction of open K-ATP channelsresults in profound APD shortening. (From Ferrero et al . 1996; used by permission)

14

A.

B.

c.


Phase3 EAD...-----

FIGURE 1.9. Classifications of afterdepolarizations: (A) Plateau EAD- an oscillation during the Phase 2 plateau;(B) Phase 3 EAD; and (C) DAD- an oscillation after complete repolarization.

that anoxia induced a 40-60% shortening of the APD while K-ATP channels demonstratedonly a 1% activation, many investigators felt that the K-ATP channel was not a majorconducive factor to APD shortening. However, several investigators (Ferrero et al. 1996;Shaw and Rudy 1997) were able to quantitatively model the K-ATP channel with conductance [srr. being dependent on the amount of intracellular ATP. By adding this individualchannel to the model, they were able to show that even a .4% channel activation can actuallyshorten the APD by 50%. Thus, this channel has been implicated as the major factor in APDshortening and thus may be a crucial factor in arrhythmogenesis.

1.1.3.2 Early afterdepolarizations (EADs) and delayed afterdepolarizations (DADs)

Early afterdepolarizations (EADs) and delayed afterdepolarizations (DADs) (Figure1.9) are single-cell arrhythmogenic triggering events, typically depending on Ca2+ alterations and the interactions between the intracellular and sarcoplasmic compartments withinthe myocyte (Marban et al. 1986; Priori and Corr 1990). Because of the dependence onintracellular calcium, which can accumulate or depreciate from beat-to-beat, mulitiple beatmodels (paced at a basic cycle length) are required to reach a steady state. Simply stated,


afterdepolarizations are notches of depolarization that occur after the typical action potentialupstroke. By definition, the EADs occur before the completion of repolarization, whereasDADs occur after the completion of repolarization.

EADs may occur during the plateau-phase (Figure 1.9a) of the action potential (plateauEADs) (Marban et al. 1986; Priori and Corr 1990) or during the phase-3 repolarizationdownstroke of the action potential (phase-3 EADs, Figure 1.9b). The plateau EAD is highlydependent on the L-type Ca2+ current (also involved in acidosis) (January and Riddle1989), which is a non-specific cation channel permeable to Ca2+, Na+, and K+. Briefly, theformation of this current is the sum of ICa, ICa,K, ICa,Na, each of which are modulated by a[Ca2+]-dependent factor (Luo and Rudy 1994; Luo and Rudy 1994):

1fca=------=-

[ 2+.]21+ rCa ]1Km,ca

(1.15)

where Km,ca is a half-maximal constant, equivalent to .6 p.mol/L, As the intracellularcalcium concentration increases.ji-, and the L-type current decrease monotonically. Additionally, the channel is controlled by voltage dependent f-gate. During the plateau phase,when intracellular Ca2+ is elevated, the L-type Ca2+ channel is relatively inactive due toa low [c« However, due to rapid intracellular Ca2+ recovery (a phenomenon associatedwith long-duration action potentialsj.ji-, and inward ICa are elevated, resulting in a netdepolarization during an otherwise repolarizing phase.

Unlike the plateau EAD, phase-3 EADs and DADs (Figure 1.9c) are dependent onNa+-Ca2+ exchanger and Ins(Ca), the non-specific calcium current. Like the ICa(L) current,the Ins(Ca) is permeable to K+ and Na"-however, an increase in [Ca2+]i increases theIns(Ca). Thus, spontaneous Ca2+ release by the SR into the intracellular compartment furtherincreases the inward current, producing either DADs or EADs (Stem et al. 1988).

Both EADs and DADs produce links between cellular conditions and arrythmogenesis.For example, simulation studies have reproduced experimental studies demonstrating thatEADs can generate ectopic activity (Saiz et al. 1996). In addition to one-cell studies,afterdepolarizations are studied in the context of linear networks (see Section 1.2) EADshave also been implicated in the long-QT syndrome, as the triggering event to a specifictype of polymorphic reentrant tachycardia, or Torsades de Pointes (TdP) (El-Sherif andTuritto 1999; Viswanathan and Rudy 1999).

1.1.3.3 Long-QT syndrome

While myocardial ischemia results in APD shortening, other myocardial pathologiessuch as Long-QT syndrome may result in APD lengthening. The etiologies of LQTS arediverse, ranging from various genetic deficiencies at distinct loci, to acquired and iatrogeniccauses. Long-QT syndrome is characterized by a prolongation of the QT-interval in theECG, presumably due to structurally-deformed potassium and sodium channels. Impairedoutward potassium flow would tend to delay the repolarization phase (Phase 3) and increasethe duration of the action potential. Ultimately, this predisposes the patient to fatal cardiacarrythmias and the unfortunate sequel-sudden cardiac death, even at early ages.

Currently, investigations of LQTS are a prototype for blending human genomics withadvanced cardiac modeling. In the early 1990's, a considerable flurry of molecular genetics

16 N. V. Thaker, V. Iyer,and M. B.Shenai

o 50 100 1SO 200

time (ms)

IKr Reduction (LQT2)

Endocardial

GKs:GKr • 15:1

Eplcardl I

GKs:GKr = 24:1

Mldmyocardlal

GKs:GKr =7:1

-100 , i I

o 50 100 1SO 200

timo (ms)

0S'§.

·so

·100

B so

o-S'§.

·50

·100

C so

0S'§.

·50

IKs Reduction (LQT1)

A so .

CL = 300 ms : 100% Reduction

FIGURE 1.10. The effect of K, (LQTI) and K, (LQT2) mutations on action potential shape and duration ofisolated cells. Each simulation represents a 100% block of the respective current reduction. (From Viswanathanand Rudy, 2000; used by permission)

studies linked LQTS populations to mutations in three putative genes located on chromosomes 3, 6 and 11 (LQT1, LQT2, LQT3). The LQT1 and LQT2 genes represent an IKr

current (potassium delayed rectifier) and IKs current (potassium slow delayed rectifier), respectively (Barhanin et al. 1996; Wang et al. 1996). The LQT3 gene represents an enhanced(incomplete inactivation) late sodium current. From these ground-breaking bench discoveries, several modeling studies were able to place molecular genetics in the context of comprehensive myocyte electrophysiology. For example, Viswanathan and Rudy were able toshow that different myocardial cells (epicardial, mid-myocardium (M-cell), and endocardialcells) respond to LQT gene defects with differing amounts of APD lengthening, producinga transmural heterogeneity ripe for the formation of EADs (Figure 1.10) (Viswanathan andRudy 1999; Viswanathan and Rudy 2000). They modeled LQT1 and LQT2 by reducingthe density ofIKs and IKr channels (thereby reducing the maximal channel conductance per


cell ). LQT3 was simulated by a right shift in the steady-state inactivation curve, such thatthe hand j gates demonstrated incomplete inactivation, resulting in a late sodium current.

Truly, long-QT syndromes are demonstrating the cutting-edge interaction betweenmolecular genetics and advanced computer modeling. While the molecular techniques havebeen instrumental in identifying the particular channelopathy, computer models have beensuccessful in placing the channelopathy in the context of other channels and the whole cell ,producing a quantitative understanding of the disease.

1.2 NETWORK MODELS

1.2.1 CELL-CELL COUPLING AND LINEAR CABLE THEORY

While the previous section treats the cardiac myocyte as an isolated element, it actuallyexists in a densely interconnected network with other myocytes. This brings up the issue ofhow current spreads from one excitable myocyte to a neighboring myocyte .The predominantmodel of current spread among excitable elements is termed cable theory (Miller andGeselowitz 1978; Spach et at. 1981; Plonsey and Barr 1986; Malmivuo and Plonsey 1995).Cable theory provides the crucial link from one-cell to many cells . In its simplest form, itpresents a set of myocytes lined up next to each other and connected by gap junctionsforming a linear "cable" of excitable elements (Figure 1.11).

Cable theory assumes the existence of two compartments, subdivided into differentialcompartments, dx. Each differential compartment is connected to its adjacent compartmentby resistances. Each transmembrane resistance, rm, repre sents a pathway for transmembranecurrent, either passive or active (action potential producing elements like ion channels). Inthe extracellular space, r, repre sents the resistance between two extracellular differentialelements. The monodomain model assumes that this resistance is negligibly small (=0),whereas bidomain models assume re to be non-zero. Likewise, r, represents the resistancebetween two differential intracellular elements. In bulk, r i accounts for both the inherentcytoplasmic resistance and the cell-to-cell resistance (gap junctions).

E: tracellular

FIGURE l.11. A one-dimensional cable representing intracellular resistances, transmembrane resistances, andextracellular resistances.The transmembrane resistance can be replaced with an active component, such as avoltage-sensitive ion channel.

18 N. V. Thakor, V. Iyer, and M. B. Shenai

Given this description, current follows a differential equation that contains both spatial(in the x direction) and temporal derivatives (Plonsey 1969; Malmivuo and Plonsey 1995):

(1.16)

The right hand of this equation can be derived from Eq. (1.11), where the membrane currentis summed from all the ion channels. Numerical methods can then solve this differentialequation to provide a spatial profile of a propagating wavefront.

1.2.2 MULTIDIMENSIONAL NETWORKS

A thin myocardial slab is modeled by extending a linear cable conductor model intotwo dimensions (Figure 1.12), assuming a highly conducting external medium. The modelin Figure 1.12 is a monodomain, and does not represent the separate intra- and extracellularresistivities. The propagation across the slab is calculated from a set of the partial differentialequations, derived from continuity and conservation of current at each node (Barr andPlonsey 1984):

(1.17)

where a is the intracellular conductivity tensor, f3 is the surface area to tissue volume ratio,CM is the membrane capacitance per unit area, and lion (A/m2) is the total membrane ioniccurrent. In two dimensions, the total current flowing into a cell (from adjacent nodes) derivesfrom Ohm's Law and must equal the total membrane current:

(V . - \/:. i.. \/:. - V+ 1 · \/:. - V . 1 \/:. - V+ 1 )_ l,j 1- ,j + l,j 1 ,j + l,j l,j- + l,j l,j = 1

MRx Rx Ry Ry(1.18)

where 1M is the membrane current in amperes, and R is the bulk resistance of the clusterarea. The terms on the left represent current that flows from adjacent cells via intercellularconnections, or gap junctions.

Since the myocardium represents a functional syncitium, a single myocyte has properties very similar to a cluster ofmyocytes. Clustering cells (Figure 1.12) allows one to modellarger areas of tissue while minimizing the computational load. In a particular direction(longitudinal or transverse), the bulk resistances of each cluster, R, and Ry , are net resistances derived from the lumped combination of intracellular resistivities and gap-junctionalresistances in series, and can be calculated by:

~xRx = - - -

ax· h· ~y

~yRy = - - - v-: h· ~x

(1.19)

where h is the slab thickness, ~x and ~y represent the cluster dimensions in the longitudinaland transverse directions, respectively, and a x = .35 Slm, a y = .035 Sim (the electricalconductivities) at baseline conditions. Equation (1.19) defines resistances not based on

FromCellularElectrophysiology to Electrocardiography

1 X 1element.q

.I .a I:..'

»:'

.'.. '

19

Network of Indi vi <bJ a1 Cells

Cells Oustered

FIGURE 1.12. A myocardial slab with a defined ischemic locality in the center. Each cell in the model can beinterconnected to adjacent cells by gap junctional resistances. To decrease computational time, these cells can beclustered into a bulk element with bulk conductivities.

20

I

I

I-

o

N. V.Thakor, V. Iyer, and M. B. Shenai

*

FIGURE 1.13. A wave of cardiac excitation approaching an observing lead at (*). The inflection is positive as thewave approaches, becomes rapidly negative at the exact time of incidence between the lead and the wave. Finally,as the wave travels away from the lead, the recording returns to zero. See the attached CD for color figure.

the individual cell, but directly on spatial dimensions of the myocardial cluster. However,in clustering an entire patch of myocytes, one assumes that all points within this patchare isopotential, which defies the assumption of continuous current spread. Thus, whileclustering can increase the slab size at a given computational load, it does compromise theresolution of the propagating wavefront.

Figure 1.13 depicts normal propagation across a 2D network of Luo-Rudy cells.Anisotropy is evident by the preferential spread of current in the horizontal direction, whichreflects the 10:I anisotropy ratio (oja y) in this particular simulation. Figure 1.13 represents only the early portion of activation, namely the upstroke, Phase 2 and early portionsof Phase 3. The procedure to reconstruct a representative electrical signal (the extracellularelectrogram) from this type of activation pattern is discussed in the next section.

1.2.3 RECONSTRUCTION OF THE LOCAL EXTRACELLULAR ELECTROGRAM(FORWARD PROBLEM)

While studying a single cell, the obvious candidate signal to study is the action potential.However, when studying many cells in a network, a large number of action potentials exist,making it difficult to study each action potential. In addition, action potentials representthe intracellular potential, are difficult to obtain in gross studies, and are impossible toobtain clinically. The local extracellular electrogram provides a representation of all cellularactivity located within the vicinity of a lead. By reconstructing the extracellular electrogram,an investigator can get a better representation of extracellular experimental recordings.Ultimately, by solving the entire forward problem, one may be able to reconstruct the entireECG (see Chapter 2). For the purposes of examining pathology at the network level, thereconstruction of the extracellular electrogram is presented.

Given a sheet of cells with time- and stimulus-dependent transmembrane potentials,the goal is to reconstruct the extracellular lead potentials e(x, y) at selected points in the

• Hyperkalemia• Acidosis• Anoxia• Decoupling

Transition zone

Stimulation

Extreme

21

FIGURE 1.14. A slab of myocardium with pre-defined regional ischemia. The "extreme" ischemic zone issurrounded by a border zone with milder ischemia. Point stimulation initiates a propagating wavefront, allowingthe calculation of the electrogram, and analysis of its features. See the attached CD for color figure.

slab shown in Figure 1.14, with respect to a point at infinity. At any given instant of time,current injection, 10, into the extracellular space induces at a distance r a potential <1>0, whichis inversely proportional to the distance between the point at which <1>0 is measured and thepoint at which 10 is injected:

10<1>0 =-

471"0"1(1.20)

For multiple elements of an array, the extracellular potential (at any point in time) is represented by superposition and summation of discrete elements (Plonsey and Collin 1961;Plonsey and Rudy 1980):

" Im(x,y,z)<l>e(XI, vi, zi, t) = L...J

all elements 471" a . r(x, y, z)(1.21)

where Im(x, y, z) is the transmembrane current at the source element positioned at point(x, y, z) and r(x, y, z) is the distance between the element at (x, y, z) and the lead position(Xl, Yl, z/). While Eq. (1.20) uses the transmembrane current to generate the extracellularpotential, many models generate transmembrane voltage, VM. Thus, the transmembranecurrent is derived by (Spach et al. 1979):

I X _ ~ (~ oVM(x, y, Z») ~ (~ oVM(x, y, Z»)m( , y, z) - ox R, ox + oy R; oy (1.22)

where R, and R; are the cell-to-cell resistances. This equation assumes that the extracellular

22

-_/

•

N. V. Thakor, V.Iyer, and M. B. Shena!

/1

-dI'l H,h

•

,.I ,

~t',,,, , J

I· •

ir 't

(

FIGURE 1.15. (Top) A wavefront propagating in space; (middle) the first spatial derivative of the wavefrontand (bottom) the second spatial derivative of the wavefront used in Eq. 3.18. The latter is directly related to theextracellular electrogram, when extended multidimensionally.

resistance is approximately 0 (infinitely conductive in the monodomain model) and thusthe extracellular potential is approximately 0 mY. Figure 1.15 conceptually describes howa passing waveform yields the characteristic shape of the extracellular electrogram. Whenapplied to the model depicted in Figure 1.13, the use of Eqs. (1.21) and (1.22) result ina depolarization complex that conforms very well to experimentally recorded unipolarelectrograms under normal and pathological conditions (Gardner et at. 1985; Irnich 1985;Blanchard et at. 1987) (Figures 1.13, 1.16).

The electrogram can be affected in two primary ways, as demonstrated in Eq. (1.23).The shape of the action potential (in space or time) can be affected either by the directalteration of ionic channel parameters (as in hyperkalemia, acidosis, anoxia or LQTS),or by the level of coupling resistances R, and R, (decoupling). Taking Eq. (1.22), thenreducing it to one dimension and expanding by the differentiation:

(1.23)

While Eq. (1.22) describes normal current spread, this correlate reveals how pathology maymanifest in the electrogram. In addition to being dependent on the explicit shape of Vm , 1m

also depends directly on the spatial derivative of (1/Ry ) , which is large near border zones


s'11•i'1~

Elevation Notching

23

FIGURE 1.16. A 2D network model of propagation across circular inhomogeneities of various manifestationsof ischemia. Hyperkalemia, anoxia, and decoupling each produce unique features in the QRS morphology of thereconstructed extracellular electrogram. See the attached CD for color figure.

(the interface of the ischemic and normal tissues). This corresponds with experimentalobservations that premature action potentials and ectopic activity may originate in theseborder zones (l anse et al. 1980; lanse and Wit 1989).

1.2.4 MODELING PATHOLOGY IN CELLULAR NETWORKS

While the one-cell paradigm can study a pathologic ion channel in the context of otherion channels , networks (10 and 20) are required to understand how pathologic cells interactwith surrounding cells. With the ability to represent both ion channel s and intercellular resistivities, networks have been able to lend insight into the dangerous sequelae of myocardialischemia and LQTS, particularly arrhythmogenesis. A proarrhythmic mechanism to manypathologic conditions is re-entry, a phenomenon by which an propagating wave circumvents refractoriness, to re-excite repolarized tissue. Thus, by definition , the study of reentryrequires multicellular networks.

To date, several major classifications of reentry have been defined (Spach 2001) . Classically, peculiarities of membrane channels (densities and inhomogeneous distributions)may lead to re-entry, as in the long-Q'T syndrome and myocardial ischemia. Alternately,


resistive discontinuities (i.e. during ischemia or infarction) and wavefront geometry (spiralwaves) may also be substrates for reentry. The remainder of this section will survey various pathologies, the ionic characteristics relevant to cell network studies, electrographicreconstructions, and how they may initiate reentry and subsequent arrhythmogenesis.

1.2.4.1 Myocardial ischemia

In the first section , myocard ial ischemia was described at the cellular level. However.ischemia usually involves the entire spectrum, from a patch of cells (regional ischemia) tothe entire heart (global ischemia). For these more macroscopic manifestations of ischemia,network models provide tremendous insight into how the cellular level changes alter thespread of excitation, and appear in the local electrogram, and ultimately, the ECG.

These concepts can be simulated at the 2D network level (Figure 1.16) where the localelectrograms have been reconstructed (Shenai et at. 1999) from a network of Luo-Rudycells . The results from these simulations suggest that different ischemic parameters affectthe propagating wave in distinct ways. For example , hyperkalemia produces a decreasein both conduction velocity and depolarization (effects derived from Phase 1 alterations ).However, anoxia does not alter conduction velocity or depolarization, but mainly producesa more rapid repolarization (a Phase 2,3 phenomenon). Finally, decoupling changes theconduction velocity only. Likewise, these individual ischemic. mechanisms display severalelements of experimentally recognized signal distortions (baseline elevations , decreasedpeak-to-peak amplitudes, notching ). Decoupled patches , for example , produce local extracellular electrograms that closely corroborate experimental recordings (Figure 1.17)(Shenai 2000).

In addition to necrosis of non-regenerative myocardium, the electrophysiological alterations of ischemia may lead to fatal arrhythmogenesis via the mechanism of reentry.When an excitation wavefront is blocked by an ischemic zone, the propagation is divertedaround the obstacle and may invade the ischemic zone retrogradely. This can create a"Figure of Eight" 2D pattern (Figure 1.18, top) that may ultimately degenerate into ventricular fibrillation . In modeling studies, reentry can be simulated with a ring-shaped linearnetwork , by which propagation can simultaneously travel in a diametric and circumferentialpath. Ferrero et at. used this ring-shaped model of Luo-Rudy cells, with defined regionsof ischemic impairment (BZ:Border Zone, CZ: Central Zone; Figure 1.18, middle) . Witha premature stimulus at the border zone, they found that the role of acidosis and hypoxiavary in the establishment of "Figure of Eight" reentry. These models suggest that the IK-ATP

current, most prominent during hypoxia, may be crucially proarrhythmic in the short term,with hyperkalemia playing an essential role. However, the arrhythmogenic effect of acidosisis minimal , and based on preliminary results, it may actually be antiarrhythmic (Ferreroet at. 2001). Cardioprotective K-ATP channel modulators (pinacidil) enhance the IK-ATP

and desensitize (or precondition) the myocardium to ischemia-derived k ATP enhancement(Grover and Garlid 2000).

The early stages of ischemia represent a fork in the road to the development of severaldifferent sequellae. While transient ischemia may yield no permanent damage, more acutecases can lead to infarction, re-entry (by Figure-of-8 or spiral mechanisms) and arrhythrnogenesis. By identifying which cellular ischemic characteristics provoke different sequellae,and coupling these alterations with the electrographic modeling described above, a uniqueparadigm emerges for evaluating sensitive detection algorithms. Cellular level alterations

ExperimentalNormalEndocardial

Frcm (roch, 1935

ExperimentalNormal Epicardial

_J,lI

Frcm BI<ncI"a'd. 1937

ExperimentalDecoupled

Fran GaU1er, 1985

Model (NormaI)

Model (Decoupled)

25

FIGURE 1.17. Experimental versus modeled waveforms for various ischemic manifestations. See the attachedCD for color figure.

(ischemic and/or proarrhythmic) can manifest in the electrographs and characteristicallyalter detection parameters significantly before a critical event. By modeling these ischemicand proarrhythmic parameters, one can precisely study how features in the electrograms orECG reflect these cellular or molecular alterations. With a theoretical understanding of thesechanges, clinical monitoring can alarm the patient, when parameters exceed physiologicalvariation, to seek the proper interventional procedures (angioplasty, bypass) or pharmacological therapies (anticoagulants, antiarrythmics).

1.2.4.2 EADs in 1D and 2D networks

The propagation of an EAD along a linear cell-network has been studied extensivelyin the context of ectopic beats and arrhythmogenesis. Triggered activity is the initiationof an abnormal impulse, induced by an EAD or DAD, that may propagate to neighboringcells. Unlike one-cell models, network models allow the use of intracellular coupling, andobservations on how coupling parameters may alter the propagation of the EAD.

tl Z

BZ (1cm ) CZ(1c m) BZ (1cm )

NZ ...

I

_f'L__r"'---""-----

NZ

BZ

BZ

BZ

B Z

CZ

pHBZ

pHBZ

FIGURE 1.18. Top: schematic of "figure of eight reentry". BZ: borderzone. ez: ischemic central zone. NZ:normal zone. Arrows show pattern of propagation. Middle: Ring-shaped I-dimensional approx imation of one ofthe reentry circuits. Numbers indicate cell number. Bottom: Various action potential traces between cell #15 and#315 with defined regions of ischemia (eZ =central zone; BZ = border zone, NZ = normal zone) correspond ingto a premature stimulation (#). See the attached CD for color figure. (From Ferrero et al. 200 1; © 2001 IEEE)


The most suspicious areas for triggered activity occur near the border of normal andabnormal myocytes. This situation was modeled by Saiz and colleagues (Saiz et al. 1997;Saiz et al. 1999) with a two-cell LR model. An abnormal cell (C1, conditions favorable toEAD formation) was coupled to a normal cell (C2) by a coupling resistance (R). Their studysuggested that C1's abnormal conditions had a strong influence over EAD formation in C2,and that APD was heavily influenced by the coupling resistance. Another investigation oftwo-cell models, by Wagner et al. (Wagner et al. 1995), coupled a sinusoidal-generatingcell to an Luo-Rudy cell, and also found that coupling resistance had an instrumental rolein EAD propagation. Similar results implicating the role of coupling resistance in EADpropagation have been shown in larger networks of cells (Saiz et al. 1996; Nordin 1997).

Larger models are able to incorporate multiple tissue types, while still maintaining fineionic detail. For example, the propagation of ectopic activity from the Purkinje network tothe ventricular myocyte network was demonstrated in a 2D model by Monserrat et al.(Monserrat et al. 2000). In this model, a 2D sheet of ventricular myocytes was coupledto a Purkinje fiber where EADs were induced. They concluded that the EADs transferto the ventricular myocytes only when within a certain range of IK blockade and an ICa,fenhancement.

While many other applications ofEAD studies exist in the context of arrhythmogenesis,the common themes center around abnormal impulse initiation (one-cell ion channel studiesthat find cellular conditions favorable to EAD genesis) or abnormal impulse propagation(cell-network models which study how intracellular conditions interact with intercellularcoupling that may elicit ectopic beats).

1.2.4.3 The ionicbasisofspiralwaves andfibrillation

A common cardiac arrhythmia that has been intensely studied with modeling techniques is ventricular fibrillation-or the degeneration of the normal wave propagation intoirregular and asynchronous events. While a variety of initiating mechanisms may exist,Davidenko et al. were the first to experimentally observe spiral waves as a mode of re-entry(Davidenko et al. 1992). Conceptually, the "head" of these spiral activation fronts wraparound to "re-enter" and stimulate the "tail" which is already recovering from the actionpotential (refractory period). The result is an intriguing, self-sustaining rotor of activity thatmay "break-up" and give rise to fibrillation. After the initial observation, many modeling investigations were able to deduce and derive conditions in which a spiral wave may develop.Several 2D models have been used to successfully demonstrate spiral wave activity, suchas both Beeler-Reuter and Luo-Rudy models (Fishler and Thakor 1991; Leon et al. 1994).

Two absolute conditions must be met for the initiation of a spiral wave (and reentry,in general). First, unidirectional block must allow wave propagation in one direction only.Second, the time (Tre) required for the wave to cover the length (Lre) of the re-entrantpath (Tre = Lre/CV) must fall within the relative refractory period of the previous actionpotential (the vulnerable window). The effect of cellular coupling and membrane excitabilityon unidirectional block is detailed in (Quan and Rudy 1990). These conditions are generallymet by initiating propagation with a rapid series of planar orthogonal S1-S2 stimuli. Thisrapid pacing produces an APD sufficiently short for re-entry. Depending on the modelingparameters, the spiral wave may remain stationary, or meander around in space (Davidenkoet al. 1992).


(

FIGURE 1.19. Initiation of spiral wave activity in an anisotropic cardiac sheet of 2 em x 2 cm using the originalLR model (A) and a modified model exhibiting a short APD (SAPD); B. Numbers at the top of each panel indicatethe time after the S2 stimulus of the cross-field stimulation protocol. The color legend used to map the potentialdistribution is shown at the bottom. See attached CD for color figure. (From Beaumont et al. 1998; used bypermission)

While these spiral waves have been observed experimentally, and demonstrated invarious models, the ionic basis for the formation of these spirals remains unclear and atopic of cutting-edge research. The difficulty in ascertaining this ionic basis lies withinthe necessary curve-fitting and estimation techniques that define complex models. As aresult, these models cannot provide the accuracy or stability to reproduce realistic spiralwaves. Nevertheless, several groups have begun to demonstrate and deduce ionic roles. Forexample, Beaumont et at. were able to show that different spiral wave patterns (stationary,chaotic, hypocycloidal meandering, epicycloidal meandering) can be defined in differentregions of a parameter space of voltage-dependence shift and sodium channel conductance(Figure 1.19, 1.20) (Beaumont et al. 1998). Qu et at. recently suggested that chaotic spiralwave meandering and spiral wave break-up are heavily dependent on the Ca2+ and K+currents (Qu et al. 2000). More recently, Xie et al. were able to demonstrate the effects ofischemia on the characteristics of spiral wave stability (Xie et al. 2001).

14 1819 13


A

S'.§ 0

~q

·10 4--t---1r---t---t--T~r---t-----J4.0 5.5 8

B

0.05 .os .09 .15 .16 .5 1.0 2.0

29

FIGURE 1.20. Representation of various types of spiral activity. and its dependence on Gna and voltage-shiftcharacteristics. (From Beaumont et al. 1998; used by permission)

By determining the roles of ionic currents in fibrillation generation, putative targets fordefibrillatory drug therapy can be identified ("chemical defibrillation"). Qu et at. studiedthe impact of different classes of experimental current blockers (INa, IK, and lea) in alteringor terminating the course to fibrillation (Figure 1.20). They found that a combination ofcurrent blockers were most effective in extinguishing a fibrillatory state.

1.2.4.4 Cell-networks in Long-QT Syndrome

Network modeling is beginning to emerge in the study of Long-QT syndromes, andhow defined molecular defects ultimately develop into arrhythmias and sudden death.Viswanathan and Rudy studied a l D fiber of Luo-Rudy cells, oriented transmurally with predefined regions (endocardial, M-cell, epicardial). Though IKs, IKr and late INa lengthen APDand EAD formation in isolated cells, this study suggested reduced gap-junction couplingcan alter the location of EAD formation between endocardial and M-cells (Viswanathanand Rudy 2000).

While the Viswanathan and Rudy model studied preferential EAD formation, anotherrecent investigation is beginning to lend insight on the role of LQT mutations on arrhythmogenesis. Clayton et at. simulated the LQT1, LQT2, and LQT3 mutations in ID and 2Dtissue models of excitation, with the focus of investigating re-entrant mechanisms. Usingsimilar cellular definitions of each LQT mutation (reduction of maximal IKs and IKr currentsand debilitation of sodium inactivation for the late INa current), this study suggested thatLQT mutations did not increase the vulnerability of LQT tissue to reentry (Clayton et at,2001). Rather, after initiation, LQT tissue is more likely to sustain increased motion of thesere-entrant waves (Clayton et at. 2001).

1.3 MODELING PATHOLOGY IN THREE-DIMENSIONAL AND WHOLEHEART MODELS

While one-cell and network models can lend insight into detailed conduction andpathologic interaction, these models and electrical reconstructions cannot extrapolate to


Tim (s)

FIGURE 1.21. Cardiac electrical restitution properties and stability ofreentrant spiral waves: a simulation study.While various classes of current reducers do not terminate activity, a combination of Ca-channel reducers andClass III antiarrhythmics lead to termination (From Qu et al. 1999; used by permission)

variables found in routine clinical settings . The 12-lead ECG remains the diagnostic goldstandard for clinical studies , and biochemical assays are primarily used for metabolic andcytopathic assessment. At a strictly conceptual level, the ECG and assay results are undoubtedly linked . Neverthele ss, the ECG cannot diagnose cellular pathology, and biochemicalassays cannot convey the global nature of a disease . The central thesis of this chapter isthat modeling can be a valuable tool to quantitatively assess how molecular and cellularprocesses are linked to ECG changes.

From Cellular Electroph ysiology to Electrocardiography 31

Reconstruction of the ECG requires placement of cells in a realistic whole heart andtorso models . Detailed whole heart and torso models take into account geometry of theheart and smooth transmural variations in fiber orientation, tissue conduction anisotropy,distinct tissue types, and volume conduction properties (Ramon et al. 2000; Scollan et al.2000). While subsequent chapters will offer an extensive treatment of the forward problemsolution (Chapter 2) and whole heart modeling (Chapter 3), the remainder of this chapterwill conceptually focus on molecular or ionic dysfunction that has been extended to thewhole heart, finally bridging cell to ECG.

1.3.1 MYOCARDIAL ISCHEMIA

As discussed in earlier sections, regional myocardial ischemia has been studied at manylevels of analysis: in single myocytes, 2D network models , and 3D tissue slabs. Occlusionand reentry studies using 3D modeling techniques provide a fundamental understanding ofischemic localization, and its tendency towards arrhythmogenesis.

A simulation study showcasing whole heart localization of tissue impairment in ischemia was conducted by Dube, Gulrajani and colleagues (Dube et al. 1996). The authorssuccessfull y reproduced local ischemia using classical characterizations of ischemic cells(depolarization of resting potential , reduced action potential upstroke, shortened APD,and reduced conduction velocity). Realistic three-dimensional localization of ischemic tissue was simulated by delineating regions of the myocardium subject to various grades ofischemia, as determined by experimental studies of artery occlusion. The cellular characteristics of the altered action potential studied in one-cell models were thus incorporated intoa whole heart model. Surface potentials were computed to reconstruct the 12-lead ECG,yielding ECG changes similar to clinical findings for each artery occluded (Figure 1.22).

In addition to causing tissue impairment in the myocardium, ischemia and infarctionprovide a substrate for arrhythmia. While several two dimensional modeling studies haveshown arrhythmogenesis in ischemia, Leon and Horacek were the first to investigate theinducibility of reentry in the presence of ischemic regions and infarcted tissue in wholeheart models (Leon and Horacek 1991). Using a three-dimensional model of the heart , thesimulation demonstrated that ischemic conditions may give rise to re-entrant activity.

1.3.2 PREEXCITATION STUDIES

Studies of anomalous excitation in the whole heart often result from deficiencies on alower level of analysi s. In Wolff-Parkinson-White (WPW) syndrome, abnormal myocardialtissue formation bridges the fibrous tissue separating the atrium and ventricle, providingaccessory pathways by which reentry is facilitated. Because of its ability to model manytypes of tissue, whole heart models have become a tool for studying WPW, which require large scale alterations of multiple tissue types in their cellular manifestation of WPWsyndrome.

On the cellular level, the refractory period , and resultant plateau calcium channel activation, are prime determinants of susceptibility to ectopic phenomena. Lorange and Gulrajaniwere the first to simulate WPW in a 3D whole heart model and to reconstruct body surfacepotentials (Lorange and Gulrajani 1986). Following their work, Wei et al. appropriatelyaltered cellular properties such as intercellular conductivity, tissue anisotropy, and refractory


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FIGURE 1.22. Action potential shape (top) and ECG reconstructions corresponding to simulation of variouscoronary artery occlusions (a.b.c.d) in moderate ischemia (bottom). Characteristic features of ischemia, including

lead-specific ST segment depression and alterations in QRS morphology, are reproduced. (From Dube et al. 1996;used by permission)


: I

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FIGURE 1.23. Reconstruction of ECG in the simulation of a particular arrhythmia associated with WolffParkinson-White syndrome. "Delta" waves. or slow upstrokes leading into QRS (prominent in II. III. and aVF)are a result of early. non-bundle branch initiation of ventricular excitation. (From Wei et al. 1990; used bypermission)

delays for WPW myocytes (Wei et al. 1990). Regionally, the investigators characterizedcell types in different regions of the heart and included atrioventricular accessory pathways.Body surface potentials were computed (Figure 1.23) show that the reconstructed 12-leadECG corresponds closely with clinically recorded WPW ECG traces.

More recently models have been used to study the pharmacological treatment ofWPW.For example. Fleischmann et al. investigated the effects of verapamil, a calcium channelblocker at the cellular level, on simulations of WPW preexcitation. Drug administrationsignificantly affected the formation of reentrant pathways in the study (Fleischmann et al.1996). WPW preexcitation simulation studies thus offer an example of how 3D models canprovide a useful tool for disease analysis. as well as a theoretical understanding for diseasetreatment

34

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FIGURE 1.24. Changes in myocardial structure associated with disarray (top) and the reconstructed ECGs(bottom) in the simulation hypertrophic cardiomypathy. Increases in QRS amplitude are observed in the septaland left-heart precordial leads, V2 and V3. (From Wei et al. 1999 Figures 2, 5-6; used by permission)

1.3.3 HYPERTROPHIC CARDIOMYOPATHY

Several modeling studies illustrated that hypertrophic cardiomyopathy (HCM) is apathology in which local cellular events, when integrated globally, translate into detectableECG changes. HCM is accompanied by many changes on the cellular and regional level. Afetal mode of genetic upregulation is observed in hypertrophy, through which preexistingmyocytes increase in size while total cell number remains relatively constant. More microscopic findings show that the orderly arrangement of myocytes in a parallel, linear layer isdisrupted during HCM, demonstrating the phenomenon of myocardial disarray in the tissue(Figure 1.24). The faster conduction velocity along the fiber orientation thus degeneratesinto isotropic conduction in the myocardium (Varnava et at. 2000).

Wei and colleagues in 1999 incorporated hypertrophied models of myocytes in arealistic 3D model to arrive at a whole heart model of hypertrophic cardiomyopathy (Weiet at. 1999). Clinical ECG traces of HCM patients frequently show Q waves with enlargedmagnitude, presumably due to increased contribution to the QRS vector. The salient cellularfeatures of HCM were implemented in a realistic 3D whole heart model to show thatindicators of this disease are built up from more fundamental cellular causes and reflectthemselves in the ECG. The reconstructed ECGs shown in Figure 1.24 reflect changescommonly seen in clinical electrocardiography.

While Wei and colleagues investigated the effect of myocardial disarray in hypertrophiccardiomyopathy, Siregar, et at. developed a realistic anisotropic cellular automata modelof the heart (Siregar et at. 1998) to study the effects of heart size and wall thicknesson clinically observed parameters. Using a Beeler-Reuter membrane model, characteristic


action potentials were derived for the myocytes, and propagation simulated across the heart.ECGs were reconstructed that confirmed experimental findings, including increased QRSamplitude. Thus 3D models were effectively used to explore different aspects of hypertrophyin the heart.

1.3.4 DRUG INTEGRATION IN THREE-DIMENSIONAL WHOLEHEART MODELS

While pathology studies certainly provide knowledge on the nature of disease, a currentdirection of studies is to identify and validate potential drug targets through simulation. Thefirst regulatory drug-to-ECG assessment was used to dismiss the potentially proarrhythmiceffect of an anti-arrhythmia drug by the American Food and Drug Administration (FDA).Concerns were raised over mibefradil, a T-type and L-type Ca2+ channel blocker, since itshowed T-wave ECG perturbations indicative of Torsade de Pointes susceptibility. Modelsof the ventricle, placed in a torso, were used to show that the T-wave ECG perturbationsactually arise from action potential shortening on the cellular level, and not on potentiallyarrhythmogenic action potential lengthening. Thus simulation studies were instrumentalin showing that ECG changes suggesting impairment can have entirely different (indeed,beneficial) cellular roots.

While the FDA study investigated drug activity from "ECG down", other studies havebuilt from cellular effects to higher levels of analysis. Recently, promising drug action studiesin 3D ionic models were performed by Garfinkel et al. with bretylium administration, whichhas been shown to flatten AP restitution (Garfinkel et ai. 2000). Scroll waves, the threedimensional correlate to spiral waves, were induced in normal tissue and spontaneous wavebreakup into fibrillatory propagation was observed. When the treated action potentials weresimulated, the scroll waves remained intact, suggesting that treatment is protective againstdegeneration into fibrillation (see Figure 1.25). Similar studies showed predictive potentialin investigating pure L-type channel blockers (Noble et ai. 1999) and Na+ -H+ exchangerblockers used in the treatment of myocardial ischemia (Ch'en et ai. 1998).

1.3.5 GENETIC INTEGRATION IN THREE-DIMENSIONAL WHOLE HEARTMODELS

With the current emergence of a wealth of molecular data, and the concomitant expansion in computational resources, gene-to-ECG and gene-to-heart pathology studies haverecently been investigated by several groups.

One such study was performed by Okazaki et ai., in investigating the link betweenLong QT (LQT) syndrome and Torsade de Pointes (TdP) (Okazaki et ai. 1998). Okazakiet ai. were able to model myocytes with genetic LQT mutations in a three-dimensionalmodel to investigate the whole heart consequences of the syndrome. Simulation of thediseased action potential led to arrhythmia and the periodic, abnormal ECG characterizingTdP (Figure 1.26). Future studies promise to similarly carry analysis from gene to ECG.Winslow's group in 2000 arrived at a model of a failing myocyte based on experimentalfindings on genetic regulation in disease (Winslow et ai. 2000). Significant downregulationof channels carrying the transient outward current Ito! and the fast inward rectifier current IK1

was observed in end-stage heart failure. Reduced expression of SERCA2A (which encodesthe smooth ER calcium pump) and increased expression of NCXl (which encodes the

36

Spontaneous Kroll wave breakup

Administration 01c ssm 1Ig8nt

Admlnlllratlon 01 brelyllum(varying dllg"," olelfecll


Continued Ilbrillatlon-1 ke propagation

Intact Kroll wavllS

FIGURE 1.25. Scroll waves show spontaneous degeneration into fibrillatory propagation (top, left). Administration of class III anti-arrhythmic agents steepen the action potential restitution and enhance fibrillation-likepropagation (top, right). Administration of bretylium, which flattens action potential restitution, results in intactscroll waves (bottom). See the attached CD for color figure. (From Garfinkel et al. 2000; used by permission)

sodium calcium exchanger) was also observed (Winslow et at. 2000). This minimal modelof a failing myocyte was incorporated into a realistic whole-heart model to investigatewhether the resulting action potential prolongation is sufficiently arrhythmogenic on thewhole heart level. Simulations showed waves of uncontrolled propagation in the diseasedheart (see Figure 1.27). Comparison of normal and model ECGs confirms this behavior inthe failing heart.

1.4 DISCUSSION

With the simultaneous explosion of molecular biology and computational power,paradigms for studying the molecular roles in whole heart pathologies are emerging


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FIGURE 1.26. Variation in APD according to M cell distribut ion is incorpora ted into a 3D simulat ion (top). Thereconstructed ECG reflects torsade de pointes (bottom). (From Okazaki et al. 1998; used by permission).

through computer modeling. These physiological models , when coupled with electrographic reconstruction technique s can reproduce clinically accessible waveforms. So far,most studies have spanned only a few levels, from gene-to-cell , from cell-to-network, orfrom cell-to-whole heart. Several reviewers have formally defined these integrative modeling paradigms-from genome-to-physiome (Rudy 2000) and from genes-to-rotors (Spach2001). Another domain of modeling is solving the forward problem of electrophysiology (see Chapters 3), through which activation patterns, and heart/torso geometries areextrapolated to the ECG. One future of integrative cardiac modeling is to yield a geneto-ECG paradigm, by linking genome-to-physiome models (Sections l.l and 1.2) withphysiome-to-ECG models (Section 1.3). This is an extremely challenging task, requiring aprofound description of the gene/molecular dynamics, intercellular connectivities, diversetissue characteristics and heart/torso geometries, all coupled to the forward problem ofelectrophysiology. It will require a tremendous amount of experimental and computationaldevelopment.


1.2 1.5

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FIGURE 1.27. EADs evoke uncontrolled arrhythmic propagation in heart failure (top); reconstructed ECG fornormal tissue (bottom, left) versus failing tissue (bottom, right) confirms erratic excitation See the attached CDfor color figure. (From Winslow et al. 2000; used by permission).

The potential benefits of developing such a paradigm, however, would be enormous.For example, comprehensive, integrative models that can simulate the cellular level ofdrugswill emerge to direct efficient pharmaceutical development-both testing novel drugs andtesting interactions of infinite drug combinations. Even farther off in the future, clinicalmanagement of heart patients may depend on integrated models customized to a particularpatient's unique set of genetic and acquired deficiencies. Today, however, cardiac modelshave found a comfortable niche in interfacing theoretical understanding with experimentalor clinical outcomes, from cell to ECG.

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91-103.EI-Sherif, N., E. Caref, H. Yin and M. Restivo (1996). 'The electrophysiological mechanism of ventricular

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EI-Sherif, N. and G. Turitto (1999). "The Long QT Syndrome and Torsade De Pointes." PACE 22 (Pt.l): 91-110.Eyring, H., R. Lumry and 1. Woodbury (1949). "Some applications of modern rate theory to physiological systems."

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42 N. V. Thakor, V.Iyer, and M. B. Shenai

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2

THE FORWARD PROBLEM OFELECTROCARDIOGRAPHY:

THEORETICAL UNDERPINNINGSAND APPLICATIONS

Ramesh M. GulrajaniInstitute of Biomedical Engineering, Universite de Montreal

2.1 INTRODUCTION

The forward problem of electrocardiography refers to the calculation of the potentials on thebody surface due to the heart sources, using the theoretical equations of electromagnetism.As a prerequisite for this calculation, suitable representations of the heart sources and ofthe torso geometry are needed. The former is usually assumed to be a current dipole, whichmay be taken to be a current source and sink ofequal magnitude / separated by a very smalldistance 8. The dipole is then represented as p = /8. The bold font indicates that p is avector, whose magnitude is /8 and whose direction is that of the vector 8, namely alongthe line joining sink to source. The rationale behind representing the heart sources witha current dipole is taken up in Section 2.2 below. In a second approach, the question of anadequate representation of the heart sources is circumvented by calculating the torso surfacepotentials using the actual potentials on the heart's epicardial surface (or more correctlyon the surrounding pericardial sheath) as the starting representation. This second approachwill also be described.

Torso geometry is nowadays modeled as a three-dimensional computer representationof the external torso surface and its internal inhomogeneities of differing conductivities.Prior to the advent of the computer, the torso was often modeled as a sphere or a cylinder, andanalytic expressions for the potential due to a current dipole within such a sphere or cylinderused to compute the surface potential. We do not consider such analytic solutions of theforward problem here, nor do we consider other early analog solutions, in which the torso

Address for Correspondence: Ramesh M. Gulrajani, Institute of Biomedical Engineering, Universite de Montreal,P.O. Box 6128, Station Centre-ville, Montreal, (Quebec) H3C 317, CANADA. Telephone: (514) 343-5705,Fax: (514) 343-6112, E-mail: [email protected]

43

44 R. M. Gulrajani

was represented by a realistically-shaped physical analog and potentials due to an insertedartificial dipole actually measured on the surface of this analog. Reviews of these analyticand analog solutions are to be found in Gulrajani et al. (1989). Thus this chapter focusesexclusively on numerically-computed solutions of the forward problem and the theoreticalequations underlying these solutions. Generally, it is the far-field forward problem thatis considered, where the desired potentials on the body surface are assumed far enoughfrom the heart sources to permit the use of simple dipole models for the latter. A morechallenging problem occurs when near-field potentials on or within the heart are desired.There is much current interest in at least being able to obtain a first approximation to theepicardial potential distribution on the heart surface, and this estimation is also considered.The chapter concludes with a section describing the more common applications of theforward problem. Other recent reviews of the forward problem and its applications are tobe found in Gulrajani (1998a; 1998b).

2.2 DIPOLE SOURCE REPRESENTATIONS

Prior to discussing the dipole source representations used in the forward problem,we present the fundamental equations that form the biophysical underpinning of potentialcalculations.

2.2.1 FUNDAMENTAL EQUATIONS

The biological current sources in the heart are the ionic currents that flow across thesurface membrane of the individual heart cells into the extracellular space. If we denote byJs(r) the net source current density (in A/m2

) at a point characterized by the spatial vectorr, then we can write the equation

J(r) = Js(r) +aE(r) (2.1)

Equation (2.1) states that the total current density J is expressed as the sum of the sourcecurrent density Js , if present, and the conduction current density a E, where E is the electricfield and a the conductivity. It assumes that quasi-static conditions apply whereby capacitive, inductive and propagation effects are all neglected (Plonsey, 1969), and field quantitiesat a given instant are determined by just considering the source currents J, existing at theinstant in question. Under these quasi-static conditions, the divergence of the total currentV . J = 0, so that taking the divergence of Eq. (2.1) yields

V· (aE) = -V· J, (2.2)

Using the relation E = - V<1>, where denotes the potential, Eq. (2.2) can be transformedto the fundamental equation that governs the relationship between electrocardiographicpotentials and heart sources, namely

V . (aV<I» = V . Js (2.3)

The Forward Problem of Electrocardiography

p

45

FIGURE 2.1. AheartofvolumeVHandsurrounding epicardialsurfaceSE existsinan infinitemediumofuniforrnconductivity a. The potentialis soughtat an arbitrarypoint P characterized by the positionvectorr. See text foradditional details.

If now the conductivity is assumed constant everywhere, i.e. the medium is infinite andhomogeneous, Eq. (2.3) reduces to Poisson's equation with the solution

(r) = _1_ f -V' . Js(r')av'41Ta Ir - r'l

VH

(2.4)

Equation (2.4) gives the potential at a fixed observation point, P,characterized by the positionvector r. It entails performing a spatial integration over the heart volume YH (Fig. 2.1), and adummy variable of integration r' that traverses the source coordinates has been introduced.The primes on V' and dY'are used to reinforce the point that it is r' that is the variable andthat all spatial derivatives need to be evaluated with respect to r'.

The membrane source currents can be expressed in an alternative form by the relation

Isv(r') == -V' . Js(r') (2.5)

where Isv denotes the source volume current density in Nm3• This relation follows fromGauss' law for the current flux that leaves a source Isv. We have

f I,w(r')dY' = f aE(r')· odS'

VH SE

(2.6)

where 0 is the unit normal to the epicardial surface SE that surrounds the heart volume YH.

Applying the divergence theorem to the right-hand side of (2.6), and substituting forV' . (o E) from Eq. (2.2), we immediately obtain the equivalence relation of Eq. (2.5).Accordingly, Eq. (2.3) can also be written as

V . (aV<I» = -Isv (2.7)

46 R. M. Gulrajani

By using the expression for the divergence of the product of a scalar and a vector,namely V . (¢ A) = V¢ . A + ¢ (V . A), we can rewrite Eq. (2.4) as

<l>(r) = _1 If J s(r') . V' (_1_) dV' - f V' . ( Js(r') ) dV'] (2.8)4na Ir - r' j Ir - r' ]

H ~

The divergence theorem can be applied to the second integral on the right converting it toa surface integral over SE (Fig. 2.1), where the heart sources J, vanish. Thus the secondintegral on the right is zero, and we have

<l>(r) = _1_ f Js(r') . V' (_1-) dV'4na [r - r'[

VH

(2.9)

Now the potential due to a current dipole p situated at r' in an infinite homogeneous mediumof conductivity a is given by

1 , ( 1 )<l>(r) = -p . V --4na [r - r']

(2.10)

Comparing Eqs . (2.9) and (2. 10), we see that the heart current sources generate a dipolar fieldand that Js can also be interpreted as a current dipole density. Note that this interpretationhinges on the validity ofEq. (2.4), which in turn is only true if a is homogeneous everywhereand Eq. (2.3) reduce s to Poisson 's equation.

2.2.2 THE BIDOMAIN MYOCARDIUM

A major difficulty occurs in estimating the ionic current density J, which is clearlyimpossible to do in a multicellular preparation like the heart . This leads to the notion ofdetermining an equivalent source Jeq for the heart, one that can replace the true sourceJs at least as far as the calculation of the far-field torso surface potentials is concerned. Anecessary prerequisite to obtaining this equivalent source is a more macroscopic view of theheart , one that smooths out the detail of the individual heart cells and their ionic currents.The individual heart cell is an elongated structure with approximate dimensions of 15 x15 x 100 J.1m (Fig. 2.2). It is made up of "sarcomeres" (from Z line to Z line), within eachof which are the interdigitating contractile myofibrils. The cells are electrically connectedto one another in the longitudinal direction by gap junctions present in the intercalateddisks separating the cells (Fig. 2.2), in effect forming long fibers. The cells also branch atirregular intervals, and this results in electrical continuity of the intracellular space in the twotransverse directions also. In effect, the entire intracellular space forms a continuum or, inbiological terms , a syncytium. Surrounding the invididual cells is the extracellular or, what istermed in cardiac electrophysiology, the interstitial space. This interstitial space too formsa second syncytium. The macroscopic view of the heart mentioned earlier then consists ofthese two syncytia or domain s, that are both assumed to occupy the same volume (Schmitt,1969; Tung, 1978; Miller and Geselowitz, 1978). The two domains do not exist in isolation,

Mitochondria

Capillary

47

FIGURE 2.2. Diagram of cardiac muscle fibers illustrating the characteristic branching, the intercalated disks,and the internal myofibrils. Individual cells are made up of sarcomeres. A sarcomere occurs from Z line to Z line,with the M lines lying at the mid-point of each sarcomere. Cells are separated longitudinally by the intercalateddisks. Other indicated structures are the blood capillaries and the mitochondrial cells that provide the energyrequired by the contracting fibers. Reproduced, with permission, from Pilkington and Plonsey (1982). © IEEE.Modified, with permission, from Berne and Levy (1977).

however, but are coupled at every point in space by the continuity of the transmembranecurrent of the individual cells which flows out of the intracellular domain and into theinterstitial one.

2.2.2.1 Equations for an Isotropic Bidomain-the Uniform Dipole Layer

Due to the infrequent branching of the cardiac cells in the transverse directions, theelectrical conductivity in the transverse directions is much less than that in the longitudinaldirection, and the myocardium is profoundly anisotropic. Nevertheless, for simplicity, inthis subsection we assume the myocardium to be electrically isotropic. This restriction willbe removed in the next subsection.

The assumption that both intracellular and interstitial domains occupy the same totalheart volume leads to the notion of effective conductivities. In the intracellular space, theeffective conductivity gi is given by gi = I.«. where a, is the real intracellular conductivityand I. is the fraction of the total cross-sectional area occupied by the intracellular space. Thisreduced effective conductivity ensures that the intracellular conductance remains the samedespite the increase in assumed intracellular cross-sectional area. Similarly, the effectiveinterstitial conductivity ge = (1 - Ii )(Je, where (Je is the real conductivity of the interstitialspace.

With these effective conductivities, we may write the following equations for theintracellular and interstitial domains, respectively:

v . (gi V<Pi) = Imv

V· (geV<Pe) = -Imv

(2.11)

(2.12)

Equations (2.11) and (2.12) are of the form of Eq. (2.7) and govern the intracellular and

48 R. M. Gulrajani

interstitial potentials, <Pi and <Pe , respectively. A volume source current formulation hasbeen used so that Imv is the membrane current (in Nm3) that flows out of the intracellulardomain and into the interstitial one. It therefore acts as a sink in the intracellular domainand is considered negative in Eq. (2.11), but is a source of equal magnitude that appears inthe interstitial domain and is taken as positive in Eq. (2.12). This macroscopic membranecurrent, Imv , which is impossible to measure, may be eliminated by combining Eqs. (2.11)and (2.12) to yield

(2.13)

Upon adding V . (gi V<Pe) to both sides of Eq. (2.13), and utilizing the usual definition ofthe transmembrane potential Ym = <Pi - <Pe , we get a second form ofEq. (2.13), namely,

(2.14)

where g = s. + ge' A third form is obtained by multiplying Eq. (2.14) by gelg so that

(2.15)

(2.16)

Eq. (2.15) becomes

(2.17)

The similarity between Eqs. (2.17) and (2.3) suggests that Jeq can serve as an equivalentsource for computing the interstitial potential <Pe. Note that Jeq only exists whenever thespatial gradient of Ym is non-zero, i.e., only in regions of the myocardium that are undergoingexcitation or repolarization. Furthermore, Jeq, unlike the true sources Js' can be computedif the effective conductivities of intracellular and interstitial domains can be estimated andif the spatial distribution of the transmembrane potential in the myocardium is known.Assuming for the moment an infinite homogeneous myocardium, the solution to Eq. (2.17)is similar to that of Eq. (2.3) and is given by (compare with Eq. 2.4)

<P(r) = _1_ f -V' . Jeq(r')dY'4rrge [r - r']

VH

Exactly as Eq. (2.4) was rewritten as Eq. (2.9), we may rewrite Eq. (2.18) as

<P(r) = _1_ f Jeq(r') . V' (_1-) av'4rrge [r - r'l

VH

(2.18)

(2.19)

The Forward Problem of Electrocardiography 49

thereby identifying Jeq as an equivalent current-dipole density. Once again this interpretationonly holds if the interstitial conductivity ge is homogeneous. Alternative formulations forJeq based on Eqs. (2.13) and (2.14) are also possible. For the former,

(2.20)

and acts in an interstitial domain of homogeneous effective conductivity g.; for the latter,

(2.21)

and acts in an interstitial domain of homogeneous effective conductivity g.The above field equations were written assuming an infinite homogeneous extent for

the bidomain myocardium. With the finite heart in the torso, this is evidently not the case.In particular, we have a bidomain-monodomain interface between heart and torso, anddeciding which boundary conditions apply at this interface is not immediately evident.Three boundary conditions are needed. The first two are the continuity of the interstitialpotential e at the epicardial surface to the torso potential <1>0 just outside the heart, and ofthe normal component of the total current that crosses over from the heart to the torso. Inmathematical terms, these two conditions at the heart-torso interface may be expressed as

<l>e = <1>0

ae, o<l>e 0<1>0gi - + ge-- =0'0--

on on on

(2.22)

(2.23)

where the normal derivative o<l>jon denotes the component of the gradient V· n alongthe outward normal n, and 0'0 denotes the torso conductivity just outside the interface. Thethird boundary condition generally used is that the intracellular current stops at the heartsurface (Tung, 1978; Krassowska and Neu, 1994), so that we have,

(2.24a)

With this condition, only the normal component of the interstitial current in Eq. (2.23)crosses over into the torso. An alternative third boundary condition (Colli-Franzone et al.,1990) that has sometimes been employed is

ae, 0<1>0g-- =0'0--

on on(2.24b)

This formulation is particularly convenient if Jeq is given by Eq. (2.21), since the interstitialmedium in which this Jeqacts has conductivity g, and Eq. (2.24b) then simply expresses thecontinuity of the normal component of the current from this equivalent interstitial mediuminto the torso.

Equations (2.16), (2.20) and (2.21) all identify Jeq as an equivalent current-dipole density per unit volume that exists wherever a spatial gradient of transmembrane or intracellularpotential is present, for example, in the vicinity of a propagating excitation wavefront. Analternative equivalent surface current-dipole density that is placed on this excitation front

50 R. M. Gulrajani

FIGURE 2.3. Diagram illustrating the spatial distribution of the transmembrane potential as an excitation wavefront sweeps the myocardium along the direction indicated by the unit vector n. Reproduced, with permission,from Gulrajani (l998a).

can be derived using Eq. (2.16). Figure 2.3 shows such a propagating action potential wavefront of spatial extent d. Ahead of this front, the myocardial cells are at rest with Vm = V"where Vr denotes the resting transmembrane potential. Behind the front, the cells are depolarized with Vm = Vd, where Vd denotes the depolarized transmembrane potential. If weapply Eq. (2.16) to the transition region between the dashed lines, the total dipole momentp, associated with the wavefront of area A, is

p = Jeq x Volume = -geqVVm x Ad

~ geq(Vd - Vr)An = geq VpAn (2.25)

In Eq. (2.25), the gradient VVmhas been approximated by (Vp/d)n, where Vp == Vd - Vr isthe amplitude of the propagating action potential and the unit vector n denotes its directionof propagation. Equation (2.25) holds in the limit that the propagating wavefront is assumedinfinitely thin and allows us to identify a surface dipole layer associated with the wavefrontwhose density is geq Vpn. Dipole orientations within the layer are everywhere normal to thewavefront, and, if the action potential amplitude is uniform everywhere, then so is the dipoledensity, whence the term uniform dipole layer used to describe this equivalent source.

2.2.2.2 Equations for an Anisotropic Bidomain-the Oblique Dipole Layer

Equations (2.11) and (2.12) are rewritten for a homogeneous anisotropic bidomain byreplacing the scalar effective conductivities gi and ge by their tensor equivalents G i and Ge,respectively. We get,

v . (Gi VJ = Imv

V· (GeVe) = -Imv

(2.26)

(2.27)

G; and Ge are now the diagonal intracellular and inter stitial effect ive conductivity tensors,respectively, and can be written in 3 x 3 matrix form,

(2.28)

where it is assumed that the cardiac fibers are oriented along the z axis. Although not strictlytrue (see Hooks et al. ,2002), symmetry about this axis is generally assumed so that g;x = g;y

and gex = gey' Note that G;V<1>; and GeVe in Eqs . (2.26) and (2.27), respectively, arevectors, being the product ofa 3 x 3 tensor matrix with a 3 x 1column matrix representationof the gradient vector.

A major complication arises on account of the fiber rotation present in the real heart,which leads to an inhomogeneous myocardium. Under these circumstances, the conductivitytensors are diagonal only in a local coordinate system, characterized by the unit vectorse" e2, e3, where e3 is always oriented along the fiber direction. These local unit vectorsmay be expressed in terms of fixed global unit vectors ex, ey, e. , e.g.,

(2.29a)

and reciprocally, the global unit vectors expressed in terms of the local unit vectors, e.g.,

(2.29b)

Accordingly, a rotation matrix A, characterizing the transformation from local to globalcoordinates, may be written as

(2.30)

with the transposed matrix AT characterizing the reverse transformation from global tolocal coordinates. From the mathematical definition of a Cartesian tensor (Fung, 1977),it can be shown that the diagonal conductivity tensors G; and Ge in the local coordinatesystem transform to the symmetric conductivity tensors G; and G~ , respectively, in theglobal coordinate system that are given by the matrix products

(2.31)

Equations (2.26) and (2.27) continue to hold in global coordinates, but with the primedconductivity tensors G; and G~ replacing the unprimed tensors G; and Ge. We have ,

V · (G;V; ) = Im v

V . (G~ Ve) = - I m v

(2.32)

(2.33)

52 R. M. Gulrajani

Note that the primed conductivity tensors vary from point to point in the myocardium withthe change in fiber orientation.

As for the isotropic case, the volume current density I mv may be eliminated fromEqs. (2.32) and (2.33), to yield

(2.34)

Upon adding V . (G;V<Pe ) to both sides ofEq. (2.34), we get

(2.35)

where G' = G; + G~ is the bulk myocardium conductivity tensor. Equations (2.34) and(2.35) are the equivalents ofEqs. (2.13) and (2.14) for the isotropic myocardium. Since G;and G~ are tensors, there is no equivalent of Eq. (2.15). Deducing that Jeq = -G;V<Pi isan equivalent current-dipole density that may be used to compute the interstitial potential<Pe is only valid, however, under the assumption that G~ is constant, since it is only thenthat Eq. (2.34) simplifies to Poisson's equation. Similarly, from Eq. (2.35), we obtainJeq = -G;VVm as an equivalent current-dipole density that can be used to compute <Pe

under the assumption that the bulk conductivity G' is constant. Despite their obviousinvalidity, the assumptions that G~ and/or G' are constant are used for want of anythingbetter; indeed often they are not only assumed constant, but isotropic as well. Onlywith these assumptions, can we consider J eq = -G;V<Pi and/or J eq = -G;VVm as therespective equivalent current-dipole densities.

When G; is an invariant scalar, then as shown in the previous subsection, Jeq =-G; V Vm reduces to a uniform dipole layer on the action potential wavefront that is normalto the wavefront (along the direction - VVm ) . In general, however, with G; a tensor, weobtain a dipole layer that is oblique to the wavefront. Moreover, the strength of this dipolelayer is not uniform, but varies in both magnitude and orientation from point to point alongthe wavefront due to the variation in G;. The boundary conditions applicable at the hearttorso interface remain the same as before, and the equivalent forms of Eqs. (2.22), (2.23),(2.24a) and (2.24b) are, respectively,

<Pe = <Po

nT G;V<Pi + nTG~V<Pe = nTGoV<Po

nTG;V<Pi = 0

nTG'V<Pe = nTGoV<Po

(2.36)

(2.37)

(2.38a)

(2.38b)

In Eqs. (2.37) and (2.38), each of the terms is a triple matrix product, involving a rowmatrix nT, a 3 x 3 conductivity matrix, and a gradient column matrix. That these Equationsdo express the continuity of the normal components of currents is evident if we note, forexample, that the triple matrix product nT GoV<Po can also be written as the scalar productn . (GoV<po) of two vectors n and GoV<Po.

More insight regarding the oblique dipole results if we look at its components inlocal coordinates. The oblique dipole now becomes Jeq = -Gi VVm where Gi is diagonal.Assuming again transverse symmetry of the z-oriented cardiac fibers so that gix = giy = gil


and giz = gil, where gil and gil are the transverse and longitudinal effective conductivities,then in a manner similar to that in which Eq. (2.25) was derived, we can show that the dipolemoment p associated with a wavefront of area A is given by

where n I , nz and n3are the components of the normal n to the wavefront in local coordinates.The above equation can be rewritten as

(2.39)

revealing that the oblique dipole consists of a component normal to the wavefront plus asecond axial component along the fiber direction (Corbin and Scher, 1977; Colli-Franzoneet al., 1982; 1983).

2.3 TORSO GEOMETRY REPRESENTATIONS

Torso geometry may be represented either by a numerical discretization of its differentinterfaces, or by a full three-dimensional discretization of its entire volume. The choice isdictated by the solution methodology to be used (see Section 2.4). Thus in Figure 2.4, whichshows a stylized torso, with interface discretization only the outer torso surface So, the lungsurfaces Sj and Sz, the outer heart surface S3, and the blood-filled heart cavities S4 and Ss,are triangulated. The intervening regions are assumed of constant isotropic conductivity.If the anisotropic-conductivity skeletal muscle layer underlying the skin (not shown inFig. 2.4) is to be included, it is first converted to an approximately-equivalent isotropiclayer (McFee and Rush, 1968) prior to the triangulation of its interfaces. With volumediscretization, the entire three-dimensional torso volume is modeled either in point-wisefashion as a collection of equispaced points, or in piece-wise fashion by a combination oftetrahedral and hexahedral (brick-shaped) elements. The major advantage of these volumerepresentations is the ability to accurately model not only anisotropic conductivity regions,but also regions of continuously varying conductivity.

2.4 SOLUTION METHODOLOGIES FOR THEFORWARD PROBLEM

Calculation of the torso potentials from heart source dipoles is done via one of two general approaches, namely surface methods or volume methods. As may be surmised, surfacemethods employ torso models with only the interfaces discretized, and obtain the potentialsonly on these interfaces. Volume methods, on the other hand, use full three-dimensionaldiscretizations of the torso volume and obtain the potential everywhere. The number ofpotentials to be evaluated is accordingly much greater with volume methods, leading tolarge coefficient matrices. However, the potential at each point is expressed only in terms ofits nearest neighbors, so that the matrices are sparse and may be inverted via sparse matrixsolvers. With surface methods on the other hand, while the coefficient matrices are muchsmaller, the potential at any interface point is coupled to the potential at every other interface

54 R. M. Gulrajani

FIGURE 2.4. Torso with multiple regions of differing isotropic conductivity. See text for more details.

point, with the result that the coefficient matrix is fully populated and sparse matrix routinesare inapplicable. Surface methods are also often termed "boundary-element methods" in theliterature.

2.4.1 SURFACE METHODS

2.4.1.1 Solutions from Equivalent Dipoles

Surface methods usually employ integral equations for the potentials to be calculated.These integral equations may be derived by applying Green's second identity to the torsogeometry shown in Fig. 2.4, to obtain for the potential ¢k(r) at the observation point r oninterface Sk (Barr et al., 1966),

(2.40)

where Jeq(r') denotes the equivalent dipole sources present in the heart myocardium VH,the torso interfaces S/ extend from 0 to N, (with internal and external conductivities (f/- and(f/+ , respectively), and dQrr' denotes the solid angle subtended at the observation point r

by an element of the surface integral d S' at r' (dQrr, = - I~~~; . tsd S'). The term involvingthe so-called "auto solid angle" (dQrr), representing the solid angle subtended at r by thesurface element containing r, is excluded from the summation. The contribution of thisauto solid angle term is 2rr if the surface around the observation point r is smooth, and hasalready been incorporated in the derivation of Eq. (2.40). The point to note is that the firstterm on the right-hand side in Eq. (2.40) is proportional to the infinite-medium potentialdue to the equivalent sources Jeq(r') (see Eq. 2.9). The second term explicitly represents theeffect of the different torso interfaces.

The simplest assumption with triangulated torso interfaces is to consider the potentialas constant over each triangle face. With this constant potential assumption, the observationpoint r may be placed at the centroid of each triangle, and an equation such as Eq. (2.40)can be written as r is moved from triangle to triangle. The ensemble of NT equations, whereNT is the total number of triangles, may be written in compact matrix form as

(2.41 )

where is now an NT x I column matrix containing the triangle potentials, G is an NT x Imatrix containing the first terms on the right ofEq. (2.40), and A is an NT x NT weightedsolid angle matrix (with diagonal terms zero on account of the elimination of the autosolid angle term from the summation on the right-hand side in Eq. 2.40) that depends onlyon torso geometry and conductivities. Equation (2.41) can be rewritten as (I-A) = G,where 1is an NT x NT identity matrix. A straightforward solution would entail inverting thecoefficient matrix (I - A) to get = (I - A) - IG. Unfortunately, this is not possible as thematrix (I - A) is singular and does not possess an inverse. The singularity can be demonstrated mathematically (Lynn and Timlake , 1968), but it can be understood by realizing thatpotential measurements are never absolute but always with respect to a reference. Thus thepotential is always indeterminate up to the constant value chosen for the reference potential,and mathematically the singularity of the coefficient matrix ensures that a unique solutionfor q, is not possible unless a choice for the reference potential is first made. In practice, theproblem is circumvented by replacing A by a "deflated" matrix A* (the deflation procedureentails removing an eigenvalue x = 1 of A), so that we end up solving (I - A*)* = G.The coefficient matrix (I - A*) is no longer singular, and accordingly this equation can beinverted. Note, however, that * is not the original potential on account of the changedcoefficient matrix. It can be shown that * = for triangles on the outer torso surfaceSo, but for internal interfaces * and differ by a constant. It can also be shown that thedeflation procedure sets the sum of the triangle potentials on Soto zero, in effect establishinga zero reference for the potential equal to the mean of the triangle potentials on So.

A more accurate procedure with triangulated torso interfaces is to assume a linearvariation of the potential over each triangle interface, so as to better represent the varyinginterface potential. This leads to placing the observation points r at triangle vertices ratherthan at triangle centroids. One immediate advantage is that for closed triangulated surfaces,the number of vertices is approximately half the number of triangles, so that the numberof unknown potentials to be determined is also approximately half, leading to smaller

56 R. M. Gulrajani

matrix sizes . Alternatively, if the computer power is already present, we can use a finertriangulation doubling the number of triangle vertices and still keeping the number ofunknown potentials the same. This vertex approach entails a slight modification of thegoverning integral equation (Eq. 2.40) as the auto solid angle is no longer 2][. This is becausewith an observation point r at a vertex, any small selected neighborhood 5E around r is nolonger smooth, but subtends a difficult-tocalculate auto solid angle Q , s,at r . Equation (2.40)may be rewritten, but this time explicitly in terms of Q, s, . We get,

[ak- (4][ - Q,sJ - ak+Q, s,l<f>k(r) = f Jeq(r' ) . V' (Ir ~ rl) dV'

VH

(2.42)

As before, the summation on the right-hand side excludes the auto solid angle term, as thishas already been moved over to the left-hand side. An equation such as Eq. (2.42) can bewritten as the observation point moves from vertex to vertex, and the ensemble ofequationscombined in matrix form . The terms on the left-hand side of this ensemble will eventuallyform the diagonal terms of the coefficient matrix that multiplies cPo Since the auto solidangle Q,s, is difficult to compute, and it only occurs in these diagonal terms, a commonapproximation is to set each diagonal term to be equal to the negative sum of the otherterms in its row. This introduces a linear dependency between matrix columns, renderingthe coefficient matrix singular, as mandated by the non-unique nature of the potential. Otherapproximations for Q,s, are discussed in Meijs et at. (1989), Heller (1990) and Wischmannet at. (1996) . As with the earlier centroid option, deflation is also needed with this vertexoption for potential sites.

An interesting variant of the traditional approach of expressing Jeq as the gradient of thetransmembrane potential was derived by Geselowitz (1989) for the special situation of equalanisotropy, i.e., when the intracellular and interstitial conductivity tensors are proportional.In particular, for a heart with isotropic intracellular and interstitial conductivities (which is adegenerate case of equal anisotropy), Geselowitz showed that the equivalent dipole surfacedensity could be represented as

(2.43)

where DH is a unit dipole perpendicular to all heart surfaces, epicardial as well as theendocardial surfaces adjacent to the blood masses (surfaces 53,54 and 55 in Fig. 2.4).Using this equivalent source representation simply converts the volume integral on the righthand side ofEq. (2.40) to a surface integral over these three heart surfaces. Ifwe denote theunion of these three surfaces by 5H , we get

(2.44)

A similar conversion of the volume integral on the right-hand side of Eq. (2.42) also resultswhen Eq. (2.43) is used as an equivalent source.

2.4.1.2 Solutions from EpicardialPotentials

As mentioned in the Introduction, the forward problem can also be solved using theheart's epicardial potentials as the starting point (Barr et al., 1977). The governing equationsmay be determined by applying Green 's theorem to a torso model containing only the heart'sepicardial surface S3and the outer torso surface So (Fig. 2.4). By allowing the observationpoint r to first approach So, and then S3, the following two equations are obtained:

1 f 1 , ,<1>B (r ) = -- -,-V <1>E . DdS4n rBE

SE

If ,(I) , If ,(I) ,+ - <1>EV - ,- . DdS - - <1>BV -,- . DdS4n rBE 4n rBB

~ ~

1 f 1 , ,<1> E(r) = -- - ,-V <1> E . DdS4n rEE

SE

If ,( I ) , If ,( I ) ,+- <1>EV - ,- . DdS - - <1>BV - ,- . DdS4n rEE 4n rEB

~ ~

(2.45)

(2.46)

In Eqs. (2.45) and (2.46) , we have explicitly denoted epicardial and body surface potentials by the SUbscripts E and B, respectively, and the corresponding surfaces bySE(= S3) and SB(= So), respectively. The unit normals 0 to these surfaces are alwaysoutward. The scalar distance [r - r' 1is now denoted by r~E' etc. , with the first subscriptdenoting the location of r and the second that of r' , and the prime on r~E is used to simplyreinforce the fact that the variable of integration is r'. Two sets of matrix equations result asthe observation point r is moved from triangle to triangle on the body and heart surfaces,and these may be written in compact fashion as

ABB~B +ABE~E +BBErE = 0

AEB~B +AEE~E + BEErE = 0

(2.47)

(2.48)

In Eqs. (2.47) and (2.48), ~ B and ~E are column matrices of body surface and epicardial potentials, T E is a column matrix of epicardial potential gradients, and the A and Bcoefficient matrices depend solely on integrations involving epicardial and body surfacegeometries. The first subscript on A (or B) indicates the surface on which the observationpoints are selected, and the second subscript whether the integration is over the epicardialor body surface. Equation (2.47) may be used to obtain an expression for r E , which whensubstituted in Eq. (2.48) yields:

(2.49)

The elements of T BE = [A BB - BBE(BEE)-IAEBr l[BBE(BEE)-IAEE - ABE] are thetransfer coefficients relating the potential at a particular epicardial point to that at a particular

58 R. M. Gulrajani

torso surface point. As was the case for forward solutions using current dipoles , these pointsmay be selected at triangle centroids, implying a potential that is constant over each triangle,or at triangle vertices with a potential that varies linearly over each triangle.

2.4.2 VOLUME METHODS

2.4.2.1 Finite-Difference Method

The finite-difference method represents the torso volume by a three-dimensional arrayof regularly-spaced points or nodes that are connected to each other by intervening resistors, whose values are selected to best reflect the intervening resistance between the points.Kirchhoff's current law is written for each node, resulting in a large set of equations relatingthe potentials between adjacent nodes . In effect, the method represents a discrete approximation to the governing differential equation (Eq. 2.7), namely V . (aV<I» = -Isv withIsv = -V· J eq, where Jeq denotes the equivalent heart sources . The solution of the resultingset of equations evidently depends on the fineness of the node spacing, and the accuracywith which the resistors represent the torso resistances. The equations are usually solvedby Gauss-Seidel iteration with successive over-relaxation. The main drawback is the slowconvergence. As with other volume methods , the finite-difference method can handle varying anisotropic conductivities. A good illustration of its application to electrocardiographyis to be found in Walker and Kilpatrick (1987).

2.4.2.2 Finite-Element Method

Here, the torso volume is represented by contiguous three-dimensional tetrahedraand/or hexahedra (brick-like elements). The finite-element method also solves Eq. (2.7),with I s v = - V . Jeq . The starting assumption is that the potential within each element canbe approximated by

r

<l>(x ,y,z) = L,Bi(X,y, Z)ii = l

(2.50)

where <l>i denotes the potential at an element node, ,Bi is an appropriate interpolation polynomial (usually linear in x, y, and z, for tetrahedral elements), and r is the number of nodesfor the element. Each ,Bi is equal to unity at node i, and is zero at all other element nodes.If we substitute Eq. (2.50) into Eq. (2.7), then on account of the approximation, we get

V . (aV<l» + Is v = R (2.51)

where a is the element conductivity and R denotes a residual. The technique of weightedresiduals (Brebbia and Dominguez, 1992) is now invoked in an attempt to reduce R to zero ,but in a "weak form" by reducing the set of weighted integrals below to zero:

f [V . (aV<l» + Isv]WidV = f RWidV = 0

v v

i = 1,2, ... . r (2.52)


In Eq. (2.52), each Wi is a weighting polynomial, and the integration is over the element inquestion. Often Wi is set equal to the interpolating polynomial f3i, and we get the so-calledGalerkin weighted-residual formulation, namely

f [V' . (aV'¢)]f3i dV +f f3;Isv dV = 0v v

The first integral is now integrated by parts to obtain

i = 1,2, . . . . r (2.53)

-f f3i(aV'¢). odS +f (aV'¢)· V'f3i dV = f f3;Isv dV

S v v

i = 1,2, .... r (2.54)

where the surface integral is over the bounding surfaces of the element and 0 is the unitoutward normal. Consider, initially, internal volume elements that do not abut the outertorso surface So. The contribution of the surface integral in Eq. (2.54) will eventually becancelled by similar terms from contiguous elements on account of the continuity of thenormalcomponent of the current, and because the f3i'S are selected such that the potentialat a common interface is only determined by nodes on that interface. This last is done toensure the continuity of the potential across the interface between elements. Both volumeintegrals in Eq. (2.54), however, need to be considered, with that on the right-hand side onlycontributing for elements where Isv is non-zero. Assuming tetrahedral elements and linearpolynomials for the f3i, the set of r equations in Eq. (2.54) may be written in linear matrixfonn

(2.55)

where A(e) is an r x r coefficient matrix, ep(e) and F(e) are r x 1 column matrices, and the

superscript (e) is used to denote that Eq. (2.55) holds for a particular element. The matrixep(e)contains the element potentials to be computed, and F(e) is the matrix representation ofthe source terms and surface integrals of Eq. (2.54).

The coefficient matrices from the different elements, internal as well as those that abutthe outer torso surface, may be combined to result in a global matrix equation

(2.56)

where now A is m x m if there are m unknown potentials in the torso to be determined, andwhere epand F are each m x 1. Surface integrals from contiguous elements will now cancel.For elements that abut the outer torso surface So, the surface integrals over the sides thatform a part of So will remain uncancelled. These uncancelled surface integrals yield the setof surface integrals over So:

-f f3i(a V'¢) . tul S,

So

i = 1,2, . . . . m (2.57)

60

If the boundaryconditions applicableon So are of the mixed form

 = a on SOl

(aV<I». DdS = indS on S02

R. M. Gulrajani

(2.58a)

(2.58b)

where a denotes the knownpotential on a portion SOl of So and in is the injected normalcurrentdensityoverthe remainingportion S02, then sincethe fJi reduceto zeroover SOl therebeing no unknownpotentialsthere,Eq. (2.57)needonlybe integratedover S02. SubstitutingEq. (2.58b) in Eq. (2.57), the uncancelledsurface integralsbecome

i = 1,2, . . . . m (2.59)

Only the integralsfor i correspondingto surfacenodes over S02 contribute. In buildingEq.(2.56), theyform another source termdue to the injectedcurrent, and add to the appropriateterm fi in the matrix F. Thus, the so-called Neumann boundary condition of Eq. (2.58b)enters naturally into the finite-element formulation. The Dirichlet boundary conditionof Eq. (2.58a), however, has to be introduced explicitly into the global matrix equation(Eq. 2.56).Thus if the potentialat nodek is a, then all matrixelementsakl in rowk are setequal to zero, except akk, which is set equal to unity; in addition fk is set equal to a' Thisalso renders A non-singular. Since A is large, solutions to Eq. (2.56) may be obtained byiterative techniques, though many finite-element packages have direct solvers that exploitthe sparse nature of A.

Finally, the finite-element methodcan alsobe used to computebody surfacepotentialsfrom epicardial potentials, in which case since the heart region is excluded, the volumeintegral on the right-hand side of Eq. (2.54) drops out. The uncancelled surface integralsof Eq. (2.57) are also zero since no current leaves the torso. Only the volume integral onthe left-hand side of Eq. (2.54) remains, and leads to the global matrix equation A<I? = O.The matrix <I?, however, also contains the known epicardial potentials and these Dirichletboundary conditions at the epicardial surface need to be introduced explicitly, exactly asexplainedabove. In effect,we end up solvingfor the body surfacepotentialsusing thegivenepicardial potentials as source functions.

2.4.2.3 Finite-Volume Method

The finite-volume methodwas firstappliedto the bioelectricproblemby Abboudet at.(1994).It is a close relativeof the finite-element method.The governingequationis nowtheintegral form of Eq. (2.7), which integral form after applicationof the divergence theoremto the left-hand side may be written as

f (a V <1» • DdS = - f IsvdV

s v

(2.60)

Whereasthefinite-element methodsoughtto satisfyEq. (2.7),albeitinweakform,in aglobalmanner over the entire torso volume, the finite-volume method aims to satisfy Eq. (2.60)

locally, over each torso element. These local elements are usually selected to be small cubicvolumes or cells, over each of which Eq. (2.60) must be satisfied. In addition, continuityof the normal component of the current between cells must be ensured. Rosenfeld et al.(1996) describe the mechanics of solving Eq. (2.60), and of ensuring that the continuitycondition is satisfied, by approximating the gradient V required in Eq. (2.60) by its integraldefinition

(2.61)

The use of Eq. (2.61) leads to Eq. (2.60) being approximated over each cell by a linearequation involving the unknown potentials, at the center of the cell in question, and atthe centers of its nearest neighbors. Eventually, when the equations from all the cellsare combined, a linear matrix equation of the type Aep = F results, where the coefficientmatrix A is again sparse. The non-unique nature of the potential is handled by applying theadditional condition that Gauss' flux theorem should be satisfied over the entire volume. Abetter approximation than Eq. (2.61) for V can be obtained if, similar to the finite-elementmethod, we start with an approximating equation such as Eq. (2.50) for the potential (Harrildand Henriquez, 1997).

2.4.3 COMBINATION METHODS

Surface and volume methods may be combined such that the former is used wherethe volume conductor is isotropic and the latter where it is anisotropic. This aims to takeadvantage of the reduced computational load of surface methods, and at the same time beable to accurately represent anisotropic conductivities. Such a combination was illustratedby Stanley and Pilkington (1989), where torso potentials were computed from epicardialpotentials in the presence of an anisotropic skeletal-muscle layer. A transfer-coefficientmatrix between epicardial potentials and the inner surface of the skeletal muscle, usingEq. (2.49), was first determined from a boundary-element discretization ofthe torso up to thisinner layer. The finite-element method was then used to represent the anisotropic skeletalmuscle and proceed from the potentials on this inner layer to the torso surface potentials. Themethodology of a combination method employing higher-order interpolation, capable ofmatching the potential as well as potential gradients across the elements, has been describedby Pullan (1996).

2.5 APPUCATIONS OF THE FORWARD PROBLEM

Three categories of applications ofthe forward problem are described below. The first isits obvious use with computer heart models to calculate torso (and in some cases, epicardial)potentials. The second is to gauge the effects of torso conductivity inhomogeneities onelectrocardiographic potentials. The final application is the reciprocal problem ofobtainingthe currents traversing the heart due to currents injected at the body surface. One furtherapplication of the forward problem that is only mentioned here in passing is its use in the

62 R. M. Gulrajani

inverse problem of electrocardiography. Almost all inverse solutions entail a prior forwardproblem calculation.

2.5.1 COMPUTER HEART MODELS

Present-day computer heart models start by storing a realistic three-dimensional representation of the heart anatomy that, in tum, is properly positioned within a second realisticthree-dimensional representation of the human torso. A full solution would then entail thesolution of the two inhomogeneous-anisotropy bidomain equations (Eqs. 2.32 and 2.33)applicable in the heart region, Laplace's equation applicable in the torso region, plus theboundary conditions at the heart-torso interface and at the torso-air interface. Equations(2.32) and (2.33) also imply a description of Im v , the membrane currents of the cardiaccell. These should be obtained from one of the more recent models for cardiac membranecurrents (e.g., Luo and Rudy, 1991). In essence, the solution to this mix should be theintracellular and interstitial potentials, <1>; and <l>e, computed everywhere in the heart, aswell as the torso potential <1>0, computed everywhere in the torso. The potentials <1>; and<l>e then permit determination of the transmembrane potential Vm in the heart, and thus adescription of the heart's excitation. In addition, the values of <l>e at the heart surface willautomatically yield the epicardial potential distribution.

To date, on account of its complexity, only one group (Lines et al., 2003a) has achievedthis complete solution for realistic heart and torso geometries. They show a single simulationof a transmembrane potential distribution in the heart along with a simultaneously-computedbody surface distribution (Lines et al., 2003b). A second group (Buist and Pullan, 2002)has described two solution methodologies for the complete solution, but only illustratetheir results with a two-dimensional model of a realistic-geometry heart slice placed insidea torso slice. Both these groups have pioneered the numerical techniques needed for thecomplete solution, but undoubtedly computer memory and speed limitations have preventedexploiting the techniques to the full. Earlier work by many other groups used a simplifiedmethodology that entailed splitting the problem into two sub-problems. The first entailssolving Eqs. (2.32) and (2.33) in the heart region, assuming that the heart is insulated fromthe torso, i.e., neither intracellular nor interstitial currents cross over into the torso. Thiswould solve the sub-problem of determining the excitation of the heart. The second subproblem would then attempt to calculate torso surface potentials and/or epicardial potentialsfrom the now-known excitation pattern of the heart, most commonly by using equivalentdipole representations for this excitation. Both sub-problems, as well as a review of themany simulations realized, are described briefly below.

2.5.1.1 Determining the Excitation Pattern ofthe Heart

Historically, two categories of heart models may be defined, those that assume afixed activation pattern for the heart, based on the activation isochrones first measuredin the human heart by Durrer et al. (1970), and those that incorporate their own activationalgorithm. The best example of the first type of heart model, with fixed activation isochronescorresponding to normal excitation, was the one developed by Miller and Geselowitz (1978).No solutions for the heart's excitation pattern are needed here; instead a pre-determined

form of the cardiac action potential is triggered at each model point at its correspondingisochrone time. This then serves to determine the spatial and temporal distribution of Vm .

Clearly, heart models possessing an intrinsic activation algorithm are much more versatile and capable of simulating both normal and abnormal excitation. Early examples ofsuch heart models were all of the so-called "cellular automaton" type. Here, as opposed tosolving Eqs. (2.32) and (2.33), activation is determined by a set of rules that define the propagation velocities between model cells, the excitation and refractory states, and the form of theaction potential. Such cellular automaton heart models originated with Okajima etal. (1968),and some of the better known ones are those of Solomon and Selvester (1971; 1973), Horacekand van Eck (1972), Lorange and Gulrajani (1993), and Werner et at. (2000). One interestingway to solve the excitation problem in more realistic fashion is to derive an approximating"eikonal" equation (Colli-Franzone and Guerri, 1993; Keener and Panfilov, 1995) for theactivation wavefront from Eqs. (2.32) and (2.33). Solution of this simplified eikonal equation then yields just the activation time at a given myocardium point or, equivalently, thewavefront position at a given time. A pre-determined action potential is again triggered ateach point corresponding to its activation time. This eikonal approach offers an approximatesolution for the wavefronts, and was developed for want of the requisite computing poweravailable at the time to solve the anisotropic bidomain equations. Another approach, alsodictated by the lack of computing power, was that of Leon and Horacek (1991). These investigators, by assuming equal anisotropy, combined Eqs. (2.32) and (2.33) into a single equationfor the transmembrane potential Vm . We rewrite these Equations below in slightly modifiedform

v . (G;'1J = f3lm - f3Istim

V . (G~ '1e) = -131m

(2.62)

(2.63)

where 13 is the surface-to-volume ratio of the cardiac cells, and an intracellularly-injectedstimulating current Istim (in Nm2) has been introduced to permit excitation sites for themodel. The surface-to-volume ratio permits the conversion of the current per unit volumeUmv, expressed in Nm3) to the current per unit membrane surface area Um, expressed inNm2) via the relation Imv = 131m. With equal anisotropy, i.e., G; = ~G~, and expressingthe membrane current as the sum of the capacitive and ionic currents

(2.64)

Eqs. (2.62), (2.63) and (2.64) may be combined to result in a single governing reactiondiffusion equation:

aVm I [~ (I) 13]- - - --V· G '1V - I --I .at - f3Cm

1+ ~ e m 13 IOn + 1+ ~ sum (2.65)

Leon and Horacek solved Eq. (2.65) for the subthreshold case, i.e. by assuming that the ioniccurrent was passive and given by lion = GmVm, where Gmis the constant resting membraneconductance per unit area. Once Vm reached a fixed threshold value, a pre-determinedaction potential waveform was triggered. Thus, the Leon- Horacek model was a hybrid, with

64 R. M. Gulrajani

correct subthreshold excitation, but with cellular automaton characteristics above threshold.A comprehensive review of most of these heart models, and of the electrocardiographicsimulations realized with them, has been provided by Wei (1997).

Later work focused on the solution of Eq. (2.65), but with both sub- as well assupra-threshold representations for lion. Simulations with a simple FitzHugh-Nagumo representation for lion have been described by Berenfeld and Abboud (1996) and by Panfilov(1997). Huiskamp (1998) described a particularly impressive study that solved Eq. (2.65)for an 800,000 point model of the dog ventricles, using modified Beeler-Reuter equations(Drouhard and Roberge, 1986) for lion' This dog model was developed by Hunter et al.(1992) and incorporated measured fiber directions at every point. Recent work by ourgroup (Trudel et al., 2001) has employed a multi-processor computer to solve Eq. (2.65)for a 12-rnillion point high-resolution version of the earlier Lorange and Gulrajani (1993)human-heart model, with analytically-introduced fiber rotation, and with a Luo and Rudy(1991) membrane model for lion. The ventricular activation isochrones for normal excitationobtained in these simulations by Trudel et al. are shown in Figure 2.5.

Current work by our group focuses on doing away with the equal anisotropy assumption and solving Eqs. (2.62) and (2.63) together, without their combination into the singleEq. (2.65). This is now feasible with our 12-rnillion point heart model, again due to anewer generation multi-processor computer now available to us. A second advantage of thesolution of Eqs. (2.62) and (2.63) without their combination would be the automatic determination of the interstitial distribution <Pe , and hence of the epicardial potential distributionof the isolated heart. Other groups are attacking the same problem but with different andnovel approaches (Penland et al., 2002; Buist et al., 2003).

2.5.1.2 Calculating Torso and/orEpicardial Potentials

Once the spatial and temporal distribution of Vm has been determined by one of theabove-described excitation methodologies, subsequent calculation of the body surface potentials has usually assumed the myocardium to be isotropic. Accordingly, most investigators have used an equation such as Eq. (2.16), namely Jeq = -geqVVm , to compute anequivalent dipole density. This leads to individual dipoles at each model point, which arethen combined vectorially for individual heart regions to realize a "multiple-dipole" heartmodel, e.g., the Miller-Geselowitz model used 23 such regional dipoles to represent activation of its ventricles. A surface methodology can then be used to compute the bodysurface potentials due to each regional dipole, which potentials are then added to get thefinal body surface potential. Thus, Figure 2.6 shows the normal 12-lead electrocardiogramcorresponding to the activation isochrones of Fig. 2.5, computed from 58 regional dipolesof the Trudel et al. heart model, using the integral equation formulation of Eq. (2.40).

Huiskamp (1998), on the other hand, used the variant equation Jeq = -gi VmDH

(Eq. 2.43) described by Geselowitz (1989), to determine dipoles on the epicardial andendocardial surfaces of his dog heart model, prior to using these dipoles as sources to compute the potential on the torso surface. This approach may also permit the approximation ofepicardial potentials by computing potentials on a surface just outside the real epicardium,as demonstrated by Simms and Geselowitz (1995). The disadvantage with this approachis that for an injured heart containing dead tissue, this dead tissue forms an additionalinterface on which equivalent dipoles need to be placed. The more traditional route of

40

30

msec

msec

FIGURE 2.5. Isochrones corresponding to normal activation of the ventricular heart model employed by Trudelet al. (200 I). A transverse section (top) and a longitudinal section (bottom) are depicted. The colors indicate thetime of activation as per the color bars on the right. Isochrones start at 5 ms after ventricular activation and arespaced at 5 ms intervals. See the attached CD for color figure. © IEEE.

66

O.5mV

aVR

'I

aVL

'"

aVF

R. M. Gulrajani

/\

V1 2

/\

V6

FIGURE 2.6. The normal I2-leadEeGcorresponding to theactivation isochronesofFigure2.5.Reproduced, withpermission, from Trudelet al. (2001). © IEEE.

lumping dipoles computed by Eq. (2.16) into regional dipoles and then using Eq. (2.40) doesnot yield sufficiently accurate potential distributions just outside the epicardium, largely because of a loss of spatial resolution, due to the lumping, that shows up at close distances.However, as shown by Hren et al. (1998 ), reasonably-accurate approximations to the epicardial distribution can be obtained if the dipoles at the individual points are not combinedbut used individually in the computations. Hren et al. also used the oblique dipole model tocompute their epicardial potentials. They also assumed that the oblique dipoles existed ina homogeneous, isotropic myocardium. Nevertheless, the computed epicardial potentials(Fig. 2.7) for paced stimulation at different intramural depths in the right ventricular wall ofHren et al .'s heart model revealed both the characteristic one-minimum two-maxima pattern,

A

-33.50/5.52

F

·15.70/4.75

B

c

D

E

·32.84/5.08

-31.68/4.85

-29.48/4.68

-26.95/4.71

-8.24/4.96

-6.12/4.93

-4.36/5.00

-1.93/5 .29

FIGURE 2.7. Simulated potential maps on a patch of epicardium 10 ms after the onset of activation, for pacingat different intramural depths in the right ventricular free wall. Pacing sites were 0.5 mm apart, progressing fromthe epicardium (panel A) to the endocardium (panel J). The epicardial projection of each pacing site is indicatedby the black dot. Isopotentiallines are plotted for equal intervals, with no zero line; solid contours represent thepositive and broken contours the negative values of the potential; the magnitudes of the minimum and maximumare given (in mV) at the bottom of each map. Note that the axis joining the two maxima rotates counterclockwisewith increasing pacing depth following the transmural counterclockwise rotation of fibers from epicardium toendocardium. Figure reproduced, with permission, from Hren et al., 1998.

68 R. M. Gulrajani

as well as the rotation with depth of stimulation, observed experimentally by Taccardi et al .(1994).

The use of the oblique dipole model, along with the approximation of a homogeneousisotropic myocardium, in order to compute torso surface potentials may be more contentious.In a simulation study involving a layered myocardium block placed inside a larger volumeconductor block, Thivierge et al. (1997) showed that the effect of the axial dipole componenton surface potentials is reduced both on account of its orientation along the high-conductivityfiber direction as well as due to a well-known effect described by Brody (1956), wherebydipoles oriented tangential to the high-conductivity blood masses are diminished in so far astorso surface potentials are concerned, whereas dipoles oriented radial to the blood massesare enhanced. In the heart, the fibers are in the main oriented tangential to the blood masses.During normal activation of the heart from endocardium to epicardium, the componentof the oblique dipole normal to the wavefront ends up being radial to the blood massesand is therefore enhanced, whereas the axial component along the fibers is diminishedboth due to the Brody effect as well as due to its orientation along the high-conductivitydirection of the myocardium. Furthermore, the axial dipole is small to begin with duringnormal activation. This is because the endocardial-to-epicardial spread of normal activationensures that the largest component of the transmembrane potential gradient VVm lies inthe endocardium-to-epicardium direction perpendicular to the fibers, and that the axialcomponent of VVm parallel to the fibers is very small. Mathematically, this transla tes toa very small value for n 3 in Eq. (2.39), and hence a small axial component for p. Allof these reasons may well explain why torso potential simulations that have ignored theaxial component have often successfully reproduced clinical electrocardiograms (ECGs).If using the oblique dipole model for computing torso potentials is being contemplated, itmay be necessary to include both the high-conductivity intraventricular blood masses aswell as the myocardial anisotropy into the torso model so as to correctly reduce the effectof the axial component. If these inclusions are not possible, it may actually be preferableto use the uniform dipole model and an isotropic myocardium. On the other hand, forepicardial potentials, as the simulations by Hren et al. (1998) show, it is essential to includethe axial component. It may also be essential to include this component for body surfacecomputations during repolarization, which being less organized than depolarization, mayhave a larger axial component. Also abnormal stimulation of the ventricles at isolatedsites will certainly lead to initial propagation along the fiber direction, and larger axialcomponents of VVm • Intramural stimulation, such as that depicted in Fig. 2.7, thereforemay warrant use of the oblique dipole model not only for epicardial potentials but also fortorso surface potentials.

All of the above computations of torso and/or epicardial potentials described abovehave been via surface methodologies. Fischer et al. (2000) recently described a mixedboundary element-finite element route to obtaining both epicard ial and torso potentials . Thestarting point was the transmembrane potential distribution Vm, calculated by Huiskamp(1998). The heart region of Huiskamp's model was then represented via finite elements ,with the known Vm distribut ion acting as a source . Outside the heart, a boundary-elementsurface representation for the torso was employed. The governing equations were Eq. (2.35)in the heart, namely V · (G 'V <l>e) = - V · (G;V Vm), and Laplace's equation in the torso.Since Vm is known, only two boundary conditions are needed at the heart-torso interface.

The ones selected were Eqs. (2.36) and (2.37), the latter rewritten in terms of the knownpotential Vm, and the unknown potentials <l>e and <1>0 being sought, namely,

(2.66)

Solution of this mixed finite-element and boundary-element problem yielded <l>e and <1>0;the former then gives the epicardial potential distribution and the latter the torso surfacepotential distribution. Two important points to note are that due to Vm being used directlyas the source and due to the finite-element methodology used for the heart, an accuratesolution results for e and <1>0 without the need to draw on the oblique dipole interpretationsused with surface methodologies. Yet the computed solution is the equivalent of a surfacemethodology that employs an oblique dipole in an inhomogeneous anisotropic heart, i.e.,with both G; and G' varying tensor quantities. Fischer et al. went on to compare their resultswith those computed assuming G' to be an isotropic constant, i.e., equivalent to an obliquedipole but acting in an isotropic heart, and with those assuming both G; and G' isotropicconstants, i.e., equivalent to a uniform dipole layer acting in an isotropic heart. Relativeerrors in the body surface potentials were approximately 34% for G' constant and 43% forboth G; and G' constant, reflecting the importance of including myocardial anisotropy in thebody surface computations. Qualitatively, however, the surface potential distributions weresimilar (Fig. 2.8). Interestingly, the solution with G' constant, i.e, equivalent to an obliquedipole in an isotropic heart, overestimated the potential variations, while the solution withboth G; and G' constant, i.e., equivalent to a uniform dipole layer in an isotropic heart,underestimated the variations. This reinforces the finding by Thivierge et al. (1997) that theaxial component of the oblique dipole does tend to get reduced by its orientation along thehigh-conductivity fiber direction. The high-conductivity blood masses were not includedin the study by Fischer et al. so the Brody effect did not come into play in the potentialdistributions of Fig. 2.8. It would be interesting to see if the presence of blood masses

<l> [mV)

-0 5

.,. t 5

Ca) (b ) (e)

FIGURE 2.8. Body surface potential maps on the anterior torso model 48 ms after the onset of activation: (a)oblique dipole layer model in an isotropic heart, (b) full anisotropic myocardium model, (c) uniform dipole layermodel in an isotropic heart. Contours are plotted in steps of 0.5 mV.Field patterns are in qualitative agreement.but quantitative differences are large, even though the isotropic conductivities were chosen to realize the smallestdifference. Figure reproduced, with permission, from Fischer et al., 2000.

70 R. M. Gulrajani

within the heart would bring the three torso surface distributions of Fig. 2.8 closer together.On the other hand, the computed epicardial distributions (not shown here) under the threeconditions clearly revealed that the approximation of a uniform dipole layer in an isotropicheart was unable to reproduce, even in qualitative fashion, the correct epicardial potentialdistribution.

2.5.2 EFFECTS OF TORSO CONDUCTIVITY INHOMOGENEITIES

The effects of torso conductivity inhomogeneities have long been of interest to researchers. Early interest was kindled simply by the desire to know the intrinsic effect ofthe major torso inhomogeneities such as the intracardiac blood masses, lungs, ribs, spine,skeletal muscle, and subcutaneous fat on the electrocardiographic potentials. A secondmotivation today is to gauge which of the above inhomogeneities has the largest perturbing effect on inverse solutions, and therefore needs to be carefully included in the torsovolume conductor used in computing these inverse solutions. Many early forward problem studies used physical analogs of the torso, into which materials that mimicked thetorso inhomogeneities could be inserted to see their effect on the surface potentials generated by a current dipole source placed within the analog. The most informative earlystudies were, however, analytical, employing spherical or cylindrical models of the heartand torso. We have already alluded to the Brody-effect which was deduced using themodel of a spherical ventricular cavity of infinite conductivity that represented the bloodmasses, situated in an otherwise infinite homogeneous medium that represented the myocardium (Brody, 1956). The Brody model was extended by Rudy and coworkers (Rudyet aI., 1979; Rudy and Plonsey, 1980) who used an "eccentric-spheres" model in which thetorso was represented by two systems of spheres. The inner system which mimicked theheart and blood-filled cavity was eccentric with respect to the outer system which represented lungs, skeletal muscle, and subcutaneous fat. Besides confirming the Brody-effectfor radial dipoles, Rudy and coworkers described the effects of varying the conductivitiesof the lungs, skeletal muscle and fat regions. A review of their work has been written byRudy (1987). The Brody effect has been revisited recently by van Oosterom and Plonsey(1991). They reiterate that, while generally correct, the magnitude of the Brody effectis dependent on the field point, and, if the medium is bounded, also on the location ofthe reference electrode. The Brody effect can also be negative under certain conditions,where the insertion of the blood masses causes the potential to shift from positive tonegative. This corresponds to shifts of the zero isopotential line in the surface potentialdistribution.

With the development of numerical techniques for computing the torso surface potentials, attention shifted to utilizing realistic-geometry torso models to study the effectsof torso inhomogeneities. One such study was done by Gulrajani and Mailloux (1983) using a boundary-element torso model that comprised intraventricular blood masses, lungsand a skeletal muscle layer. This last was approximated as an isotropic layer of increasedthickness as first suggested by McFee and Rush (1968). No subcutaneous fat layer wasused, the approximated skeletal-muscle layer extending all the way to the surface. Equation(2.40) was used to compute the surface potentials due to the 23 individual current dipolesof the Miller-Geselowitz heart model as the inhomogeneities were introduced one-by-oneinto an otherwise homogeneous torso. Most of Gulrajani and Mailloux's findings were in


accordance with earlier work employing realistic torso models (Barnard et al., 1967;Selvester et aI., 1968; Horacek, 1971). Apart from qualitatively confirming the Brodyeffect , by activating the 23 Miller-Geselowitz dipoles in concert to generate normal activation Gulrajani and Mailloux could gauge the effects of the inhomogeneities on the normalECG and the whole body surface potential map (BSPM). The major qualitative effects wererestricted to a smoothing of notches in the ECG and of isopotentials in the BSPM due to,in descending order of importance, the blood masses, muscle layer and lungs. However,although qualitative pattern changes in the ECG and BSPM were limited to these smoothingeffects, there were large quantitative changes in both, notably magnitude increases due tothe blood masses and magnitude decreases due to the muscle layer. The latter is due to theincreased distance of the torso surface from the dipoles on account of the increased effectivethickness of the muscle layer, and the former is due to the Brody-effect enhancement on thepredominantly radial orientation of the current dipoles associated with normal activation.

The advent of powerful computers has seen more elaborate finite-element and finitedifference torso models being used for the study of inhomogeneity effects. One example isthe finite-element model developed at the University of Utah (Johnson et al., 1992), whichwas constructed on the basis of magnetic resonance images of the torso, and incorporatedlungs, an anisotropic skeletal-muscle layer, subcutaneous fat, as well as secondary inhomogeneities such as epicardial fatpads, blood-filled major arteries and veins, sternum, ribs,spine, and clavicles. This model has been used in a study by Klepfer et aI. (1997) on theeffects of the inhomogeneities and anisotropies on a known, fixed, epicardial potential distribution. Klepfer etaI. estimated 11 to 15% changes in the BSPM due to addition or removalof either the lungs, anisotropic skeletal-muscle layer, or subcutaneous fat. No major BSPMpattern changes were noted. Klepfer et al. , however, may have lessened the impact of theinhomogeneities since the starting epicardial distribution was kept fixed and not "loaded"by the changing inhomogeneities. Changes in the amplitudes of epicardial potentials withchanges in the torso conductivity have been reported by MacLeod et al. (1994) in an experimental study that used measured epicardial and surrogate torso potentials from a dogheart placed inside a human-shaped torso tank. Data from the Utah model was also usedin a study by Bradley et at. (2000) that used boundary elements for the epicardial surface,lung surface and torso cavity, but finite elements for the anisotropic-conductivity skeletalmuscle and subcutaneous fat layers. Bradley et at. used a single time-varying current dipolederived from a Frank vectorcardiographic signal to represent the excitation of the heart.This dipole was placed within the heart region and served as the source. Consequently,the calculated epicardial potentials are subject to loading effects as the inhomogeneitiesare removed or added. Again pattern changes in the BSPM were not noted, but up to 30%magnitude changes in the BSPM could be seen. Bradley et al. found that the effect of thesubcutaneous fat to be more important than that of the skeletal muscle. This was in contrastto the earlier studies of Gulrajani and Mailloux (1983) and of Stanley and Pilkington (1986)who found that the effect of the skeletal muscle layer was important. However, in both theseearlier studies the skeletal muscle layer extended all the way to the outer torso surface,there being no subcutaneous fat layer. It could well be that it is the layer that abuts the outertorso surface that needs to be correctly represented for more accurate torso potential magnitudes. On the other hand, Hyttinen et aI. (2000) in a study involving a finite-differencemodel of the torso, constructed on the basis of the US National Library of Medicine'sVisible Human Man data, gauged the effect of a 10% increase in the conductivities of the

72 R. M. Gulrajani

individual inhomogeneities on torso surface potentials due to a heart dipole . They foundthat the effect of the heart muscle was largest, followed by the intracardiac blood , skeletalmuscle, lungs and subcutaneous fat, in that order. This protocol of a 10% conductivitychange, by standardizing the extent of the change, may be a better indication of the effectof a particular inhomogeneity, than adding or removing a homogeneity altogether.

While the inhomogeneities mentioned above only affected potential magnitudes withlittle effect on BSPM patterns, Bradley et al. also found that the position and orientationof the heart in the torso made the most difference for both torso potential magnitudes anddistributions. This is in accordance with recent work by Ramanathan and Rudy (2001a) thatused measured epicardial and surrogate torso potentials from a dog heart placed inside ahuman-shaped torso tank. They found that torso inhomogeneities have a minimal effect ontorso patterns computed from the measured epicardial potentials, but that for a good matchwith measured torso potentials it was essential that heart and torso geometry be accuratelyrepresented. In an accompanying paper (2001b), they show that even a homogeneous torsocan be used in the inverse computation of epicardial potential distributions, provided heartand torso geometry are correctly represented. This was also shown in the study by Hyttinenet al. (2000) cited earlier who, having access to heart geometries corresponding to both systole and diastole, found that the error in inversely-computed dipoles increased significantlyif the wrong heart model was used in the inverse computations.

Clearly the best way to judge the effect of torso inhomogeneities on surface potentialsis to use a fully-coupled complete heart-torso solution. Buist and Pullan (2003) have donesuch a study but with their two-dimensional heart-torso slice mentioned earlier. They reportthat in none of their tested situations did the two-step equivalent dipole approach completelyreproduce the fully-coupled results further supporting the above assertion.

2.5.3 DEFIBRILLATION

Defibrillation consists of applying a high-energy shock to the fibrillating heart , the ideabeing to simultaneously depolarize all the ventricular cells thereby halting the fibrillatoryactivity. Upon recovery from the shock, the sinus node often regains control of the heart anda normal heartbeat ensues. This is the "total extinction" hypothesis for defibrillation, firstput forward by Wiggers (1940). Later, Zipes et al. (1975) proposed the "critical mass" hypothesis, whereby halting the fibrillatory activity in a certain critical mass of myocardium,thought to be greater than 75% of the total mass was sufficient for successful defibrillation.The remaining mass would then be incapable of sustaining the residual fibrillatory activity. More recently, Chen et al. (1986) suggested that even if a shock was strong enoughto halt the fibrillatory activity everywhere or in a critical mass of myocardium, it couldstill reinitiate fibrillation upon removal, and therefore had to be somewhat larger. Thishypothesis was based on two observations, first that fibrillation was only induced in dogsfor shocks between a lower and an upper limit, and second that shocks above this upperlimit of vulnerability never failed to defibrillate an already fibrillating heart, presumablybecause it never reinitiated fibrillation. Chen et al. 's hypothesis has come to be known as the"upper limit of vulnerability" hypothesis. Whichever of the above three hypotheses holds,it is clear that a certain minimum level of excitation is required at the heart for successful defibrillation which translates to a minimum value of the applied current density (or

voltage gradient) everywhere in the heart. The number used in modeling studies is between12.5 - 35 mNcm2 or 5-6 V/cm (Karlon et aI.,1994; Panescu et al., 1995; Min and Mehra,1998). At the same time, it is necessary that the current density anywhere in the heart doesnot exceed approximately 500 mNcm2

, since at these densities tissue damage is likely tooccur. Thus, it is important that the defibrillation electrodes ensure a reasonably uniformcurrent distribution in the heart. For transthoracic defibrillation, the shock is applied via external paddle electrodes, but in internal defibrillation intraventricular catheters or epicardialpatches are used for shock application.

The finite-element method is the most direct way to calculate the current density in theheart due to the defibrillatory shock. The governing equation is obtained by replacing thesurface integral in Eq. (2.54), by the uncancelled component of this integral over S02 thatremains once Eq. (2.54) is applied to all volume elements. This uncancelled component isgiven by Eq. (2.59), so that we get,

f (O'V<l». Vf3id V = f f3i lsvd V +f f3;Jn d S

v V ~2

i = 1,2, . . . . m (2.67)

Note that the number of equations spans the number of unknown potentials in the entirevolume conductor. Since during these current density calculations, the heart is treated aspassive, the first volume integral on the right-hand side is zero. Moreover, during defibrillation it is incorrect to assume that the electrodes inject a uniform current density In' Thus, itis more appropriate to use a Dirichlet boundary condition, setting the potential at the nodescorresponding to the electrodes equal to the applied electrode voltage. This eliminates thesurface integral in Eq. (2.67) also. Only the left-hand side remains in Eq. (2.67), whichmay then be reduced to global matrix form. Due to the complex torso models used, thematrices are large, and Ng et al. (1995) discuss the use of parallel computers in obtainingthe solution. Once all unknown node potentials are calculated, the current density within theelements representing the heart is obtained from the gradient of the potential distribution<1>. By using a finite-element representation for the intraventricular catheter electrode, thefinite-element approach can also be used for studying the heart current densities due toimplantable defibrillators (Jorgenson et al., 1995).

External or transthoracic defibrillation can also be studied with the boundary-elementformulation (Claydon et al., 1988; Oostendorp and van Oosterom, 1991; Gale et al., 1994;Gale, 1995). While matrix sizes are smaller, the major disadvantage is that anisotropicconductivity variations, especially those in the myocardium, cannot be taken into account.The governing equation for the potential (r) anywhere in the heart is again obtained froman application of Green's second identity to the torso geometry ofFig. 2.4 and is (Oostendorpand van Oosterom, 1991):

~f +' f In(r') ,41l'O'(r)(r) = L.J (0'/- - a/ )(r )dQrr,+ --,dS/=0 [r - r I

SI So

(2.68)

In Eq. (2.68), O'(r) is the conductivity at the observation point, and In(r') is the normal

74 R. M. Gulrajani

component of the injected current density at the defibrillation electrode. Oostendorp andvan Oosterom discuss the numerical solution of Eq. (2.68) for the unknown potentials andcurrent densities in terms of the known potentials at the defibrillation electrodes.

Simulation studies of defibrillation have concentrated on determining the optimalpositioning and size of defibrillation electrodes in order to ensure an adequate andapproximately-uniform current density everywhere in the heart (Gale et aI., 1994; Gale,1995; Camacho et al., 1995; Panescu et al., 1995; Schmidt and Johnson, 1995). More recent work has focused on determining the defibrillation threshold, namely the electrodevoltage or energy needed for a given percentage of the myocardial mass (usually 95%)to attain a voltage gradient of at least 5 V/cm (Aguel et al., 1999; De Jongh et al., 1999;Eason et al., 1998; Kinst et al., 1997; Min and Mehra, 1998). This is in line with the critical mass hypothesis for defibrillation. One early study focused on the sensitivity of thecurrent-density distribution in the heart to variations in skeletal muscle anisotropy (Karlonet al., 1994). It was found that in transthoracic defibrillation, the anisotropy made littledifference to current flow patterns in the heart, but simply affected current magnitudes. Onthe other hand, the same study showed that other inhomogeneities such as the lungs, ribsand sternum affected both magnitudes and current patterns. Another study (Eason et aI.,1998) found that the voltage defibrillation threshold, for internal defibrillation between aright ventricular catheter electrode and the defibrillator can in the pectoral region of theleft chest, differed by only 4.5% if the realistic fiber architecture in the heart model wasreplaced with an isotropic conductivity myocardium.

It is being widely recognized today that defibrillation is an immensely complex phenomenon, and that the above simulation studies yield, at best, an estimate of the extracellularpotential gradient and current density inside a monodomain passive heart. What really countsis the transmembrane potential distribution in the heart, and the response of the active cellsin the heart to this transmembrane distribution. Theoretical analysis has revealed that evenwith extracellular stimulation of the passive bidomain heart, the unequal anisotropies ofthe intracellular and interstitial space result in contiguous regions of large transmembranepotential depolarization and hyperpolarization in the heart (Sepulveda et al., 1989). Localcurrents flowing between these regions have the ability to re-initiate fibrillation and negatethe findings of studies that assume a monodomain heart. Experimental work and theoreticalsimulations both tend to support this re-initiation mechanism (Efimov et al., 2000; Skouibineet al., 2000). A second perturbing factor that affects real transmembrane voltage gradientsin the heart is the effect of local conductivity discontinuities (e.g., gap junctions, fiber curvature, random clefts). Sobie et at. (1997) have shown how, using a "generalized activationfunction," the effects of both extracellular stimulation and of conductivity discontinuitiescan be taken into account in determining the transmembrane potential distribution. Thisgeneralized activation function suggests that more than the gradient of the extracellularpotential, it is the second spatial derivative of the extracellular potential that determines thetransmembrane potential change. Although Sobie et at. illustrated their approach in passivemyocardium, it can just as easily be applied to active myocardium if sufficient computationalresources are available. This represents the final, and most important step, in translating anapplied extracellular potential to the actual response of the cardiac cells. Already, Skouibineet at. (2000) illustrate such active responses in simulations in a two-dimensional bidomainsheet of active myocardium subjected to a defibrillation shock.


2.6 FUTURE TRENDS

75

This review of the forward problem of electrocardiography has presented its theoretical underpinnings, the solution methodologies employed, and finally its major applications.The greater accessibility of multiprocessor computers means that we shall likely see moreaccurate heart models capable of simulating more complex heart pathologies, with simultaneous computation of heart activation and torso potentials via a complete solution. Torsorepresentations are also likely to be more accurate with better accounting of anisotropic conductivity, and with the effect of torso inhomogeneities gauged with a complete fully-coupledthree-dimensional heart-torso solution. Finally, successful defibrillation is more than justa matter of knowing the extracellular potential or current density everywhere in the heart.Rather it is the interaction of this potential or current density with the cardiac cell, and thesubsequent effect on the cellular action potentials that determines whether the defibrillation shock succeeds or fails. While the defibrillation simulations described above may beuseful in electrode design, a better understanding of defibrillation necessitates simulationsin which the heart is not treated simply as a passive conductor, but as an active bidomain.Already, Efimov et al. (2000) have described the transmembrane potentials generated in athree-dimensional realistic-geometry rabbit-heart model with varying fiber directions, following uniform electric-field application. Although they represented the myocardium withpassive bidomain equations, it is only a matter of time before such whole-heart defibrillationsimulations will be realized with an active membrane representation for the heart's ioniccurrents.

ACKNOWLEDGMENT

Work supported by the Natural Sciences and Engineering Research Council of Canada.

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3

WHOLE HEART MODELING ANDCOMPUTER SIMULATION

DamingWeiGraduate Department of Information System,The University of Aizu, Japan

3.1 INTRODUCTION

Bioelectrical models of the heart are studied in three levels : the single-cell model, thecell-network (tissue ) model and the whole heart model (Wei, 1997). The single-cell modeldescribes ionic current flow across myocardial cell membranes. The cell-network modeldescribes ionic current flow between aggregates of myocardial cells in temporal and spatialdomains. Details of these model s have been described in the previous chapters. The development and spread of ionic currents throughout the heart and body volume conductor resultin electrical potentials that can be measured on the body surface, called electrocardiogram(ECG). A whole heart model describes three major mechanisms: the propagation of activation in the heart, the cardiac electrical sources, and the extracellular potentials within andon the body surface. Thi s kind of model is able to relate the body surface ECG waveformsto the action potential, conduction velocity of cardiac tissue and other electrophysiologicalproperties of the heart and, thus, yield clinically comparable ECG waveforms. For this reason, the whole heart model offers a unique means to bridge the clinical applications withthe single-cell or cell-network models .

Conventionally, a whole heart model refers to a 3-dimentional (3D) heart-torso modelthat contains realistic geometry of the heart and torso. Some heart models contain both atriaand ventricles to represent the entire heart , but some only contain a portion of the heart suchas the ventricles, or just the left ventricle. A computer heart model is usually representedby a 3D voxel array. The voxel in each grid is called an element. As the array increases insize, the model exhibits higher resolution. In most of the existing whole heart models, theelement is of an order of 1 mm in diameter. It is usually several thousand times larger thana true cardiac cell . In simulation studies, the element is usually treated as a cardiac cell,called a model cell in this chapter, but it is important to keep in mind that the element ormodel cell in a whole heart model is actually a lumped model of tissue.

Tsuruga, Ikki-machi, Aizu-Wakamatsu City, Fukushima 965-8580, Japan, Tel. +81-242-37-2602, Fax. +81-24237-2728, [email protected]

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Whole heart modeling and computer simulation are typical topics associated withthe electrocardiographic forward problem. Since the main theory and methodology for theforward problem have been described in detail in Chapter 2, this chapter will focus on how toconstruct a 3D heart-torso model, how a whole heart model can be applied to understandingmechanisms of bioelectric phenomena of the heart, and to relating the electrical activity ofthe heart to the surface ECG.

3.2 METHODOLOGY IN 3D WHOLE HEART MODELING

The main issues in whole heart modeling shall include: heart-torso geometry modeling,specialized conduction systems, rotating fiber direction and myocardial anisotropy, actionpotential and propagation, cardiac electric sources, and body surface potential calculation.These topics are described in detail in the following sections.

3.2.1 HEART-TORSO GEOMETRY MODELING

Descriptions of heart anatomy can be found in a vast number of textbooks. For heartmodeling purposes, one should acquire some basic knowledge of the heart's anatomicfeatures. First of all, we should have a quantitative concept of the heart's size. One pointis that the left ventricle including the left free wall and the interventricular septum (wallthickness 9-11 mm) constitute the major mass of the heart. The right ventricular wall isabout 1/3 the thickness of the left and, therefore, about 1/27 of the total myocardium involume. The wall of the atria is even thinner. This means that the left ventricular myocardiumprovides the major contribution to the generation of ECG.

To construct a heart model with a realistic shape, the first step is to map the heartgeometry onto a matrix of a regular 3D grid. In the early stage of model development(Okajima et al., 1968), the heart geometry was constructed using traditional digitizationtechniques. With such a method, a heart is first embedded in a gelatin solution and thenfrozen. After that, the frozen heart is sliced and photographed, and then pictures of eachslice are digitized and input into a computer.

For construction of high-resolution heart models, it is necessary to use CT imagingtechnology. In the study of Rorange et al. (1993a), a human heart obtained from autopsywas first inflated with air at end-diastolic pressures and submerged in liquid nitrogen. Thefrozen heart was then CT-scanned in 1 mm intervals resulting in 132 slices of 512*512image data. These data were processed to extract the heart edges and eventually to obtaina 3D matrix of about 250,000 points spaced 1 mm apart.

Recently, several databases provide accurate and anatomically detailed images ofthe human body available for model studies. One of the most commonly used databasesfor this purpose is the Visible Human Project of the U.S. National Library of Medicine(http://www.nlm.nih.gov/research/visible/visibleJmman.html). It provides 1,871 sectionimages of a human cadaver at 1 mm intervals. The original images are 1048 by1216 pixels with 24-bit color. These data have been used in recently published models(Balasubramaniam et al., 1997; Kauppinen et al., 1999; Ramanathan and Ruddy 2001a).

The main techniques to model the heart-torso shapes include image segmentation andelement generation. To the author's knowledge, there is no full automation technology

Whole Heart Modeling and Computer Simulation 83

currently available for image segmentation for the purpose. In most studies, the segmentation is performed in a semi-automatic manner, which includes steps of digital filtering,image enhancement, region growing processing, and segment decision (Heinonen et al.,1996; Hsiao and Kao, 2000). After these procedures, the boundary contours of distinctanatomical regions are obtained. Based on these data, the anatomical regions are filled withdiscrete elements representing distinct cardiac cell types. The cell types usually includecells of the atria, ventricle and specialized conduction system. For an inhomogeneous torsomodel, the volume is usually divided into piecewise homogeneous regions that may includethe heart with blood mass, the lung, the fat, and the bones, with distinct regions being assigned different values of electrical conductivity. Cell classification can only be performedmanually with the current techniques.

An important issue in whole heart modeling is spatial and temporal resolutions ofthe model. Qualitatively, the model's resolution mainly affects the propagation details,especially if rotating fiber directions are taken into account. For surface ECG calculations,the resolution of most existing heart models is sufficient, considering the fact that manystudies reduce the total number of elemental dipoles to a few multiple dipoles beforecalculating the surface ECG potentials. A heart model of 1 mm spatial resolution hasenabled the simulation of reentrant propagation such as spiral waves (Panfilov 1993, 1995).In a recent model of Hren and Horacek (1997, 1998), a high resolution 0.5 mm is adaptedwith 1.7 million cells. The heart model has enclosed more anatomic details including themyocardial wall, trabecular tissue, and papillary muscles.

3.2.2 INCLUSION OF SPECIALIZED CONDUCTION SYSTEM

For studying bioelectrical phenomena, the heart can be simply thought of as an electricgenerator comprising a specialized conduction system (SCS) and excitable myocardialtissues, as diagrammed in Fig. 3.1. Understanding the SCS is important for whole heartmodeling because it is one of the key issues that affect the propagation sequence of activationin normal and abnormal hearts. In the following paragraphs, preparative knowledge for theactivation of the heart and the role of the SCS are briefly introduced while referring to Fig.3.1. Detailed references can be found in other references (Guyton, 1986; Van Dam, 1989).

For a normal heart, the electric pulse is spontaneously generated by the sinus node,the pacemaker (located in the upper right atrium as shown in Fig. 3.1). The pulse stimulatesthe neighboring atrial cells to activate them. The activating cells in tum stimulate theirneighboring cells so that the excitation propagates throughout the atria. Although somestudies suggest that there may exit a specialized conduction system in the atria, there is nodefinite evidence to support this assumption.

As clearly illustrated in Fig. 3.1, there is no direct "electric connection" between theatria and the ventricles, except in some abnormal hearts such as those suffering from theWPW syndrome (described later in Section 3.4). For normal hearts, the only pathway forconduction of the excitation between the atria and the ventricles is the atrioventricular node(AV node) and a specialized conduction system consisting ofthe His bundle, the left and rightbound branches, and the Puikinje network, as illustrated in Fig. 3.1. The most important feature of the specialized conduction system is the high conduction velocity in comparisonto that of normal myocardial fiber. The typical conduction velocity in the His bundle andthe main branches is 2 mS- I , while that in the ventricular myocardium is 0.5 mS- I . The

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FIGURE 3.1. A schematic diagram of the heart and the specialized conduction system.

D.Wei

distribution of the Purkinje network influences the heart excitation sequence the most, andtherefore is very important to the model study. The Purkinje fibers penetrate the septal myocardium near the apex and are distributed in the subendocardium. In general, the Purkinjenetwork is distributed on the apical half of the subendocardium and absent on the basal half.This configuration avoids outflow obstruction.

In whole heart modeling, distributing the Purkinje network in the ventricular modelgreatly influences the excitation process of the heart model and the resulting body surfaceECG . Because it is not possible to identify the Purkinje fiber from image data, modelingthe Purkinje network is usually based on the excitation sequence of the ventricles observedby experiments. The most cited literature regarding human heart excitation is that of Durreret al. (1970). The measured excitation isochrones in isolated human hearts are shown inFig . 3.2. According to Durrer et aI., the early excited areas in the left ventricle were observedin three endocardial sides : high on the anterior paraseptal wall just below the attachmentof the mitral valve, central on the left surface of the interventricular septum, and posteriorparaseptal about one third of the distance from the apex to the base. Early excitation of theright ventricle was found near the insertion of the anterior papillary muscle. Septal activationwas found to start at the middle third of the junction of the septum and posterior wall. Thesedata are useful for arranging the Purkinje fiber in a heart model and for evaluating thesimulation results of the model.

Inclusion of the SCS to a heart model is usually a time-consuming task in the construction of a model. The positions and distributions are usually adjusted repeatedly until

Whole Heart Modeling and Computer Simulation

o 5 10 15 20 25 30 35 40 45 50 55 60 65 ms

8S

FIGURE 3.2. Excitation sequence of the human heart. (Reproduced with permission from Durrer et aI., 1970).

simulated excitation isochrones fit the experimental data. In the adjustment, the distributionof the Purkinje network, the position of joint points connecting bundle branches to thePurkinje network, and the activation time arriving at these points have the most importanteffects on simulation results.

In most whole heart models, the Purkinje network is represented by a one-layer sheetand the bundle branches are represented by cables (Aoki et al., 1987; Rorange et al., 1993).The sheet model of the Purkinje fiber network is supported by some experimental studies.In the model of Aoki et aI., the left bundle branch terminates at three points: the central regionof the septum, the antero-basal region, and the postero-apical region of the left endocardium.The right bundle branch terminates at one point on the antero-apical region of the rightendocardium. A more detailed model of the specialized system can be found in Al-Nashashand Lvov (1997), where the His-Purkinje electrogram is the target of simulation. In thisstudy, a His-Purkinje model with a 3D curvature resembling the ventricular endocardialsurface is built and the His-Purkinje system electrogram is simulated using the volumeconductor theory.

3.2.3 INCORPORATING ROTATING FIBER DIRECTIONS

Myocardial anisotropy is an important issue in studying cardiac phenomena. Nowadays, a whole heart model without inclusion of myocardial anisotropy would hardlybe accepted. Inclusion of myocardial anisotropy in a heart model involves three aspects: anisotropic geometry of the myocardial muscle, anisotropic propagation, andanisotropic cardiac sources. Anisotropy geometry means that a heart model should have

86 D.Wei

location-varying fiber directions for all discrete elements, based on the fact that the ventricular myocardium has a spiral structure with fiber orientations rotating from the epicardialsurface to the endocardial surface in a total angle of 900 to 1200 (Streeter et aI., 1969). Thisis known as rotational anisotropy. Anisotropic propagation requires a direction-dependentconduction velocity in controlling the excitation process of the model. The anisotropiccardiac source arises from the fact that both the intracellular and extracellular domainsare anisotropic, and the anisotropic ratios are essentially different from point to point inmyocardial muscle.

There are several ways to incorporate rotating fiber directions in a heart model. Thesimplest way is called stylized representation, as used in Lorange et al. (1993). With thismethod, a family of nested ellipsoids of revolution extending from the endocardium to theepicardium is used, where the fiber angle varies 1200 from the endocardial to the epicardialellipsoids. The fiber direction at a model point is obtained by determining which ellipsoidpasses through the point. A similar method is used in other models (Adam et al., 1987;Leon et aI., 1991a, b). The advantage of this method is the simplicity in calculating thepoint-by-point fiber direction. The disadvantage is that the fiber directions are too simplein comparison with actual heart anatomy.

The most precise way to incorporate fiber direction is through microscopic determination of fiber orientation as reported in Panfilov et al. (1993). In this study, detailed dataof the ventricular geometry and fiber orientation were microscopically measured on anintact canine heart to produce a finite element model (Nielson , 1991). These data werethen mapped onto a regular 93*93 *93 grid with 1 mm of distance between the grid points .Compared to the stylized representation, this method provides more geometric details ofthe heart.

Microscopic determination of fiber orientation is not always possible for model studies .Wei et al. (1989, 1995) proposed a discrete method to calculate point-by-point fiber direction.This method can generate intermediate precision between that of microscopic determinationand stylized representation. This method uses the following assumption based on Streeteret al. : (1) the myocardial fibers of the ventricles have a layered structure, (2) all fiberorientations are parallel to each other in one layer but different from layer to layer, (3) thefiber orientations rotate counterclockwise over 90° to 1200 with increasing depth from theepicardium to the endocardium.

Fig. 3.3 (a) shows a longitudinal cross-section of the heart model. The spatial configuration of the elements is illustrated in Fig. 3.3(b). To assign each element a fiber direction,the model is layered as shown in Fig. 3.3(c). The myocardial fibers in one layer are mathematically described as intersection curves by cutting the layer with a group of parallelplanes, called fiber planes, as illustrated by Fig. 3.3(d) and (e). Obviously, the fiber planesin one layer have a common normal direction, called fiber plane direction (FPO). Following experimental data (Spaggiari , 1987), an FPO perpendicular to the geometric heart axisalong the apex-to-base direction is assigned to the epicardial layer. Then , the FPO is rotatedcounterclockwise in the septal plane to determine FPOs for all layers in the epicardium-toendocardium sequence. The rotating angle for a layer I with respect to that of the FPO ofepicardium is given by

Aa(l) = I·

N(3.1)

WholeHeartModeling and Computer Simulation

(a)

87

FIGURE 3.3. Heart model of Wei et al. (a) Longitudina l cross-sect ion of the heart model. P denotes the Purkinjefiber. (b) 3D configuration of model elements . (c) Model elements layered for solving point-by-point fiber directions. (d) Illustration of rotational fiber orientations in the layers. (e) Mathematical relationships of fiber plane,fiber plane direction, and fiber plane rotatio n. FPo: fiber plane of the outermost layer; FPj: fiber plane of the i-thlayer; FPDj: fiber plane direction of the i-th layer; ali): rotating angle of layer i ; PS: septal plane (Reproduced withpermission from Wei et al., 1995).

88 D.Wei

..................·....". .. . ...........· , .... .... ..... .• If • •• I I . · ··

~'NtY~w..'.'.'.'.'.' .' . ' . ' . ' .' . ' .' .• # •• • • • • ••• • •••

•••• If • • I t • • • • ••

• ••••• • • • ••• 10 •••... ..... ...... ... ... ...... ...... .....

~~~~illllll~~ .... ..... ............ .. .. .... , .. .. . .. . .. .. .. .. .. . .. ..." ...... ............................ ... .. .. .. - ...... "" I • • .... .. .. . .. .. .. .. .. .. .. .. . .... .. ... .. .. ....

..::::::::::.'::: .:::~'::: : : :~.:::::::::::::::::::::::: :::::::::::::::: :::::::: :::.-:::: ::::

, ,. ,

11111111111111111 layer ~I '. ".lnyei ]

(e)

FIGURE 3.3. (cont.)

where N is the total number of layers of the model and A is the total rotating angle betweenthe epicardial and endocardial layers, having a value between 90° and 120°.

In implementation, the heart model is layered from the epicardium to the endocardiumby applying simultaneous stimuli to all model elements on the outermost layer (the epicardium) of the model and solving the propagation sequence of "virtual excitation" for theentire model. The septum of the heart model is specially treated in the propagation processso as to make the septal fibers natural extensions of the left ventricular fibers. As a result,


(c)

FIGURE 3.3. (cont.)

the sequence number of such a propagation process corresponds to the layer number ofthe model element. Then , the outer product of two unit vectors determines the local fiberdirection at each element. One is the fiber plane direction of the layer to which the unitbelongs, and the other is the normal direction of the layer at that element. If an element(i, i .k) belongs to layer l and the layer has a fiber plane direction of P(l) and a normaldirection N (i, j , k) at (i, i .k), the local fiber direction is given by

F(i , j, k) = P(l ) x N(i , i .k) (3.2)

To confirm the rotating fiber directions of the heart model, the model is stimulated at theleft ventricular wall at three depths : at the epicardial layer, the intramural layer, and theendocardial layer.The isochrones are shown in Fig. 3.4. The long axes ofthe isochrones showapproximately 90° of total rotation from the epicardial (Fig. 3.4b), intramural (Fig. 3.4c),and endocardial (Fig. 3.4d) layers. For comparison, propagation in the isotropic model isshown in Fig. 3.4a.

Determination of fiber direction can also be performed in terms of analytical methods.In the study of Hren and Horacek (1997), the epicardial and endocardial surfaces are analytically obtained with sampling data from CT images using surface harmonic expansion(Hren and Stroink, 1995). Then, the principal fiber direction at a point lying in a tangentialplane at an element is determined so that the fiber direction rotates counterclockwise fromthe epicardial to the endocardial surface.

3.2.4 ACTION POTENTIALS AND ELECTROPHYSIOLOGIC PROPERTIES

In simulation studies , the model cell mimics the actual cardiac cell to reconstruct thecardiac process . Like an actual cell , the model cell should own the action potentials andother electrophysiologic properties. These properties are the main input data for simulationstudies .

(a) .

.. ... .. .. . .• ••• • : • • • • • • • • . • . • ::: •.• • : : . • : ••••• •• • : • .• , 0 ••••••••• '0' '"0 0°: •

• • 0 .

.. ... ..... . . . .......... .. .. ..... .... ..... . .... .. .. .. . . ,0 0 ••

........ . . . .. ... .. .. .. . . 0 . 0 0

.. .. . . . .. . . . . . . . 0.0 ...... .. .. .. .. .. .. .. .. . . .

..... 0 ••• •••• • •• •

................ 0 . ... •• •• .. • • • • ..

•••••• • 0 • • •• ..

' 0' ••• ••••• 0 0 •

(h)

FIGURE3.4. Simulated isochrones created by applying stimuli at different depths in the left ventricle of the heartmodel. The stimulus is denoted by '+'.Time sequence is identified by the thickness of the circle and its centerpoint: the thicker the circle and the center point, the more advanced the time sequence. Each increase in thicknessof the circle represents 6 ms, and that of the center point represent s 36 ms (circles with no center points representthe first 36 ms).(a) Epicardi al isochrones of the left ventricle with the isotropic propagation(b) Epicardial isochrones of the left ventricle with anisotropic propagation(c) Intramurallayers beneath the left epicardium with anisotropic propagation(d) Intramural isochrones 9 layers benea th the left epicardium with anisotropic propagationThe isochrones from (b) through (d) show rotating fiber directions in the heart model. (Reproduced with permissionfrom Wei et al., 1995).


•• • • •• • • •. . . . • . ••.... . • . ; • ••••• •• •• ~ t:"l. .. .. ......... . . .... . .......... .... . .. . .. .. .... .. .... . . . .... ... .. . . . . .... . .... .. .. .. . .. . .. .. .. . . . .. .. .. ....... .. ... .... ... ... .. .. .... .... .. . .. , .... .. . .. . .. .... .... .. . . . . . . . . . . . ...... . .. .. .. ...... ... .... . .. .. .. . .. . .... .. . ... .. .. ... . .. .. .. .. .. . ... . ..... .. . ...... . .. .. . .. .. . .. .. .

.. . . .. . . . .. . .. . . ... .

91

(c). ....... .. .... . .. .... .. . . .. .. .. .. .. .. ..

i: ·1;]'

(d) ~

FIGURE 3.4. (cant.)

The ideal way to assign action potentials to the model cell is to use the single-cellmodel to generate action potential as a function of time and distances. However, since thecalculation of action potentials with single-cell models is computationally demanding andthe scale of a whole heart model is quite large, most existing whole heart models do notdirectly calculate the action potentials, but assign pre-defined action potentials to the modelcells.

For an electrophysiological simulation study, one should take into account at leastthe following basic nature of action potential in modeling. First, different kinds of cardiaccells have different action potentials. Second, at least for the ventricular cells, the actionpotential duration (APD) is location-dependent in the heart, longest on the endocardium andthe base, and shortest on the epicedium and the apex (Harumi et al., 1964). Third, the APD

92 D.Wei

is time-varying during premature excitation, depending on the restitution property with thecoupling interval (Harumi et al., 1989a).

In many studies, the cardiac cell is represented by finite state automaton. This is calleda cellular automata (CA) model (Siregar et aI., 1996). In fact, the CA model was first usedin an early study by Moe et al. (1964) to simulate atrial fibrillation . In this study, a discretetime step of 5 ms and a unit diameter of 4 mm were used. One time step was the propagationtime in a fully recovered unit , corresponding to a normal conduction velocity of 80 cmlsfor the atrial tissue. The excitability of the units was represented by 5 states. The first statecorresponded to the absolute refractory period, having a duration of R = K JC,where Cwas the preceding cycle length and K was a property of the unit, different from unit tounit. States 2, 3 and 4 together corresponded to the relative period. Dividing the relativeperiod into different states incorporated the time-varying conduction velocity during thepropagation process. From state 2 to state 4, the conduction velocity increased from 1/4to 1/2 of the normal conduction velocity. In state 5, the conduction velocity returned tonormal. State 5 lasted until the next excitation and corresponded to the recovery period.

Because the model of Moe et al. was aimed at studying the propagation of activation,the waveform of the action potential was not needed . For models that need to calculate ECGpotentials, the waveform of the action potential should be defined. A simple representation isthe piecewise linear approximation as used in Siregar et al. (1996). In this model , the musclecell is resented by four states : resting, depolarization, absolute refractory, and absoluterefractory. The function of the action potential, Y, is expressed by

(3.3)

where e and i denote model cell and state, respectively, a and f3 define a line for the celland state. As functions oftime, a and f3 are context-dependent so that the action potential istime-varying according to the prematurity of the incoming impulse. This model is capableof simulating typical types of arrhythmias.

In assigning action potentials to the model cells, there is a technical difficulty due tothe fact that a 3D heart model usually has a huge number of cells . If each model cell hasto be accompanied with a dataset containing an action potential and other parameters, thememory space for modeling would be extremely large. One method for solving the memoryproblem used by Wei et al. (1995) is to assign the action potentials and electrophysiologicparameters to cell types, instead of individual cells. In this way, model cells are organizedin limited cell types. Action potentials and other electrophysiologic properties are linked tocell types so that limited memory is required. There are sixteen cell types available in themodel of Wei et al. Each of them is associated with a parameter table containing the actionpotential waveform and other electrophysiological parameters as listed in Table 3.1. Theavailable cell types include normal cardiac cells like the atria, ventricles, and a specializedconduction system. Besides, any special cell types, such as ischemia, infarction, or ectopicbeats , can be defined by parameter settings suitable to the study.

The following paragraphs show how the action potential and other electrophysiologicaldetails are assigned to the model cells in Wei et al. ( 1990, 1995). See Table 3.1 for meaningsand abbreviations for the parameters, and see Fig. 3.5 for the action potential definition.

The basic parameter relative to action potential is action potent ial duration (APD) inms, taken as the sum of phases °through 4. The parameters relative to time are given as a

TABLE 3.1. Electrophysiologic Parameters

Parameter

Action PotentialAPDTOTlT2T3T4VOVIV2V4GRDDVTDC

Conduction:CVLCRDC

Automaticity:ICLPRTDLYACCBKP

Pacing:BCLBNICN

Definition

action potential duration (ms)duration of action potential phase 0 (% of APD)duration of action potential phase 0 (% of APD)duration of action potential phase 0 (% of APD)duration of action potential phase 0 (% of APD)duration of action potential phase 0 (% of APD)potential of phase 0 (resting potential) (mv)maximum action potential (mv)potential of phase 2 (mv)potential after full recovery (mv)gradient of the APD distribution (mslIayer)deviation of APD for random distribution (mv)APD change to coupling interval (%)

conduction velocity along the fiber axis (m1sec)anisotropic ratio (m1sec)conduction velocity after ARP (m1sec)

intrinsic cycle length (ms)protected or non-protected (yes/no)maximum delay of the phase response (% of ICL)maximum acceleration of the phase response (% of ICL)break point position of the phase response (%)

basic cycle length of pacing (ms)beat number of pacing for the simulationincrement of cycle length per cycle (% of BCL)

YO

T2

V2

T3

4

FIGURE 3.5. Definition of action potential used in the simulation (Reproduced with permission from Wei et aI.,1995).

94 D.Wei

percentage of the APD. Lines specified by time intervals and voltages are used to representphases 0, I, 2 and 4 of the action potential waveform. Phase 3 is defined by a curve throughinterpolation (second order Lagrange interpolation) of sample data. The action potentialwaveform is linked to each cell type and used as a look-up table during simulation.

To distribute the pre-defined action potential waveforms over the ventricles so thatthe APD lengthens from the epicardial to the endocardial and from the base to the apex,a parameter GRD (see Table 3.1) is defined to specify the gradient interval along theepicardial-to-endocardial and base-to-apex sequence. Thus, the value of APD for a cell atlocation (i, j, k) is given by

APD(i, i. k) = APo, + GRD· SQ(i, j, k) (3.4)

where APDd is the defined value of APD for the cell type, and SQ(i, i. k) is a sequentialnumber of the cell sorting along the epicardial-to-endocardial and base-to-apex directions.In this model, a positive GRD value of about 5 ms per sequence yields a normal T wave inthe simulated ECG. Adjusting the parameter GRD in a simulation study is useful to simulateT wave abnormalities.

To make the APD adaptive to the coupling interval so that it is dynamically modifiedduring the simulation, a parameter called dynamic coefficient (DC) is defined as the ratioof the change in APD to the change in coupling interval. During simulation, the value ofAPD at any time, t, is dynamically modified by

APD(t) = AP D(t - 1) + DC . fj.C l(t) (3.5)

where fj.C/ is the change in coupling interval at time t.In addition to the action potentials, many other electrophyosiologic properties are

defined with parameters. These parameters are concerned with conduction velocity, pacing,and automaticity. The conduction velocity is made adaptive by assuming a piecewise linearfunction of the coupling interval. During the absolute refractory period, the conductionvelocity is zero. After full refractory, the conduction velocity is assumed constant specifiedby a parameter, CVL (see Table 3.1). During the relative refractory period, the conductionvelocity is a linear incremental function of the coupling interval. To define a cycle lengthand pacing times, anyone or more cells can be paced in the simulation. The pacing ratecan be increased or decreased with the changing rate specified by a parameter INC, whichis defined as a percentage increment to the basic cycle length. The parameters relative toautomaticity define the electrotonic interaction between normal and ectopic pacemakersand is used to simulate cardiac arrhythmias (Wei, 1992).

3.2.5 PROPAGATION MODELS

There are three types of propagation models used in whole heart modeling. These arepropagations based on Huygens' principle, Hodgkin-Huxley (HH) formulism (Hodgkinand Huxley, 1952; Plonsey, 1987; Leon et al., 1991a, b; Beeler and Reuter, 1977; Luo andRudy, 1991, 1994a, b), and the FitzHugh-Nagumo (FN) model (FitzHugh, 1961; Panfilovand Keener 1995; Berenfeld, 1996). The model using Huygens' principle provides thesimplest way to simulate the propagation of the action potentials. It requires calculations to


generate wavelets around activating cells at each discrete instant. Cells within the waveletsare to be activated at the next instant. Constructing the wavelets for each instance yieldsthe excitation sequence of the heart model. The effectiveness and the details of this type ofmodel depend on the way the action potential and other electrophysiologic properties aredefined. If the definitions contain sufficient electrophysiologic details, the heart model isefficient enough to describe the dynamic process of the heart such as the cardiac arrhythmias. The propagation model based on HH formulism is theoretically most correct. It hasthe capability to respond to ionic current, membrane potentials, and other factors to providedetailed information that cannot be provided by the former. However, the computation complexity of such a model limits the application in large-scale models. While most publishedstudies using HH formulism in simulating propagations are two-dimensional models, manyresearchers currently show increasing interest in applying the HH type propagation to 3Dwhole heart modeling. A recent work is discussed in Chapter 2. Propagation using theFitzHugh-Nagumo (FN) model is a compromise between the implementation complexityand the electrophysiologic details. The FN type cellular dynamics is an electrodynamicmodel that reconstructs the action potential by reaction state equations. Compared to HHtype equations, numerical solutions for FN type propagation need much less computationtime.

3.2.5.1 Propagation model ofHuygens' type

As an example, the propagation of the Huygens' type is introduced with the model ofWei et al. In this model, the excitability of model cells is one of two basic types: conductiveor non-conductive. Non-conductive activation is concerned with pacemaker cells. For thesecells, the activation is obligatory whenever the firing time comes. Conductive activation istreated in three steps. The first step is to calculate the extent of propagation in one time steparound each excited model unit by constructing an ellipsoidal wavelet based on local fiberdirection as shown in Fig. 3.6. The long serniaxis of the wavelet is along the fiber direction,and the other two short serniaxes are along the two transversal directions. The lengths ofthe long and short serniaxes are

and

RI = Vt(t)· T (3.6)

(3.7)

respectively, where T is the time step, Vt(t) is the longitudinal conduction velocity of thecell at time t, and k, is the conductivity ratio. The extent of propagation in one time step isdescribed by

(3.8)

where I, n, t are the distances along and across the fiber direction in the local coordinatesystem established at the excited element.

96

(;70

(a)

Fiber direction

D.Wei

(b)

FIGURE 3.6. An illustration of the propagation wavelet.

Cells within the wavelet can be activated if they are excitable at that time. The excitability is checked by the principle of refractoriness as

(3.9)

where Tpre is the starting time of the previous excitation of the model unit and TA R P(t) isthe absolute refractory period at time t.

If the model cell is recognized as excitable, the third step is to assign a conductionvelocity to it for propagation at the next time step. Note that the conduction velocity in thismodel is time-varying, depending on the coupling interval.

In this model, the automaticity of a pacemaker cell can be either protected or unprotected, depending on the parameter setting. For unprotected automaticity, stimuli fromneighboring cells unconditionally activate the unit and reset it automatically after activation. For the protected automaticity, the model unit is not directly activated by surroundingstimuli, but its intrinsic cycle length is modulated based on a so-called phase responsecurve (PRe). Fig. 3.7(a) shows a PRC assigned to the cells of an ectopic pacemaker tosimulate the ectopic firing of the ventricular parasystole . The biphasic line of the PRC was


Delay (%)

SO

40

30

20 DLY

10

0

-10

-20

-30

-40

-SO(a)

(b)

97

FIGURE 3.7. (a) The phaseresponse curveassigned to cells of an ectopic pacemakerin simulatingectopic firingof ventricularparasystole (Reproduced with permission from Wei et aI., 1995). (b) SN: sinus node, the normalpacemaker; PVC: the ectopic pacemakercausing premature ventricular contraction. (c) From top to bottom aresimulated ECGsof bigeminy, trigeminy, quadrigeminy and an operativeratio of the 5:I type.

98

(e)

~.omv

O.2sec

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II

II

II

FIGURE 3.7. (cont.)

used to approximate the experimental data (Jalife, 1976; Moe, 1977). By assigning propervalues of intrinsic cycle length to sinus and ectopic pacemaker cells (Fig. 3.7(b», ECGwaveforms of bigeminy, trigeminy, quadrigeminy, and an operative ratio 5:1 type were simulated as shown in Fig. 3.7(c) . The simulated ECG waveforms are comparable with thosein clinical findings. They are also in agreement with a theoretical model where the cardiacdynamics are represented by difference equations (Ikeda , 1983). The example demonstrates that a Huygens' type model is able to dynamically reproduce the electrophysiologicprocess of the heart when adaptive action potential and other properties are used in thesimulation.

3.2.5.2 Propagation ofHodgkin-Huxley type

The HH equation is expressed as

(3.10)

where L; is the transmembrane current density; Vm = <1>; - e is the transmembrane potential; em is the membrane capacitance; gk, gNa, g/ and Ei ; E Na , E/ represent the conductivityand Nemst potential of potassium, sodium and leakage, respectively. Originated from theHH model, the membrane ion kinetics has been modified based on new experimental findings in recent years, leading to a wide range of single-cell models such as the sinus cellmodel (Noble, 1989), the atrial cell model (Earm and Noble, 1990), the Purkinje fibermodel (DiFrancesco and Noble, 1985), and the ventricular model (Beeler and Reuter, 1977;Luo and Rudy, 1991, 1994a, b). For simulating 2D or 3D propagations of activation in theventricular myocardium, the Beeler and Reuter model (referred to as the BR model) andthe Luo and Rudy (referred to as the LR model) are widely used. The BR and LR modelsare general mammalian ventricular cell models derived from experimental data and mathematically represented by HH type formulism. These models are expressed in physiologicalparameters so that they are well suited for simulation studies. Use of BR model takes a

smallercomputationload than the LR modeldoes, but the LR modelhas incorporatedmorerecent informationand providesmore options such as the ability to chang the extracellularion concentrations.

To model HH type propagation, it is necessary to link the transmembrane currentdensity, L; to an excitable tissue network arising from the electrical structure. For a onedimensionaltissue model, the propagation of the action potential is described(PlonseyandBarr, 1987)by

1 av,lm=-----

Lnair, + re) az (3.11)

wherea is the radius of the fiber, r, and r; are the intracellularand extracellularresistancesper unit length along the axial coordinate of z. The equation expanded to 20 and 3D withanisotropy canbefoundinPlonseyandBarr(1987).Usually, thepartialdifferentialequationis solved numerically with discrete steps in time and space domains.

One difficulty in the implementation of HH type propagation is that the numericalsolutionfor Vm(x, y, z, t) in a 3D model sometimestakes an impracticalcomputationload.This is the reason why early studies of HH type simulation used 20 tissue models (Viraget al., 1998). On the other hand, however, HH type propagation can relate surface ECG tocellularinformation whenusedin 3Dwholeheartmodeling. Recently, someresearchgroupsare trying to realize HH type propagation in 3D whole heart modelingin different ways. AJapanese group is establishing such a model using a supercomputer(Suzuki et aI., 2001).Gulrajaniand coworkershave recently simulatedthe surface ECG successfullyduring thenormal cardiac cycles using a 3D whole heart model employing the LR model using theparallel computing technique(see Chapter2 for details).

An alternative way to apply HH formulism to a 3D heart model is the combinationof HH and Huygens' type models, as reported by Leon et al. (1991). In this study, whentransmembrane potential is less than a potential threshold, the propagation in progresswithin the model cell is governedby

avCm - = V . DVv - i;on(v) + iappat v < Vth (3.12)

wherev is the transmembrane potential; i;on,the ioniccurrent; iapp, the appliedcurrent; Cm,the membrane capacitance; and D, the conductivity tensor.

When the transmembrane potential is greater than the potential threshold, the modelreduces to a conventional Huygens' type and the propagationprocess is controlled by

V=!Ct,T ,C) (3.13)

where! is a pre-assignedactionpotentialwaveformfor a cell type r and its valuevarieswithtime t, and the coupling interval C. In this manner, both subthresholdand suprathresholdphenomenawere simulatedat a whole heart level.

100 D.Wei

3.2.5.3 Propagation using Fitzllugh-Nagumo model

Taking the model of Berenfeld (1996) as an example, the action potential using theFitzHugh-Nagumo model is represented by state equations of V and U:

dv- = c(-U +bV - V 3)+ zdt

dU 1- = - (-U +a + V)dt c

(3.14)

(3.15)

where a and b are constants, and parameter c is an adaptive function representing thereaction to the cellular mechanism of the depolarization and repolarization processes. As inthis equation, the variable of state V is able to represent the property of membrane potential.It can be linked to the membrane current density by giving z the physiologic meaning ofstimulus intensity, and thus obtain

dv 3- = c(- U + bV - V') + V' . (DV'V)dt

(3.16)

where D is the diffusion tensor dependent on fiber orientation. The term z = V' . (DV'V)behaves the same as with the transmembrane current term in the cable equation. Equations(3.16) and (3.14) constitute a so-called reaction-diffusion system. See details in Berenfeld(1996) , where a complete cycle of ECG is simulated with the reaction-diffusion systemincorporating rotational anisotropy.

3.2.6 CARDIAC ELECTRIC SOURCES AND SURFACE ECG POTENTIALS

With any of the propagation strategies described above, we can estimate transmembranepotential Vm(x, y, z, t) at any voxel of the heart model. Therefore, we are able to determinethe cardiac sources based on the spatial distribution of transmembrane potential at any instantof simulation (see Chapter 2 for details). It is usually convenient to calculate elementalsources at individual cells of the model, and then integrate them throughout the heart .From the cardiac sources, the ECG potentials on the surface of the torso model can becomputed. Simulation of surface ECG potentials is one of the main purposes for whole heartmodeling.

The basis to determine the cardiac electric source is the bidomain theory. Solving theECG potentials on the torso model is a typical volume conductor problem. Both of thesetopics are fully covered in Chapter 2. Therefore, the general principles and methodology willnot be repeated in this chapter. Instead, we introduce the algorithms and implementationsin our own model study as an example .

We start from the Miller and Geselowitz (1978) formula that is based on bidomaintheory:

(3.17)

where (J is the conductivity, ¢ is the membrane potential , and subscript i refers to theintracellular domain . It says that the current dipole density is proportional to the spatial

gradient of intracellular potential distribution. In fact, the cardiac sources can be equivalentlyexpressed in terms of either intracellular or extracellular membrane potentials dependingon the form of effective interstitial conductivity used in simulation. Equation (3.17) canalso be modified (Geselowitz, 1989) as

(3.18)

where <Pm represents transmembrane potential. In this case, the effective interstitial conductivity becomes a = a, +a.; called bulk conductivity. Because the spatial gradient can beapproximated by the potential difference between a cell and its neighbors, this expressionis convenient for the implementation in whole heart models with discrete elements.

The cardiac source expression of Miller and Geselowitz applies to isotropic media.If the anisotropy in both intracellular and interstitial domains is taken into account, thesolution will become complicated. In Wei et al. (1995), the anisotropy was simplified byassuming an anisotropic intracellular domain and an isotropic interstitial domain. With thisassumption, the expression of the cardiac dipole source has a similar form to (3.17):

(3.19)

where D, is the intracellular conductivity tensor.The simplification of assuming an anisotropic intracellular domain and an isotropic

interstitial domain is reasonable if we analyze the measured values of the conductibilityanisotropy. The conductivity ratios measured by Clerc (1976) are approximately 9.0 for theintracellular space and 2.4 for the extracellular space. Those measured by Roberts et al.(1979) are approximately 5.8 and 1.5, respectively. Clearly, as compared to the intracellularconductivity ratio, the extracellular conductivity ratio is quite small. In other words, theeffect of conductivity anisotropy is predominantly caused by intracellular anisotropy.

To calculate the conductivity tensor, it is convenient and reasonable to assume anaxially symmetric anisotropy. Thus, if we establish a local coordinate system with one axisI being along the fiber direction, and the other two perpendicular axes, t} and tz, beingwithin the transversal plan, the conductivity tensor can be expressed as

(3.20)

where a/ and at represent intracellular conductivity along and perpendicular to the fiberdirection. Then D, is expressed in the global system as

(3.21)

where I, t1 and ti are unit vectors (column vectors) along each axis in the local coordinatesystem. If we further assume an equal conductivity ratio and let the ratio along and acrossthe fiber axis be r, the conductivity tensor can be simplified as

(3.22)

102

where

C = I + (r - l)U T

D.Wei

(3.23)

In computing the surface ECG potentials , we (Aoki et al., 1987; Wei et al., 1995)developed an algorithm that transforms Poisson equation to Laplacian equation to simplifythe volume conductor problem, and used the boundary element method (BEM) for solutions.

The problem originated from the Poisson equation with respect to the surface potential¢ is expressed as:

(3.24)

Since the surface potentials are results of the primary source owing to the current dipolesin the heart , and the secondary source owing to the boundary effect, we introduce anintermediate variable 1/J = ¢ - ¢o with the boundary condition of

a1/J a¢o °-=-=-qan an (3.25)

where ¢o is the potentials in an infinite medium, and n represents normal direction to thebody surface. Applying the BEM with respect to 1/J leads to

where G is the Green function of

G= _4nlr - r'l

(3.26)

(3.27)

where r is the distance from a dipole source and the integration is performed with respectto ; ' . Discretization of the integral equation (3.26) leads to linear equations

N N

L hij1/Jj = - LgijqJj = ] j =l

fori = 1,2, . . . , IV (3.28)

where hij and gij are coefficients depending on the torso geometry, and N is the number ofelemental triangles which approximates the torso surface. To ensure a unique solution, an(N + 1)th equation

N N

'L aj1/Jj = - 'Laj¢Jj = ] j=1

(3.29)

is added to (3.28), where aj is proportional to the area of an elemental triangle which hasan indexj. This condition is used to define a potential reference so that the surface potential


integral is zero. Rewrite the (N + 1) simultaneous equations of (3.28) and (3.29) in a matrixnotation, we have

Hlp = _GQo - A

The final solution for surface potentials is obtained as

where

and

3.3 COMPUTER SIMULATIONS AND APPliCATIONS

(3.30)

(3.31)

(3.32)

(3.33)

If we look at the details of publications in the past, it is very clear that whole heartmodels and simulations have been extensively employed in research investigating cardiacmechanisms. Part of the significance of whole heart modeling is the capability of linking theundergoing pathology to the ECG features used in clinical applications. Typical examplesare models of abnormal ST-T waves (Harurni, 1989b; Dube, 1996; Hyttinen et aI., 1997;Abildskov and Lux, 2000), myocardial infarction (StarttiSelvester et al., 1989, Zenda etaI., 2000), WPW syndrome (Lorange et aI., 1986; Wei, 1987, 1990), hypertrophy and cardiomyopathy (Harumi, 1989b; Wei et aI., 1999), reentrant and scroll waves in ventriculararrhythmias and fibrillations (Panfilov, 1993; Gray and Jalife, 1996; Okazaki et aI., 1998;Clayton et al., 2001a, b). Some new topics in the past five years are long QT syndrome(Clayton et al., 2001, Okazaki and Lux, 1999), T-wave altemans (Abildskov and Lux, 2000),and late potentials (Yamaki et aI., 1999). Furthermore, recent studies have extended modelapplications to the development ofnew instrumentation techniques. Examples are pace mapping (Xu et al., 1996, Hren and Horacek, 1997), epicardial mapping (Ramanathan and Rudy,2001), Laplacian ECG mapping (Wu et aI., 1998; He and Wu, 1999; Wei and Mashima, 1999;Wei, 2001), artificial heart (Zhang et aI., 1999), and impedance CT (Kauppinen et al., 1999).

In the following sections, some typical applications of the whole heart model are introduced. In each section, the undergoing pathology is briefly introduced before the descriptionof the model study. For details of pathology, please refer to medical textbooks. Three major reference books used in the description are Goldman (1986), MacFarlane and Lawrie(1989), and de Luna (1993).

3.3.1 SIMULATION OF THE NORMAL ELECTROCARDIOGRAM

A correct simulation of the normal ECG is usually the first step for whole heartmodeling, before applying the model to special pathologic conditions. The normality of

104

TABLE 3.2. Definition of l2-lead Electrocard iogram

D.Wei

Lead

Bipolar limb leads

IIIIII

Augmented unipolar limb leadsaVR

aVLaVF

Precord ial leadsVj(I = 1, 2, .. . , 6)

Definit ion

1= EL - ERII = EF - ER

III = EF - EL

aVR = (3j2)(ER - Ewe'>aVL = (3j2)(EL - Ewe,)

aVF = (3j2)(EF - Ewe,)

Ewo! is called Wilson Central Terminal.Ewct = (EI.+ ER+ EF)/3 .E denotes potential, subscripts L. R, and F denote left arm, right arm

and left foot, respectively.See Fig. 3.8 for electrode positions of V;.

TABLE 3.3. Electrode Positions for Recording 12-lead Electrocardiogram

Electrode

Left arm (R)Right arm (L)

Left foot (F)Right foot (G)

VIV2

V3V4

V5

V6

Position

Left wristRight wrist

Left ankleRight ankle

Right sternal margin, fourth intercostal spaceLeft sternal margin, fourth intercostal spaceMidway betwee n V2 and V4

Left midclavicular line, fifth intercostal space

Left anterior axillary line, V4 levelLeft midaxi llary line, V4 and V5 level

the simulation results can be evaluated with the excitation sequence of the heart model, thesimulated vectorcardiogram (VCG) , l2-lead ECG, and body surface isopotential maps. The12-lead ECG is the most popular lead system used in clinical practice. To evaluate modelsand simulations with different pathologies, the results are usually compared with clinicalrecorded l2-lead ECGs. Recording the l2-lead ECG requires 10 electrodes. Four electrodesare placed on limbs to record six limb leads and the other six are placed on the precordialchest wall to record the precordial leads. The definition of l2-lead ECG is summarized inTable 3.2, and the electrode positions for recording are shown in Fig. 3.8(a) and describedin Table 3.3 (Horacek, 1989). In model studies, torso models usually do not include thelimbs . The limb leads are moved to the closed positions on the torso . Because the potentialdifference on the limb is sufficiently small , this does not sign ificantly change the simulationresults. Most torso models are represented by polygon meshes, and the nodal point s maynot exactly overlap the electrode position. In this case, interpolation is usually needed tocalculate either the electrode position s from positions of the surrounding nodal points, orthe ECG potentials from the potentials on surrounding nodal points.

A typical ECG waveform is illustrated in Fig. 3.8(b). The waves of the ECG aredesignated by Einthoven as P, Q, R, S, T, as shown in this figure . It is well known that the P


' .

. ~ ,. ' .,,; ': " .r ' •. '

-,

"....... ...·.·.\ '1 V2-·. · · .

....·-,f -. V3 -:: .: '.

(a)

QRSInterval

R

1.0

u

STse~ent

0.4 0.6 0,8time (1mm=O,04s)

0.2

l 1---+---1-+---+---+-----4

8,~

~

(b )

FIGURE 3.8. (a) Electrode positions for the precordial leads. (b) A typical electrocardiogram.

wave corresponds to atrial depolarization, the QRS complex to ventricular depolarization,and the T wave to ventricular repolarization. The U wave is occasionally recorded after theT wave, and its mechanism remains unclear (di Bernardo and Murray, 2002). The criteriaof normality for the ECG include time and amplitude standards. Clinically, the followingparameters are evaluated for the normality of ECG:

• Cardiac rhythm and heart rate• PR interval and segment• QRS interval• QT interval• Pattern and amplitude of P wave• Pattern and amplitude of QRS complex

106

• Pattern and amplitude of T wave• ST segment and T wave• Mean electrical axis (frontal plane)

D.Wei

The limit of normal ECG can be found in MacFarlane and Lawrie (1989) and thetextbook of de Luna (1993). The following are typical values: P wave interval = 100ms, QRS interval = 90 ms, PR interval = 150 ms, and QT interval = 380 ms. Becausemost heart models do not yield absolute values for ECG potentials, the amplitudes can beevaluated by patterns and relative amplitudes. For example, the aVR wave should have aninversed waveform. The R wave should gradually increase from VI to V4 or V5 and fallfrom V5 or V6. In simulation studies, the use of VCG is very convenient in evaluatingthe simulation results. By summing the cardiac dipole sources in all model cells at eachinstance into a single dipole and plotting it in the frontal, horizontal and saggital planes,the VCG is obtained. It is theoretically close to the clinical VCG. From these figures, themean electrical axises can easily be estimated. The normal P, QRS, and T axises should bebetween 0° to +900

, - 30° to + 110°, and 0° to +90°. The body surface isopotential maps arealso easily used for evaluation. The typical patterns for a normal heart can be found in manyarticles and textbooks (e.g., MacFarlane and Lawrie, 1989). Fig. 3.9 shows the simulatedresults of a normal heart mode by Wei at al. In Fig. 3.9(c), potential distributions on thetorso during the QRS period are shown by isopotential countour maps. In most instances,the distributions show dipole fields with one positive maximum on the anterior chest andone negative potential minimum on the back at the early QRS and gradually reversed in thelate QRS. In a short period of the middle QRS (time 183-189), we can find a multipole fieldrepresented by one potential maximum and two negative minimums. This is a typical patternin the isopotenials maps representing the right ventricular breakthrough, as experimentallyobserved by Taccardi (1963).

Simulation of a normal heart is generally a repetitive procedure to adjust the modeluntil simulated excitation isochrones fit the experimental data, and the simulated vectorcardiogram and surface ECGs fall within the normal range. Generally, obtaining a normalP wave is not difficult. The excitation from the sinus node spreads leftward and downwarddoes reproduce a normal P wave in the simulated ECG. Getting a normal QRS lead in the12-lead ECG is more difficult. To keep normal waveforms in most leads on the surfaceof the torso model, including the normal time periods and normal amplitude relationshipsamong the leads, it is important to correctly mount and adjust the specialized conductionsystem in the heart model.

The simulation ofthe T wave requires a deep understanding of the T wave mechanismthe reason why positive T waves are measured in most of the 12-lead ECG. Suppose theaction potential is uniform at all ventricular myocardium, the depolarization and repolarization would be along the same direction, and thus the T waves would have the oppositepolarity as that of the QRS waves. But actually, positive T waves are observed in most leads.The mechanism can be interpreted with a theoretical model that assumes longer action potential duration in the sides of the endocardium and apex than toward the epicardium andthe base (Harumi et al., 1964). In this way, the repolarization spreads along the oppositedirection to the depolarization. As a result, the T wave polarity is the same as that of the QRSwave. The T wave model is consistent to the experimental measurement on the epicardium(Spack et al., 1977). A detailed description of a T wave mechanism can be found in Barr


~ I~I~avtv-av~av~(a) vt- r vt- vl v:k- v~

107

Horizontal frontal sagittal

(b)

114~ 117 ms

~ e I [J~) °1

(c )

I(~6:~ II ~68 ~ II ~1 1 ~ 8 II

pT~9~~~~195 ms 198~ 201~ 204 ~ 201 ~

f'Pr8JlJ I~A'\ II~Mil (@) 01[ @) 01FIGURE 3.9. Simulation results of a normal heart model: (a) ECG ; (b) VCG; and (c) body surface isopotent ialmaps. In (c), "+"and "-" show positions of potential maximum and minimum , respectively. Lines of light blackrepresent contours of zero potential. The left area of six tenths correspond s to the anterior torso chest, and theright of four tenths corresponds to the back. Time is counted from the onset of the P wave. (Figures of a and barereproduced with permiss ion from Wei et al., 1995).

(1989). The algorithmused in simulation to distribute the action potential in the 3D wholeheart model is introducedin Section 3.2.4.

3.3.2 SIMULATION OF ST-T WAVES IN PATHOLOGIC CONDITIONS

Studyingthe pathologicchangesin theST segmentand the T wavesare typicaland importantapplications of wholeheartmodels. This is becausethe ST-T changesare associatedwithseriousheartdiseases such as myocardial ischemia, hypertrophy, and cardiomyopathy.

108

>'E-

• J. 1St

TImezoo :&SO

(ms)

D.Wei

FIGURE 3.10. Pre-defined act ion potentials assigned to cells in normal and middle, moderate , and severelyischemic regions in the simulation of Dube et al. (Reproduced with permission. from Dube et al., 1996).

Myocardial ischemias arise from insufficient blood flow due to an occlusion of coronary arteries. In recent years , the percutaneous trasluminal coronary angioplasty (PTCA)provides an opportunity to precisely confirm the relationship between the ECG features andthe sides of the blood block by controlling the balloon inflation during the PTCA operation.Dube et al. (1996) simulated clinical body surface potential maps and ECGs using the PTCAprotocol. Because the action potential change of the ischemic tissue is the direct cause of theST segment changes in the ECG, the way of setting action potential in the heart model is akey to the simulation. Three transmural zones , middle, moderate and severe ischemia, wereset to the heart model, located in the vicinity of the left anterior descending, left circumflex,and right coronary arteries . The action potentials, as shown in Fig. 3.10, were set to theseregions, representing action potentials under middle, moderate and severe ischemias. Thesimulation produced ECG maps quantitatively similar to clinical maps.

Fig. 3.11 shows another example that simulates "giant negative T waves" known asthe main feature of the apical hypertrophic cardiomyopathy (Harumi, 1989b). The simulated ECG and VCG give surprisingly similar results to clinical findings. The results wereobtained by modifying the APD gradient and the conductivity value for the pathologiczone. Unlike the ischemia, the heart with hypertrophic cardiomyopathy is impossible tomake with experimental animals . In this sense , computer simulation is the only way of invivo experimentation to study the unknown mechanism.

3.3.3 SIMULATION OF MYOCARDIAL INFARCTION

Myocardial infarction is a typical concern in heart modeling. The example introducedin this section demon strates that model study is not only useful for understanding themechanism, but also it is useful for helping to develop a diagnosis tool for clinical practice.


Frontal r---r-----, r---r-- --, Sagittal

109

FIGURE 3.11. Simulated ECG and VCG in the heart modelof apical myocardial cardiomyopathy (Reproducedwith permission from Harumi et aI., 1989).

Myocardial infarction is caused by the occlusion of the coronary artery. The development of myocardial infarction is usually classified in three phases by ECG patterns (deLuna, 1993). The early phase of ischemia is characterized by T wave changes. The laterphase of injury is characterized by ST segment changes. The final phase of necrosis is characterized by Q wave changes. The ST and T wave changes can be simulated with a wholeheart model by changing the action potentials for the model cells. The action potentialsduring the reduced blood flow can be measured experimentally.

The location and size of myocardial infarction is an important aspect in clinical diagnosis. In clinical practice, the locat ion and size of infarction are qualitatively interpretedwith the theory of vectorcardiogram. With a whole heart model , StarttlSelvester et al. (1989)systematically simulated the infarcts due to three major coronary artery distributions, andexpected Q waves were obtained in each case. They found that in any case the degree ofQRS change was proportional to the degree of local infarction. Dividing the ventricles intofour walls by 12 segments, they developed a quantitative method to estimate the locationand size of the infarction. The result led to a practical tool known as the ECGNCG scoringsystem, where each point scored was set up to repre sent 3% of the left ventricular myocardium. The scoring system predicted a distribution of damage in the 12 left ventricular

110 D.Wei

segments in good correlation with the average of planometric pathology found in the samesubdivisions. They further developed a monogram based on the same simulation study,which relates the VCG changes to the infarct size. If the duration and magnitude of QRSdeformity are measured before and after infarction, the infarct size can be simply found onthe monogram. Details can be found in Startt/Selvester et al. (1989).

3.3.4 SIMULATION OF PACE MAPPING

Pace mapping is a new technique that is used to speed up the procedure for localizing ectopic focus in catheterization (SippensGroenewegen et aI., 1993). In SippensGroenewegenet aI., body surface potential maps in patients with cardiac arrhythmias but with no evidenceof structural heart disease were recorded during ectopic beats by catheter stimulation at different endocardial sites in the ventricles. Based on these data, they were able to classify theQRS integral map patterns with respect to the location of ectopic beats.

The same procedure was reproduced by Xu et aI. (1996) with the heart model ofLorange et aI. In the simulation, 38 selected endocardial sites (25 on the left ventricleand 13 on the right ventricle) corresponding to SippensGroenewegen et aI. were pacedto initialize the excitation process of the heart model. With a more detailed heart model(0.5 mm spatial resolution), Hren and Horacek (1997) generated a database of 155 QRSintegral maps by pacing the epicardial surfaces in the left and right ventricles. This databasewould be useful in catheter pace mapping during treatment of ventricular arrhythmia. Thesimulation of pace mapping is a good example to show the clinical usefulness of wholeheart models.

3.3.5 SPIRAL WAVES-A NEW HYPOTHESIS OF VENTRICULARFIBRILLATION

Reentrant excitation is recognized as a mechanism of life-threatening arrhythmias.Among several hypotheses to explain the reentrant excitation, spiral wave is the latest oneand is getting more and more support from experimental studies (Gray, 1995, 1996). Inaddition to experimental studies, computer simulation of the spiral wave using the wholeheart model with realistic geometry and anatomy is a useful tool because it is capable ofproducing comparable results with experiments. Fig. 3.12 shows an example of spiral wavessimulated with a whole heart model (Gray and Jalife, 1996). The simulated isochrones andECGs (right) are comparable with the measured results.

3.3.6 SIMULATION OF ANTIARRHYTHMIC DRUG EFFECT

The example introduced in the following demonstrates how the drug effect is confirmedby a simulation study with a whole heart model (Wei et aI., 1992). The study was basedon an experiment that investigated the relationship (Harumi, 1989a) between the restitutionof premature action potential and the stimulation coupling interval for Purkinje fiber andventricular muscle in dogs before and during the infusion of antiarrhythmic drugs. In thesimulation, lines through linear regression as shown in Fig. 3.13(a) were obtained to approximate the experimental data. The slopes of these lines corresponding to a parameter calleddynamic coefficient (DC) were input to the heart model. Note that, the figure shows different slopes for the ventricular muscle and the Purkinje fiber before and during the infusion


FIGURE 3.12. Simulation of spiral wave.(Reproducedwith permission, from Gray and Ja1ife, 1996).

111

of antiarrhythmic drugs. In the simulation, ten successive extra-stimuli of 170 ms intervalstarted at 300 ms after the first sinus pacing were applied to the epicardium of the ventricle.The simulated ECG shown on the top of Fig. 3.13(b) corresponds to the case where the APDchanges follow the lines before the drug infusion. In this case, the stimulation caused twotachycardia-like waveforms followed by sustained VF. When APD changes follow the linesduring the drug infusion, normal waveforms are restored after the stimulation, as shown bythe waveform in the bottom of Fig. 3.13(b). Figure 3.13(c) shows a picture of animationdeveloped for visualizing the propagation of excitation during the fibrillation. A number ofpropagation wavefronts developed due to a number of reentries can be seen in this picture.This simulation supported the assumption that different ratios of restitution in prematureAPD between the ventricular muscle and the Purkinje fiber may playa role in the inductionof VF. It also demonstrated the antiarrhythmic drug effect in suppressing VF.

3.4 DISCUSSION

Principles, methodology and applications in 3D whole heart modeling are described inthis chapter. The significance of 3D whole heart models is that, as compared to experimentalstudies, computer simulations with whole heart models are always in vivo so as to provideinformation relating intracardiac events to the body surface electrocardiogram in different

Ventricular musclePurkinje fiber

Ventricular musclePurkinje fiber

,

<::)lI'\N

<::)<::)N

Q

~<::)lI'\N

<::)<::)N

(a)

300 400 500Coupling interval (rns)

300 400 500Coupling interval (ms)

(b)

Lomv

0 2 Se C

FIGURE 3.13. (a) Approximation of experimental results describing restitutions of premature action potentialto the stimulation coupling interval for Purkinje fiber and the ventricular muscle in dogs before (left) and during(right) the infusion of antiarrhythmic drug. (b) Simulated ECGs. The waveform on the top shows ventricularfibrillation induced by applying successive stimuli to the left ventricular wall. The waveform on the bottom showsthat the fibrillation stops after the infusion of the antiarrhythmic drug. (c) An image of 3D animation showingexcitation propagation during ventricular fibrillation.


pathologic processes. Such information is sometimes difficult to obtain by experimentalstudies. Other advantages arise from the facts that the computer simulation is low-cost , fastand repeatable. As an experimental tool, a whole heart model can be used in laboratoriesfor research purposes and used in classroom for computer-aided instruction in medicaleducation.

The main limitation of current whole heart models is the insufficiency of the electrophysiological details, due to the large scale of the whole heart and the computationalcapability related to the problem. In the present stage, the heart models in single-cell, cellnetwork and whole heart levels are still separately studied . More time is needed to realizeand spread the 3D whole heart models of next-generation that can generate action potentialsbased on cellular mechanisms, that can simulate propagation based on HH type formulisrn ,and reconstruct clinically comparable body surface electrocardiograms. This kind of modelwould be able to directly relate basic experimental findings to clinical applications.

In a search of related abstracts on Medline in the U.S. National Library for the twodecades till the end of 1980s, Malik and Camm (1991) found that both the absolute andrelative numbers of publications in computer modeling and simulation have increased muchfaster than the total increase in cardiology as a whole . We have searched the same databasefor publications in the 1990s using the keywords "heart +model + simulation +computer."The result is shown in Fig. 3.14. Publication activity continues to increase.

Nowadays, computer modeling and simulation have become more attractive to researchers than ever before . Because the biological system is so complex, no single experiment can synthesize the system as a whole .As the computation ability of computers becomesmore and more powerful , computer modeling and simulation technologies are making itpossible to study the biological system of humans as a whole. This is recently knownas "Physiome" ("physi" means life, "orne" means whole , see http://www.physiome.orgl).Whole heart modeling is one of the common concerns in Physiome.

Key wordr-'Hea:t+model+S1mulahon+computer'100

'0 40

a::,Q

§ 20

io

1990 1991 1992 1993 1994 199~ 1996 1997 1998 1999 2000

YE-:U'

FIGURE 3.14. Publications searched on MedPu b. (Keyword = "heart + model + simulation + computer").

114

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D.Wei

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4

HEART SURFACEELECTROCARDIOGRAPHIC

INVERSE SOLUTIONS

Fred Greensite1

Department of Radiological ServicesUniversity of California, Irvine

4.1 INTRODUCTION

In this chapter, we will review the problem of noninvasive and minimially invasive imagingof cardiac electrical function. We use the term "imaging" in the sense ofmethodology whichseeks to spatially resolve distributed properties of cardiac muscle electrophysiology suchas extracellular potential, or features of the action potential. Thus, we do not consider theproblems of computing properties ofan "equivalent" cardiac multipole, moving dipole(s), orany other source model that does not satisfy such criteria. We will further restrict ourselvesto resolving such electrophysiological features on the epicardial or endocardial surfacesa reasonable restriction, since measurements currently accessed by invasive proceduresare obtained on these surfaces, and also because the spatial dimension of the "source"domain then nominally matches the spatial dimension of the data domain. Thus, we willnot consider the earliest distributed source model, representing intramural current densityimaging (Barber and Fischman, 1961; Bellman et al., 1964), on which work continues (e.g.,see (He and Wu, 2001), or the recent heart-excitation-model based 3D inverse imagingapproach (Li and He, 2001) in Chapter 5 in this book).

Following an historical perspective, we will discuss in some detail the inherent difficulties of this imaging problem (principally mathematical), and strategies developed tocircumvent them. We will not attempt to comprehensively cite the voluminous work doneon these formulations of the inverse electrocardiography problem. Excellent reviews for theperiod prior to 1990 exist (Gulrajani et al., 1989; Rudy and Messinger-Rapport, 1988), andmore recent shorter reviews can be found in (MacLeod and Brooks, 1998; Gulrajani, 1998).

I Mailing Address: Fred. Greensite, Department of Radiological Sciences, University of California-Irvine MedicalCenter, Trailer 11, Route 140, 101 The City Drive South, Orange CA 92868 USA. E-mail: [email protected]: (714) 456-7404, FAX: (714) 456-6380.

119

120 F. Greensite

Our principal objectives are to provide a meaningful presentation of the issues involved inthis inverse problem, and to survey some work of the last several years.

4.1.1 THE RATIONALE FOR IMAGING CARDIAC ELECTRICAL FUNCTION

Coordination of the heart's mechanical functioning is accomplished electrically. In fact,there is a tight coupling of mechanical and electrical function both at the microscopic cellular and macroscopic organ level. Global contraction of the heart proceeds from a sequence oflocal changes in cell membrane conductances to various ionic species (the opening and closing of voltage-gated species-specific membrane channels). These membrane conductancechanges, resulting in transmembrane currents, generate local source currents which lead toOhmic currents throughout the remainder of the body volume conductor. In this manner,the sequential cardiac muscle membrane changes, which characterize the heart's electricalfunctioning, are reflected in electrical potentials that can be measured at the body surface.

The local changes in intracellular ionic concentrations, resulting from the transmembrane electrical current, trigger the local subcelluar mechanical events (the "sliding" offilaments over each other) which on a global scale summate to coordinated contractionof the muscle mass. Since the local mechanical events must be coordinated satisfactorilywith each other to produce an effective organic contraction, a corollary is that a mechanical catastrophe can only be avoided if the inciting local electrical events are themselvescoordinated properly on a global scale. Even when mechanical function appears adequate,the spatiotemporally distributed electrical functioning of the heart is of the greatest interestmedically, since well characterized disturbances of cardiac electrical functioning are knownto predispose to such mechanical catastrophes, and are known to be amenable to various interventions (pharmcological, catheter-based, or surgical). Evidently, it is useful to considerhow such cardiac electrical distubances might be effectively diagnosed noninvasively.

Although indications of electrophysiological dysfunction may be apparent from the unadorned clinical electrocardiogram (tracings of electrical potential at certain body surfacelocations), successful treatment often requires "imaging" of such disturbances-usuallyaccomplished by laborious transvascular electrode catheter manipulations within a cardiacchamber. The prospect that such imaging might be effected in a less invasive and efficient manner, is the principal motivation for present efforts to develop a practical clinicaltechnology for imaging cardiac electrical functioning.

4.1.2 A HISTORICAL PERSPECTIVE

In our context, imaging could be looked at as a global depiction of local features. Thus,it can be understood in terms of microscopic and macroscopic cardiac electrophysiology.

Microscopic: Action Potential

Sensory-neuro-muscular tissue is distinguished by its ability to generate and conductaction potentials. The latter represent cellular transmembrane potential evolution during thecharacteristic functioning of the tissue (figure 4.1). The concept was inherent in the work ofBernstein in the late 1800s, wherein functional neuronal cellular electrical activity was attributed to sequential membrane permeability changes. Muscle is particularly distinguished

Heart Surface Electrocardiographic Inverse Solutions

mV

121

-90

FIGURE 4.1. The action potential in heart Purkinje cells resembles a pulse (i.e., square wave). The upstroke(phase 0, "activation") corresponds to rapid intracellular influx of sodium. A limited fast repolarization (phase 1),and subsequent plateau (phase 2), correspond to an interval for which the location is refractory to stimulation. Thedownstroke (phase 3) is over an interval for which stimulation will lead to weakened activation. The horizontalaxis can be either time (action potential duration is on the order of hundreds of milliseconds) or space (althoughsimultaneous recording of transmembrane potential along a path in the heart muscle at a single time instant wouldusually only reveal the full action potential shape in the setting of re-entry type arrhythmias). The -90 millivoltbaseline defines phase 4.

in that arrival of an action potential at a given location triggers a cascade of events leadingto contraction. Wilson et al. (1933) expressed great insights into the significance of thistransmembrane functioning in cardiac muscle, in a seminal work from the 1930's.

All aspects of the action potential are potentially of interest. For example, phase 0(signifying local activation) initiates the events leading to the local contraction (slidingof filiments). At the phase 0 time, the location "depolarizes" (transmembrane potentialchanges from roughly -90 millivolts to roughly zero millivolts). The amplitude of phaseohas important implications as regards local muscle integrity. During Phase 1 (transient,limited, fast repolarization) and Phase 2 (plateau), the location cannot be further stimulated(absolute refractory period). Phase 3 (progression through to "repolarization") is a relativerefractory period, during which attenuated responses to further stimulation are possible.The durations of the absolute and relative reftractory periods govern when the location willbe susceptible to being triggered again (for the next heartbeat), and how strong that nextcontraction might be locally. During phase 4, the location is fully repolarized, i.e., maintainsthe baseline -90 millivolt transmembrane potential difference.

Given the significance of these features, an obvious goal would be to noninvasivelyimage (spatially resolve) the action potential at every location in the heart muscle (i.e.,reconstruct the time series of the transmembrane potential at each location). However,articulation of such a goal has been long in coming.

Macroscopic: Electrocardiogram

Recognition of the electrical functioning of muscle predates the nineteenth century(and possibly the eighteenth century) (Malmivuo and Plonsey, 1995). By the mid-nineteenth

122

bn--_ _

I;

F. Greensite

FIGURE 4.2. An illustration adapted from (Waller, 1889). The cardiac-generated potential field depicted isessentially dipolar in nature. [Adapted from figure 5, p. 186: Waller, A., 1889, On the electromotive changesconnected with the beat of the mammalian heart, and of the human heart in particular, Phil. Trans. R. Soc. Lond.

B. 180: 169-194. Used by permission.]

century, the epicardial potentials had been invasively accessed. Later in the century, Wallerhad the brilliant insight that the limbs could be viewed as electrode leads emanating from theheart-so that coupling them to an appropriate electrical apparatus should allow noninvasiveassessment of the previously established cardiac electrical functioning. From his publishedillustrations, it appears that he conceived of modeling the heart as a current dipole source,begging the question of inversely "imaging" this source from his body surface potentialmeasurements (Waller, 1889) (figure 4.2).

Interestingly, Waller (1911) was not optimistic that such information would be of muchuse medically. However, nearly coincident with his cautionary remarks, Einthoven (1912)was engaged in the work of demonstrating the clinical efficacy of improved instrumentationto accomplish just such a goal. Indeed, early in the twentieth century Einthoven popularized(via the "Einthoven triangle") the concept of the cardiac dipole as (qualitatively) estimatedfrom measurements at standardized electrode locations-the electrocardiogram. In ensuingdecades, a number of investigators attempted to make such calculations more quantitative(e.g., (Wilson et al., 1947; Frank, 1954». In this context, Wilson's earlier work (Wilsonet al., 1933) provided the conceptual link between the microscopic and macroscopic. Inparticular, it was observed that macroscopic cardiac activation wavefronts, arising fromcellular membrane events, effect measurable remote potentials whose magnitude is roughly

Heart Surface Electrocardiographic Inverse Solutions 123

proportional to the solid angle subtended by the activation wavefront and the electrodelocation.

However, the implication that cardiac electrical functioning might be remotely imagedhad to await the diffusion of computer technology to the biomedical community in the early1960s (Gerlernter and Swihart, 1964; Bellman et al., 1964; Barnard et al., 1967).

4.1.3 NOTATION AND CONVENTIONS

The imaging formulas for cardiac electrophysiology derive from linear partial differential equations. When they are "discretized" for numerical solution (see Chapter 2 in thisbook), one encounters large scale algebraic manipulations. We will find the following vectorand matrix conventions useful: We denote a matrix composed of zeros as 0, and an identitymatrix as I. Superscript t applied to a matrix denotes the transpose, so when it is applied toa column vector it therefore denotes the corresponding row vector (and vice-versa). Thus,for column vectors a, b, the inner product will be written as either a . b or atb, while theouter product is written as either a ® b or ab' (the outer product of a and b is the matrixwhose (i, j) entry is the product of the i-th component of a with the j-th component of b).A vector in three dimensions will be denoted by a bold face font (e.g., n will denote a unitnormal to a surface, andj will denote a current density). Matrices will be denoted by capitalletters, e.g., M. The i-th row of M will be denoted as Mi : , and the j-th column will bedenoted M: j • Random variables (and random vectors and random matrices) will sometimesbe denoted with a superscripted bar, e.g., a. The probability (or probability density) of someevent will be denoted p(.), so that p(alb) is the (conditional) probability of event a giventhat event b has been observed.

As is customary in physics, elemental sources (e.g., electrical monopoles) are expressedvia the Dirac delta function 8(x)-a convenient shorthand for the limiting notion of afunction whose nonzero portion is localized to a tiny region, but whose integral over allspace is unity (Jackson, 1975). We note that such entities are rigorously treated in the contextof the theory of distributions (e.g., see (Keener, 1988) for details).

4.2 THE BASIC MODEL AND SOURCE FORMULATIONS

In this section, we will derive imaging equations relating body surface potentials toheart surface potentials. We do not present the analogous equations for magnetic field data,which can be derived in a similar manner from the equations of magnetostatics.

As noted, description of muscle electrophysiology can be organized around the concept of an action potential, which reflects the opening and closing of ionic channels inthe cell membrane-which in tum lead to local intracellular ionic environmental changesthat trigger the local mechanical function, and the return to the resting state. By definition, an action potential is the transmembrane potential that occurs in sensory-neuromuscular tissue during its characteristic functioning. It encompasses the phenomena ofself-propagation to adjacent locations, meaning that the form ofthe transmembrane potential at a fixed location as a function oftime is recapitulated in a tracing oftransmembranepotential at a fixed time as a function of location (along the path of activation). Propagation of the action potential coordinates local contraction over the full three-dimensional

124 F.Greensite

extent of the muscle, and implies its link to the global mechanical functioning of thetissue.

Of course, the action potential is measured historically via microelectrodes insertedinto cells. Our objective is to resolve features of the transmembrane potential remotely fromthe tissue. Thus , we must formulate the relationship between cardiac muscle action potential,and remote measurements (such as on the body surface) . This is conveniently accomplishedvia the so-called "bidornain" model , a concept first suggested by Schmitt (1969), and latergiven mathematical form by others (reviewed by Henriquez (1993)) . Although the model isdiscussed in Chapter 2, we re-present it here as we wish to derive our source formulationsfrom "first principles", as well as provide a consistency of notation within the presentchapter.

At a level of resolution appropriate to this problem, it can be assumed that every bodylocation consists of a small amount of intracellular space and a small amount of extracellularspace. We can write

(4.1)

where ¢i, ¢e, and ¢m, are intracellular potential, extracellular potential, and transmembranepotential, respectively. In the context of our remote sensing problem, it has been shown thatcapacitive, inductive, and electromagnetic propagative effects are "negligible" (the "quasistatic assumption") (Plonsey, 1969). Therefore, the intracellular or extracellular currentdensities at the bidomain point are each linearly dependent on the gradient of the intracellularor extracellular potential (Ohm' s Law). Thus, these respective current densities are givenby

j i = -GiV¢i

je = -GeV¢e,

(4.2)

(4.3)

where G, and G, represent conductivity in the intracellular and extracellular componentsof the bidomain point, and these conductivities are (in practical terms) independent of thepotential (the medium in linear) (Plonsey, 1969). It is important to note that cardiac muscletissue is "anisotropic", in that it is composed of (interconnected) muscle fibers whoseintracellular conductivity along the fiber direction is much greater than the conductivitynormal to the fiber direction. Due to the geometrical constraints imposed by the interveningfibers, the extracellular conductivity is also anisotropic. Therefore, G i and Ge are tensors (inthis case, symmetric 3 x 3 matrices) which thereby linearly map a gradient of potential (ata point) to a current density vector. If G i and G e were proportional to each other (implyingthat the principal axes, and the conductivity ratios for different pairs of principal axes, arethe same for intracellular space as for extracelluar space), we could speak of there being"equal anisotropy".

According to Eq. (4.2) and Eq. (4.3) , total current density j = j i + je at any of thebidomain "points" must satisfy

(4.4)

The quasi-static assumption and charge conservation imply that any excess current (nonzero


divergence) appearing in the extracellular component of a bidomain point must come fromthe intracellular component of the bidomain (via the cell membrane), and vice versa. Thatis, V' . j = O. Thus, Eq. (4.4) implies

(4.5)

The divergences on both sides of the above equation express the location's role as a sourceof extracellular current (there is net transmembrane current if the divergence is nonzero).Transmembrane currents capable of influencing body surface potentials only occur in excitable tissue (i.e., sensory-neuro-muscular tissue)-the predominant one being the heart.Thus, outside the heart there is no propagation of action potentials, and no source current,since G i is zero there. Writing the extracellular potential e, as simply 4>, then using Eq. (4.1)we can rearrange Eq. (4.5) as

- V' . [(Ge + Gi)V'4>] = V' . [G i V'4>m]. (4.6)

We have the further condition that no current leaves the body, so the component of currentdensity normal to the body surface is zero. Thus, for y on the body surface and Dy a unitnormal to the body surface at y, we have D y . GeV'4>(y) = o. A uniformly zero boundarycondition (such as this) is referred to as "homogeneous".

Equation (4.6) is a partial differential equation, specifically, Poisson's equation. Letus imagine that we are given V' . [G i V'4>m] (the "source"), and we wish to compute theresulting potential 4>-the so-called "Forward Problem" (see Chapter 2 of this book). Onevery important feature of the Poisson equation is its linearity. That is, if a solution to- V' . [(Ge + Gi) V'4>] = f satisfying the homogeneous boundary condition is known as 4>f'and a solution to - V' . [(Ge + Gi)V'4>] = g satisfying the homogeneous boundary conditionis known as 4>g, then it is easy to verify that the solution to a solution to -V'. [(G e +GJV'4>] = c.] + C2g is given by cl4>f + c24>g (for Cl, C2 constants), and this solution alsosatisfies the homogeneous boundary condition. This means that if we know the solutionsto Eq. (4.6) for a source localized to any single location in the heart, then to determine thesolution for any more geometrically complex source within the heart volume V we onlyneed to add up (integrate) the solutions that would be obtained for each single locationcomprising the complex source.

Thus, consider Eq. (4.6) where its right-hand-side (the source) has unit strength whenintegrated over all space, but is zero everywhere except at x (i.e., the source is an electricmonopole). Denoting the solution as 1/r(x,y) (where 1/r(x, y) as a function of y satisfies thehomogeneous boundary condition), we have

-V' . [(G e + GJV'1/r(x, y)] = 8(x - y), (4.7)

where the divergence and gradient operators are with respect to the field point y in threespace. Thus, 1/r(x, y) is the potential at point y in the body that would be induced by a unitstrength source that was zero everywhere in the heart except at the location x (a unit strengthsource localized to a point is mathematically represented by a delta function). 1/r(x,y) isknown as a Green's function. Since Poisson's equation is linear, we can now write the

126 F. Green site

solution to Eq. (4.6) for the geometrically complicated source (on its right-hand-side) as

cPe(Y) = Iv 1fr(x , y )V . [Gi(x)VcPm(x)]dVx, (4.8)

where V is the heart volume.Following Yamashita and Geselow itz (1985), integration by parts applied to Eq. (4.8)

(i.e., application of the Divergence Theorem and the identity V . (fVg) = fV . Vg + Vf .Vg ) leads to

q,(y ) = L1fr(x , y) [G;VcPm(x ) , llx]dSx - Iv V1fr (x , y) . [G;VcPm (x )]dVx (4.9)

= - Iv[G;V1fr(X,y)] . VcPm(x)dVx (4.10)

where S is the surface surrounding the heart muscle volume V . Equation (4.10) followsfirstly because the surface integral in Eq. (4.9) vanishes, i.e., G;VcPm' n, is zero for x onthe heart surface (G;VcPmis the source current, and therefore confined to the heart , so that itwill have no component normal to the heart surface ). The integrand on the right-hand-sideof Eq. (4.10) results becau se of the symmetry of G; (i.e., for vectors a, b and symmetricmatrix C, a' Cb = b'Ca). A second integration by parts , now applied to Eq. (4.10), gives

Given that y is not in V (e.g., we typically consider y to be on the body surface ), thevolume integral on the right-hand side ofEq. (4.11) is zero if G, is proportional to Ge (equalanisotropy). This is because in that case we would have for some scalar a

V · G;V1fr(x , y) = aV · (G; + Ge)V1fr(x, y) = 0, (4. 12)

where the second equality ofEq. (4.12) follows from Eq. (4.7) since y on the body surfaceis external to the source domain V. In this case, the imaging equation (4.11) becomes

cP (y) = - L[G;V1fr (X , y)] . ll xcPm(x )dSx. (4.13)

Since 1fr (x , y ) satisfies Eq. (4.7), as a function of y it can be thought of as the field generatedby a monopole at x. Thus, [G;V1fr(x , y) ] . n, = V1fr(x , y) . [G;llx] can be thought of asthe field generated by a current dipole at x pointing in the G;llx direction. This can beverified by introducing the second source -8(x' - y) where x' is a point close to x with theline between x and x ' oriented as G;llx' This monopole of opposite polarity is associatedwith a second Green 's function -1fr(x' , y ), so that the composite of monopole sources ofopposite sign at x and x ' approach a dipole . The appropriate limiting procedure leads to afield determined by (G;n) . V1fr as in the integrand above.


Thus, we have dervied a linear relationship between transmembrane potential at thecardiac surface S (endocardium plus epicardium), and measurable body surface potential.In the practical setting where the geometry is discretized, the function ¢ over body surfacepoints is expressed as a column vector whose components are measured potentials at variouselectrode sites, while ¢mis a vector whose components are transmembrane potential at someset of locations on the heart surface, and

-l[G;V1/J(X, y)]. nA·)dSx (4.14)

is a matrix.The forward problem associated with the above transmembrane potential formulation

requires knowledge of the anisotropic conductivity of the heart in construction of the operator Eq. (4.14). For example, solution of Eq. (4.7) for any given source point x requiresknowledge of G; + G, throughout the body volume, including in the heart (where wehave to know G; and Ge as tensors). There is an additional complication in that the equalanisotropy assumption is not accurate, so one must also consider the second integral on theright-hand-side of Eq. (4.11).

However, in the portion of the body external to the heart muscle we have that G; V¢; =0, since G; is zero outside the heart. In that volume, we have from Eq. (4.5) that

(4.15)

i.e., Laplace's equation. The relevant volume is bounded by the epicardium and the bodysurface. Thus, the boundary conditions are divided into two parts: 1) the zero normalcomponent of current density at the body surface, 2) the (unknown) epicardial potentials.Suppose we know the solution to this equation for the situation where the epicardial potentialis identically zero except for having unit strength concentrated at location x-and call thissolution kj(x, y). From the linearity of Eq. (4.15), we can then find the solution for anygeometrically complex epicardial potential distribution by simply adding together suchelemental solutions, just as was done for Poisson's equation. This again defines a linearrelationship as

(4.16)

where S is now the epicardial surface and ¢ep; is the epicardial potential. Thus, we have thelinear relationship between epicardial potentials and (measured) body surface potentials.The Green's function k, (x, y) is provided by solution of the forward problem (see Chapter2 in this book). This formulation avoids having to consider the anisotropic myocardium inthe construction of k, (x, y), since the volume under consideration for the partial differentialequation does not include the heart volume.

The last source formulation we will consider is that of the endocardial potentials.If we design a transvenous catheter such that its tip is embedded with many electrodes(the "probe"), pass it into a cardiac chamber, and register its location with respect to theendocardium (e.g., via ultrasound or electronic means), then we can consider the volumebetween the catheter probe and the endocardium. Laplace's equation (4.15) still holds forthis volume (it is source-free). The boundary conditions are the zero component of current

128 F.Greensite

density normal to the probe surface, and the (unknown) endocardial potentials. Analogously,we can again use Green's functions to derive a linear relationship between the endocardialpotentials and the probe electrode potentials, and we again have an equation of the form asabove, i.e.,

</J(y) = ik2(x , y)</Jendo(x)dSx • (4.17)

That is, S is the endocardial surface, </Jendix) is potential at point x on the endocardium, and k2(X ' , y) is the solution to Eq. (4.15) subject to the endocardial potential asdescribed by 8(x - x'). Data </J(y) is the potential measured at point yon the electrode probesurface.

Figure 4.3 presents a diagram which underlines the similarities and differences, andrelative advantages and disadvantages, of the endocardial and epicardial potential reconstruction methods.

Equation (4.13) (for transmembrane potential), Eq. (4.16) (for epicardial potential)and Eq. (4.17) (for endocardial potential), constitute the basic imaging equations for thetechniques we will consider. As noted, there are very important distinctions between theabove source formulations, in that computation of the Green's function for the transmembrane potential source formulation on the right-hand-side of Eq. (4.6) requires knowledgeof the anisotropic conductivity of the heart as well as knowledge of the torso tissue conductivities external to the heart. In this regard, we note that strategies exist for patient-specifictissue conductivity tensor imaging via MRI (Reese et al., 1995; Ueno and Iriguchi, 1998),which might possibly be applied to the forward problem associated with Eq. (4.6). In theepicardial potential formulation Eq. (4.15), only knowledge of tissue conductivity externalto the heart is required-although this still requires knowledge of the heterogeneous torsoconductivity. However, recent work by Ramanathan and Rudy (2001) suggests that it isunnecessary to precisely include the torso tissue conductivities for the epicardial potentialimaging problem (the opposite conclusion was reached by Huiskamp and van Oosterom(1989) in the setting of activation imaging). In the endocardial potential formulation, oneonly needs to have a value for the blood conductivity-a striking simplification (that must,however, be paid for by the invasiveness of the approach).

After discretization of the relevant surfaces and volumes (see Chapter 2 in this book),Eq. (4.13), Eq. (4.16), and Eq. (4.17) become equations of the form

h = Fg, (4.18)

where h and g are data and source vectors, respectively, and F is a (transfer) matrix.

4.3 HEARTSURFACE INVERSE PROBLEMS METHODOLOGY

Some significant mathematical issues are involved in providing optimal solutions forthe imaging equations of the last section. Firstly, one must deal with the concept of an "illposed problem" (intuitively, a problem whose nominal solution is unstable to small changesin data-e.g., unstable to small noise variations). Accordingly, it is necessary to introduce"regularization" formalisms. Since the problem is inextricably bound up with noise, we

myocardium

"il·[G,"il¢] = 0

cavity G, a constant

"il . [G."il4>] = 0

(8<1»_ ~O

an p,.obe

endocardium:

epicardium: ¢'P'

body G, com plicated

129

'--------------- body su rface: ¢b.

FIG URE 4.3. A diagram containing the elements used to define the endocardial versus epicardial potentialimaging problems . Epicardial potential imaging problem: The outer box contains the body. The outermostellipse represents the epicardial surface, on which potentia l 1>el' i exists. Electrical potential satisfies Laplace'sequation in the body volume external to the epicardium. The mixed boundary condit ions of the (unknown)epicardial potential 1>el' i , and absent current component normal to the body surface, fully determine potential 1> inthe body volume external to the heart. Green's second identity can be used to derive a linear dependence betweenpotential at the body surface and potential at the epicardium. However, computation of this linear dependen ce (theforward problem) is dependent on knowledge of the extracellular conductivity Ge - which is very heterogeneous,e.g., due to the lungs, fat, and muscle (however, there is recent evidence that the impact of these inhomogeneities onan inverse solution may be small (Ramanathan and Rudy, 200 1» . Endocardial potential imaging problem: Theinnermost ellipse represents the surface of a catheter electrode probe, and the next ellipse represents the endocardialsurface. The region between these two surfaces (the blood-filled lumen of a cardiac chamber) contains no currentsources, so Laplaces equation holds in this volume. The boundary conditions are the endocardial potential, and theabsent component of current normal to the probe surface. Again, a linear relationship can be derived-this timebetween measured probe potentials, and the (unknown) endocardi al potential. The latter linear operator is mucheasier to compute than the corresponding operator for the epicardial potential problem, since the conductivity inthe relevant volume is uniform (simply being the conductivity of blood), and the geometry is easily measured(i.e., it does not require CT or MRl ). However, the attendant advantages are tempered by the fact the the methodis invasive.

emphasize the statistical approach, from which other methods (e.g., those of Tikhonov) canbe interpreted as special cases . Secondly, one must be prepared to deal optimally with atime seriesof such problems-i-i.e., stochastic processes.

4.3.1 SOLUTION NONUNIQUENESS AND INSTABILITY

As noted in the last section, we are required to deal with equations of the form h = Fg,where the components of vector h are our noninvasive or minimally invasive measurementsat various spatially acces sible locations, F is a transfer matrix (supplied by the forward

130 F. Greensite

problem solution), and g is the image of transmembrane, epicardial, or endocardial potential. Assuming that F has an inverse, one would presume that we could supply theimage as

g = F-1h.

To understand the naivity of such an approach, we can consider that our situation issimilar to being given a highly blurred image of a scene (our electrodes are located remotefrom the sources, and all the sources potentially contribute to what is measured at eachelectrode). We have some knowledge of the "point spread function", and can accordinglyattempt some "image enhancement" with the objective of "deconvolving" the source imagefrom the point spread function. But a blurring operator attenuates the information responsible for higher resolution. Thus, the signal power of the high resolution information relativeto the power of the low resolution information is much smaller in the blurred image than inthe unblurred source image. At the same time, noise (not described by the blurring operator)is also present in the resulting blurred image. That is, the noise (being added in additionto the blurring) will not be blurred away (much of it can be thought of as being addedafter the blurring operation, e.g., due to electrode noise or imprecisions in computation ofthe transfer matrix F). Thus, it will be typical that the noise power exceeds the signal powerin the high resolution subspaces of the data. If one naively applies the inverse of the blurringoperator, the noise will continue to dominate the high resolution subspaces-thus assuringcontinued absence of identifiable high resolution features. Furthermore, since the blurringoperator severely attenuates high resolution, its inverse must involve a marked amplificationrelevant to the high resolution subspaces-which then also markedly amplifies the noisein these subspaces, so that noise will dominate the entire solution (i.e., contribute most ofthe power to the resulting image). This means that a nonsense solution estimate will beobtained, also characterized by its instability to small changes in the data. Because of thedominance of noise in the high resolution subspaces, one must simply forgo restorationof the high resolution subspaces and be content with restoring those subspaces where theblurred signal outweighs the noise-unless physiologically meaningful and valid extrinsicconstraints can be imposed.

In other medical imaging modalities, one does not have this difficulty. In MR!, forexample, the spins (the hydrogen protons in the body water or fat) are induced to produce asignal (picked up by an antenna) whose frequency reflects their spatial location. Thus, theamplitude ofthe signal at a particular frequency reflects the number of spins at correspondinglocations. In fact, following selective excitation of the spins in a single slice of the body,the spins at a particular location in the slice are cleverly given two frequencies in differentbands (one from their spatially-varying NMR gyroscopic precessional frequency resultingfrom a magnetic field gradient applied along one spatial direction, and another one viamanipulations of their relative phase via application of magnetic field gradient pulses in thedirection orthogonal to the first). The relationship of frequency to magnetic field strength(which varies in space due to the applied gradients) means that an image of the tissue can beobtained by applying a two-dimensional Fourier Transform to the antenna data, so that themagnitude of the resulting function is an image of spin density in the tissue slice. Unlike ablurring matrix, a (discrete) Fourier Transform does not attenuate information in any sourcesubspace more or less than in any other subspace. Thus, inverting the effects of the operatordoes not involve any differential noise amplification.


To intelligently approach our dilemma, one ultimately needs to make the above notionsmore quantitative. For this, it is useful to introduce the singular value decomposition (SVD)of a matrix. A matrix represents a linear transformation, mapping a domain vector to arange vector. For any linear transformation, it can be shown that there exists a particularorthogonal coordinate system in the domain space, and a particular orthogonal coordinatesystem in the range space, such that a vector pointing along a coordinate axis of the domainspace, is mapped to a vector pointing along a coordinate axis of the range space, and whosemagnitude is amplified by a nonnegative scalar depending only upon which domain axis itwas pointing along. Since any vector in either space can be written as a linear combinationof unit vectors in the above coordinate axis systems, the preceding wordy statement isequivalent to the assertion that any matrix F can be written as the SVD,

F = USV I, (4.19)

where the columns of matrices V and U are the requisite orthogonal bases of the domainand range coordinate systems alluded to above, and

S = diagts}, ... , sn)

is a diagonal matrix whose diagonal entries (the singular values) are the amplificationconstants referred to above (they are arranged in order from largest to smallest). U and Vare each orthogonal matrices, and S is referred to as the singular value matrix. Note thateach singular value s, is associated with corresponding one-dimensional domain and rangesubspaces (the i-th columns of V and U). By convention, we will take U to be an (m x m)matrix, V to be an (n x n) matrix, so that S is a (m x n) matrix.

We are now in a position to understand the severe mathematically determined difficultyof our problem. In any noninvasive imaging technique for cardiac electrophysiology, F isalways severely ill-conditioned-because the field ¢ diminishes with distance from thesource, and the field at a point has contributions from all sources (there is "blurring"). Asa result, the ratio of its largest and smallest positive singular values is "large" (the value ofthe smallest positive singular value is "small" compared to the value of the largest singularvalue). That is, F is "ill-conditioned". In particular, the noise in the data in many of thesingular subspaces (columns of U) dominates the signal in those subspaces. However,F- 1 = V S-I U I (assuming the inverse exists). A solution of the form F- 1h thereby entailsapplication of l/si to the data component of h in the Uu subspace. If this is one of themany subspaces for which 1[s, is very large and in which the noise dominates the signal,we can appreciate that the noise is this subspace will be markedly amplified in the solutionestimate (this will also imply that the solution estimate will be very unstable to small noiseperturbations). Intuitively, we would thereby expect that it will be necessary to somehowattenuate the solution components associated with many (or most) of the subspacesmeaning that there will be a severe limit on the number ofdegrees offreedom in a meaningfulestimate for g in Eq. (4.18) (essentially given by the number of singular values large enoughnot to attenuate signal components below the noise amplitude in the subspace defined by thecorresponding column of U). Thus, much of the structure of an estimate for g must come inthe form ofa priori constraints-s-either by default (imposed as artifacts of the regularizationprocedure), or by design (constraints that truly reflect the class ofphysiologically meaningful

132 F.Greensite

solutions). Without such constraints, the solution estimate would be nonunique-since theaddition of any vector in the supressed high resolution subspace to any solution estimategives a new estimate that is also consistent with the accessed data.

The field of Inverse Problems typically deals with situations where one is given datareflecting the effect of some operator on a "source" we would like to estimate, but wherethe inversion procedure (undoing the effect of the operator) is inherently unstable (e.g.,highly noise amplifying), and (in practical terms) solution estimates are not unique. Suchproblems are loosely referred to as "ill-posed" (the latter term has a quite precise meaningin general Hilbert space settings, that we will not go into further).

4.3.2 LINEAR ESTIMATION AND REGULARIZATION

Taking noise vector v into account, Eq. (4.18) becomes

h=Fg+v, (4.20)

where it is required to estimate g given hand F.It might first occur to us that a useful estimate for g would be the one which maxi

mizes p(hlg)-i.e., the choice that maximizes the conditional probability (density) that themeasured h would occur given a particular candidate for g. This is known as the "maximumlikelihood estimate". Assuming F has an inverse F- 1, and the noise has zero mean, thisis given by gml = F- 1h-because the likeliest value for the noise (its mean) is the zerovector (under conditions of zero mean Gaussian noise, the maximum likelihood solutioncoincides with the least-squares solution, i.e., if F- 1 does not exist, F- 1 is replaced by thepseudoin-verse). If F is ill-conditioned, we have seen that this is in general a very poorestimate for g, and is very unstable to noise variations.

But one could alternatively take the estimate for g to be the choice which maximizesp(glh)-i.e., the choice that maximizes the chance that a particular g will be present giventhe observed noisy data h. This is referred to as the "maximum a posteriori" estimate,gmap. The relationship between the two conditional probability densities above is given bya version of Bayes Theorem: p(glh) = [p(h Ig)p(g)]/p(h).

gmap has the great advantage of being stable to noise variations-but the disadvantagethat its calculation requires one to first supply nontrivial information concerning statisticalproperties of g (in fact, the entire "posterior" probability density p(g)).1f such is availableand reliable, the methodology is referred to as "Bayesian". If one supplies the statisticalproperties as "drawn out of the air", or perhaps estimated from the given data h itself("noninformative"), the methodology is referred to as "empirically" Bayesian. The generalapproach is also referred to as "Satatistical Regularization" (evidently, in discrete settings,the maximum likelihood estimate is equivalent to the Bayesian approach in a minimuminformation setting where every realization of g is considered equally likely to occur).

All of this suggests that, once noise is introduced as in Eq. (4.20), it is useful tocarefully consider statistical notions. We will endeavor now to make these a little moreprecise. Specifically, each component of noise vector v is a "realization" (outcome) of a"random variable", the composite of which defines random vector v. For our purposes,a random variable is an entity that associates a probability density value with every realnumber. From this, we can compute the probability that some realization of the random


variable (outcome of a measurement) will yield a value falling in some given interval.Accordingly, the expectation tTl of some expression involving a random variable is theintegral of the expression over all possible values of the random variable weighted by theprobability density associated with each value (a zero mean random variable ais such thatEra] = 0). Furthermore, a "Gaussian" random variable ahas a Gaussian probability density(the familiar bell-shaped curve), and is fully characterized by its particular expectation £[a]and variance £[(a - £[a])2]. Similarly, a zero mean Gaussian random vector W is a columnvector of zero mean jointly Gaussian random variables Wi, i.e., W = (WI, ... , wnY. A zeromean random vector is further characterized by its autocovariance matrix £[ww t

], whichdescribes the dependence between all different pairs of components of W (note that theproduct of jointly Gaussian random variables is Gaussian). Similarly, the cross-covariancematrix of zero mean random vectors v, w is given by £[Dw t

] , and describes the mutualdependence of ii and W.

Just as v is a realization of a random vector v, so too can g be considered to be an(unknown) realization of random vector g. For notational simplicity, in this subsection wewill suppress the superscript" - ", and denote a random variable and its realization by thesame symbol. However, we will resume the notational distinction in the next subsection.

In approaching Eq. (4.20), a good objective is to find gopt such that £[lIg - gopt 112]

is minimum (this being the "minimum-mean-square-error" estimate). If g and v are realizations of zero mean Gaussian random vectors, gopt is obtained via the Wiener filter.Under these conditions, the maximum a posteriori estimate gmap is equivalent to gopt. Thislinear estimation procedure develops as follows.

A linear estimate of g is given by application of an "estimation matrix" Mest to datah, i.e., gest = Mesth. Ideally, we desire the solution estimate

(4.21)

such that £[llg - Mopth 11 2] is minimum. Thus, it is sufficient to calculate Mopt' The way to

proceed follows from the "Orthogonality Principle", which asserts that gopt minimizes themean-square-error when

(4.22)

i.e., when the cross-covariance matrix of the "error of the estimate" (g - gest)and the "datavector" h is the zero matrix (so that the error and the data have no dependence). Intuitively,the Orthogonality Principle assures that every bit of useful information is extracted fromthe data h in making the solution estimate gopt. Substitution of Eq. (4.20) and Eq. (4.21)into Eq. (4.22) immediately gives

(4.23)

Thus, assuming that g and v are independent (i.e., £[gvt] = 0), Eq. (4.23) can be writtenas

where Cg == £[ggf] and C; == £[vvt] are the autocovariance matrices of signal g and

134

noise v. Hence,

F.Greensite

(4.24)

Thus, the optimal solution estimate for Eq. (4.20) is provided by Eq.(4.24) and Eq. (4.21)assuming we know the autocovariance matrices of signal and noise.

It is interesting to express this estimation matrix MopI as a modification of F- 1,

assuming the latter exists. We have from Eq. (4.24) that

where Ci; = (FCgFI + Cv ) is the autocovariance matrix of h (as is seen via Eq. (4.20)).Thus, Mop I involves an initial preprocessing of the data h via (Ch - CV)Ch1 (the classicalWiener filter (Papoulis, 1984)) followed by application of F- 1 (note that providing Ci;nominally requires knowledge of Cs- although one can attempt to estimate Ch using the givenmeasurement h itself-essentially the problem of spectral estimation (Papoulis, 1984)).

If Cv = a; I (i.e., if the noise is white), Eq. (4.24) becomes

(I 2 -1)-1 IMop l = rrs «;c; F, (4.26)

since (FIF+a;C;-I)CgFI = FI(FCgFI+a;l). If Cg=aiI, using the SVD, F=USV I

, we can write Eq. (4.26) as

(4.27)

An alternative applicable formulation for treating Eq. (4.20) is provided by themaximum likelihood method in concert with a deterministic constraint (such as that thesignal power is equal to some a priori vaule E). Maximum likelihood then corresponds tominimizing IIFg - hll2 subject to IIgl1 2 = E (assuming Gaussian noise). This also leads toa linear estimation matrix of similar form to the above. Ultimately, the distinction betweenthe Bayesian and constrained maximum likelihood methods are that the latter treat only thenoisy measurements in a statistical fashion, while the former also treats the underlying signal statistically. Thus, the Bayesian methods potentially allow (or require) introduction of alarger class of a priori information. A third alternative, Tikhonov regularization (Tikhonov,1977), could be viewed as either a hybrid or an empirical Bayesian method. The linearestimation matrix supplied by it can typically be interpreted as resulting from the assumption of white noise and selection of a term proportional to Cg /a2 in Eq. (4.26), where theproportionality scalar (the regularization parameter) is selected using either a deterministicor data-dependent constraint.

Comparing the right-hand-side of Eq. (4.27) to F- 1 = VS-I U I (assuming the inverse exists), we can appreciate the violence that estimation theory does to the notion of ahigh resolution reconstruction of g. Reversal of the effects of F would require applicationof 1/s i to the data component in the U; subspace. Instead, for the regularized estimate,the factor sd(s; + y) is applied. As s, becomes small, this factor bears no resemblanceto 1/Si-SO there is no attempt at faithful reconstruction of components of the associated source subspaces. Another way of looking at the situation, is that a square-integrable


(i.e., well-behaved) function (or image) must be such that its higher order Fourier coefficients tend to zero (we imagine the Fourier coefficients to be with respect to the SVD domaincoordinate system of F, given by the columns of V in the discretized approximation). Inthe presence of white noise, whose Fourier coefficients therefore do not tend to zero, it isclear that higher order Fourier coefficients of data h are hopelessly noise-corrupted. TheWiener filter, and Tikhonov regularization, achieve stable results by removing any attemptat meaningful reconstruction of the high resolution components.

There is only one way out of the dilemma of resolution loss. If physiological constraintsexist which effectively reduce the dimension of the solution space to be commensurate withthe number of useful data Fourier coefficients, one can anticipate that it will be possible topreserve spatial resolution. For example, for the inverse electroencephalography problem(where one wishes to image the brain sources of the scalp electrical potentials), it might betrue that only a single focus is responsible for inciting an epileptic seizure, and that this focuscan be modeled as a single current source dipole located at some unknown location in thebrain. In that case, one is searching for an entity with six degrees of freedom (reflecting itslocation, orientation, and magnitude). High spatial resolution could conceivably be possibleassuming there are six or more data Fourier coefficients (with respect to the transfer matrixSVD-derived coordinate system) that are not dominated by noise. At first blush, such anobvious constraint does not appear to be physiological in the heart, since the heart is notfaithfully modeled as a single dipole. However, a deeper look at the geometry revealsthat such constraints do in fact apply (in principle) for the "critical points" of ventricularactivation-from which an activation map can be fashioned (see Section 4.6).

4.3.3 STOCHASTIC PROCESSES AND TIME SERIES OF INVERSE PROBLEMS

The data available in our problem are distributed in time as well as space. Thus,Eq. (4.20) would be more appropriately written as

hi = Fig, + Vi, (4.28)

i = 1,2, ... , n, where i indexes the time instants at which measurements are made (notethat we leave open the possibility that the transfer matrix is time-varying, thus we write itas F;). Underlying Eq. (4.28) are the time series of random vectors, hi, gi, and vi-i.e.,stochastic processes. The important additional feature of a stochastic process is that the i-thrandom vector may have correlations with the j-th random vector for j =1= i. However, a"state variable model" which embodies known correlations between gi and gj, for i =1= j, isnot explicitly available in our problems. Given the lack of such explicit accurate constraints,it is usual to adopt a "minimum information" perspective. Though such an approach seemsreasonable, and suggests that the equations ofEq. (4.28) might best be treated independentlyof each other (by simply applying the methods of the prior section to each one), the realityis more subtle.

For convenience, let us define matrices H, G, N such that Hi = hi, G:i = gi, Ni =Vi-SO that Eq. (4.28) becomes

Hi = FiG:i + Ni' (4.29)

136 F.Greensite

i = 1, ... , n. We then have the underlying random matrices as if, G, N such that theiri-th columns are hi, gi, Vi, respectively. The usual assumption is that the entries of G areindependent and identically distributed random variables (and similarly for the entries ofN). This is equivalent to the statement that all row autocovariance matrices are proportionalto the identity matrix (with the same proportionality constant), and all row cross-covariancematrices are the zero matrix (this is also equivalent to the statement that all column autocovariance matrices are proportional to the identity matrix, with the same proportionalityconstant, and all column cross-covariance matrices are the zero matrix). This minimuminformation assumption would imply that each member of equation sequence Eq. (4.28)can be treated independently of every other member of the sequence.

However, if we leave open the possiblity that there are correlations between the different gi, the members of the equation sequence can no longer be considered necessarilyindependent-and an optimal processing of the data is subject to specification (or identification) of appropriate choices of the cross-covariance matrices of the columns of G. Thus,suppose we continue to assume that the row cross-covariance matrices of G are the zeromatrix and the row autocovariance matrices of G are identical, but that the latter are notnecessarily proportional to the identity matrix (thus, we will be rejecting the minimum information approach). This means that the column autocovariance matrices are proportionalto the identity matrix-but we still have not specified the column cross-covariance matrices.Estimates for these will be derived from the data, i.e., empirically. This is actually not aradical thing to do, since even in the minimum information approach one typically derivesthe signal power (or signal-to-noise ratio) from the given data (thus, the minimum information approach is by no means "pure" in this respect). In fact, under the present conditions,there is a favored nontrivial choice of each cross-covariance matrix £[Gi 0 G:j].

For the purposes of linear estimation, Eq. (4.29) can be equivalently written in blockmatrix form as

(~:l) = [~' ..~....~ .. :] (~l) + (~:1).H:n 0 0 . . . F; G'; Nn

(4.30)

The Wiener filter (detailed in the last subsection) supplies an optimal estimate of the entriesof G as given by

(4.31)

where diag(Fi ) is the block diagonal matrix on the right-hand-side of Eq. (4.30), and

c., = (£[G:i 0 G:j])i,j

CN = (£[N: i 0 s.»;(4.32)

(4.33)


i.e., Cc is the block matrix whose (i, j) block entry is the cross-covariance matrix of ifiwith c.; etc. Thus, Cc is the (large) autocovariance matrix ofthe random vector consistingof the entries of G. Equation (4.31) is simply the composite of Eq. (4.21) and Eq. (4.24)applied to Eq. (4.30).

In the "Standard Method" (the minimum information approach), one takes E[G:i 0G: j ] to be the zero matrix when i #- j. In the "New Method", for i #- j, one takes

(4.34)

assuming the trace of F! F, is not zero. It can be shown that the mean-square-error in theresulting estimate of signal autocovariance matrix Cc is smaller than the estimate used inthe Standard Method (Greensite, 2002).

For the case of white noise, and assuming the F, are identical (i.e., F, = F, for all i),it can also be shown that the New Method reduces to the following procedure: Instead ofindividually treating the equations

Hi = FG:i + Ni,

i = 1, ... , n, we instead individually treat the equations

HX: i = FGX: i + NX: i ,

(4.35)

(4.36)

i = 1, ... , n, where the columns of n x n matrix X are the eigenvectors of HI H. Denotingthe solution to the i-th equation ofEq. (4.36) as (GX):i (the i-th column of a matrix (GX)),we take the solution estimate for G to be

(4.37)

The method generalizes to the case where there are nontrivial spatial correlations,and also to the case where a priori constraints are available regarding time correlations.However, if nonwhite characteristics of the noise are known, the route of Eq. (4.37) isunavailable, and one is left with the computationally complex method resulting directlyfrom Eqs. (4.30)-(4.34) (Greensite, 2002).

Underlying the New Method is the recognition of a fundamental asymmetry regarding H = F G + N. That is, the signal G undergoes a spatial transformation, but does notundergo a time transformation. For an equation of the form h = Fg + v, where we consider h, g, v to be spatial vectors, it is quite reasonable in a filtering context to impose asignal (g) autocovariance matrix proportional to the identity matrix. This would imply anautocovariance matrix for the noiseless portion of h as given by F I F -implying nontrivialfiltering of noisy h (see the second equality in (4.25)). However, for an equation of the formh = ho+ v, where the h and v are considered time series, it is quite unreasonable to set thesignal (ho) autocovariance matrix proportional to the identity, since the resulting (Wiener)filter is no filter at all (assuming white noise). Thus, since F is a spatial transformation, whileG is spatiotemporal, one cannot simply impose the minimum information condition thatthe entries of G are independent and identically distributed-assuming that one wishes to

138 F.Greensite

effectively filter in the time domain. In the setting of "minimum constraints" as opposed to"minimum information", the New Method is a means of performing spatiotemporal filteringin a manner dictated by the broken-symmetry of the problem , and the desire to minimizemean-square-error in the utilized signal autocovariance matrix .

4.4 EPICARDIAL POTENTIAL IMAGING

The source formulations for the inverse problem of electrocardiography have includedthose of a single moving dipole (Gabor and Nelson , 1954), two moving dipoles (Gulrajaniet al., 1984), dipole arrays (Lynn et al., 1967; Barber and Fischman, 1961; Bellman et al.,1964; He and Wu, 2001), multipole expansion coefficients (Geselowitz, 1967), and a heartexcitation model (Li and He, 2001). But in a very influential letter to the editor, Zablow(1966) asserted the need to reconstruct an actual anatomically-based entity that was alreadybeing accessed invasively-so that artifacts of the source model might be minimized, andthe result could be thought of as representing some sort of verifiable physiological truth.In essence, he noted that a linear relationship existed between the epicardial potentialsand measurable potentials at the body surface, and suggested the former as the sourceformulation to be reconstructed. This was particularly attractive, since physiologists werealready engaged in measuring the epicardial potentials invasively, and these were deemeduseful. Over the next several decades many investigators pursued the objective of epicardialpotential imaging. The many proposed refinements of technique can be divided into thosepertaining to

• Statistical regularization,• Tikhonov regularization,• Truncated SVD regularization,• Constrained least squares regularization,• Nonlinear regularization methods ,• Augmented source formulation ,• Different methods for selecting regularization parameters,• Preprocessing the data,• Introduction of spatiotemporal constraints.

Before embarking on a discussion of these refinements, we observe that there is noconsensus regarding which methods are the most worthy of employment-and we do notattempt such value judgements here . A comprehensive approach to this question is itself asizable objective that has not yet been achieved. Ultimately, the difficulty is in the experimental setup required-i.e., the need for (ideally) simultaneous collection of body surfaceand epicardial data , with coincident anatomical imaging and electrode registration, in aseries of animals and human subjects with a variety of pathological conditions (Nash et al.,2000).

4.4.1 STATISTICAL REGULARIZATION

In what was apparently the first serious treatment of the epicardial potential source formulation , Martin and Pilkington (1972) reported on the dismal prospects of "unconstrained"


inverse epicardial potenti al imaging , identifying implications of the problem ill-posednessdiscus sed in Section 4.3. They subsequently applied the Weiner filter, and reported moreencouraging results in followup simulations (Martin et aI., 1975). This approach requiresestimates of both the signal and noise autocovariance matrices. While the noise might beconsidered white (ignoring inaccuracies in the forward problem construct F), a choice forthe signal autocovariance matrix is less obvious. They proposed two ways of choosing one.The first was based on estimating the spatial autocovariance from time ensembles of epicardial potential maps supplied from a representative set of activation sequences. The secondwas a Monte Carlo method, whereby each epicardial location was given some a prioriprobability of being activated at any given time, and epicardial maps were then generatedby random numbers assigned to each location-thus leading to a computation for the signalautocovariance matrix .

Following the innovat ions of Barr et al. (1977) on the forward problem, Barr and Spach(1978) reported on inverse calculation of epicardial potential in twelve dogs with chronicallyimplanted epicardial electrodes. In applying the Wiener filter, they simply opted to take thesignal autocovariance matrix as proportional to the identity (i.e., as random variables , theepicardial potential s at all locations on the epicardium were presumed to be independentand identically distributed). They concluded that some features of the epicardial potentialdistribution through time can be imaged, particularly in dogs for which detailed geometrymeasurements.were available (postmortem).

Recently, van Oosterom (1999) has re-examined the statistical regularization approach,concluding that impressive improvements in accuracy (compared with other regularizationmethods ) are possible if a nontrivial accurate signal covariance matrix is available. Hesuggested that the signal autocovariance matrix could be based on prior estimation of theactivation sequence via other techniques (e.g., as in Section 4.6).

4.4.2 TIKHONOV REGULARIZATION AND ITS MODIFICATIONS

Despite the presence of noise v in Eq. (4.20), we are still tempted to view our problemas one of applying a kind of "inverse" of F . Indeed, given that the magnitude of v is "small",we are even tempted to pretend that we are dealing with h = Fg . Our prior discussion hassurely revealed that one cannot expect to apply the actual inverse of F to h becau se ofinstability problems, but it is also useful to specifically address the (typical) setting whereF doesn 't even have an inverse (which will always be the case if F is not square). Firstly,it could be that h is not even in the range of F. In that case , we might (naively) choose thesolution estimate g est that minimizes

(4.38)

over all points g in m-space (the expres sion Eq. (4.38) is known as the "residual" or"discrepancy"). But a second problem could be that the residual may not have a uniqueminimizer (as occurs when the dimension of h is smaller than the dimension of g). Onecould then ask for the estimate gest that is of minimum norm among all the minimizers ofthe residual. From convexity arguments, it can be shown that the minimizer of the residualEq. (4.38) of smallest norm is unique . In fact, it can be shown that the minimum-norm

140

least-squares solution for gin h = Fg is given by

F. Greensite

where Ft is the "pseudoinverse" of F. For the SVD of F as in Eq. (4.19), the pseudoinverseis given by

where st is the diagonal matrix whose i-th diagonal entry 1/s, if s, -=I- 0, and zero otherwise.Intuitively, it is easy to see why this works: The null space of Ft is the subspace orthogonalto the range of F. Thus, Ft h does not burden the estimate with any component that doesn'tcontribute to fitting the data. Otherwise, Ft simply undoes the attenuation s, that componentsof g experience when F is applied.

However, the above is simply a fix for the situation where F does not have an inverse.From Section 4.3, we know that, even if F has an inverse, we are faced with solutionestimate instability if F is ill-conditioned-because of the subspaces corresponding to smallpositive singular values. This consideration obviously will still hold for the pseudoinversebased solution. We have already encountered the Wiener filter regularization approach,which requires a priori estimates of at least the form of the signal and noise autocovariancematrices.

Colli-Franzone et at. (1985) introduced the Tikhonov regularization method to theepicardial potential imaging problem, ostensibly avoiding the problem of providing estimates for the signal autocovariance and noise autocovariance matrices. This approach skirtsthe usual notions of stochastic processes, and instead begins with the desire to minimizeEq. (4.38). But instead of simply searching for a solution estimate gest which minimizes theresidual (which would lead to an estimate that is exceedingly noise sensitive and unstable),in the Tikhonov approach one searches for the solution estimate that minimizes

(4.39)

where R is some matrix. Thus, one seeks an estimate for which the residual II Fgest - h f is"small", while at the same time some other property of the estimate, measured by II Rgest [[2,is also small. For example, R might be the identity, in which case one is looking for anestimate with small residual as well as a small norm (unstable nonphysiological solutionswill tend to have large norms). Alternatively, R could be such that II Rgest 11 2 reflects the firstor second spatial derivative of the solution estimate-so that the regularized estimate wouldhave to be relatively "smooth". These approaches require selection of a "regularizationparameter" y, which regulates how strong an influence the co-minimized second propertyhas in determining the solution estimate.

In the Tikhonov approach, the regularized estimate is given by

(4.40)

obtained by setting to zero the derivative of Eq. (4.39) with respect to g (a gradient),assembling the simultaneous equations into a matrix equation, and solving for g (note


that Eq. (4.39) is a function of the variable g-a point in m-space; thus, its gradient isa vector in m-space). In the sense of Section 4.3, Eq. (4.40) evidently describes a linearestimation method-the estimation matrix being the expression in brackets on the righthand-side of Eq. (4.40). Many alternative regularization operators R can be used. With"zero-order Tikhonov", the estimation matrix results from the choice R = I-so the technique corresponds to statistical regularization under the assumption that the signal andnoise covariance matrices are both proportional to the identity matrix, with the regularization parameter presumptively being the inverse of the square of the signal-to-noise ratio(i.e., compare Eq. (4.40) to the first equality in Eq. (4.27)). First-order and second-orderTikhonov regularization correspond to a choice of R derived from the gradient and Laplacianoperators. Thinking of these in the context of statistical regularization (compare Eq. (4.40)to Eq. (4.26)), these symmetric higher-order Tikhonov reguarization operators correspondto a signal autocovariance matrix that is a smoothed version of the sharp "ridge" represented by the identity matrix (with the assumption of white noise). That is, there is nowa nonzero covariance between spatially proximate locations-instead of these being takento be independent (as with a signal autocovariance matrix proportional to I). AlthoughColli-Franzone et at. (1985) suggested that first-order Tikhonov regularization was moreaccurate in in vitroexperiments, Messenger-Rapport and Rudy (1988) found no significantdifferences in the results obtained with zero-order, first-order, or second-order Tikhonovregularization.

When expressed in terms of an SVD, F = Usv', the zero-order Tikhonov solutionestimate is given by

(4.41)

which employs a linear estimation matrix comparable to that in Eq. (4.27). A so-called"regional regularization" scheme was suggested by Oster and Rudy (1997), whereby thesolution is given as

where D is a diagonal matrix whose diagonal elements take on a few different values-ineffect, the diagonal values of D represent multiple regularization parameters. Again, thiscan be interpretted as an attempt to supply the signal autocovariance matrix.

The "spatial regularization" method (Velipasaoglu et al., 2000) selects R in a mannerinspired by the fact that the noisy data fails to satisfy the Discrete Picard Condition (Hansen,1992; Throne and Olsen, 2000). The latter condition asserts that a stable solution requiresthat the squares of the Fourier coefficients of the data with respect to the eigenvectors ofFP, should on average decay faster than the eigenvalues of FFt. A means of modifyingthe data to conform to this Condition is the basis of this approach.

4.4.3 TRUNCATION SCHEMES

As noted earlier, the minimum-norm least-squares solution for gin h = Fg is givenby gest = Ft h = Ysto' h, where st is the diagonal matrix whose i-th diagonal entry I/siif s, =J 0, and zero otherwise. With this in mind, one could consider a regularized solution

142

estimate given by

F. Greensite

(4.42)

where st is the diagonal matrix whose i-th diagonal entry is 1/ s, if s, > f and 0 otherwise.This regularization method is known as Truncated SVD, or TSVD regularization (Hansen,1992). The value f functions as a regularization parameter. TSVD performance is usuallyvery similar to that of zero-order Tikhonov regularization.

The truncation idea is also incorporated in the "Generalized Eigensystems" approachof Throne and Olson (Throne and Olsen, 1994), which is relevant to a finite element discretization of the body (rather than a boundary element discretization). Instead of truncatinga solution expanded in the singular vectors of the transfer matrix F (as in Eq. (4.42)), theyconsider a set of generalized eigenvectors defined over the entire finite element mesh, having the properties that each generalized eigenvector satisfies the boundary conditions on theforward problem, as well as Laplace's equation within the volume, and the "subvectors"consisting of the components on the epicardial surface are orthogonal. One then constructsa linear combination of the generalized eigenvectors such that the body surface potentialdata h is fitted by the components that correspond to locations on the body surface . Thecomponents corresponding to the epicardial potential locations then take on values determined by this linear combination-which would be the presumed inverse solution desired .However, this will nominally lead to an unstable noise-dominated solution . Therefore, onetruncates the linear combination, using only the generalized eigenvectors associated withthe largest generalized eigenvalues-thus achieving a stable solution estimate . Instead ofexpanding (and truncating) the solution series in terms of the eigenvectors of F F' as with aTSVD (a sequence which most efficiently represents the effects of F for a given (truncated)number of terms), one is truncating a series derived from a set of field vectors that mostefficiently pack the power of the field over the entire body volume in a given (truncated) setof components.

Truncation is also employed in the "local regularization" scheme of Johnson andMacLeod (Johnson, 2001). In this approach, it is recognized that F is expressed in termsof the "inverses" of three different submatrices, when a finite element discretization of theforward problem is employed. Since these matrices have much different condition number,the implication is that they should each be receive different degrees of regularization (e.g.,individualized SVD truncation).

4.4.4 SPECIFIC CONSTRAINTS IN REGULARIZATION

The Tikhonov type formulation Eq. (4.39) also suggests a way in which other typesof constraints could be formulated . For example, suppose one has knowledge that thesolution shares some features with a preliminary estimate grough ' One could then suggestthe minimization of

(4.43)

The explicit expression for g esl is then obtained by setting the gradient of the above expression to 0, and solving for g. This is known as the Twomey method (Oster and Rudy,1992).


Iakovidis and Gulrajani (1992) introduced a method whereby a deliberately overregularized estimate (to be used to estimate the location of the epicardial zero-potentialline, but otherwise having too few interesting features ) was used to constrain the solutionfor what would otherwise be an under-regularized estimate (i.e., the second regularizationparameter would have been too small if the constraint had not been present, in the sensethat the estimate obtained would have been unstable and "noisy").

4.4.5 NONLINEAR REGULARIZATION METHODOLOGY

Overwhelmingly, linear estimation methods have been applied to obtain regularized(i.e.•noise-stable) estimates of linear formulations of the inverse electrocardiography problems. However, nonlinear (e.g., information theoretic) methods have also been applied (aparticularly recent example is provided in (He et al., 2000)). We will not describe suchapproaches here.

4.4.6 AN AUGMENTED SOURCE FORMULATION

If one treats Eq. (4.5) via a Green's function approach along the lines of what wasdone with Eq. (4.6) (i.e., performing two integrations-by-parts analogous to Eq. (4.9) andEq. (4.11)), one obtains an expression relating body surface potentials to the compositeof epicardial potentials and the normal component of epicardial current density (as in(Greensite, 2001, p. 151-152),

Noting that such a formulation occurs as an intermediate step in the development of theforward problem method expounded by Barr et al. (1977) , Horacek and Clements (1997)investigated solutions obtained where both the epicardial potential and epicardial normalcurrent density are inversely computed. They suggested that this problem might be slightlybetter posed than the traditional epicardial potential formulation, and they also investigatedrefinements in the regularization technique.

4.4.7 DIFFERENT METHODS FOR REGULARIZATION PARAMETERSELECTION

The Tikhonov estimate Eq. (4.40) requires selection of a value for the regularization parameter. Some guidance in this regard is provided by the "Discrepancy Principle"(Hansen, 1992). Here the reasonable assumption is made that the discrepancy (or residual) IIFg - h 11 2 is not zero-because of the noise present. Rather, the discrepancy (for thetrue solution g) is most likely to be the noise power. Thus, it makes sense to look for asolution estimate which produces this discrepancy. So, consider the constrained minimization problem: "minimize IIgll 2 such that II Fg - hll 2 = f". Again , IIgl1 2 and II Fg - hll 2 areeach functions of g (which varies over points in m-space). We know from Calculus thatminimization of the first function subject to the constraint on the second function implies(under rather general conditions) that the gradients of the two functions at the constrainedminimum gm in lie on the same ray-i.e., the gradients are proportional at gmin' Another

144 F. Greensite

way of saying this is that there exists a scalar y such that the gradient of Eq. (4.39) is zeroat gmin (in this case R = 1). As we know, the requisite gmin is given by the right-hand-sideof Eq. (4.40), where the regularization parameter y was to be determined. But now, theregularization parameter is simply given as that which produces a Tikhonov solution estimate gmin satisfying the discrepancy expression [IFgmin - h 11 2 = E, where E is the noisepower.

However, the error in the data (noise power E) is not known (being a composite ofelectrode noise and modeling errors in F). There are actually several other methods forregularization parameter selection. In the so-called "Lcurve method" (Hansen, 1992), onecomputes a log-log plot of the first term in Eq. (4.39) versus the second term in Eq. (4.39)(the residual versus the solution estimate seminorm). The solution estimate is chosen as theone corresponding to the "comer" of the above L-shaped graph (a balance between smallseminorm and small discrepancy). It should be noted, however, that a comer on the L-curvedoes not always exist.

The Composite Residual and Smoothing Operator (CRESO) method (Colli et al., 1985)chooses the smallest positive value of the regularization parameter for which the secondderivative of the first term in Eq. (4.39) with respect to the regularization parameter equalsthe second derivative of the second term in Eq. (4.39) with respect to the regularizationparameter. The cross-validation method (Whaba, 1977) is another important regularizationparameter selection method, though it has not been prominently applied in the context ofthe inverse electrocardiography problem.

4.4.8 THE BODY SURFACE LAPLACIAN APPROACH

As we have previously noted (see equation (4.25)), statistical regularization can beviewed as the composite of a particular "preprocessing step" followed by application of thestandard inverse (if such exists). This suggests the possible application of other preprocessing steps. He and Wu (1997) proposed preprocessing the data to be in the form of bodysurface Laplacian measurements, and solving the inverse problem using this processed dataas input. The resulting transfer matrix is of a different nature (evidently, it results fromleft multiplying the usual transfer matrix by the same matrix that is used to preprocess thebody surface data), in that the surface Laplacian of potential at a body surface location issignificantly influenced by many fewer source locations than potential itself (i.e., there isless "blurring"). As a trade-off, the effect of the source on the "data" falls of more rapidly(with the fourth power of the distance from the source, rather than the second power of thedistance, as with body surface potential), and there is a theoretical noise amplification innumerical differentiation of the body surface potential.

Ignoring the curvature of the body surface, the body surface Laplacian of body surfacepotential h with respect to an (x, y) body surface coordinate system is given by

Ultimately, we can write this as L[h], where L is the Laplacian-a linear operator. After theproblem is discretized for numerical treatment, L[·] is simply a matrix. Thus, h = Fg + v


becomes

Lh = (L F) g + L v,

145

Our data is now Lh, and we now need to "invert" (L F) to estimate epicardial potentialg. The regularization tools remain the same as before. L is a differential operator-andthus can be thought of as akin to a high pass filter. With respect to a Fourier expansion ofh , application of L amplifies the high frequency terms. In particular, the high frequencycomponents of the noise are greatly amplified (in the nondiscretized setting, the noise is"unboundedly" amplified).However, the latter is in principle taken care of by the fact that LFis more "singular" than F -meaning that it's "inverse" will be smoother (i.e., applicationof the inverse is more stable, and tends to smooth noise). If one has an expectation ofsomehow reducing the L v contribution to data Lh prior to application of the "inverse",one might expect greater stability and fidelity in the estimate of g than that obtainable withthe direct treatment of h = Fg + v. Such an expectation could be reasonable with use ofLaplacian electrodes (He and Cohen, 1992), though it has not been established that these canbe accurately designed in practice. In the absence of such, investigations have proceededwith direct application of L to measured data h . The approach seems to have potential inidentifying and spatially distinguishing cardiac sources close to the body surface electrodes(Johnston , 1997; He and Wu, 1997).

4.4.9 SPATIOTEMPORAL REGULARIZATION

Ultimately, one is faced with a time series of problems

(4.44)

i = 1, 2, ..., where the subscript i now refers to the source and data at the i-th time pointin the cardiac cycle. Oster and Rudy (1992) suggested using preliminary (e.g., zero-orderTikhonov) estimates at time points i - I and/or i + 1 to constrain the regularization atthe i-th time step. This was done using a Twomey regularization formalism via equat ionEq. (4.43).

On the other hand, Eq. (4.44) evidently describes a stochastic process, which begs thequestion of applying Kalman filter theory. This requires that a stochastic model be applied,defining the presumed interdependence of the epicardial potentials between different times(Joly et al., 1993)-is itself a not entirely trivial problem.

Temporal and spatial constraints can also be joined by the the method of Brookset al. which employs two or more regularization parameters in a traditional constrainedminimization format (Brooks et al., 1999). Thus , Eq. (4.44) is written as

(4.45)

One now writes a functional to be minimized, consisting of the residual (for this augmented

146 F. Greensite

problem), a spatial regularizing operator (e.g., expressing the sum of the norms of thesolution estimates at each time point), and a temporal regularizing operator (e.g., the magnitude of the discretized "time derivative" of the solution estimates over all the time points)where the latter two operators are given their own regularization parameters. The solutionestimate is ultimately expressed as

where diag(F) is the block matrix on the right-hand-side ofEq. (4.45), B is the discretizedversion of a temporal differential operator, and Yl, Yz are the two regularization parameters.

The "admissible solution" approach of Ahmed et al. (1998), posits that any solutionsatisfying a sufficiently robust composite of constraints is deemed satisfactory, and suchconstraints can include those related to time. The solution algorithm requires the constraintsto be convex, e.g., the "ball" of vectors g satisfying IIg liZ < C is an example of a convex set(any line joining any two members of the set consists only of members lying in the set). Theneed for regularization parameters is replaced by the need for bounds defining the requiredconvexity.

Finally, the approach of Greensite (1998; 2002) (described in Section 4.3.3) effectively replaces the original sequence of (nonindependent) Eq. (4.35) (or Eq. (4.44)) witha smaller number of mutually independent equations Eq. (4.36)-without the impositionof any extrinsic temporal constraints (or temporal regularization parameter). The methodderives from the recognition that there is something intrinsically wrong with the assumptionthat the entries of G are realizations of independent and identically distributed random variables. Indeed, this symmetry condition is broken once one poses the problem described byEq. (4.44). Given the assumption that the rows of G are independent and have identical autocovariance matrices, the solution mechanism uses a more accurate signal autocovariancematrix estimate than the other methods (in a mean-square-error sense).

4.4.10 RECENT IN VITRO AND IN VIVO WORK

There has been a significant amount of in vitro work done with the Utah torso tank. Theexperimental setup employs a heart suspended in a torso shaped electrolytic tank, perfusedby an anesthetized dog external to the tank (Oster et al., 1997). Electrodes are present on theouter margin of the tank, and also in proximity to the epicardium. Such work has addressedinverse reconstruction of epicardial potentials and activation in the setting of sinus rhythm,pacing, and arrhythmias (Burnes et al., 2001), and has also been used to assess the impactof torso inhomogeneities (Ramanathan and Rudy, 2001).

Perhaps the most impressive demonstration of the potential promise of the epicardialpotential imaging formulation is in the quasi-in vivo work where densely sampled epicardialpotential data was accessed from infarcting dogs, and used in simulations with numericalhuman torsos to test the fidelity of the zero-order Tikhonov inversion under realistic conditions of modeling and electronic noise (figure 4.4) (Burnes et al., 2000). This approach hasalso been applied to demonstrate the feasibility of reconstructing repolarization propertiesof interest (e.g., increased dispersion of repolarization (Ghanem et al., 2001)).

On the other hand, earlier results of a different group using invasive data from patientsundergoing arrhythmia surgery, suggested that the usual epicardial potential regularization


FIG URE 4.4. Maps of epicardial potential at two different times during ventricular activation. in the study of(Burnes et al., 2000). The invasively measured epicardial potential data (from dogs) is used to forward generatebody surface potentials on a numerical human torso surface. Geometri cal and "electronic" noise is added to thesebody surface potentials , and inversely reconstructed epicardial potential maps are computed. [From: Burnes, J. E.,Taccardi , B.• Macleod, R. S., and Rudy, Y, 2000, Noninvasive ECG imaging of elec trophysiologcially abnorma lsubstrates in infarcted hearts, a model study, Circulation. 101: 533- 540. Used by permission.]

methodology was able to usefully image epicardial potential during the QRS inteval onlyin its initial portions (Shahidi et ai., 1994).

Finally, a report by Penney et ai. (2000) (extending work by MacLeod et al. (1995))identified local changes in inversely computed epicardial electrograms in patients whosedata was accessed during coronary catheterization, preceeding and following angioplastyballoon catheter inflation. In the eighteen study patients, the predicted region of ischemiafollowing balloon inflation correlated with the expected region of perfusion deficit basedon the vessel occluded.

4.5 ENDOCARDIAL POTENTIAL IMAGING

Interventional cardiologists employ transvenous catheter procedures to treat arrhythmogenic foci and aberrant conduction pathways. Such treatment first requires mapping theendocardial potential. This initial invasive imaging is cumbersome, tedious, and lengthy.Typically, a roving probing trasvenous electrode catheter is brought into contact with manyendocardial locations over the course of many heartbeats, and a depiction of an endocardialactivation map is thereby inferred (an improved technology along these lines is given byGepstein et ai. (1996)). The number of sites accessed is limited , and there is no accountingfor beat-to-beat activation variability when reconstructing the maps from the many beats.Recently introduced expandible basket electrode arrays (Schmitt et ai. , 1999) have theirown problems related to limited numbers of electrodes, the need to contact (and perhaps

148 F. Greensite

irritate) the endocardium, and the possibility of difficulties in collapsing the basket at theend of the acquisition.

These problems can be potentially addressed by the use ofa transvenous catheter whosetip is studded with multiple electrodes, and which is placed somewhere in the midst of acardiac chamber (without contacting the endocardium). Once the catheter location relativeto the endocardium is registered, it becomes theoretically possible to inversely computethe endocardial potentials from a single heartbeat-indeed, to follow dynamic isopotentialmaps within a single beat, as well as beat-to-beat changes in activation maps. For thisinverse problem, the volume is bounded by the endocardial surface and the multielectrodeprobe surface. Laplace's equation holds in this volume, and the boundary conditions are the(unknown) endocardial potentials, and the zero normal current density at the multi-electrodeprobe surface. As in Section 4.2, a linear relationship is derived between the endocardialpotentials and the catheter electrode potentials.

Notwithstanding the inconvenience of the required cardiac catheterization, there aretwo very significant advantages of this formulation over the technique of imaging theepicardial potentials from the body surface. First, the electrodes are relatively close to allportions of the surface to be imaged (e.g., as opposed to the distance between body surfaceelectrodes and the posterior wall of the heart). Second, the relevant volume is composed onlyof blood in the lumen of the cardiac chamber. Therefore, the geometric modeling requiredfor estimation of the transfer matrix is vastly less, and the uncertainties in the values of keycomponents of the model (i.e., tissue conductivities) are markedly diminished (the bloodhas uniform isotropic conductivity).

The initial proposal and work on a multielectrode noncontact array, placed in a cardiacchamber for purposes of accessing endocardial potentials, was due to Taccardi et al. (1987).In the past few years there has been much significant work reported on successors to thisidea. For example, in experiments on dogs, Khoury et al. (1998) used a 128 electrodecatheter, inserted via a purse string suture in the left ventricular apex, and showed thatfaithful renditions of endocardial activation, both with paced and spontaneous beats, waspossible by solving the inverse problem. Ischemic zones were also well defined. A spiralcatheter design has also been investigated (Jia et al., 2000).

An impressive series of experiments has been performed with a competing system,developed by Endocardial Solutions, Inc. In addition to a 64-electrode 7.5 ml inflatable balloon catheter, a second transvacular catheter is passed and dragged along the endocardium.As it is dragged, a several kHz signal is passed between it and the electrode catheter, localizing its position with respect to the electrode catheter. In this way, a rendition of theendocardium with respect to the electrode catheter is produced. Following construction ofa "virtual endocardium" via a convex hull algorithm applied to the above anatomical data,the inverse problem is then solved, generating several thousand "virtual electrograms" onthe virtual endocardium (figure 4.5 and figure 4.6).

The literature on this subject is growing rapidly, and we site only a few of examples.Overall, very impressive utility and fidelity is being established. For example, a report bySchilling et al. (2000) describes the classification of atrial fibrillation in humans in termsof numbers of independent reentrant wavefronts identified. A report by Strickberger et al.(2000) describes the successful ablation of fifteen instances of ventricular tachycardialguided by this catheter system. A recent report by Paul et al. (2001) describes the utility ofthe system in directing catheter ablative therapy in subjects with atrial arrhythmias refractoryto pharmacologic therapy.


ECGI ~~ rC '\,.-"'""-..rVvr-_'\_\J,(~---...,..N/~

R -'-----~~/\.../"'v'\~

ECGI v---,·r.-...._v.... ......./___.f'-

J\ J '\ rC ""~------ ,~ .>

R~ ~j"\~/'~,/'/~r~ECGI-~ JC ~r-'v~~{/r'~~~-"V\j)R "J""/\V'-'---\jl \,/"''"'\

FIGURE 4.5. Surface ECG from lead I (ECGI), endocardial electrogram via a contact electrode (C), and inverselyreconstructed electrogram using input from a noncontact multielectrode probe in the atrium (R), with C and Rfrom the same location, in three patients with atrial fibrillation, in the study of (Schilling et al., 2000, [From:Schillin g, R. J., Kadish, A. H., Peters, N. S., Goldberger, J., Wyn Davies, D., 2000, Endocard ial mappin g of atrialfibrillation in the human right atrium using a non-contac t catheter, European Heart Journal. 21: 550-564. Usedby permission of the publisher, WB Saunders .]

4.6 IMAGING FEATURES OF THE ACTION POTENTIAL

4.6.1 MYOCARDIAL ACTIVATION IMAGING

Epicardial and endocardial potential imaging addresses the need to reconstruct something that is currently accessed invasively, and is thus of evident interest. However, suchpotentials are not themselves a clinical endpoint. Ultimately, clinicians are interested in theaction potential---or at least , feature s of the action potential. The most important featuresof the action potential are activation time (time of arrival of phase zero at every location,the aggragate of which globally describe conduction disturbances), phase zero amplitude(reflecting ischemia), and action potential duration (reflecting refractory periods, potentiallyassociated with propensity for re-entrant arrhythmias).

The marker for activation in an electrogram (a tracing of epicardial or endocardial potential at a given cardiac site) is the "intrinsic deflection"-defined as the steepest downwarddeflection of the electrogram. Recall that the source is the gradient of transmembrane potential (e.g., Eq. (4.10» , and that during cardiac activation this is usually appreciably nonzeroonly at the locus of points undergoing action potential phase O. This locus is approximatelya surface (the interface between depolarized and nondepolarized muscle). Electrically, thisbehaves approximately as a propagating surface of dipole moment density (a double layer).There is a discontinuity of potential as the double layer is crossed. Ideally, as an extracellularlocation is passed over by the activation wavefront, there will then be a sharp downwarddeflection in the extracellular (electrogram) potential-the intrinsic deflection. However,

150 F.Greensi te

FIGURE 4.6. Time sequential views of a portion of the "virtual endocardium" depiction of the atria in thestudy of (Paul et al., 2001), showing isopotential maps at six successive times. Spreading endocardial activationwavefronts can be appreciated (e.g., two of these collide in E and F) during an atrial reentrant tachycardia. Seethe attached CD for color figure. [From: Paul, T., Windhagen-Mahnen , B., Kriebel, T., Bertram, H., Kaulitz, R..Korte, T., Niehaus, M., and Tebbenjohanns, 1., 200 I, Atrial Reentrant Tachycardia After Surgery for CongenitalHeart Disease Endocardial Mapping and Radiofrequency Catheter Ablation Using a Novel, Noncontact MappingSystem, Circulation. 103: 2266-227 1. Used by permission.]

the reality is that it is not infrequent that there is more than one reasonable candidate for theintrinsic deflection within a given location's electrogram. Furthermore, the intrinsic deflection is often rather lengthy, so the selection of a single activation time within the intrinsicdeflection is to some extent arbitrary (Ideker et al., 1989; Paul et al., 1990). The activationtime is presumably the inflection point of the deflection (which itself is poorly defined inthe noisy setting). To a large extent, these problems are inherent in the source formulation :The epicardial (or endocardial) potential at a location actually reflects contributions fromelectrical activity at all surrounding locations , when in fact we desire to resolve results ofthe membrane function at a single location-i.e., the local action potential. In this sectionwe examine work done on imaging the myocardial activation feature of the action potential ,rather than the epicardial potential.

Enthusiasm for immedi ately attacking the problem of reconstructing the transmembrane potential </>m(x), or its gradient, is tempered by recogniti on of a dimensionality problem: our measurements are confined to a surface (of the body), while the source V . (G i V</>m)permiates a volume (the heart). Inherently, we are faced with a "projection" of the three

dimensional source on the two dimensional body volume (a further exacerbation of thealready described ill-posedness of the problem). However, building on the work of Wilsonet at. (1933), Frank (1954) noted that the source during the QRS inteval was roughly adouble layer (i.e., a surface), which tends to mitigate the above dimensionality problem.While Frank was interested in quantifying the inaccuracy of the single moving dipole modelof the heart via forward computations (rather than imaging the double layer), two decadeslater Dotti (1974) made an interesting observation: Neglecting anisotropic conductivity ofthe heart, assuming uniform action potential amplitude, and recognizing the fact that thegradient of transmembrane potential propagates as a dipolar wavefront (double layer), henoted that the source surface at any time is electrically equivalent (as regards points externalto the heart) to a double layer consisting of the portions of the endocardium and epicardiumalready depolarized (figure 4.7).

This is a consequence of the well-known fact from electrostatics that a closed uniform double layer in an isotropic medium generates no external potential. This means thatone can derive a relationship between the body surface potential at a given time, and thelocus of points on the cardiac surface that have been activated. Thus, the dimensionalityproblem resolves, and the surface of interest is actually fixed. Dotti presented a very smallscale two dimensional simulation illustrating this concept. Similar observations were madeindependently by Salu (1978) a few years later.

However, the concept can be said to have been formally introduced in a more completeengineering context by Cuppen and van Oosteroom in the early 1980s. They presented theimaging equation as

¢(y, t) = iA(x, y)H(t - r(x»dS, (4.46)

where S is the composite of the endocardial and epicardial surfaces, and the action potential(during the QRS interval) is modeled using the Heaviside function H(t) (zero for t < 0,unity for t > 0). Thus, the action potential (figure 4.1) is taken to be the step function

a(x) + b(x)H(t - r(x». (4.47)

The action potential amplitude b(x) is assumed to be constant over the ventricles, and issubsumed into the transfer function A(x, y). The offset a(x) is also assumed constant, andthus has no effect since Is A(x, y)dS = 0 (i.e., a uniform closed double layer generatesno external potential). Note that in the absence of reentrant arrhythmias there is no repolarization during the QRS interval, so the action potential can then be modeled as a stepfunction in that interval. Thus, Eq. (4.46) is fully consistent with Eq. (4.13) (dervied fromthe bidomain). Using Eq. (4.46), one wishes to determine rex), the time that point x on thesurface surrounding the heart undergoes action potential phase zero. Note that the equationis nonlinear.

Equation Eq. (4.46) achieves a superficially satisfying form upon integration over theactivation QRS interval,

f¢(y, t)dt = [ A(x, y)f H(t - r(x»dtdSx = - [ A(x, y)r(x)dSx . (4.48)

QRS 1s QRS 1s

152

B

F. Greensite

c

FIGURE 4.7. An electrical double layer in an infinite homogeneous volume conductor of infinite extent generatespotential at a point proportional to the solid angie subtended by the point and the double layer. The above diagramdepicts how ventricular intramural depolarization wavefronts generate potential equivalent to that generated by"virtual" double layers on the epicardium and/or endocardium. [From: van Oosterom, A., 1987, Computing thedepolar ization sequence at the ventricular surface from body surface potentials, in: Pediatric and Fundamental

Electrocardiography , (1. Liebman, R. Plonsey, and Y. Rudy,eds.), Martinu s Nijhoff, Zoetermeer, The Netherlands,pp. 75-89. Used by permission.]

Apparently, the imaging of myocardial activation is also a linear problem. But attemptsto solve Eq. (4.48) soon run up against the problem that the computed activation timesare entirely unrealistic-because regularization schemes typically favor solution estimateswith lower norm (even with higher order Tikhonov regularization). Thus, the computed


QRS intervalbecomeshighlycontracted. Furthermore, there is the impressionthat one hasbeen wasteful of the temporally resolved (dynamical) information inherent in Eq. (4.46),by integrating it all away in Eq. (4.48). This is unacceptable in an already very ill-posedproblem.Huiskampand vanOosterom(1988)addressedthis objectionablefeature by usingthe regularized solution to Eq. (4.48) as a seed for a quasi-Newton routine for solving aregularizedversion of the full nonlinear expression Eq. (4.46). As with the basic Newtonprocedure from Calculus, which extracts the root of a nonlinearfunction nearest the seed,the quasi-Newton procedureapplied here is a means of finding a root (i.e., the appropriater(x» fora regularizedversionof¢(y, t) - Is A(x, y)H(t - r(x»dS = O.However,aswiththe basic Newtonmethod, one is dealing with an intrisically local procedure that does notperform a global optimization. The solution estimate obtained is highly influencedby theinitial seedfrom Eq. (4.48).On the otherhand, there is no reasonwhy a globaloptimizationroutine such as simulatedannealing, could not be used (in fact, this is proposed in a veryrecentpaperon activation timeand actionpotentialamplitudeimaging(Ohyuet al., 2002».However, a furtherproblemis thatEq. (4.46) is validonlyunder the assumptionthat cardiacmuscle has isotropicconductivity, or satisfies equal anisotropy.

A differentapproachwas takenby Greensite(1994; 1995).The general idea is that themyocardial surface activation function r(x), like any (nominally differentiable) function,is greatly characterized by its relative extrema-e.g., its relative maxima and minima.Predominantly, these are the epicardial breakthrough points and activation sinks of thetransmuraldepolarization wavefront. Indeed, since t (x) is definedover a compact domain(the heart surface), and has a finite range (the QRS interval), knowledgeof these "criticalpoints" reduces the space of admissible solutions to that of a compact set of functions.The problem of reconstructingthe rest of r(x) from Eq. (4.48) is nominally a well-posedproblem.In simpleterms,if therelativemaximaandminimaof r (x) are known,theproblemof determining the rest of r(x) becomes simply a matter of optimized interpolation-forwhich the constraintsembodiedby Eq. (4.46) should be sufficient. An efficientmeans forcomputing the critical points was given in the Critical Point Theorem (Greensite, 1995).Consider the "data operator"

¢[.] = 1 ¢(y, t)(·)dy.QRS

In the practical setting, ¢ is a space-time matrix, each of whose rows is the body surfacepotential time series (ECG) at a particular electrode location. The Critical Point Theoremstates that x' is a critical point of r(x) if and only if A(xl

, y) is in the space spanned bythe eigenfunctions of ¢¢t. In fact, the Theorem holds even in the case of an anisotropicmyocardium. Complications ensue once noise is added to the formulation, but an efficient algorithm employingthese ideas in a noisy context was proposed in (HuiskampandGreensite, 1997).

Oostendorpet at. at the University of Nijmegen/University of Helsinkihaveproducedwork evaluating the latter approach both in vitro (Oostendorp et aI., 1997) and in vivo(Oostendorp and Pesola, 1998) (validation in hearts removed at the time of cardiac transplantation, figure 4.8).

Work on invasive validation of these latter ideas has also recently been undertakenby a group at the Technical University of Graz, (Tilg et aI., 1999; Modre et aI., 2001a,

154 F. Greensite

FIGURE 4.8. One of a series of four hearts, removed at transplantation, in the study of (Oostendorp and Pesola,

1998). The two upper images of the anterior and posterior ventricular epicardium show the activation maps obtained

at the time of surgery (prior to cardiac transplantation) via application of an epicardial electrode sock (epicardial

electrode locations indicated by circles). The lower two images show the corresponding preoperative activationmap, inversely computed from body surface potential electrode data. [From: Oostendorp, T., and Pesola, K., 1998,

Non-invasive determination of the activation time sequence of the heart: validation by comparison with invasive

human data, Computers in Cardiology. 25:313-316. Copyright IEEE. Used by permission].

Wach et ai., 2001; Tilg et ai., 2001; Modre et ai., 2001b), and a group at the University ofAuckland/University of Oxford (Pullan et ai., 2001).

4.6.2 IMAGING OTHER FEATURES OF THE ACTION POTENTIAL

Let tl be a time during the TP interval of the ECG-i.e., a time during which allventricular locations are in action potential phase 4 (fully repolarized). Let t: be a timeduring the ST interval of the ECG-i.e., a time during which all ventricular locations arein phase 2 (fully depolarized). From Eq. (4.13) and the action potential model Eq. (4.47),

¢(y, t2) - ¢(y, t1) = L[GiV1/r(X, y)]. nA¢m(x, t2) - ¢m(x, tl»dS

= L[GiV1/r(X, y)]. nxb(x)dS. (4.49)

Note that if b(x) is a constant, both sides of the above equation will be zero (e.g., a closeduniform double layer generates no external potential). Indeed, the body surface potential

during the TP and ST segments have the same value in healthy subjects. However, in thecase of cardiac ischemia, the action potential amplitude is spatially varying. In that setting,one can imagine solving the above integral equation to obtain the spatially-varying actionpotential amplitude-up to a spatial constant (the null space of the operator is the spaceof constant functions). Since the phase 0 amplitude in healthy myocytes is already knownto be approximately 90 millivolts, one can then (in principle) image the action potentialamplitude h(x) fully.

Reflecting on the approach of Cuppen and van Oosterom (1984), Geselowitz (1985)noted that it would be possible to image the area under the action potential (i.e., the integralwith respect to the baseline of the action potential) by simply extending the time interval ofintegration in Eq. (4.48) to be the interval (encompassing the time period of activation andrepolarization). Thus,

{ </>(y, t)dt = - { ([G;V1/f(x, y)]. Dx</>m(X, t)dSdtJQRST JQRST Js

=- ([G;V1/f(X,y)].Dxb(X) ( H(t-r(x))dtdSJs JQRST

= -l[G;V1/f(X, -n DxfL(X)dS,

where fL(x) is the area under the action potential at x.Now one can use knowledge of h(x) and fL(X) to create an image of action potential

duration as fL(X)jb(x). Thus, one might anticipate imaging action potential attributes suchas action potential amplitude and action potential duration, in addition to phase 0 time(activation imaging). Apparently, this joining of the prior two paragraphs has not beeninvestigated, and the practicality of such manipulations is speculative.

4.7 DISCUSSION

Among the many engineering challenges posed by the imaging problem treated inthis chapter, the necessity of a proper mathematical understanding of the computationaldifficulties (and their optimal treatment) has some pre-eminence. In this regard, thereare lively controversies regarding which is the favored source formulation to be imaged(epicardial/endocardial potentials versus action potential features), the possible role of preprocessing the raw signals (e.g., Laplacian electrocardiography), the reductionist role inactivation imaging (e.g., the Critical Point Theorem), and the desirability of integratingthe temporal data from a stochastic processes standpoint. Recent history has shown thatthere is surely room for improvement in algorithmic technique. Methodological refinementscontinue to be proposed by many different groups.

At the same time, the biophysical understanding, technical apparatus, and mathematicalmethodology, are clearly already in place to create images of extracellular potential andaction potential features on the epicardial and endocardial surfaces. The principal questionis whether the resulting images are either too blurred to be of much use, or are otherwiseunreliable and misleading (e.g., due to the inherent ill-posedness of the problem, lack

156 F. Greensite

of sufficiently powerful mitigating constraints, or insufficiently accurate forward problemsolutions due to uncertainties in knowledge of body tissue conductivities and anisotropies).Thus, image validation is presently a major question of interest in this field. Such validation isfairly well advanced in the case of the minimally invasive techniques of imaging endocardialpotential via transvascular catheter probe electrode arrays. However, for the epicardialimaging approaches, it is particularly difficult to address the validation question adequately,because validation ideally requires the simultaneous acquisition of epicardial signals, bodysurface signals, and anatomical body imaging (via CT or MRI). That is, the body imagingshould be conducted closed chest (otherwise, the body surface signals would be subject toan unrealistic transfer matrix), despite the simultaneous need for "gold standard" invasivelyobtained epicardial potentials for the validation. Nevertheless, the latter validation goal isbeing aggressively pursued by a number of groups worldwide.

It is likely that the field is maturing to the extent that the next several years will seeclarification of the true potential and promise of the methodologies discussed in this chapter.Accordingly, the era of Noninvasive Imaging of Cardiac Electrophysiology (NICE) couldsoon be at hand.

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Martin, R. 0., Pilkington, T C., and Morrow, M. N., 1975, Statistically constrained inverse electrocardiography,

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single beat data under clinical conditions, Biomedizinishe Technik 46:213-215.Nash, M. P, Bradley, C P., Cheng, L.K., Pullan, A. J., and Paterson, D. J., in press, An in-vivo experimental

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5

THREE-DIMENSIONALELECTROCARDIOGRAPHICTOMOGRAPHIC IMAGING

Bin He*University of Illinois at Chicago

5.1 INTRODUCTION

Cardiac electrical activity is distributed over the three dimensional (3D) myocardium.It is of significance to noninvasively image distributed cardiac electrical activity throughoutthe 3D volume of the myocardium. Such knowledge of the source distribution would play animportant role in our effort to relate the electrocardiographic inverse solutions with regionalcardiac activity.

Historically, attempts to noninvasively obtain spatial information regarding cardiacelectrical activity started from body surface potential mapping by using a larger numberof recording leads covering the entire surface of the body (Taccardi, 1962). From suchmeasurements, instantaneous equipotential contour maps on the body surface have beenobtained and shown to provide additional information when compared to a conventionalelectrocardiogram (See Flowers & Horan, 1995 for review). Since body surface potentialmaps (BSPMs) are manifestation of cardiac electrical sources on the body surface, effortshave been made to solve the electrocardiography inverse problem-to seek the generatorsof BSPMs.

Equivalent dipole solutions have been investigated with the aim of extracting useful information regarding cardiac electrical activity. Such efforts included (l) MovingDipole Solutions (Mirvis et al., 1977; Savard et al., 1980; Okamoto et al., 1983; Gulrajaniet al., 1984), in which one or more current dipoles are estimated at the location(s)that best describe the body surface recorded electrocardiograms; and (2) Fixed Dipoles

* Present address for correspondence: University of Minnesota, Department of Biomedical Engineering, 7-105BSBE, 312 Church Street, Minneapolis, MN 55455 E-mail: [email protected]

161

162 B.He

Solutions (Barber & Fischman, 1961; Bellman et aI., 1964; He & Wu, 2001), in which anarray of dipoles are arranged at fixed locations where their moments are determined byminimizing the difference between the model-generated and the measured body surfaceelectrocardiograms.

It has been demonstrated that the single moving dipole solution can provide a goodrepresentation of well-localized cardiac electrical activity (Savard et aI., 1980). Efforts havealso been made to estimate two moving dipole solutions, although technical challenges existwhen the number of equivalent dipoles increases from one to two (Okamoto et aI., 1983;Gulrajani et al., 1984). Due to the ill-posedness of the inverse problem, currently there is nowell established method to estimate three or more moving dipoles. In addition to equivalent dipole approach, equivalent multipole models have also been investigated (Geselowitz1960; Hlavin & Plonsey, 1963; Pilkington & Morrow, 1982) during the early stage of electrocardiography inverse solutions, in an attempt to obtain equivalent 3D information oncardiac electrical activity. The limitation of the multipole approach, however, is its inabilityof localizing cardiac electrical activity.

In the past decade, most research on the electrocardiography inverse problem has beencarried out in the line of heart surface inverse solutions. As reviewed in Chapter 4, theseresearch efforts are mainly related to epicardial potential inverse solutions or heart surfaceactivation imaging.

Three dimensional electrocardiographic tomographic imaging has received much attention since 2000. He and Wu reported their effort on electrocardiographic tomography intheir presentation at the World Congress on Medical Physics and Biomedical Engineeringheld in Chicago in 2000. In this work, He & Wu demonstrated the feasibility in a computersimulation study to image the 3D distribution of cardiac dipole source distribution fromnoninvasive body surface electrograms by using the Laplacian weighted minimum normapproach (He & Wu, 2000, 2001). In subsequent work, He and coworkers developed aheart-model based 3D activation imaging approach (He & Li, 2002; He et al., 2002), and a3D transmembrane potential (TMP) imaging approach (He et al., 2003), which were introduced in a presentation at the 4th International Conference of Bioelectromagnetism in2002 (He & Li, 2002). Ohyu et al. (2002) have developed an approach to estimate the activation time and approximate amplitude of the TMP from magnetocardiograms using theWiener estimation technique. Skipa et al. presented their effort to estimate transmembranepotentials from body surface electrocardiograms at the 4th International Conference onBioelectromagnetism (2002).

In this Chapter, we review the principles and methods of performing 3D electrocardiographic tomographic imaging, with a focus on introducing the recently developed distributed3D electrocardiographic tomographic imaging techniques.

5.2 THREE-DIMENSIONAL MYOCARDIAL DIPOLE SOURCE IMAGING

5.2.1 EQUIVALENT MOVING DIPOLE MODEL

Equivalent dipole inverse solutions were among the earliest efforts ofobtaining electrocardiography inverse solutions. Early efforts were made on moving dipole inverse solutions

Three-Dimensional Electrocardiographic Tomographic Imaging 163

where one or two equivalent dipoles were used to represent cardiac electrical activity in thesense that the dipole-generated body surface potential maps (BSPMs) matches the measuredBSPMs well (Mirvis et aI., 1977; Savard et al., 1980; Okamoto et al., 1983; Gulrajani et aI.,1984). In the moving dipole model, the locations of the equivalent dipoles vary from timeto time which provide information on the centers of gravity of electrical activity within theheart. Such location information offers an important capability for moving dipole solutionsto localize the regions of myocardial tissues which are most responsible for the measuredBSPMs.

A limitation of this approach, however, is that the inverse solution is sensitive tomeasurement noise and thus limiting the number of moving dipoles that can be reliablyestimated from the measured BSPMs. For this reason, the moving dipole inverse solutionmay be useful in localizing a focal cardiac source during the initial phase ofcardiac activationfor a single activity. For general cardiac activation, the moving dipole inverse solution failsto represent the complex cardiac electrical activity. A detailed review on an equivalentmoving dipole solution can be found in reference (Gulrajani et al., 1984).

5.2.2 EQUIVALENT DIPOLE DISTRIBUTION MODEL

As early as the 1960's, Barber and Fischman suggested the possibility of modelingcardiac electrical activity using an array of current dipoles located at fixed locations withinthe myocardium (Barber & Fischman 1961; Bellman et aI., 1964). In this model, the dipolesare not moving butfixedover time, while its moments remain variable. Yet this model did notreceive much attention in the field of electrocardiography inverse problem in the past threedecades, partially due to the dominance of the epicardial potential (Barr et aI., 1977; Frazoneet aI., 1978; Shahidi et aI., 1994; Throne & Olson, 1994, 1997; Johnston & Gulrajani, 1997;Oster et aI., 1997; He & Wu, 1997; Greensite & Huiskamp, 1998; Burnes et al., 2000) andthe heart-surface activation time (Cuppen and van Oosterom 1984; Huiskamp & Greensite1997; Greensite 2001; Modre et al., 2001; Pullan et al., 2001) inverse solutions developedduring the same period.

Recently, the fixed dipole array model has been expanded into a volume distributionof current dipoles for the purpose of tomographic dipole source imaging (He & Wu, 2000,2001). In this work, He & Wu modeled cardiac electrical sources by means of a large numberof current dipoles over the 3D volume of the ventricles. Each of the dipoles was locatedat a particular position, representing the local electrical activity, while the moments of thedipoles were varied over time. The magnitude function of the regional dipoles provideda spatial distribution of current strength with the 3D myocardial volume. Estimation ofthis current dipole moment provides a means of imaging the spatial distribution of currentsources.

5.2.3 INVERSE ESTIMATION OF 3D DIPOLE DISTRIBUTION

The key hypothesis under the 3D dipole distribution imaging is based on the assumptionthat the electrical sources located at a small region of myocardial tissue are coherent and

164 B.He

can be approximated by a current dipole. By assigning one such current dipole to each"small" region of the myocardium, the following mathematical model, which relates thecurrent dipole distribution inside the myocardium to the body surface ECG measurements,can be obtained:

V=AX (5.1)

where V is the vector consisting of m body surface-recorded ECG signals, X is the unknown vector consisting of the moments of the current dipoles, which are located at n sitescovering the entire myocardial volume, and A is the transfer matrix. The measurement ateach electrode sensor is produced by a linear combination of all dipole components, withcolumns in A serve as weighting factors. By solving (5.1), one obtains an estimation of3D current dipole source distribution corresponding to each measured BSPM. Since thenumber of measurement electrodes is always far less than the dimension of the unknowndipole source vector X, this problem is an underdetermined inverse problem and a properregularization strategy is necessary for obtaining a reasonable solution to (5.1).

The minimum norm (MN) solution is one of the feasible solutions (Hamalainen &Ilmoniemi, 1984)

(5.2)

where (*)+ denotes the Moore-Penrose inverse. As the minimum norm solution is intrinsically biased towards the superficial position, the weighted minimum norm solution (Jeffset aI., 1987) and the Laplacian weighted minimum norm solution (LWMN) (Pascual-Marquiet aI., 1994) has been proposed to solve the linear inverse problem. He & Wu investigated3D electrocardiography dipole source imaging using the principles of LWMN (He & Wu,2000,2001). LWMN utilizes a weighting operator LW, where L is a Laplacian operator andW is a diagonal 3n by 3n matrix with Wii = I[Ai II and Ai is the z-th column of the transfermatrix A. Assuming the weighting factor is nonsingular, then

(5.3)

and the LWMN solution of (5.1) becomes

(5.4)

When using LWMN the resulting solution tends to be over-smoothed due to the constraints of minimizing the Laplacian of the signal. For well-focused cardiac sources, suchas the sites of origins of cardiac arrhythmias, a recursive weighting strategy, which waspreviously developed for improving the performance of MN MEG imaging (Gorodnitskyet aI., 1995), has been used to search for focal sources in the heart from initial LWMNestimates. This algorithm recursively enhances the values of some of the initial solutionelements, while decreasing the rest of the elements until they become zero. In the end, onlya small number of winning elements remain non-zero, yielding the desired type of localizedenergy distribution of the solution.

Three-Dimensional Electrocardiographic Tomographic Imaging

5.2.4 NUMERICAL EXAMPLE OF 3D MYOCARDIAL DIPOLESOURCE IMAGING

165

Computer simulation results for 3D myocardial dipole source imaging have recentlybeen reported (He & Wu, 2001). A 3D heart-torso inhomogeneous volume conductor model(Wu et aI., 1999) was used in the simulation. Considering the low conductivity of the lungs ,the conductivity ratio of torso to lungs was set to 1:0.2; and the conductivity for myocardialmuscle was assumed to be the same as the torso (Mulmivuo & Plonsey, 1995; Gulrajani ,1998). The ventricles were divided into an equi-distant lattice structure of 1,124 nodeswith a resolution of 6.7 mm. A regional current dipole was assigned on each of the nodes,resulting in 1,124 regional dipoles.

Fig. 5.1 illustrates an example of3D cardiac dipole source imaging .Two dipole sources ,oriented from the waist towards the neck, were used to approxiate two localized cardiacsources , which were located close to the endocardium at the right ventricle and the epicardium at the left ventricle (Fig. 5.l(a)). Gaussian white noise of 5% was added to thebody surface potentials calculated from assumed cardiac dipole sources to simulate noisecontamin ated body surface ECG measurements. The inverse imaging algorithm describedin Section 5.2.3 was used to attempt to reconstruct the source distribut ion within the myocardium , without a priori knowledge of the number of primary current dipoles.

The LWMN solution is illustrated in Fig. 5.1(b), where the red and yellow colorsillustrate the strength of the equivalent dipole source distribution throughout the ventricles.Fig. 5.1(b) shows that the LWMN solution reached maxima in both the right ventricle andthe left ventricle, overlying with the locations of the source dipoles. The LWMN solutionshowed a stronger source distribution over the left ventricle probably because the dipolein the left ventricle is located closer to the chest when compared with the dipole locatedin the right ventricle. In addition, Fig. 5.1(b) shows that there is another major area ofactivity appearing over the posterior ventricular wall in the LWMN solution . The recursivelyweighted LWMN solution is illustrated in Fig. 5.l (c) where after 20 iterations the sourcestrength distribution is well focused at two locations. One of the source localization resultswas consistent with the "true" dipole at the right ventricle , while the other was located atthe left ventricle, but shifted about 1 em towards the direction of the endocardium from the"true" diople position .

Previous studies have shown that the equivalent dipole solution suffers from existingexperimental noise if the number of the moving dipoles increases to two or more (Okamotoet al., 1983). In a clinical setting, however, it is necessary to localize and image sites oforigins of arrhythmias even without knowing in advance how many dipoles should be used.Hence , there is a need to develop a technique, which can localize and image sites of origins ofcardiac arrhythmias without a priori constraints on the number of equivalent moving dipoles(such as one dipole). The LWMN approach which He & Wu (2000, 2001) have applied tocardiac dipole source imaging , on the other hand, does not attempt to make assumptionson the number of focal cardiac sources. Some of the a priori information being taken intoaccount in the 3D cardiac dipole source imaging approach is that the myocardial electricalactivation is smooth over a reasonably small region. With such constraints , the estimatedinverse solution provides a smoothed distribution of current density over a large area ofmyocardium (Fig. 5.1(b)). For focal sources, as illustrated in Fig. 5.1(a), additional strategy

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such as recursive focusing is needed to obtain the inverse solution, in which it is assumed that the sites of origins of cardiac arrhythmias are localized over small regionsinside the myocardium. With this additional constraint, the inverse dipole source distribution shows localized distribution of current density close to the original "true" dipolesources.

Note, however, that the LWMN inverse solution with a recursive weighting strategy still shows certain shift towards the "interior" of the myocardium from the "true"solution positions (Fig. 5.1). Both the LWMN algorithm and the recursive weighting algorithm may contribute to such "shift." Although the work reported by He & Wu suggeststhe promise of imaging cardiac electrical activity using LWMN approach, a systematicstudy should be conducted to evaluate the reconstruction results for a number of sourceconfigurations including sources located in various regions of the heart with various orientations.

5.3 THREE-DIMENSIONAL MYOCARDIAL ACTIVATION IMAGING

Myocardial activation imaging received much attention in recent years, in which localactivation time over the heart surface is estimated from BSPMs (Cuppen and van Oosterom1984, Huiskamp and Greensite 1997, Greensite 2001, Modre et al., 2001, Pullan et al2001). As reviewed in Chapter 4, this approach is based on the bidomain theory, whichallows direct linking of the heart surface activation time with body surface potentials underthe assumption of the electrical isotropy (or "equal anisotropy") within the myocardium(Greensite 2001).

Recently, the concept of myocardial activation imaging has been extended from 2Dheart surface to 3D myocardial volume (He et al., 2002; He & Li, 2002; Ohyu et al., 2002).In these approaches, the activation time throughout the 3D myocardium is estimated frombody surface electrograms by means of a heart-excitation-model (He et al., 2002; He & Li,2002) or a Wiener inverse filter (Ohyu et al., 2002). In this Section, the heart-model-based3D activation imaging approach (He et al., 2002) is presented.

5.3.1 OUTLlNE OF THE HEART-MODEL BASED 3D ACTlVATION TIMEIMAGING APPROACH

The 3D distribution of activation time throughout the ventricles has been estimatedwith the aid of a heart-model based approach, in which the a priori knowledge on cardiacelectrophysiology is embedded. Fig. 5.2 illustrates a schematic diagram of this approach. Arealistic geometry computer heart-torso model is used to represent the relationship between3D activation sequences within the myocardium with BSPMs. The a priori knowledge oncardiac electrophysiology and the detailed anatomic information on the heart and torso areembedded in this heart-torso model. The 3D myocardial activation sequence is estimatedas the parameter of the heart-model and obtained by means of a nonlinear estimationprocedure.

168 B.He

FIGURE 5.2. Schematic diagram of 3D electrocardiography tomographic imaging. See attached CD for colorfigure (From He et aI., Phys Med & BioI, 2002 with permission)

A preliminary classification system (PCS) is employed to determine the cardiac status based on the a priori knowledge of the cardiac electrophysiology and the measuredBSPM by means of an artificial neural network (ANN) (Li & He, 2001). The output of theANN based PCS provides the initial estimate of heart model parameters to be used laterin a nonlinear optimization system. Using these initial parameters as the result of PCS, theoptimization system then minimizes objective functions that assess the dissimilarity between the measured and heart-torso-model-calculated BSPMs. The heart model parameterscorresponding with the calculated BSPM are employed to produce 3D myocardial activation sequence if the measured BSPM and the heart-torso-model-calculated BSPM matcheswell. Before the objective functions satisfy the given convergent criteria, the heart modelparameters are adjusted with the aid of the optimization algorithms and the optimizationprocedure proceeds.

5.3.2 COMPUTER HEART EXCITATION MODEL

Numerous efforts have been made to develop computer heart models that can simulatecardiac electrophysiological processes as well as the relationship between cardiac activitiesand BSPMs (See Chapter 2 for review). Although the more detailed information incorporated the better, the cellular automaton heart excitation model (Aoki et al., 1987; Lu et al.,


1993) has been used in the 3D activation time imaging research due to its capability ofsimulating cardiac activation, BSPMs, and computational efficiency.

In this work (He et aI., 2002), we used a cellular automaton ventricle model that wasconstructed as a 3D array of approximately 42,000 myocardial cell units with a spatialresolution of 1.5 mm. The ventricles consisted of 50 layers with inter-layer distance ofbeing 1.5 mm, and were divided into 53 myocardial segments. Each segment is comprisedof approximately the same number of myocardial cell units. The action potential of each ofheart units was already determined according to the cardiac action potential experimentallyobserved and stored in the action potential data file. From the epicardium to the endocardium,the refractory period of the action potential of cardiac cellular units gradually increased forthe T-wave simulation. The primary current dipole sources are proportional to the gradientof the transmembrane potentials at adjacent cardiac units (Miller & Geselowitz, 1978).

The anisotropic propagation of excitation in the ventricular myocardium was incorporated into this heart model (He et aI., 2003) in order to obtain more accurate simulation ofthe body surface ECG and myocardial activation sequence (Nenonen et al., 1991, Lorangeand Gulrajani 1993; Wei et al., 1995; Franzone et al., 1998; Huiskamp, 1998; Fischer et aI.,2000). Ventricular myocardium was divided into different layers with thickness of 1.5 mmfrom epicardium to endocardium. The myocardial fiber orientations were rotated counterclockwise over 1200 from the outermost layer (epicardium, -60°) to the innermost layer(endocardium, +60°) (Streeter et aI., 1969) with identical increment between the consecutive layers. All units on a myocardial layer of ventricles from epicardial layer to endocardiallayer had identical fiber orientation. For each myocardial unit, a fiber direction vector, whichis located on its local tangential plane, was determined by its fiber angle. The fiber orientations of all myocardial units of ventricles were determined, and put in the realisticallyshaped inhomogeneous torso model for calculating the body surface ECG. Excitation conduction velocity of myocardial units was set to 0.6 m/s and 0.2 m/s along the longitudinaland transverse fiber direction, respectively. Electrical conductivity of myocardial units wasset to 1.5 mS/cm along the longitudinal fiber direction and 0.5 mS/cm along the transversefiber direction (Nenonen et aI., 1991).

Fig. 5.3 shows the realistic geometry inhomogeneous heart-torso model (a), an exampleof simulated sinus rhythm (b), and an example of paced activity (c). Fig. 5.3(b) shows theactivation sequence corresponding to sinus rhythm (left), and an example of the anteriorBSPM and a chest ECG lead simulated during sinus rhythm (right). Fig. 5.3(c) shows anexample of the simulated BSPM on anterior chest (middle-bottom) at 30 ms followingpacing the anterior wall of the ventricle, and the ventricular excitation sequence over theepicardium (middle-top) corresponding to the pacing site (left), and a chest ECG lead (right).The pacing site is shown on the left.

5.3.3 PRELIMINARY CLASSIFICATION SYSTEM

There are a large number of parameters associated with the cellular automaton heartmodel that need to be determined in order to estimate the activation sequence. A PreliminaryClassification System (PCS) is used to approximately classify, from BSPMs, cardiac statuswith the a priori knowledge on cardiac electrophysiology and the mapping relationshipbetween cardiac activation and resulting BSPMs. An ANN has been used to serve as a PCS.In this implementation (Li & He, 2001), a three-layer feed-forward ANN was used, with the

170

Sin Rhythm

~Pa in

l

B.He

(a)

(c)

FI GURE 5.3. Illustration of computer heart-torso modeling and simulation. (a) Realistic geometry heart torsomodel. (b) Simulation of sinus rhythm: Left panel-Intracardiac activation sequence over three slices withinthe ventricles; Right panel-An example of simulated anterior BSPM and chest ECG lead. (c) Simulation ofepicardial pacing: Left panel- Heart model and a pacing site at middle anterior epicardium of ventricle. Middlepanels- Anterior view of simulated epicardial isochrone (top), and an example of the BSPM over the anteriorchest following pacing (bottom). Righ panel-A simulated chest ECG lead. See attached CD for color figure.(Modified from He et aI., Phys Med & BioI, 2002 with permission)

number of neurons in the input layer being set to the number of body surface electrodes, andthe number of neurons in the output layer being set to the number of myocardial segmentsbeing studied. Gaussian white noise (GWN) was added to the BSPMs to simulate noisecontaminated body surface ECG measurements . The BSPM maps during 25 to 50 ms afterinitial activation were used as inputs to train the ANN.

5.3.4 NONLINEAR OPTIMIZATION SYSTEM

The heart model parameters associated with myocardial activation sequence are estimated by minimizing dissimilarity between the measured and heart-torso-model-generated(referred to as "simulated" below) BSPMs. The dissimilarity between the measured andsimulated BSPMs is described by appropri ate characteristic parameters extracted from theBSPMs. The following three objective functions (Li & He, 200 1) have been used to reflect the dissimilarity between the measured and simulated BSPMs for paced ventricularactivation:

(a) Ecd x), which was constructed with the average correlation coefficient (CC) betweenthe measured and simulated BSPMs from instant T1 to instant T2 of the cardiac excitationafter detection of initial activation, is defined as:

TzEccCx) = L [l - CCms(x , t ))/ (T2 - T J)

I=T,

(5.5)

where CCms(X, t) is the CC between the measured and simulated BSPMs at instant t. x is aparameter vector of the spatial location of initial activation in the computer heart-excitationmodel.

(b) Eminp(x), which was constructed with the deviation of the positions of minima ofthe measured and simulated BSPMs from instant T] to instant T2, is defined as:

T2

Eminp(X) = L II P~n(O - r:»,0[1t=T,

(5.6)

where P~n(O and P~in(x, t) represent the positions of the minima in the measured andsimulated BSPMs at instant t, respectively. The definition of x is the same as that in Eq.(5.5).

(c) ENPL(x), which was constructed with the relative error of the number of bodysurface recording leads, at which the potentials are less than a certain negative threshold,in the measured and simulated BSPMs from instant T1 to instant T2, is defined as:

(5.7)

where L%(t) = L~~l U(¢T - ¢(t, i» and L~(x, t) = L;:;] U(¢T - ¢(x, t, i), are thenumbers of recording leads, at which the potentials are less than a given threshold ¢T(<0), in the measured and simulated BSPMs at instant t, respectively. ¢(t, i) and ¢(x, t, i) arethe ith-lead measured and simulated potentials at instant t, respectively. u(e) is the unit-stepfunction, which gives a unity output if the potential at a lead is less than the pre-set threshold.NL is the number of body surface recording leads. The definition of x is the same as that inEq. (5.5).

Combining the above objective functions, the mathematical model of the optimizationfor this heart-model-based electrocardiographic imaging can be represented as the followingminimization problem:

min(Ecdx» = E~c' Eminp(X) < £minp, ENPL(x) < £NPLXEX

(5.8)

where X is the probable value region of the parameters in the computer heart-excitationmodel. x is a vector of heart model parameters. E~c is the optimal value of the objectivefunction Ecdx). £minp and £NPL are the allowable errors of the objective function Eminp(x)and ENPL(x), respectively. Eq. (5.8) was solved by means of the Simplex Method.

5.3.5 COMPUTER SIMULATION

The feasibility of 3D myocardial activation imaging has been suggested in a computersimulation study (He et al., 2002), and presented in this section. In this simulation study,pacing protocols were used to simulate paced cardiac activation. By setting pacing sitesin different myocardial regions of the heart excitation model, sequential pace maps wereobtained by solving the forward problem using the heart-torso computer model. Two pacingprotocols, single-site pacing and dual-site pacing, were used to evaluate the performance of

172 B. He

the 3D myocardial activation imaging approach. Gaussian white noise (GWN) of 10 !.LV wasadded to the BSPMs at each time instant after the onset of pacing, to simulate the noisecontaminated body surface potential measurements. The maximum value of the BSPMduring the QRS complex was set to 3 mV.

The performance of activation time imaging was tested by single-site pacing in 24different sites throughout the ventricles. The CC and RE between the vector of simulatedactivation time and the vector of estimated activation time were calculated for each of the24 pacing sites. The vector of activation times consists of the activation time of each voxelwithin the ventricles. Averaged over all 24 sites, the RE and CC between the "true" andestimated activation times are 0.07 ± 0.03 and 0.9989 ± 0.0008, respectively, suggestingthe high degree of fidelity of the inverse estimation of activation time in the ventricles.

Fig. 5.4 shows two typical simulation examples. The top rows show the simulated"true" activation sequence, and the bottom rows show the inversely estimated activationsequence. Each row shows the activation sequence in 5 longitudinal sections «b)'""-'(f))and I transverse section of ventricles (a). Five horizontal lines in the transverse section

True

(A)

Estimated

(8) (b) (c) (d) (e) (I)

True

(B)

Estimated

(8) (b) (c) (d) (e) (I)

FIGURE 5.4. Two examples of activation time imaging results during dual-site ventricular pacing. The activation sequence within the ventricles was inversely estimated from the BSPM with !Of!.V Gaussian white noisebeing added. First row shows the simulated "true" activation sequence , and second row the activation sequencecorresponding to the inversely estimated result. Each row shows the isochrone in 5 longitudinal sections «b)- (f»and I transverse section of the ventricles (a). Five horizontal black lines in the transverse section of the ventriclesfrom top to bottom respectively indicate the positions of 5 longitudinal sections from (b) to (f). The unit of colorbar is millisecond. The distance between both neighboring sections is 4.5 mm. (A) One pacing site at septumendocardium of left ventricle, and another at intramural of left-anterior wall (marked by yellow dots). (B) Bothpacing sites at left-posterior intramural adjacent to endocardium (marked by yellow dots). See attached CD forcolor figure. (From He et aI., Phys Med & Bioi, 2002 with permission)

Three-Dimensional Electrocardiographic Tomographic Imaging

TABLE 5.1. Effectsof heart and torso geometry uncertainty on theactivation time imaging

Pacing Region BA BRW BLW BS BP Mean ± SD

NM 0.08 0.03 0.06 0.04 0.05 0.05 ± 0.02LSX 0.08 0.03 0.06 0.10 0.06 0.06 ± 0.03RSX 0.06 0.03 0.06 0.09 0.07 0.06 ± 0.02FSY 0.08 0.03 0.07 0.10 0.06 0.05 ± 0.02BSY 0.08 0.06 0.07 0.12 0.05 0.07 ± 0.03NTM+ 10% 0.09 0.11 0.08 0.05 0.07 0.08 ± 0.02NTM - 10% 0.09 0.10 0.06 0.10 0.07 0.08 ± 0.02

Note:

NM: Normal Model.

LSXlRSX: Left/Right shift along the x-direction.

FSYIBSY: Front/Back shift along the y-direction.

NTM + IO%/NMT- 10%: 10% enlargement and reduction of the normal torso model.

173

of ventricles from top to bottom respectively indicate the positions of the 5 longitudinalsections from (b) to (f). In panel (A), one pacing site is located at the left ventricular septalendocardium, and another pacing site is located at the left anterior intramural wall. In panel(B), both pacing sites are located at left-posterior intramural adjacent to endocardium. Inboth cases, 10 f.LV GWN was added to the BSPMs to simulate noise-contaminated bodysurface ECG recordings. Fig. 5.4 suggests that, for dual site pacing at two separate locations(A) or adjacent locations (B), the 3D myocardial activation imaging can reconstruct wellthe activation sequence, although the estimated activation sequence showed little delayedactivation as compared with the "true" activation sequence.

Effects of heart-torso geometry uncertainties were tested by selecting five pacing sites,in five different regions adjacent to the AV-ring(BA: basal-anterior; BRW: basal-right-wall;BP: basal-posterior; BLW: basal-left-wall; BS: basal-septum). The modified (enlarge orreduce by 10%) torso models or position-shifted heart models (in 4 directions) were usedin the forward BSPM simulation, and 10 f.LV GWN was added to the simulated BSPMs. Byusing the modified heart-torso models in the forward simulations while the standard model inthe inverse calculation, the effect of inter-subject geometry variation was initially evaluated.Table 5.1 shows the RE between the simulated "true" ventricular activation sequence andthe estimated activation sequence following a single-site pacing. NM refers to normal case,in which only measurement noise is introduced without geometry uncertainty. Note that theheart-torso geometry uncertainty showed little effect on the activation sequence estimationas determined by the RE measure. For example, 2% increase in RE was obtained for thebackward shift of the heart along the y-direction (BSY), as compared with the NM case. Theestimation errors associated with the 10% enlarged or reduced torso models have averagedREof8%.

Effects of conduction velocity of ventricular activation in the heart model were assessedby varying the conduction velocity in the forward heart model. The BSPMs were simulatedby using this altered ventricle model and noise was added to simulate noise-contaminatedBSPM measurements. The standard heart model, in which the average conduction velocitywas used, was then used to estimate the inverse solutions. Fig. 5.5 shows a typical simulationexample, with the same format as in Fig. 5.4. GWN of 10 f.LV was added to the BSPMs

174 B.He

126ll'll'

(0(e)(d)(c)(0)(a)

Troe

FIGURE 5.5. An example of activation time imaging results with variation in the conduction velocity. Sameformat of display as in Fig. 5.5. The top rows show the simulate d "true" activation sequence with altered conductionvelocity in the forward heart model, and the bottom rows show the inversely estimated activation sequence. Theresults correspond to 10% increase in conduction velocity following a single-site pacing. See attached CD forcolor figure. (From He et aI., Phys Mcd & Bioi, 2002 with permission)

to simulate noise-contaminated body surface ECG recordings. The top row shows the simulated "true" activation sequence with altered conduction velocity in the forward heartmodel, and the bottom row shows the inversely estimated activation sequence, following asingle-site pacing when the conduction velocity of the forward heart model was increasedby 10%. When the average conduction velocity is different in the forward heart model whencompared with that in the heart model used in the inverse procedure, the estimated activationtime at specific region s within the ventricles differs from the original activation time distribution in the forward solution. In particular, the early activation moved down toward theapex direction. Nevertheless, the overall distributions of the activation time are not affectedsubstantially. The RE and CC between the "true" and estimated activation sequences withinthe ventricles are 0.0916/0.106 and 0.996/0 .998, respectively, corresponding to 5%110%increase in the conduction velocity.

5.3.6 DISCUSSION

In this Section, a new approach for noninvasive 3D cardiac activation time imagingby means of a heart-excitation-model is reviewed. This approach is based on the observation that a priori information regarding cardiac electrophysiology should be incorporatedinto the cardiac inverse solutions in order to obtain useful information on the 3D cardiacactivation from the two-dimensional electrical measurements over the body surface . In thisapproach, the a priori information on cardiac electrophysiology is incorporated into theheart-excitation-model, which is not an equivalent physical source model but an equivalentphysiological source model. By linking this physiological source model with body surfaceECG measurements, physiological parameters of interest are estimated from body surfaceECG recordings. A unique feature of such an approach is that the rich knowledge we havegained in the forward whole heart modeling (See Chapters 2 and 3) can be directly appliedto the 3D cardiac imaging. Furthermore, the anisotropic nature of myocardial propagationcan also be incorporated in the 3D myocardial activation imaging , as shown in this Section. Such a priori electrophysiological information serves as constraints when solving theinverse problem, leading to robust 3D inverse solutions.

Three- Dimensional Electrocardiographic Tomographic Imaging 175

Although a cellular-automaton heart-excitation-model has been used for the 3D activation time imaging, it is anticipated that more sophisticated 3D heart-excitation-models (e.g.Gulrajani et al., 2001) can be realized in 3D activation time imaging in the future, providingneeded spatial resolution for more complicated cardiac activation sequences. With rapidadvancement in computer technology, this goal seems to be more feasible than expected.

In parallel to the heart-excitation-model based 3D activation imaging approach, Ohyuet al. (2002) has reported another 3D activation imaging approach in which the actionpotentials are approximated by a step function with the initial activation being varied fromsite to site within isotropic ventricles. The distribution ofactivation time was then connectedto the BSPMs and estimated by means of Wiener inverse filter. Both this activation imagingapproach (Ohyu et al., 2002) and the dipole source distribution imaging approach (He & Wu,2001) reviewed in Section 5.2 use linear systems connecting the inverse solution parametersdirectly with the BSPMs. It would be of interest to evaluate the heart-model-based activationimaging reported by He et al. (2002), with linear system based activation imaging reportedby Ohyu et al. (2002).

5.4 THREE-DIMENSIONAL MYOCARDIAL TRANSMEMBRANEPOTENTIAL IMAGING

In association with activation time, the transmembrane potential (TMP) reflects important electrophysiological properties on the local myocardial tissues. The TMP has beenestimated over the heart surface from magnetocardiograms (Wach et al., 1997). Recently,efforts have been extended from the heart surface to the 3D myocardium. He and coworkersreported (2002, 2003) their effort to estimate TMP distribution within the 3D ventriclesfrom body surface electrocardiograms by means of a heart-model based imaging approach.Ohyu et al. has developed an approach to estimate the activation time and approximateamplitude of the TMP from magnetocardiograms using Wiener inverse filter (2002). Skipaet al. reported their initial results on estimation of TMP distribution within the heart fromBSPMs (2002).

In this Section, we present the heart-mode-based 3D TMP imaging approach and itsapplications to imaging TMP distributions associated with paced ventricular activity, andacute myocardial infarction. The whole procedure of the heart-model-based TMP imagingapproach may also be illustrated in the schematic diagram in Fig. 5.2, except that thereconstructed 3D cardiac sources are spatio-temporal distribution of TMP.

Similar to the 3D heart-model based activation imaging, the following procedures areused. A realistic geometry 3D heart-torso-model is constructed based on the knowledgeof cardiac electrophysiology and geometric measurements via CTIMRI. The anisotropicnature of myocardium can be incorporated into this computer heart model. The BSPMs arelinked with the 3D TMP distribution by means of the heart-torso-model. To reduce the dimensionality of the parameter space, a preliminary classification system (PCS) is employedto classify cardiac status based on the a priori knowledge of cardiac electrophysiologyand the BSPM, by means of an ANN. The output of the PCS provides the initial estimateof heart model parameters which are employed later in a nonlinear optimization system.The nonlinear optimization system then minimizes the objective functions that assess thedissimilarity between the "measured" and model-generated BSPMs.

176 B.He

If the "measured" BSPM and the heart-torso-model-generated BSPM match well, the3D distribution of transmembrane potentials is determined from the heart model parameters corresponding with the resulting BSPM. If the results do not match, the heart modelparameters are adjusted with the aid of the optimization algorithms and the optimization procedure proceeds until the objective functions satisfy the given convergent criteria.When the procedure converges, the TMP distribution throughout the 3D myocardium isdetermined.

The feasibility of imaging 3D TMP distribution has been suggested in computer simulation in paced activities (He et al., 2003). The performance of the above TMP imagingapproach was tested by single-site pacing in 24 different sites throughout the ventricles.GWN of 10 J.LV was added to the BSPMs, and GWN of 10 mm was added to the bodysurface electrode positions, to simulate noise-contaminated body surface ECG recordings.Fig. 5.6 shows the TMP amplitude distributions (a-b) of ventricular depolarization following a single-site pacing at the septum in 5 longitudinal sections within the ventricles (c).The TMP distribution in each longitudinal section at 8 typical instances (from 6 ms to 48ms with a time step of 6ms) after the onset of pacing is shown in one row. The 5 longitudinalsections within the ventricles arc illustrated in Fig. 5.6(c) by 5 horizontal black lines in the

Layel'-3

(a) 6ms 12ms 18ms 24ms 3Jms

•

35ms 42ms 48ms

FIGURE 5.6. An example of TMP imaging results during single-site ventricular pacing. (a) and (b) illustratethe forward and inverse solution of the TMP distributions in 5 longitudinal sections within the ventricles, duri ngventricu lar depo larization following a single-site pacing at the septum . The TMP distri bution in each longitud inalsection at 8 typical instances (from 6 ms to 48 ms with a time step of 6 ms) after the onset of pacing is shownin one row. The 5 longitudinal sections within the ventricles are illustrated in (c) by 5 horizontal black lines inthe transverse section of the ventricles from top to bottom indicating their positions. The gray regions in thelongitudina l sections indicate the resting cell units. The Max and Min of color bars correspond to the maximumand minimum values of the TMP amplitude during the first 60 ms from the onset of activation. See attached CDfor color figure.


La 3

JSms12ms6nu

uyu.1S

(b)

(c)

FIGURE 5.6. (cont.)

transverse section of the ventricles from top to bottom indicating their positions. The grayregions in the longitudinal sections indicate the resting cell units . The Max and Min of colorbars correspond to the maximum and minimum values of the TMP amplitude during the first60 ms from the onset of activation . Fig . 5.6 suggests that the inverse TMP distribution captures well the overall spatio-temporal patterns of the forward TMP distribution following asingle site pacing at the septum, but with a slight shift of the area of initial activation towardsthe base (as observed in Layer 3 in the inverse TMP distribution). Averaged over 24 sitesfor single-pacing, the RE and CC between the "true" and estimated TMP distributions are0.1266 ± 0.0326 and 0.9915 ± 0.0041 , respectively, indicating that the 3D TMP imagingapproach can reconstruct well the TMP distributions within the ventricles corresponding toa well-localized ventricular activation.

178

(a) (b)

B.He

FIGURE 5.7. A numerical example of myocardial infarction imaging. (a) Green shows preset acute myocardialinfarction. (b) Red shows estimated infarcted area over the same layer in the heart model. See attached CD forcolor figure.

The effects of heart-torso geometry uncertainties on the performance of the 3D TMPimaging approach has also been evaluated with torso size, heart position uncertainties beingconsidered. The 3D TMP imaging approach was found to be robust for up to 10% torso sizevariation and up to 10 mm heart position shift. See reference (He et al., 2003) for detaileddescriptions of the results for TMP imaging of paced activities.

The 3D TMP imaging approach has also been applied to image acute myocardialinfarction (MI) (Li & He, in press). Fig. 5.7 illustrates an example of myocardial infarctionimaging. In this case, GWN of5 f..LV was added to the BSPMs to simulate noise-contaminatedbody surface ECG recordings. Fig. 5.7(a) shows the "true" MI that is marked by green color,and Fig. 5.7(b) shows the inversely estimated MI that is marked by red color. A "true" MIis located in the middle left wall (MLW) of ventricle and distributes in 7 myocardial layers(not shown here). The estimated MI is located close to the "true" MI site, and has similarshape with the "true" MI, except that some small MI areas occurred in layers adjacent tothe 7 "true" MI layers (not shown here). Fig. 5.7 suggests the feasibility of applying the 3DTMP imaging approach to imaging the spatial location and extent of the acute MI in theventricles .

5.5 DISCUSSION

In this Chapter, we reviewed the recent progress in the development of noninvasive 3D electrocardiography tomographic imaging approaches, including 3D dipole sourcedistribution imaging , 3D activation imaging , and 3D transmembrane potential imaging.These activities may be classified as two general approaches in terms of methodology. Oneis to solve the system equations connecting electrophysiological characteristics (such ascurrent density, activation time and TMP) to BSPMs. This approach involves solving thesystem equations using inverse techniques such as weighted minimum norm (He & Wu,2000,200 I ; Skipa et al., 2002) and Weiner technique (Ohyu et al., 2002). Another approachis to solve the electrocardiography tomographic imaging problem indirectly, by means ofa heart-model. In this heart-model based approach, we have developed a localization approach to localize the site of origin of activation from body surface ECG recordings (Li &He, 2001), an activation imaging approach to image the activation time distribution (He etal., 2002) , and an TMP imaging approach to image distribution of transmembrane potential s


throughout the ventricles from BSPMs (He et al., 2003). The 3D TMP imaging approachhas been applied to image dynamic spatiotemporal patterns of activation induced by pacing(He et al., 2003), and to image and localize the site and size of acute myocardial infarction(Li & He, in press).

The heart-model based approaches are based on our observation that a priori information regarding the distributed cardiac electrophysiological process should be incorporatedinto the cardiac inverse solutions in order to obtain useful information on the distributed3D cardiac electrical activity from the two-dimensional BSPMs. In the present approach,the a priori information on cardiac electrophysiology is incorporated into the distributedheart-model, which is not an equivalent physical source model but an electrophysiologicalsource model, in which knowledge of electrophysiology and pathophysiology is imbedded. The distributed electrophysiological process within the heart is represented by cellularautomata, on each of which the site of origin of activation, activation time, or transmembrane potential are determined based on the knowledge of cardiac electrophysiology. Insuch approaches, since substantial electrophysiology a priori information is incorporatedinto the inverse solutions, more accurate inverse solutions are anticipated as compared withother approaches without taking this information into account. Such electrophysiology a

priori information not only includes more accurate forward solution at each time point, butalso a more realistic time-varying dynamics as set by the heart electrophysiology model.Therefore, it is not surprising that good matches between "true" cardiac electrical activityand estimated inverse solutions are obtained by means of the heart model based approaches.On the other hand, the system equation approach has the benefit that there is no need tolimit the search space for heart model parameters, as currently being practiced in the heartmodel based approaches. The inverse solutions are obtained directly by solving the systemequations that link the electrophysiological properties with BSPMs via biophysical relationships. It would be of interest to compare the performance of these two approaches for3D electrocardiography tomographic imaging.

The inverse problem of electrocardiography has been solved by means of equivalentpoint sources (dipole localization), distributed two-dimensional heart surface imaging methods (epicardial potential imaging, and heart surface activation imaging), and 3D distributedsource imaging approaches. While the 3D distributed source imaging, as reviewed in thischapter, represents an important advancement in the field of electrocardiography inverseproblem, all 3D electrocardiography tomographic imaging approaches have only been evaluated, up to date, in computer simulations. It is of ultimate importance and significance toexperimentally validate the 3D distributed source imaging approaches, in order to establishelectrocardiography tomographic imaging as a useful means for imaging noninvasive threedimensional distribution of cardiac electrical activity, for aiding clinical diagnosis and management of a variety of cardiac diseases, and for guiding radio-frequency catheter ablativeinterventions.

ACKNOWLEDGEMENT

The author wishes to thank his postdoctoral associates and graduate students, Dr.Guanglin Li, Dr. Dongsheng Wu, and Xin Zhang, with whom this work was conducted.This work was supported in part by NSF BES-0201939, a grant from the American HeartAssociation #0140132N, and NSF CAREER Award BES-9875344.

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Huiskamp, G., Greensite, E: A new method for myocardial activation imaging. IEEE Trans. Biomed. Eng., 44:

433-446,1997.Huiskamp, G.: Simulation of depolarization in a membrane-equations-based model of the anisotropic ventricle.

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Oster, H.S., Taccardi, B., Lux, R.L., Ershler, P.R, Rudy, Y: Noninvasive electrocardiographic imaging: reconstruction of epicardial potentials, electrograms, and isochrones and localization of single and multiple electrocardiac events. Circulation, 96: 1012-1024, 1997.

Pascual-Marqui, R.D., Michel, C.M., Lehmann, D.: Low resolution electromagnetic tomography: a new methodfor localizing electrical activity in the brain. Int. J. Psychophysiol., 18: 49-65, 1994.

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182 B.Ue

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6

BODYSURFACE LAPLACIANMAPPING OF BIOELECTRIC

SOURCES

Bin He* and lie LianDepartment of Bioengineering, University of Illinois at Chicago

6.1 INTRODUCTION

6.1.1 HIGH-RESOLUTION ECG AND EEG

Targeting two of the most life-critical organs, the heart and brain, the electrocardiogram(ECG) and the electroencephalogram (EEG) are the two important bioelectric recordingsto study the cardiac and neural activity.

Conventional ECG and EEG have many advantages. First, they are noninvasive measurement. Second, they are very convenient for application and have relatively low cost.More importantly, they have unsurpassed millisecond-scale temporal resolution, which isessential for revealing rapid change of dynamic patterns of heart and brain activities. However, the major limitation of the conventional ECG and EEG is their relatively low spatialresolution as compared to some other imaging modalities, such as the computed tomography(CT) or the magnetic resonance imaging (MRI).

One reason contributing to the low spatial resolution is the limited spatial sampling.Conventional EEG uses the standard international 10-20 system, which has about 20 electrodes over the scalp, with corresponding inter-electrode distance of about 6 em (Nunezet al., 1994). For the ECG measurement, the most commonly used configuration in a clinicalsetting is the 12-lead ECG. Despite its great success in many clinical applications, it has amajor limitation in that it contains very little spatial information, and doctors have to inferthe cardiac status mainly based on temporal analysis of the ECG waveforms. Therefore, oneway to enhance the spatial resolution of ECG and EEG is to increase the spatial sampling,by using larger number of surface electrodes in ECG and EEG measurement.

*Address all correspondence to: Bin He, Ph.D. University of Minnesota, Department of Biomedical Engineering,7-105 BSBE, 312 Church Street, Minneapolis, MN 55455. E-mail: [email protected]

183

184 B. He and J. Lian

However, even with very high-density spatial sampling, the spatial resolution of theEEG and ECG is still limited, because of the volume conduction effect. In other words, theelectrical signals will get smeared as they pass through the media between the bioelectricsources and the body surface sensors. For the brain, it's the head volume conductor, particularly the skull layer, which has low conductivity (Nunez, 1981, 1995). For the heart, it'sthe torso volume conductor, including the effects of lungs, the ribs and other tissues (Mirviset al., 1977; Spach et al., 1977; Rudy & Plonsey, 1980).

Therefore, advanced techniques are desired in order to compensate for the volumeconduction effect and enhance the spatial resolution of the ECG and EEG. As reviewed inChapters 4 and 5 with applications to the heart, one of such methods is to solve the so-calledinverse problem, which attempts to estimate the bioelectric sources from the body surfacepotential measurements. Another method is the surface Laplacian, which will be thoroughlydiscussed in this chapter.

6.1.2 BIOPHYSICAL BACKGROUND OF THE SURFACE LAPLACIAN

The concept of the Laplacian originated centuries ago, and the Laplacian operator hasbeen widely used in digital image processing as a spatial enhancement method. Similarly,the Laplacian technique can also be used for high-resolution bioelectric mapping.

By definition, the surface Laplacian (SL) is defined as the 2nd order spatial derivative ofthe surface potential. Due to its intrinsic spatial high-pass filtering characteristics, the SL canreduce the volume conduction effect by enhancing the high-frequency spatial components,therefore can achieve higher spatial resolution than the surface potentials (Figure 6-1).

Consider the non-orthogonal curvilinear coordinate system on a general surface Q,

u= x, v= y, and z = f (u, v), where f(u, v) is a continuous function whose 2nd orderpartial derivatives exist. If V(u, v) is the analytical surface potential function (whose 2nd

order partial derivatives exist) on Q, the SL of V (u, v) can be written in tensorial formulation(Courant & Hilbert, 1966; Babiloni et al., 1996):

2 1 {a [ (av av)] a [ (av av)]}Vs V =.y'g au "fi gIla-;; + g12a;;- + av "fi g21a-;; + g22a;;-

where the components of the metric tensor are given by (Babiloni et al., 1996):

afaf---

au avgl2 = g21 =

g

1+(~rg22 =

g

(6-1)

(6-2a)

(6-2b)

(6-2c)

(6-2d)

Body Surface Laplacian Mapping of Bioelectric Sources 185

FIGURE 6-1. Schematic illustration of the SL as a spatial enhancement method. The cardiac activity located atthe anterior apex (black circle) is sensed by potential measurement over the larger area on the chest (light grey)but by Laplacian measurement over the smaller area on the chest (dark grey). (From Tsai et aI., Electromagnetics,2001 with permission)

For the plane model where z = j(u, v) = 0, u = x and v = y, the SL is reduced to:

(6-3)

For the sphere model, assume z = .Jl - u2 - v2, U = x and v = y, the SL is then givenby (Perrin et al., 1987a):

(6-4)

The Laplacian electrogram (we refer electrogram to either ECG or EEG when theheart or the brain is concerned) shall be defined as the negative SL of the surface potentialelectrogram (He, 1999; He & Wu, 1999), to facilitate the interpretation of the Laplacianmaps in comparison to the potential maps. As stated in equation (6-3), assuming a planarsurface in the vicinity of the observation point, a reasonable approximation of the localarea of the body surface would be the tangential plane at the point of interest, over which alocal Cartesian coordinate system (x, y, z) can be considered. Assuming z to be normal tothe tangential plane, the Laplacian ECGIEEG at the observation point becomes (He, 1999;He & Wu, 1999):

where J denotes the current density and Jeq is an equivalent current source (He & Cohen,1992a, 1995; He, 1997, 1998a, 1999; He & Wu, 1999).

Unlike the ECG and EEG inverse problems, the SL approach does not attempt to locate the bioelectric sources inside the heart and brain. Instead, the Laplacian ECGIEEG


can be viewed as a two-dimensional (2D) projection of the three-dimensional (3D) bioelectric source onto the body surface. Therefore, as shown in equation (6-5), the LaplacianECG/EEG can be interpreted as an equivalent current source density on the body surface,which has the similar physical units as the primary bioelectric source density. On the otherhand, compared to the ECG and EEG inverse approaches, the SL approach does not requireexact knowledge about the conductivity distribution inside the torso and head volume conductors and has unique advantage of reference-independence as compared with the potentialmeasurement.

6.2 SURFACE LAPLACIAN ESTIMATION TECHNIQUES

6.2.1 LOCAL LAPLACIAN ESTIMATES

If the body surface in the vicinity of the measurement point can be approximatelyrepresented by a planar surface, the SL can be calculated using equation (6-3). In practice,the second order derivatives can be approximated by means of finite difference (Hjoth,1975).

Consider a grid of unipolar electrodes with equal inter-electrode distance b on the bodysurface (Figure 6-2A), the regular Laplacian electrogram at each non-boundary electrode(the cross-hatched circle) can be estimated by using the regular finite difference representation (Hjoth, 1975; He & Cohen, 1992a; Wu et al., 1999; Lian et al., 2002):

LR(i, j) ~ ~ {V(i, j) - ~ [V(i - 1, j) + V(i + 1, j) + v«, j - 1)+ V(i, j + I)]}

(6-6)

where V(i, j) and LR(i, j) represent the potential and the regular Laplacian electrogram atthe electrode (i, j), respectively. For each non-boundary electrode, equation (6-6) uses thepotential measurement at five electrodes (the cross-hatched circle and its four neighboringopen circles in Figure 6-2A) to estimate the Laplacian electrogram at the center electrode(the cross- hatched circle). Similarly, the Laplacian electrogram at electrode (i, j) can also beestimated from the potential recorded from this electrode and those recorded from its otherfour neighboring electrodes in the diagonal direction (neighboring black circles surroundingthe cross-hatched circle in Figure 6-2A). Denote the distance from the center electrode to itsdiagonal neighboring electrodes as d (for uniform grid, d = .fib), the diagonal Laplacianelectrogram can also be estimated by (Wu et al., 1999; Lian et al., 2002):

LD(i, j) ~ :2 {V(i, j) - ~[V(i - 1, j - I) + V(i - 1, j + 1)

+V(i+l,j-l)+V(i+l,j+I)]} (6-7)

where L D(i, j) represents the diagonal Laplacian electrogram at the electrode (i, j).


••

,........'• •

{ . .• •, I

• • •. . /- .--

•••••••••••••••••••••••••••••••• 0 ••••••••••••• 0 0 ••••••••••••• 0 ••••••••••••

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

A B

FIGURE 6-2. Schematic illustration of the local Laplacian estimates. (A) Regular or diagonal 5-point localLaplacian estimation. (B) Circular finite difference local Laplacian estimation.

A more general form of the finite difference reprentation of the SL utilizes the potential information from more local electrodes to realize the circular Laplacian electrode(He & Cohen, 1992a). As illustrated in Figure 6-2B, to estimate the Laplacian electrogram at the center electrode, the unipolar potential data are obtained from this electrode as well as from n electrodes located along a small circle (with radius r) surronding it, and the finite difference representation of the Laplacian electrogram is given by(Le et aI., 1994; Wei et al., 1995; He, 1997, 1998a, 1999; Wei & Mashima, 1999; Wei,2001):

4 ( 1 n )u, ~ 2" Vo - - LVir n i=!

(6-8)

where Vo and La represent the potential and circular Laplacian electrogram at the center electrode, respectively, and V;(i = 1, 2, ... , n) represents the potential at one of thesurrounding electrodes.

Another local Laplacian estimate uses bipolar concentric electrode that consists of twoparts: a conductive disk at the center and a surrounding conductive ring (Fattorusso et al.,1949; He & Cohen, 1992a). In the bipolar approach, the Laplacian electrogram may beestimated as (He & Cohen, 1992a, 1995; He, 1997):

Lo ~ i (Vo __1_ 1. Vdl)r2 2nr r (6-9)

where the integral is taken around a circle of radius r.In addition, some other local-based Laplacian algorithms were proposed in order to

achieve more acurate numerical estimates, for instance, by local modeling of the scalpand the potential distribution (Le et aI., 1994), or by means of the local polynomial fitting(Wang & Begleiter, 1999).

188

6.2.2 GLOBAL LAPLACIAN ESTIMATES

B. He and J. Lian

From equations (6-1) and (6-2), it can be seen that both the analytical models of thepotential distribution V (u, v ) and the surface geometry f eu, v) are required to calculate theSL. However, in real applications, the only data available are the limited number of potentialrecording s and body surface geometri c coordinate samplings. Thus, V(u , v) and f eu, v )

must be interpolated or approximated, and the most widely used interpolation scheme isthe spline interpolation. Among the investigations on the spline SL, of noteworthy is thespherical spline SL (Perrin et al., 1987a,b, 1989), the ellipsoidal spline SL (Law et al.,1993), and the realistic geometry spline SL (Babiloni et al., 1996, 1998; He et al., 2001,2002 ; Zhao & He, 2001).

Considering the non-planar shape of the body surface , a global SL estimation usingspline technique will be more accurate than the local-based SL estimates. Furthermore, asa major advantage over the local-based SL estimation, the spline SL has been shown toprovide a more robust characteristic against noise (Perrin et al., 1987a; Law et aI., 1993;Babiloni et al., 1996; He et al., 2001, 2002).

In the following, a recently developed realistic geometry spline Laplacian estimationalgorithm is presented (He et al., 2001, 2002). Estimation of the parameters associatedwith the spline Laplacian is formulated by seeking the general inverse of a transfer matrix.The number of spline parameters, which need to be determined through regularization, isreduced to one in the present approach , thus enabling easy implementation of the realisticgeometry spline Laplacian estimator.

6.2.2.1 Spline interpolation ofthe surface geometry

Given body surface sampling points (Xi, Yi, z.) . i = 1, ... , M, where M is the number ofsampling points for coordinate measurement, the mathematical model of the body surfacegeometry Z = f (x , y) can be described by 2D thin plate spline (Harder & Desmarais, 1972;Perrin et al., 1987a,b; Babiloni et al., 1996; He et al., 2001, 2002):

M M m- I d~ ~ 2(m- l) ( 2 2) ~~ d-k k

z = f(x , Y ) = ~ Pi Km- I + Qm- l = ~ p;d; log di +W + ~~qdkx Yi=1 i=l d=Ok=O

(6-10)

where m (spline order) is set to 2 (Perrin et aI., 1987a,b; Babiloni et aI., 1996; He et aI.,2001,2002 ), d; = (x - Xi)2+ (y - Yi)2, Km- I and Qm - I are basis function and osculating function , respectively, and w is a constant which accounts for effective radius of therecording sensor (Harder & Desmarais, 1972; Perrin et al., 1987a). The coefficients Pi andqdk are the solutions of following matrix equation s (Duchon, 1976; Perrin et aI., 1987a,b;He et al., 2002):

KP +EQ=Z

ETp =0

(6-11a)

(6-11b)


where P, Q, and Z are the vectors containing Pi , qdk. and Zi respectively, the matrices K andE are composed of elements of basis function and sampling coordinates, respectively.

6.2.2.2 Spline interpolation ofthe surface potential distribution

Similarly, given body surface potential recordings Vi at positions (Xi, Yi, z. ), i =1, 2, . . . , N, where N is the number of recording electrodes, the body surface potentialdistribution over the 3D space at an arbitrary point (x, y, z) can be modeled by the 3D spline(Babiloni et al., 1996, 1998; He et aI., 2001 , 2002):

N N m- I d k

V ( ) ~ H R ~ (2m- 3)/2 ~~~ d-k k-g gx,y, z = ~ti m-I + m-I = ~tiri +~~~rdkgX Y Z

i=1 i=1 d= O k=O g= O

(6-12)

where m (spline order) is set to 3 (Law et aI., 1993; Babiloni et aI., 1996, 1998; He et aI.,2001, 2002), rl = (x - Xi)2 + (y - Yi)2 + (z - Zi)2, Hm- l and R m-l are basis and osculating functions, respectively, and the coefficients t i and r dkg can be determined by solving thematrix equations (Law et aI., 1993; Babiloni et aI., 1996; He et al., 2002):

(6-13a)

(6-13b)

where T, R, and V are the vectors containing t. , r dkg, and Vi , respectively, the matrices Hand F are composed of elements of basis function and electrode coordinates, respectively.

6.2.2.3 Determination ofthe spline parameters

In an attempt to overcome the ill-poseness of the systems, approximation instead ofinterpolation of the surface geometry and potential distribution are used by introducing correction terms in equations (6-11a) and (6-13a), which are respectively changed to (Babiloniet aI., 1996, 1998; He et aI., 2001, 2002):

(K +wI)P + EQ = Z

(H + AI)T + F R = V

(6-14a)

(6-14b)

where 1 is the identity matrix, parameters wand A. are used to improve the numerical stabilityof the systems. The optimal values of these two parameters need be determined separatelyby either "tuning procedure" or other regularization techniques (Babiloni et al., 1996, 1998).

Instead of searching the optimal parameters in two dimensions, the above equationscan be reformulated by combining equations (6-11a,b) and (6-13a,b) into one linear systemequation (He et aI., 2001 , 2002):

AX=B (6-15)


where

[ff, E 0

~]0 0

(6-16a)A-- 0 0 H0 0 FT

x= [P Q T Rf (6-16b)

B = [Z 0 V O]T (6-16c)

Then the problem becomes seeking the solution of equation (6-15). Applying theconcept of the general inverse, we have (He et al., 2001, 2002):

(6-17)

where A# is the pseudo-inverse of A. Matrix A is ill-posed, thus regularization methodsmust be used to improve the stability of the system. Notably, after reformulating the matrixequations into one unified linear system in equation (6-15), only one single regularizationparameter needs to be determined when seeking the general inverse A#. Therefore, thepresent method not only can significantly reduce the computation effort and improve theefficiency and stability of the spline SL algorithm, but also can be combined with manyregularization techniques which have been extensively studied to determine the parameter in seeking the general inverse A# (for details on the regularization techniques, seeChapter 4).

6.2.3 SURFACE LAPLACIAN BASED INVERSE PROBLEM

The SL-based ECG or EEG inverse problem has also been explored to achieve highresolution heart or brain electric source imaging. One of the approaches is to estimate theepicardial potentials from the body surface Laplacian ECG (He, 1994; Wu et al., 1995, 1998;He & Wu, 1997, 1999; Johnston, 1997; Throne & Olson, 2000), or estimate the corticalpotentials from the scalp Laplacian EEG (He, 1998; Babiloni et al., 2000; Bradshaw &Wikswo, 2001).

As illustrated in Figure 6-3, if Vis an isotropic homogeneous volume conductor surrounded by an outer surface S1 and an inner surface Sz,and there is no current source existingwithin V, the potential on the inner surface can be related to the potential or Laplacians onthe outer surface.

Applying Green's second identity to the volume V results in (Barr et al., 1977):

U("'l = -.L If U . dQ - -.L If u . dQ - -.L If 1. . 2E..dSr ~ ~ ~ r ~

51 52 52

(6-18)

where u(r'l-the electrical potential at the observation point r*dQ-the solid angle of an infinitesimal surface element ds as seen from r*aau-the first derivative of potential u with respect to the outward normal to dS

r;


Sl

FIGURE 6-3. Schematic illustration of the arbitrarily shaped volume conductor.

By discretizing the surfaces Sl and S2 into triangular elements, and taking the limitof observation point approaching the surface element on St and S 2, respectively, from theinside of V, the following matrix equations can be obtained:

~ ~ -PIIU, + P'2U2+ G I2r2 =0

~ - -P21U j + P22U2+ G22r2 = 0

(6-19)

(6-20)

where Uk is the vector consisting of the electrical potentials at every surface elementon Si , and r k is the vector consisting of the normal derivatives of the electrical potentials at every triangle element on Sk but just inside of VI. P lI , P12, P21 , P22, G12, andG22 are coefficient matrices (Barr et aI., 1977). Solving equations (6-19) and (6-20)leads to the followingequation that relates the inner surface potential U 2 to the outer surface

potential U,:

(6-2 1)

where Tl2 = (P lI - Gl 2G221 P2d-l(G 12G221 P22 - P12 ).The surface Laplacian of the potential at the position r *at the outer surface Sl can be

written as follows (Wu et aI., 1998; He & Wu, 1999):

4* 1 If (a2Q I ) 1 If (a2

Q I ) 1 If au ( a2

1 I )L s(r ) = - 4rr U • d a,;T r* + 4rr U • d an2 r* + 4rr ar. · an2( ,:) r* dS

SI S2 S2(6-22)

where n is the normal direction of the surface Sl at r*.Similarly, by discretizing the surfacesSl and S2 into triangular elements, the following matrix equation can be obtained:

........ .... --->. ......

Ls=AU,+BU2+Cr2 (6-23)


~

where L, is the vector consisting of the surface Laplacians at every surface element on S"and A, B, and C are coefficient matrices (Wu et aI., 1998; He & Wu, 1999). From equations(6-19) , (6-:?l) , and (6-23), we can relate the inner surface potential U2 to the outer surfaceLaplacian L, by transfer matrix H:

(6-24)

where H = A · Tl2 + B - C · G22' . ( P2, . Tl2 + P22)The potential-based inverse problem seeks the inner surface potentials from the outer

surface potentials by solving equation (6-21):

(6-25)

On the other hand, the SL-based inverse problem seeks the inner surface potentials basedon solving equation (6-24 ):

(6-26)

where # denotes the general inverse of the transfer matrix.In addition, the hybrid potential-Laplacian-based inverse solution can also be solved

by minimizing the error function (He & Wu, 1999; Throne & Olson, 2000):

(6-27)

where ex is a weighting coefficient. The resulting inverse solution is given by (He & Wu,1999):

(6-28)

Equation (6-28) suggests that the inner surface potentials can be estimated from both theouter surface Laplacians and out surface potentials.

6.3 SURFACE LAPLACIAN IMAGING OF HEARTELECTRICAL ACTIVITY

6.3.1 HIGH-RESOLUTION LAPLACIAN ECG MAPPING

By applying the SL technique to the potential ECG , body surface Laplacian mapping(BSLM) was first proposed by He and Cohen (l992a,b). Theoretical and experimentalstudies have been carried out, demonstrating the unique feature of BSLM in effectivelyreducing the torso volume conduction effect and enhancing the capability of localizingand mapping multiple simultaneously active myocardial electrical events (He & Cohen,


1992a,b; He et aI., 1993, 1995, 1997,2002; Oostendorp & van Oosterom, 1996; Umetaniet aI., 1998; Wei & Harasawa, 1999; Wu et aI., 1999; Tsai el aI., 2001; Besio et al., 2001 ;Wei et aI., 2001; Li et aI., 2002).

6.3.2 PERFORMANCE EVALUATION OF THE SPLINE LAPLACIAN ECG

Through human experiments and computer simulations, we have systematically evaluated the signal to noise ratio of the Laplacian ECG, during ventricular depolarization andrepolarization, and demonstrated the feasibility of recording the Laplacian ECG, using the5-point local SL estimator (Wu et aI., 1999; Lian et aI., 2001,2002). Further improvement ofthe Laplacian ECG estimation may be achieved by using the global-based spline Laplaciantechnique.

In this section, we present the performance evaluation of the 3D spline SL algorithm(see Section 6.2.2). Computer simulations were conducted using both a spherical modeland a realistic geometry heart-torso model, and comparison studies were also made withthe 5-point local SL estimator (He et aI., 2002).

Given the torso surface geometry coordinates and the potential measurement, theLaplacian ECG was estimated by using the realistic geometry spline SL algorithm as detailedin Section 6.2.2. The linear inverse problem in equation (6-17) was solved by using thetruncated singular value decomp osition (TSVD) (Shim & Cho , 1981), and the truncationparameter was determined by means of the discrepancy principle (Morozov, 1984).

6.3.2.1 Effects ofnoise

We first evaluated the effects of noise on the SL estimation, by approximating thetorso volume conductor as a homogeneous single-layer unit-radius sphere model , withnormalized interior conductivity of 1.0. A radial dipole or a tangential dipole was usedto represent a localized cardiac electrical source. Three eccentricities (0.5, 0.6, 0.7) wereused to assess the effect of the source depth on the SL estimation. The Gaussian whitenoise (GWN) of different noise levels (5%, 7%, 10%) was added to the dipole-generatedsurface potentials sampled from 129 surface electrodes, simulating noise-contaminatedpotential ECG measurement. Two cases of geometry noise were also considered (2% geometry noise plus 10% potential noise, and 5% geometry noise plus 5% potential noise).For each noise level, ten trials of noise were generated and simulations were conducted.The correlation coefficient (cq values between the estimated SL and the analytical SLfor all ten trials were averaged and shown in Table 6-1. The SL was estimated by threedifferent methods : (1) 5-point local SL (5PL), (2) two-parameter spline SL (2SL), and (3)the recently developed one-parameter spline SL (1SL). The two-parameter spline SL wasestimated optimally by using the tuning procedure (Babiloni et aI., 1996, 1998) to findthe optimal values of (J) and A in equation (6-14) and the one-parameter spline SL wasestimated optimally by searching the optimal truncation parameter in TSVD procedure (Heet aI., 2002). The optimal parameters correspond to the maximum CC between the analyticalSL and the estimated SL. The quant ity w in equation (6-10) was set to 0.16 in spline SLestimation.


TABLE 6-1. The CC values between the analytical and estimated SL under different levels of noisein one-sphere model

GWN5%PN 7%PN 10%PN 2% GN+IO%PN 5%GN+5%PN

5PL 2SL ISL 5PL 2SL ISL 5PL 2SL ISL 5PL 2SL ISL 5PL 2SL ISL

I-RD, r=0.5 0.79 0.98 0.96 0.73 0.97 0.95 0.63 0.96 0.94 0.58 0.93 0.92 0.72 0.91 0.90I-RD, r = 0.6 0.90 0.97 0.98 0.85 0.96 0.97 0.77 0.95 0.95 0.58 0.91 0.93 0.70 0.91 0.92I-RD, r= 0.7 0.96 0.96 0.96 0.93 0.95 0.95 0.89 0.93 0.92 0.76 0.89 0.90 0.80 0.92 0.93I-TD, r = 0.5 0.71 0.98 0.96 0.62 0.97 0.95 0.50 0.96 0.94 0.65 0.90 0.91 0.67 0.88 0.90I-TD, r = 0.6 0.84 0.95 0.98 0.78 0.95 0.95 0.67 0.94 0.95 0.60 0.92 0.92 0.66 0.92 0.92I-TD, r = 0.7 0.94 0.96 0.96 0.91 0.95 0.95 0.85 0.91 0.93 0.71 0.88 0.88 0.79 0.91 0.92

Note: RO-radial dipole, TO-tangential dipole, PN-potential noise, GN-geometry noise.

Three findings are obvious from Table 6-1. First, for all three different SL estimators,the higher the noise level, the smaller the Cc. Second, for all the cases studied, the spline SLhas superior performance (higher CC) than the 5-point local SL, while the two-parameterspline SL and one-parameter spline SL have similar performance. Third, the 5-point localSL has the best performance for superficial sources and under low potential noise level,but its performance degrades dramatically as the source moves to deeper position or underhigher noise levels. On the other hand, the spline SL generally has good performance overa broader source depths (from 0.5 to 0.7), and shows more robust characteristics against thenoise in potential measurement. Specifically, for the one-parameter spline SL estimator, theCC values for all cases studied are greater than 0.92 under 10% potential noise, and equalor greater than 0.90 under 5% potential noise plus 5% geometry noise.

6.3.2.2 Effects ofnumber ofrecording electrodes

Table 6-2 shows the effects of number of recording electrodes on the SL estimation.One or multiple dipoles with varying orientations were placed in the spherical conductormodel (see note under Table 6-2 for dipole configurations), and 5% GWN was added to thedipoles-generated potentials. The CC values between the analytical SL and the estimatedSL with different electrode numbers and different dipole configurations are shown in Table6-2. Similarly, three different SL estimators were evaluated and compared. The quantity win equation (6-10) was set to 0.16, 0.18, 0.20, and 0.24 corresponding to 129,96,64, and

TABLE 6-2. The CC values between the analytical and estimated SL corresponding to differentelectrode numbers and dipole configurations in one-sphere model

Electrode Number129 96 64 32

5PL 2SL ISL 5PL 2SL ISL 5PL 2SL ISL 5PL 2SL ISL

Config. A 0.84 0.95 0.98 0.78 0.97 0.97 0.88 0.96 0.97 0.72 0.95 0.84Config.B 0.90 0.97 0.98 0.92 0.97 0.98 0.92 0.97 0.97 0.85 0.96 0.96Config. C 0.77 0.96 0.99 0.82 0.94 0.97 0.80 0.94 0.96 0.59 0.92 0.88Config.D 0.87 0.96 0.98 0.88 0.97 0.95 0.88 0.97 0.90 0.49 0.73 0.71Config.E 0.74 0.94 0.98 0.75 0.89 0.94 0.77 0.92 0.92 0.47 0.67 0.83

Note: Configurations A: I-TO at r= 0.6; B: I-RO at r = 0.6; C: two +z-direction dipoles at (±0.3, 0.0, 0.5); 0: one -l-x-directiondipole at (0.0, 0.0, 0.7) and two -l-z-direction dipoles at (0.0, ±OA, 0.5); E: 4-RO at r = 0,6, each one is ttl] with respect to the

z-axis. RO-radial dipole, TO-tangential dipole.


32 electrodes, respectively. The spline SL was estimated optimally by seeking the optimalregularization parameter(s).

Table 6-2 clearly indicates the correlation between the goodness of SL estimation andthe number of surface electrodes. In general, the more electrodes being used, the higherCC of the SL estimation. The CC values drop significantly when 32 electrodes are used,which is consistent with the fact that a minimum sampling in the space domain is neededto restore the spatial frequency spectrum. Again, Table 6-2 indicates that two-parameterspline SL and one-parameter spline SL have comparable performance, and are more robustagainst measurement noise than the 5-point local SL estimation.

6.3.2.3 Effectsofregularization

In simulations, the optimal SL can be estimated by means of a priori information ofthe analytical SL, i.e., by seeking the optimal parameter that maximizes the CC betweenthe analytical SL and the estimated SL. In real applications, the SL can also be estimatedwithout using the a priori information of the analytical SL, for example, by using thediscrepancy principle (Morozov, 1984). In Table 6-3, the upper rows show the CC betweenthe analytical SL and the optimal estimated SL. The lower rows show the CC between theanalytical SL and the estimated SL obtained by means of the discrepancy principle, withoutthe a priori information on the analytical SL. In this simulation, multiple dipoles withvarying orientations were placed in the spherical conductor model (see note under Table6-3 dipole configurations), and 5% GWN was added to the analytical surface potentialssampled at 129 recording electrodes. The quantity w in equation (6-17) was set to 0.16 inspline SL estimation.

Table 6-3 indicates that the regularization results always have lower CC than theoptimal results (by definition). However, the results obtained via regularization (by usingthe discrepancy principle in this case) are comparable to the optimal SL estimates. Out offour source configurations, the CC values for configurations B and D are almost similarfor these two types of results. For Configuration C, the CC of the regularization is smallerthan the optimal result by 1%. For configuration A, the CC of the regularization result issmaller than the optimal result by less than 3%. However, the absolute CC is 96% or above,suggesting the feasibility of the estimation of the SL through regularization.

Figure 6-4 depicts one typical example of the normalized surface potential maps andthe SL maps corresponding to source Configuration C in Table 6-3. In this figure, (A) is thenoise-contaminated surface potential map, (B) is the analytical Laplacian ECG map, (C) isthe optimal spline Laplacian ECG map estimated by means of a priori information, (D) is

TABLE6-3. Comparison of the optimalestimated spline SL and the splineSL estimated by usingthe discrepancy principle in one-sphere model

Dipole Configuration

Optimal spline SLRegularized spline SL

A

0.990.96

B

0.980.98

C

0.980.97

D

0.980.98

Note: Configurations A: two +z-direction dipoles at (±O.3, 0.0, 0.5); B: one +x-direction dipole at (0.0, 0.0, 0.7) and two +zdirection dipoles at (0.0, ±O.4, 0.5); C: 4-RD at r = 0.6, each one is nl] with respect to z-axis; D: 4-TD at r = 0.7, each one isrr/4 with respect to z-axis. RD-radial dipole, TD-tangential dipole.


.r~ .~8:J _~8:J -~8:J .~~·1 0 1 - 0 1 -I 0 1· 0 1 ·1 0 1

A BC D E

FIGURE 6-4. A typical example of the normalized potential ECG map (A) and the Laplacian ECG maps (B-E).See text for details. See the attached CD for color figure. (From He et al., IEEE-TBME, 2002 with permission)

© IEEE

the spline Laplacian ECG map estimated by means of the discrepancy principle, and (E)is the Laplacian ECG map estimated by the 5-point SL estimator. Figure 6-4 indicates that,from the viewpoint of imaging and mapping, the regularization spline SL estimate is almostidentical to the optimal spline SL estimate, and similar to the analytical SL result, for thecase studied. This is consistent with the high CC values obtained (Table 6-3) between theanalytical SL, the optimal spline SL, and the regularization spline SL estimates. Also noted,the 5-point local SL is more sensitive to the measurement noise as compared to the splineSL, especially at the border regions.

6.3.2.4 Simulationin a realistic geometry heart-torso model

The performance of the present 3D spline SL estimator was further examined usinga realistic geometry heart-torso model, where cardiac electric activity was simulated bypacing one or two sites in the ventricles of the heart model (He et al., 2002) (Figure6-5A). The potential ECG induced by ventricular pacing was simulated by means of theboundary element method (Aoki et al., 1987). The GWN was added to the simulated surfacepotentials to simulate noise-contaminated potential BCG measurement. The Laplacian ECGwas estimated from the noise-contaminated potential ECG using the 3D spline SL algorithm,and comparison was also made with the conventional 5-point local SL estimation.

Figure 6-5B depicts the simulation results when simultaneously pacing two sites (site#1 at the free wall of right ventricle, site #2 at the ventricular anterior) in the ventricularbase, and the single site pacing results corresponding to these two sites are shown in Figure6-5C and Figure 6-5D, respectively. In these figures, (i) shows the activation sequenceinside the ventricles induced by the pacing. (ii) shows the 5% GWN contaminated bodysurface potential map over the anterior chest immediately following the pacing. (iii) and(iv) respectively show the estimated body surface Laplacian ECG maps over the anteriorchest, by using the 5-point local SL estimator and the one-parameter spline SL estimator.Note that in Figure 6-5B, the estimated Laplacian ECG maps provide multiple and morelocalized areas of activity overlying the two pacing sites, whereas the body surface potentialmap does not reveal the spatial details on this source multiplicity, due to the smearing effectof the torso volume conductor. Also noted, that the spline SL estimate is less noisy andcan separate the two areas of activity more efficiently than the 5-point local SL estimate. InFigures 6-5C-D, both the potential ECG and Laplacian ECG maps corresponding to singlesite pacing reveal one pair of negative/positive activity, while the Laplacian ECG maps

Body Surface La placian Mapping of Bioelectric Sources 197

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FIGURE 6-5. Computer simulation of the spline Laplacian mapping in a realistic geometry heart-torso model.(A) Heart-torso model and the locations of two pacing sites. (B) Dual-site pacing example. (C) Single site (#1)pacing example. (D) Single site (#2) pacing example. See text for details. See the attached CD for color figure.(From He et aI., IEEE-TBME, 2002 with permission) © IEEE

provide much more localized spatial pattern. The two pairs of negative/posit ive activitiesrevealed in Figure 6-5B correspond well to the activities observed in Figures 6-5C-D.Consistently, the 5-point local SL estimates are noisier than the spline SL estimates.

6.3.2.5 Spline Laplacian ECG mapping in Humans

Applying the spline Laplacian algorithm we have developed , body surface Laplacianmapping has been explored in a group of healthy male subjects during ventricular and atrialdepolarization. Ninety-five channel body surface potential ECG was recorded simultaneously over the anterolateral chest in the subjects. The Laplacian ECG was estimated from

198

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(A) (B) (C) (D)

FIGURE 6-6. Body surface potential and Laplacian maps of a healthy human Subject around the peak ofR-wave.

Time instant is referenced to the onset of QRS and illustrated by a vertical line labeled in the Lead I ECG tracing(A). The BSPM map is shown in (B). The spline BSLM map is shown in (C) with spatial details denoted by letter'P' and 'N' followed with a numerical number for positive and negative activities, respectively. The corresponding

BSLM map estimated by the 5-point SL estimator is shown in (0). The physical units of the color bars in theBSPM and the BSLM maps are mVand mv/cmr, respectively. See the attached CO for color figure.

the recorded potentials during QRS complex and the P-wave by means of the one-parameterspline SL estimator. For all subjects, more spatial details were observed in the SL ECGmaps as compared with the potential ECG maps, with spline SL more robust against noisethan the 5-point SL (Li et al., 2003). Figure 6-6 shows one example of the SL ECG mapover the anterolateral chest of a healthy male subject around the peak of R-wave (Figure6-6A). Figure 6-6B shows the potential map, which shows a pair of positivity and negativityover the anterolateral chest. The corresponding spline BSLM map is shown in Figure 6-6C,illustrating a localized negative activity, N2, located over the central chest, a positive activityP2 slightly shifted toward the left lateral chest with respect to the position of N2, anotherpositive activity to the left of P2, and another negative activity N3 appeared in left-superiorarea. Figure 6-6D shows the SL ECG map estimated using the 5-point local SL estimator.Note that the local 5-point SL estimate (Figure 6-6D) shows more focused activities ascompared with the potential map (Figure 6-6B), but failed to reveal the spatial details asillustrated in the spline SL map (Figure 6-6C). The negative and positive activities observedin the group of human subjects have been related to the epicardial events (Li et al., 2003).

Figure 6-7 shows an example of spline SL mapping in a healthy human subject duringatrial depolarization (Lian et al., 2002b). Compared with the diffused potential map (Figure6-7B), the corresponding spline BSLM map (Figure 6-7C) clearly shows two major positiveactivities, PI and P2, representing the local maxima on the right and left anterior chest,

42 ms

A (8)

Nl2

(C)

FIGURE 6-7. Body surface potential and Laplacian maps during the mid P wave in a healthy human subject.(a) The potential P wave recorded from the left lower anterior chest, and the time instant for constructing the maps.(b) The instantaneous BSPM map shows smooth pattern of potential distribution (colorbar unit: j..LV). (c) Theinstantaneous BSLM map shows two major positive activities PI and P2, associated with three negative activitiesN I, N2, and N3 (colorbar unit: j..LV/cm"), See the attached CO for color figure.

Body Surface Lapl acian Mapping of Bioelectric Sources 199

respectively.Correspondingly, three associated negative activities can also be observedanddenoted as Nl , N2, and N3, representing the local minima on right, middle and left chestseparated by PI andP2, respectively. Dataanalysis andcomputer simulation studies suggestthat the positivities PI and P2 may correspond to the activation wavefronts in the right andleft atria, respectively (Lian et al., 2002b).

Compared to the smooth pattems of the BSPMs, more spatial details are revealed inthe BSLMmaps during ventricular and atrial activation, whichcould be correlated with theunderlying multiple myocardial activation wavefronts.

6.3.3 SURFACE LAPLACIAN BASED EPICARDIAL INVERSE PROBLEM

The feasibility on solving the ECG inverse problem by means of the Laplacian ECG(He, 1994; Wu et aI., 1995, 1998; He & Wu, 1997, 1999; Johnston, 1997), and a hybridapproach using both potential ECG and Laplacian ECG for epicaridial inverse problem(He & Wu, 1999;Throne & Olson, 2000) have also been explored (see Section 6.2.3).

As an example, Figure 6-8 shows the simulation results for testing the feasibilityof the Laplacian ECG based epicardial potential inverse solution in a realistically shapedheart-torso model. Current dipoles located inside the anterior myocardium pointing fromendocardium to the epicardium were used to simulate anterior sources. To simulate thenoise-contaminated experimental recordings, up to20%GWNwasadded to theheart-modelgenerated body surface ECG signals before the epicardial potentials were reconstructed.Figure 6-8 plots the relative error (RE) and CC between the epicardial potentials calculatedfrom two anteriordipoles using the boundary element method, and the epicardial potentials

1.0 tt::a:::::::::::;t ~.......................................................................

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11.30.211. 111.11 +-- - - ----- .----- - - - - .--- - -------.

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FIGURE 6-8. RE and CC values between the forward epicard ial potent ials and the epicardial potentials reconstructed from the potential ECG and the Laplacian ECG in a realistically shaped heart- torso model with two dipoleslocated in the anterior ventricular wall. (From He & Wu, Crit Rev BME, 1999 with permission from Begell House)


reconstructed from the noise-contaminated potential ECG and surface Laplacian ECG overthe whole torso. Note that the Laplacian ECG based epicardial inverse solutions alwaysprovide smaller RE as compared with the potential ECG based epicardial inverse solutionsfor the same noise level. For a larger noise level of 20% in the Laplacian ECG, the LaplacianECG based epicardial inverse solutions still show a comparable performance as comparedwith the potential ECG based inverse solutions at a lower noise level of about 3%. Due toincreased noise level in Laplacian ECG as compared with the potential ECG, the real meritsof the Laplacian ECG based inverse solution would depend on how accurate one may recordor estimate the Laplacian ECG in an experimental setting. The data comparing the epicardialinverse solutions obtained from the body surface potentials with those obtained from thebody surface Laplacians, which are estimated from the noise-contaminated potentials, arecurrently lacking in the literature.

6.4 SURFACE LAPLACIAN IMAGING OF BRAINELECTRICAL ACTIVITY

6.4.1 HIGH-RESOLUTION LAPLACIAN EEG MAPPING

As a spatial enhancement method, the SL technique has been applied for many yearsto high-resolution EEG mapping. The SL has been considered an estimate of the localcurrent density flowing perpendicular to the skull into the scalp, thus it has also beentermed current source density or scalp current density (Perrin et aI., 1987a,b; Nunez et aI.,1994). In addition, the relationship between the SL and the cortical potentials has also beeninvestigated (Nunez et aI., 1994; Srinivasan et aI., 1996). Since Hjorth's early exploration onscalp Laplacian EEG (Hjorth, 1975), a number of efforts has been made to develop reliableand easy-to-use SL techniques. Of noteworthy is the development of spherical spline SL(Perrin et aI., 1987a,b), ellipsoidal spline SL (Law et aI., 1993), and the realistic geometryspline SL (Babi1oni et al., 1996, 1998; He, 1999; Zhao & He, 2001; He et al., 2001).

6.4.2 PERFORMANCE EVALUATION OF THE SPLINE LAPLACIAN EEG

In this section, we present the performance evaluation of the realistic geometry splineLaplacian estimation algorithm in high-resolution EEG mapping (He et al., 2001). Theevaluation was conducted by computer simulations using both a 3-concentric-sphere headmodel (Rush & Driscoll, 1969) and a realistic geometry head model. In addition, we examined the performance of the spline SL algorithm in high-resolution mapping of neuralsources using experimental visual evoked potential (VEP) data.

The realistic geometry spline SL estimation algorithm is detailed in Section 6.2.2. Thelinear inverse problem in equation (6-17) was solved by the TSVD (Shim & Cho, 1981), andthe truncation parameter was determined by means of the discrepancy principle (Morozov,1984).

6.4.2.1 Effects ofnoise

The analytical SL was used to evaluate the numerical accuracy and reliability of thespline SL estimation from the scalp potentials. The accuracy of the spline SL estimator wasevaluated by the CC and RE between the estimated and analytic SL distributions.

Body Surface Laplacian Mapping of Bioelectric Sources

1

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201

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FIGURE 6-9. Effects of noise on the spline SL estimation in a 3-sphere inhomogeneous head model. (From Heet aI., Clin Neurophy, 200 1 with permission)

Figure 6-9 shows an example of the simulation results for the effects of noise in the3-concentric-sphere head model. A radial dipole (Rad) or a tangential dipole (Tag) locatedat an eccentricity of 0.6 was used to represent a well-localized areas of brain electricalactivity. The scalp potential and the scalp SL at 129 electrodes were calculated analytically(Perrin et al., 1987a). Noise of up to 25% was added to the analytical potentials to simulatenoise-contaminated scalp potential measurements. For each noise level, ten trials of GWNwere generated and simulation conducted . The RE and the CC between the estimated SLand the analytical SL for all ten trials were averaged and displayed in Figure 6-9. The SL wasestimated by minimizing the RE between the analytical SL and the SL estimate obtained bysolving equation (6-17) for a regularization parameter. The parameter w in equation (6-10)was set to 0.16.

Figure 6-9 indicates that the higher the noise level the larger the RE (or smaller theCC). The CC value was greater than 96% for the radial dipole , and greater than 91% forthe tangential dipole, for up to 25% noise level. Figure 6-9 suggests that the one-parameterrealistic geometry SL estimator is robust against the additive white noise in the scalppotential measurements.

6.4.2.2 Effects ofnumber ofrecording electrodes

Table 6-4 shows an example of the simulation results with different number of surfaceelectrodes in the 3-concentric-sphere head model. The RE and CC between the analyticalSL and the estimated spline SL with different electrode number and different dipole configurations are shown in Table 6-4. One or multiple dipoles with unity strength were placedin the spherical conductor model (see note under Table 6-4 for detailed description of theparameters of the dipole configurations). The GWN of specified noise level was added tothe scalp potentials to simulate noise contaminated EEG measurements. The parameter win equation (6-10) was set to 0.16,0.18,0.20 and 0.24 corresponding to 129,96,64 and 32electrodes, respectively. The SL was estimated by minimizing the RE between the analyticSL and the SL estimate.

Table 6-4 clearly indicates the correlation between the goodne ss of the SL estimationand the number of surface electrodes. In general , the more electrodes used, the higher


TABLE 6-4. The RE and CC values between analytical and estimated SL correspondingto different electrode numbers and dipole configurations in 3-sphere model

Electrode Number 129 96 64 32

Configuration ARE 0.25 0.27 0.29 0.32CC 0.97 0.96 0.96 0.95

Configuration BRE 0.30 0.31 0.40 0.51CC 0.95 0.95 0.92 0.86

Configuration CRE 0.37 0.40 0.46 0.51CC 0.93 0.92 0.89 0.86

Configuration DRE 0.28 0.33 0.36 0.64CC 0.97 0.94 0.93 0.77

Configuration ERE 0.20 0.22 0.28 0.42CC 0.98 0.98 0.96 0.91

Note: Configurations A: I-RO at r = 0.5, with 15% GWN; B: I-TO at r = 0.7, with 5% GWN; C: I dipole at (0.0,

0.1,0.75) pointing to +x direction, another dipole at (0.0, -0.1,0.6) pointing to +z direction, with 5%GWN; 0:

2-RO at r = 0.7, and I-TO at r = 0.6, each has an angle of rr/6 with respect to z-axis, with 5% GWN; E: 2-RO at

x-axis and 2-TO at y-axis, all at r = 0.65, and each has an angle of rr/6 with respect to z-axis, with 10%. RO: radial

dipole. TO: tangential dipole.

CC (or lower RE) of the SL estimation. This phenomenon is consistent with the fact that aminimum sampling in the space domain is needed to restore the spatial frequency spectrum.As can be seen from Table 6-4, the CC for the SL estimation is 92% or larger for allcases when 96 or more electrodes were used. The CC for Configuration C was about 89%when 64 electrodes were used and 86% when 32 electrodes were used. This relativelylow CC values, as compared with other configurations may be explained by the largeeccentricity of the dipoles in the Configuration C. The closer the dipole is located to thescalp, the sharper the spatial distribution of the scalp Laplacian. Thus higher spatial samplingrate is desired. This phenomenon is further observed when only 32 electrodes are used, theCC values dropped lower than 90% for Configurations B, C and D. Of interesting is thelow CC value for Configuration D when 32 electrodes were used. The CC dropped from93%, when 64 electrodes were used, to 77% when 32 electrodes were used. Table 6-4suggests that a high-density electrode array of 96 or more is desirable for scalp spline SLmapping.

6.4.2.3 Effects ofregularization

Table 6-5 shows examples of the simulation results comparing the SL estimate obtainedby means of a priori information of the analytical SL (denoted below as the optimal SLestimate), and the SL estimate without using the a priori information of the analytical SL.In this simulation, the discrepancy principle (Morozov, 1984) was used to determine thetruncation parameter of the TSVD procedure. In Table 6-5, the upper rows show the REand CC between the analytical SL and the optimal SL estimate. The lower rows showthe RE and CC between the analytical SL and the estimated SL obtained by means of thediscrepancy principle. In this simulation, 129 electrodes were uniformly distributed over theupper hemisphere, the parameter w in equation (6-10) was set to 0.16, and GWN of varyingnoise level was added to the analytical scalp potentials to simulate noise-contaminatedpotential measurements.


TABLE6-5. Comparison of the optimal estimated spline SL and the spline SL estimated by using the discrepancy principle in 3-sphere model

Dipole Configuration A B C D

Optimal RE 0.36 0.34 0.20 0.29TSVD CC 0.95 0.94 0.98 0.98

Estimated RE 0.38 0.40 0.25 0.44TSVD CC 0.94 0.93 0.97 0.91

Note: ConfigurationsA: 2-RD at r = 0.70, each has an angle of '!f/8 with respect to z-axis, with 5% GWN; B:2-RDat r = 0.6, and I-TD at r = 0.7, each has an angle of '!f/6 with respect to z-axis, with 10%GWN;C: 2-RDand 2-TD, all at r = 0.65, and each has an angle of n16with respect to z-axis, with 10% GWN; D: 4-RD at r =0.75, each has an angle of '!f17 with respect to z-axis, with 5% GWN. RD: radial dipole. TD: tangential dipole.

o

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BFIGURE 6-10. Two examples of the normalized potential EEG and Laplacian EEG maps. See text for details.See the attached CD for color figure. (From He et a!., CEn Neurophy, 2001 with permission)

Table6-5 indicatesthat the regularizationresults are always worse than the optimal SLestimates (by definition), since the optimalSL estimates are obtainedby minimizingthe REbetweenthe analyticaland estimatedSL.However, Table6-5 showsthat the resultsobtainedvia regularization (by using the discrepancy principle in this case) are comparable to theoptimalSLestimates.Outof four sourceconfigurations, the CC for Configurations A, BandC are almost similar for these two types of results. For Configuration D, the CC of the regularizationis smaller than the optimal SL estimate by about 6.5%. Howeverthe absolute CCis above 91%, suggesting the feasibility of the estimation of the SL through regularization.

Figure6-10depictstwoexamplesof thenormalizedpotentialandLaplacianEEGdistributionscorrespondingto Configuration A and Configuration B in Table6-5. Foreach sourceconfiguration, the first panel shows is the noise-contaminatedscalp potential map, the second panel showsthe analyticalsplineLaplacianEEG map, the thirdpanel showsthe optimal

204

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B. He and J. Lian

Dipole #1

L RDipole #2.

FIGURE 6-11. A realistic geometry head model built from one subject with two simulated dipoles located withinthe brain. (From He & Lian, Crit Rev BME, 2002 with permission from Begell House)

estimated Laplacian EEG map by means of a priori information, and the last panel showsthe regularization estimated Laplacian EEG map by means of the discrepancy principle.

6.4.2.4 Simulation in a realistic geometry head model

The computer simulation was also conducted by using a realistic geometry head modelbuilt from one healthy subject (male, 34 years old) who was later involved in the VEPexperiment. The CT images of the subjects were obtained, and the BEM models of the scalp,the skull, and the brain surfaces of the subject were constructed (Figure 6-11). Two artificialdipoles were used to simulate two simultaneously active brain electrical sources. One dipolesource is located in right medial temporal lobe with orientation tangential to the corticalsurface, and another dipole is located in right inferior frontal lobe with radial orientation.

Figure 6-12 shows the result of spline Laplacian imaging of the simulated dipolesources in realistic geometry head model. The scalp potentials generated by the two artificialdipoles were contaminated with 5% GWN to simulate the measurement noise, and showa blurred dipole pattern of distribution with frontal positivity and posterior negativity. Thespline Laplacian EEG map, however, effectively reduces the blurring effect caused bythe head volume conductor, and clearly reveals two localized activities corresponding tothe underlying dipole sources.

6.4.2.5 Surface Laplacian imaging ofvisual evoked potential activity

Besides simulations, human VEP experiments were carried out to examine the performance of the spline SL estimator. The same above subject who gave written informedconsent was studied in accordance with a protocol approved by the UIC/IRB. Visual stimuliwere generated by the STIM system (Neuro Scan Labs, VA). The 96-channel VEP signalsreferenced to right earlobe were amplified with a gain of 500 and band pass filtered from 1Hz to 200 Hz by Synamps (Neuro Scan Labs, VA), and were acquired at a sampling rate of


FIGURE 6-12. Spline Laplacian mapping of the simulated dipole sources in a realistic geometry head model.(A) Scalp potential map. (B) Estimated spline Laplacian EEG map. See the attached CD for color figure. (FromHe & Lian, Crit Rev BME, 2002 with permission from Begel! House)

1,000 Hz by using SCAN 4.1 software (Neuro Scan Labs, VA). The electrodes' locationswere measured using Polhemus Fastrack (Polhemus Inc. , Vermont). Full or half visual fieldpattern reversal checkerboards (black and white) with reversal interval of 0.5 sec served asvisual stimuli and 300 reversals were recorded to obtain averaged VEP signals . The displayhad a total viewing angle of 14.30 by ILl 0 , and the checksize was set to be 175' by 135'expressed in arc minutes . The SL was estimated at the peak of the P100 component.

Figure 6-13 shows the recorded scalp potential maps and the estimated spline LaplacianEEG maps at the PIOO peak time point of the pattern reversal VEP recorded from 96

A B c o

FIGURE 6-13. Spline Laplacian mapping of YEP activity in a human subject. (A) Scalp potential map elicitedby the full visual field stimuli. (B) Spline Laplacian EEG map in response to the full visual field stimuli. (C) Scalppotential map elicited by the left visual field stimuli. (D) Spline Laplacian EEG map in response to the left visualfield stimuli. See the attached CD for color figure. (From He et aI., Clin Neurophy, 200 1 with permission)


electrodes over the scalp. As shown in Figure 6-13A , the scalp potential map elicitedby the full visual field stimuli is characterized by the strong but diffused activity thatdistributed symmetrically over the occipital area. The estimated spline Laplacian EEGmap (Figure 6-13B), on the other hand, greatly improves the spatial resolution and clearlyreveals two dipole-like sources located in the visual cortices in both hemispheres. Notably,the distribution of the positivity and negativity on each part of the scalp suggests orientationof cortical current sources , pointing toward the contralateral hemisphere. As shown in Figure6-13C, in response to the left visual field stimuli, a dominant positive potential componentwas elicited with a widespread distribution on the left scalp . However, the estimated splineLaplacian EEG map (Figure 6-13D) shows a dominant dipole-like current source locatedin the right visual cortex. Similarly, the positivity-negativity distribution on the right scalpin the spline Laplacian EEG map suggests orientation of cortical current sources, pointingtoward the contralateral hemisphere.

6.4.3 SURFACE LAPLACIAN BASED CORTICAL IMAGING

Using the procedure as detailed in Section 6.2.3, the application of using the scalp SLto reconstruct the cortical potentials has also been explored (He, 1998b). Figure 6-14 shows

(a)

(c)

(b)

(d)

FIGURE 6-14. An example of cortical imaging from scalp Laplacian EEG. (a) The "ture" cortical potentialdistribution generated by four radial dipoles located at eccentricity of 0.8 in the 3-sphere head model. (b)-(d)cortical potential distributions reconstructed from scalp Laplacian EEG with (b) 10%, (c) 30%, and (d) 50%Gaussion white noise, respectively. See the attached CD for color figure. (From He, IEEE-EMB, 1998 withpermission) © IEEE


FIGURE 6-15. Cortical imaging of the SEP activity over a realistic geometry head model in a human subject,using standard and the SL pre-filtered WMN estimate. See text for details. See the attached CD for Color figure.(From Babiloni et aI., MBEC, 2000 with permission)

an example of the simulation study based on the 3-concentric-sphere head model (Rush &Driscoll, 1969). Figure 6-14(a) shows the "true" cortical potential distribution generatedby four radial dipoles. Figures 6-14(b)-(d) show the inverse cortical potential distributionsestimated from the scalp SL with 10%, 30%, and 50% GWN added to the scalp SL,respectively. Notice that the higher the noise level in the scalp SL, the higher the backgroundnoise level in the estimated cortical potentials. However, reconstruction of the multipleextrema of the cortical potential distribution is quite robust, suggesting the feasibility andunique feature of the cortical potential imaging from the scalp SL. In a separate study,by investigating the spatial filter characteristics of the source-Laplacian relationship,Bradshaw and Wikswo (2001) demonstrated that dramatic improvement is evident inthe SL-based inverse solution, as compared with inverse reconstruction from the rawdata.

In another approach, the SL pre-filtered EEG data was used as input for the weightedminimum norm (WMN) linear inverse estimate of the cortical current sources, in orderto remove subcortically originated EEG potentials from the scalp potential distribution(Babiloni et aI., 2000). As an example, Figure 6-15 shows the application of this techniqueto the cortical imaging of the somatosensory evoked potentials (SEPs) over the realisticgeometry head model of one subject. For all the SEP components being examined (P20N20, P22, N30-P30), the cortical imaging inverse solutions have enhanced spatial resolutionthan the scalp potential maps. Moreover, with respect to the WMN estimate, the SL prefiltered WMN estimate presented enhanced spatial information content, in that the potentialmaxima over the cortical surface were sharper and more localized.

208

6.5 DISCUSSION

B. He and J. Lian

The SL, as demonstrated by many investigators, enjoys enhanced spatial resolution andsensitivity to regional bioelectrical activity located close to the surface recording electrodes,and has unique advantage of reference independence. Conventionally, the local-based SLoperators have been used to estimate the Laplacian ECG or Laplacian EEG, by approximating planar surface at the recording electrode. More accurate estimation can be achievedby taking into account the realistic geometry of the body surface using the spline interpolation scheme. On the other hand, due to the high-pass spatial filtering characteristics of thelocal SL operator, amplification of the noise associated with the potential measurementsis unavoidable. Spatial low-pass filters, such as the Gaussian filter (Le et al., 1994) andWiener filter (He, 1998a), have been shown to be useful in improving the signal-to-noiseratio of the SL. The spline SL, on the other hand, has been shown to provide an intrinsicspatial-low-pass filtering in addition to its spatial-high-pass filtering characteristics (Nunezet al., 1994; Srinivasan et al., 1998).

Estimation of the spline SL from potentials does not require the information on theconductivity distribution inside the volume conductor. On the other hand, the spline SL estimation techniques do need a mathematical model describing the geometry of the surfaceover which the SL is to be estimated. The spline SL has been estimated over a spherical,ellipsoidal, and a realistic geometry surface (Perrin et al., 1987a,b, 1989; Law et al., 1993;Babiloni et aI., 1996, 1998; He, 1999; Zhao & He, 2001; He et al., 2001, 2002). Furthermore, the recently developed 3D spline SL algorithm (see Section 6.2.2) eliminates theneed of determining two spline parameters, and provides a rational determination of thespline parameter through regularization process (He et al., 2001,2002). Such new approachprovides comparable computational accuracy and stability, while substantially reduces thecomputational burden of optimizing two independent regularization parameters, as requiredin the previously reported approaches.

The performance of the present realistic geometry spline SL estimator has been evaluated through a series of computer simulations. Based on the one-sphere homogeneousvolume conductor model, the simulation results demonstrate that the performance of theone-parameter spline SL algorithm is comparable with that of the traditional two-parameterspline SL algorithm (Tables 6-1-6-3), but with much greater computational efficiency. Thesimulation study also demonstrates that the spline SL estimators are more robust againstadditive noise in both potential and geometry measurements as compared to the 5-pointlocal SL estimator (Table 6-1), and consistent results are found for different numbers ofrecording electrodes (Tables 6-2). An interesting finding is that the 5-point local SL hasgood performance only for shallow sources and under low noise level, while the spline SLhas good performance over a broader source depths and under variant noise levels (Table6-1). This can be explained by the high-pass spatial filter property of the 5-point local SLestimator, versus the band-pass spatial filter property of the spline SL estimator (Nunezet al., 1994; Srinivasan et al., 1998). Table 6-3 and Figure 6-4 further suggest that the SLcanbe estimated by using the well established regularization techniques, such as the discrepancyprinciple (Morozov, 1984), without apriori information of the "true" SL. In addition, basedon the 3-sphere inhomogeneous volume conductor model, consistent evidence is shownthat the spline SL estimation algorithm is robust against noise in potential measurements(Figure 6-9), provides consistent performance for different number of recording electrodes(Table 6-4), and can be estimated by means of the discrepancy principle (Table 6-5).


The application of the 3D spline SL estimator to the realistic geometry volume conductor models further suggests the potential use of the realistic geometry spline Laplacian ECGIEEG estimation. Figure 6-5 indicates that the Laplacian ECG maps provide amuch-localized projection onto the body surface in the areas directly overlying the heart,and are especially useful in identifying and characterizing the source multiplicity as compared with the potential ECG maps. Similarly, Figure 6-12 indicates that the Laplacian EEGmap can effectively reduces the smoothing effect of the head volume conductor, and clearlylocalizes the underlying brain electrical sources. The human YEP experiments further suggest the usefulness of the Laplacian EEG mapping. The full visual field stimuli elicitedsymmetrical potential distribution about the midline of the occipital scalp. The estimatedLaplacian EEG map showed much more localized dipolar current sources in both visualcortices, with dipolar orientations pointing toward the respective opposite hemisphere. It iswidely accepted that the half visual field stimuli activates the visual cortex on the contralateral hemisphere of the brain. But paradoxically, the left visual field stimuli elicited strongerpositive potential distribution over the ipsilateral side of the scalp, which might be misinterpreted as left visual cortex activation. However, by using the 3D spline SL method, theestimated Laplacian EEG map clearly indicated that the right visual cortex was activated,and these results are consistent with previous reports (Barrett et aI., 1976; Blumhardt et aI.,1977; Towle et aI., 1995).

In summary, the SL, as a spatial enhancement method, can enhance the high-frequencyspatial components of the surface EEG and ECG. The 3D spline SL algorithm can take intoconsideration of the realistic geometry of the body surface, and is applicable to both brainand heart electrical source imaging. Only one spline parameter needs to be determinedthrough regularization procedure in this spline SL algorithm, thus enabling easy implementation of the spline SL in an arbitrarily shaped surface of a volume conductor. Bothcomputer simulations and preliminary human experiments have demonstrated the excellentperformance of the 3D spline SL in high-resolution ECG and EEG mapping, suggesting itmay become an alternative for noninvasive mapping of heart and brain electrical activity.

ACKNOWLEDGEMENT

The authors would like to thank their colleagues Dr. G. Li and Dr. D. Wu for usefuldiscussions. This work was supported in part by a grant from American Heart Association#0140132N, NSF CAREER Award BES-9875344, and NSF BES-0201939.

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7

NEUROMAGNETIC SOURCERECONSTRUCTION AND INVERSE

MODELING

Kensuke Sekihara' and Srikantan S. Nagarajarr'I Departmentof ElectronicSystems and Engineering, Tokyo Metropolitan Instituteof Technology,

Asahigaoka 6-6, Hino, Tokyo191-0065, Japan2Department of Radiology, University of California,San Francisco, 513 Parnassus Avenue, S362

San Francisco, CA 94143, USA

7.1 INTRODUCTION

The human brain has approximately 1010 neurons in its cerebral cortex. Their electrophysiological activity generates weak but measurable magnetic fields outside the scalp.Magnetoencephalography (MEG) is a method which measures these neuromagnetic fieldsto obtain information about these neural activitie s (Hamalainen et al., 1993; Roberts et al. ,1998; Lewine et al., 1995). Among the various kinds of funct ional neuroimaging methods,such a neuro-electromagnetic approach has a major advantage in that it can provide finetime resolution of millisecond order. Therefore, the goal of neuromagnetic imaging is to visualize neural activit ies with such fine time resolution and to provide functional informationabout brain dynamics. To attain this goal, one technical hurdle must be overcome. That is,an efficient method to reconstruct the spatio-temporal neural activities from neuromagneticmeasurements needs to be developed. Toward this goal, a number of algorithms for reconstructing spatio-temporal source activities have been investigated (Baillet et al., 2001) .

This chapter deals with this neuromagnetic reconstruction problem . However, we donot provide a general review of variou s algorithms for this reconstruction problem. Instead,we describe a particular class of source reconstruction techniques referred to as the spatialfilter, which allows the spat io-temporal recon struction of neural activities without assumingany kind of source model. Furthermore, among the spatial filter techniques, we focus onadapti ve spatial filter techniques. The se techniques were origi nally developed in the fieldsof array signal processing , including radar, sonar, and seismic exploration , and have been

Corresponding author: Kensuke Sekihara Ph.D., Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6,Hino, Tokyo 191-0065, Japan, Tel:81-42-585-8642, Fax:81-42-585-8642, E-mail: [email protected]

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214 K. Sekihara and S. S. Nagarajan

widely used in such fields (van Veen et al., 1988). Nonetheless, the adaptive spatial filtertechniques are relatively less acknowledged in the MEG/EEG community.

In this chapter, we also formulate reconstruction techniques based on linear leastsquares methods (Hamalainen and Ilmoniemi, 1984) as the non-adaptive spatial filter. Thisformulation enables us to compare the least-squares-based techniques with the adaptivespatial filter techniques on a common, unified basis. Actually, we compare these two typesof techniques by using the same figure ofmerit called the resolution kernel, and show that theadaptive techniques can provide much higher spatial resolution than the least-squares-basedmethods.

The organization of this chapter is as follows: Following a brief review of the neuromagnetometer hardware in Section 7.2, we describe the forward modeling and somebasic properties of MEG signals in Section 7.3. Section 7.4 presents the formulation of thelinear least-squares-based methods as the non-adaptive spatial filter. Section 7.5 describesthe adaptive spatial filter techniques. In Section 7.6, we present a quantitative comparison between the adaptive and non-adaptive methods using the resolution kernel criterion.Section 7.7 presents a series of numerical experiments on the adaptive spatial filter performance. In Section 7.8, we demonstrate the effectiveness of the adaptive spatial filtertechniques by applying them to two sets of MEG data.

7.2 BRIEFSUMMARY OF NEUROMAGNETOMETER HARDWARE

It is generally believed that the neuromagnetic field is generated by the post-synapticionic current in the pyramidal cells of cortical layer III. Here, neuronal cells are organizedinto so-called columnar structure, and the synchronous activity of these cells results insuperimposed magnetic fields strong enough to be measured outside the human head. Theaverage intensity of this neuromagnetic field, however, is around a few hundred femtoTesla (fT)*. To measure such an extremely weak magnetic field, a neuromagnetometer usesa special device called a super-conducting quantum interference device (SQUID) (Clarke,1994; Drung et al., 1991). This device is so sensitive that it can in principle measure a singlequantum of magnetic flux.

When measuring such a weak neuromagnetic field, a major problem arises from thebackground environmental magnetic noise. Background magnetic noise is generated byelectronic appliances such as computers, power lines, cars, and elevators. Such noise iscommon at a site where the neuromagnetometer is installed, and its average intensity isusually five to six orders of magnitude greater than the neuromagnetic field. One obviousway to reduce it is to use a magnetically-shielded room. However, a typical medium-qualityshielded room, most commonly used in MEG measurements, can reduce the backgroundnoise by up to only three orders of magnitude.

To further reduce the background noise, neuromagnetometers are usually equippedwith a special type of detector configuration called a gradiometer (Hamalainen et al., 1993;Lewine et al., 1995). The first-order gradiometer consists of two coils of exactly the samearea; they are connected in series, but wound in opposite directions. Therefore, the gradiometer cancels the electric current induced by the background noise fields because the

* One IT is equivalent to I x 10- 15 T.

Neuromagnetic Source Reconstruction and Inverse Modeling 215

sources of such noise fields are generally far from the gradiometer and induce nearly thesame amount of electric current in both coils. The gradiometer can achieve two to threeorders of magnitude reduction in the background noise, and can remove the influence ofthe residual noise field within the magnetically-shielded room. The reduction performance,however, depends on the manufacturing precision of the two coils .

Aside from the gradiometer, several other methods of removing the external noise havebeen investigated (Vrba and Robinson , 2001; Adachi et ai., 2001). Many of these methodsuse extra sensors that measure only the noise fields. (Such sensors are usually located apartfrom the sensor array which measures the neuromagnetic fields.) Quasi-real-time electronicsthen perform the on-line subtraction between the outputs from the extra sensors and fromthe regular sensors. A method that does not require additional sensor channels has alsobeen developed. This method applies a technique called signal-space projection, and canbe implemented completely as a post-processing procedure (Parkkonen et ai., 1999). Theseexternal noise cancellation methods make it possible to use a magnetometer as a sensorcoil instead of the gradiometer. They also permit the measurement of neuromagnetic fieldsoutside a magnetically-shielded room.

The most remarkable advance in neuromagnetometer hardware over the last ten yearshas been the rapid increase in the number of sensors. Since neuromagnetometers witha 37-channel sensor array became commercially available in the late 80s (Lewine et al.,1995), the number of sensors in commercially available neuromagnetometers has constantlyincreased . The latest neuromagnetometers are equipped with 200-300 sensor channels, withwhole-head coverage of the sensor array.

7.3 FORWARD MODELING

7.3.1 DEFINITIONS

Let us define the magnetic field measured by the rnth detector coil at time t as bm(t),and a column vector bet) = [b,(t) , b2(t) , . .. , bM(t)]T as a set of measured data where Mis the total number of detector coils and the superscript T indicates the matrix transpose . Aspatial location is represented by a three-dimensional vector r : r = (x, y , z), The sourcecurrent density at r and time t is defined as a three-dimensional column vector s(r, t) .

The magnitude of the source current is denoted as s(r, t) (= js(r, t)[), and the orientationof the source is defined as a three-dimensional column vector "'l(r, t) = s(r, t)/s(r, t) =[1JxCr, t), 1Jy(r, t), 1Jz(r, t)] T, whose ~ component (where ~ equals x, y, or z in this chapter),is equal to the cosine of the angle between the direction of the source moment and the ~

direction.Let us define l!n(r) as the rnth sensor output induced by the unit-magnitude source

located at r and directed in the ~ direction. The column vector I I;(r) is defined asI I;(r) = [fr(r) , l~(r) , . . . , It(r)] T . Then , we define a matrix which represents the sensitivity of the whole sensor array at r as L(r) = [IX(r ), per ),F(r)]. The rnth row of L(r),Im(r) = [f~(r), l~(r), l~(r )], represents the sensitivity at r of the rnth sensor. Then, usingthe superposition law, the relationship between b(t) and s (r , t ) is expressed as

bet) = f L(r)s(r , t)dr +net)· (7.1)


Here, n(t) is the noise vector at t. The sensor sensitivity pattern, represented by the matrixL(r), is customarily called the sensor lead field (Hamalainen et al., 1993; Sarvas, 1987),and this matrix is called the lead-field matrix. We define, for later use, the lead-field vectorin the source-moment direction as l(r), which is obtained by using l(r) = L(r)'Y/(r).

7.3.2 ESTIMATION OF THE SENSOR LEAD FlEW

The problem of the source reconstruction is the problem of obtaining the best estimateof s(r, t) from the array measurement b(t). It is thus apparent that, to solve this reconstruction problem, we need to have a reasonable estimate of the lead-field matrix. In thissubsection, we describe how we can obtain the lead field using the spherically symmetric homogeneous conductor model, which is most commonly used in estimating the MEG sensorlead field. Also, we briefly mention a realistically-shaped volume-conductor model, whichcan generally provide more accurate lead field estimates, particularly for non-superficialbrain regions. Estimation of the lead-field is called the forward problem, because this isequivalent to estimating the magnetic field from a point source located at a known location;this problem stands in contrast to the inverse problem in which the source configuration isestimated from a known magnetic field distribution.

Let us define the electric potential as V, the magnetic field as B, and the electriccurrent density as j. The source current s(r) defined in Section 7.3.1 is called the primarycurrent, (alternatively called the impressed current), which is directly generated from theneural activities. There is another type of electric current called the return current or volumecurrent. It results from the electric field in the conducting medium, and it is not directlycaused by the neural activities. Defining the conductivity as p, the return current is expressedas - pVV where - VV is equal to the electric field. Thus, the total electric current j isexpressed as

j(r) = s(r) - pVV. (7.2)

The relationship between the total current j and the resultant magnetic field B is givenby the Biot-Savart law,

/.Lo f . r - r' IB(r) = - itr': x ,dr,4n Ir - r'l·

(7.3)

where /.Lo is the magnetic permeability of the free space. In order to derive the analyticalexpression for the relationship between the primary source current and the magnetic field,we first consider the case in which a whole space is filled with a conductor with constantconductivity p. In this case, it is easy to show that the following relationship holds:

/.Lo fir - r' IBo(r)=- s(r)x ,dr.4n Ir - r'l'

(7.4)

Here the magnetic field is denoted as Bofor later convenience. Note that Eq. (7.4) is similarto Eq. (7.3). The only difference is that the total current density j(r) is replaced by theprimary current density s(r).


We then proceed to deriving a formula for the magnetic field outside a sphericallysymmetric homogeneous conductor, B. To do so, we make use of the fact that the radialcomponent of the magnetic field B is equal to the radial component of Bo in Eq. (7.4)(Cuffin and Cohen, 1977; Sarvas, 1987), i.e.,

B . I, = Bo . I" (7.5)

where I, is the unit vector in the radial direction defined as I, = r Ilrl. (Note that we setthe coordinate origin at the sphere origin.) The relationship V x B = 0 holds outside thevolume conductor, because there is no electric current. Thus, B can be expressed in termsof the magnetic scalar potential U(r),

B = -J),oVU(r).

This potential function is derived from

1 100

1 100

U(r) = - B(r + TI,)· l,dT = - Bo(r +TI,)· f .d»,~ 0 ~ 0

(7.6)

(7.7)

where we use the relationship in Eq. (7.5). By substituting Eq. (7.4) intoEq. (7.7), we finallyobtain

1 f s(r') x r' . r ,U(r) = -- dr

4;rr A '

where

A = Ir -r'I(lr -r/lr + Irl2 -r' -r).

The formula for B is then obtained by substituting Eq. (7.8) into Eq. (7.6), i.e.,

B(r) = J),o f ~[AS(r/) x r' - (s(r /) x r' . r)VA]dr',4;rr A

where

(7.8)

(7.9)

Ir-r/12 (r-r')·r [ (r-rl).r]

VA = [ + +21r -r/l +2Irl]r - Ir -r/l +21rl + r',Irl Ir - r'l Ir - r/l

To obtain the component of the lead-field matrix, l~(ro), we first calculate B(rm )

(where r m is the mth sensor location) by using Eq. (7.9) with s(r') = ft;o(r ' - ro), whereI ~ is the unit vector in the ~ direction. When the sensor coil is a magnetometer coil, (which

measures only the magnetic field component normal to the sensor coil), l~(ro) is calculatedfrom l~(ro) = B(rm) . I,::il where I'::il is a unit vector expressing the normal direction ofthe mth sensor coil. When the sensor coil is a first-order axial gradiometer with a baselineof D, l~(ro) is calculated from l~(ro) = B(rm) . I,::il - B(rm + D I,::il) . rr. This l~(ro)


represents the sensitivity of the mth sensor to the primary current density located at r o anddirected in the ~ direction.

One of the important properties of the lead field obtained using the sphericallysymmetric homogeneous conductor model is that if s(r o) and r o are parallel, i.e., if theprimary current source is oriented in the radial direction, no magnetic fields are generatedoutside the spherical conductor from such a radial source. Also , we can see that when r oapproaches the center of the sphere, l~ (r 0) becomes zero, and no magnetic field is generatedoutside the conductor from a source at the origin.

The spherically-symmetric homogeneous conductor is generally satisfactory in explaining the measured magnetic field when only superficial sources exist , i.e., when allsources are located relatively close to the sensor array. This is because the curvature of theupper half of the brain is well approximated by a sphere. However, for sources located inlower regions of the brain, the model becomes inaccurate because the curvature of the lowerbrain regions significantly differs from a sphere. Such errors caused by misfits of the modelmay be reduced by using a spheroidally-symmetric conductor model (Cuffin and Cohen ,1977) or an eccentric sphere conductor model (Cuffin, 1991).

More fundamental improvements can be obtained by using realistically-shapedvolume-conductor models. Such conductor models can be constructed by first extracting thebrain boundary surface from the subject's 3D MRI. We denote this surface ~. We then usethe following Geselowitz formula (Geselowitz, 1970; Sarvas , 1987) to calculate magneticfields outside the volume conductor:

J.lo l ' r - r'B(r ) = Bo(r ) - -p V Cr )JzJr') x '3 dS,4rr :E Ir - r I

(7.10)

where the integral on the right-hand side indicates the surface integral over ~; r ' representsa point on ~, and!:E(r' ) is a unit vector perpendicular to ~ at r ' . Here , we assume that theconductivity within the brain is uniform, and denote it p. The second term on the right-handside of Eq. (7.10) represents the influence from the volume current, and to calculate thisterm, we need to know VCr) on ~ , which is obtained by solving (Sarvas, 1987)

where

p p 1 ' , r -r'-V(r)=Vo(r )-- V(r)!:E(r) · 3dS,2 4rr :E Ir - r 'l -

1 / ' r -r' ,Vo(r ) = - s(r ). :J dr ,4rr Ir - r 'I '

(7.11)

(7.12)

and rand r' are points on the boundary surface.Because the brain boundaries are irregular, the magnetic field B(r ) can only be ob

tained numerically using the boundary element method (BEM ). In this calculation, wefirst estimate the electric potential V (r) on the brain boundary surface ~ by numericallysolving Eq. (7.11). We then calculate magnetic fields outside the brain using Eq. (7.10).The details of these numerical calculations are out of the scope of this chapter, and theyare found in (Barnard et al., 1967; Hamalainen and Sarvas, 1989). The numerical method


mentioned so far assumes uniform conductivity within the brain boundary, and is calledsingle-compartment BEM. It is usually used in estimating MEG sensor lead fields (Fuchset al., 1998). It can be extended to the multiple-compartment BEM and such models areusually used for estimating the EEG sensor lead fields. The BEM-based realistically-shapedvolume conductor models generally provide significant improvements in the accuracy ofthe forward calculation particularly for deep sources (Fuchs et al., 1998; Cuffin, 1996), although they are computationally expensive. Improvements in the computational efficiencyof the BEM have been reported (Bradley et al., 2001; van't Ent et al., 2001).

7.3.3 LOW-RANK SIGNALS AND THEIR PROPERTIES

Let us consider specific cases where the primary current consists of localized discretesources. The number of sources is denoted Q,and we assume that Q is less than the numberof sensors M. The locations of these sources are denotedr , .r-, ... ,r Q. The source-momentdistribution is then expressed as

Q

s(r, t) = LSD(rq, t)8(r -rq),q=]

(7.13)

where sD(rq) = f s(r)dr and this integral extends the small region around r q where theqth source is confined. This type of localized source is called the equivalent current dipolewith moment sD. Here, sD is called the moment because it has a dimension of current xdistance. The basis underlying the equivalent current dipole is physiologically plausible(Okada et al., 1987), and sources of neuromagnetic fields are often modeled with the currentdipoles.

Since there is little advantage to explicitly differentiating the current density s(rqv t)and the current moment sD(rq, t), for simplicity we keep the same notation s(rq, t) toexpress the current moment. Then, the Q-dimensional source magnitude vector is definedas vet) = [s(r], t), s(r2, t), ... , s(rQ' t)]T. We define a 3Q x Q matrix that expresses theorientations of all Q sources as Wet) such that

[TJ(r~ , t)

lJi"(t)=

o

o

o

The composite lead-field matrix for the entire set of Q sources is defined as

(7.14)

Then, substituting Eq. (7.13) into Eq. (7.1), we have the discrete form of the basic relationship between bet) and vet) such that

bet) = [LclJi"(t)]v(t) + net). (7.15)


Let us define the measurement covariance matrix as Rb ; i.e., Rb = (b(t)b T (t»), where(.) indicates the ensemble average. (This ensemble average is usually replaced by the timeaverage over a certain time window.) Let us also define the covariance matrix of the sourcemoment activity as Rs ; i.e., R, = (!li(t)v(t)v T (t)I]i"(t)T). Then, using Eq. (7.15), we get therelationship between the measurement covariance matrix and the source-activity covariancematrix such that

(7.16)

where the noise in the measured data is assumed to be white Gaussian noise with a varianceof (J2, and I is the M x M identity matrix.

Let us define the kth eigenvalue and eigenvector of Rb as Ak and ei, respectively. Letus assume, for simplicity, that all sources have fixed orientations. Then, unless some sourceactivities are perfectly correlated with each other, the rank of R, is equal to the numberof sources Q. Therefore, according to Eq. (7.16), Rb has Q eigenvalues greater than (J2

and M - Q eigenvalues that are equal to (J2. The signal whose covariance matrix has suchproperties is referred to as the low-rank signal (Paulraj et al., 1993; Sekihara et al., 2000).Let us define the matrices Es and EN as Es = [er, ... , eQ] and EN = [eQ+]' ... ' eM].The column span of E s is the maximum-likelihood estimate of the signal subspace of Rb ,

and the span of EN is that of the noise subspace (Scharf, 1991). In the low-rank signals, themeasurement covariance matrix Rb can be decomposed into its signal and noise subspacecomponents; i.e.,

Here, we define the matrices As and AN as

As = diag[AI, ... , AQ] and AN = diag[AQ+l, ... , AM],

(7.17)

(7.18)

where diagl- ..] indicates a diagonal matrix whose diagonal elements are equal to the entriesin the brackets.

The most important property of the low-rank signal is that, at source locations, thelead-field matrix is orthogonal to the noise subspace of Rb. This can be understood by firstconsidering that

(7.19)

Since L; is a full column-rank matrix, and we assume that R, is a full rank matrix, theabove equation gives

(7.20)

This implies that the lead-field matrices at the true source locations are orthogonal to anynoise level eigenvector; that is, they are orthogonal to the noise subspace (Schmidt, 1981),

Neuromagnetic Source Reconstruction and Inverse Modeling

i.e.,

221

(7.21)

Since the equation above holds, the relationship 1T (r q)EN = 0 also holds. These orthogonality relationships are the basis of the eigenspace-projection adaptive beamformer described in Section 7.5.2, as well as the basis of the well-known MUSIC algorithm (Schmidt,1986; Schmidt, 1981; Mosher et al. , 1992).

7.4 SPATIAL FILTER FORMULATION AND NON-ADAPTIVE SPATIALFILTER TECHNIQUES

7.4.1 SPATIAL FILTER FORMULATION

Spatial filter techniques estimate the source current density (or the source moment) byapplying a linear filter to the measured data. Because the source is a three-dimensional vectorquantity, there are two ways to implement the spatial filter approach in the neuromagneticsource reconstruction. One is the scalar spatial filter and the other is the vector spatial filter.

In the scalar approach , we use a single set weights that characterizes the propertiesof the spatial filter, and define the set of the filter weights as a column vector w(r , 7]) =[WI (r, 7]) , w2(r , 7]) , . . . , wM(r, 7])]T . Here the weight vector depends both on the locationr and the source orientation 7]. This weight vector w(r , 7]) should only pass the signal froma source with a particular location r and an orientation 7]. The weight vector rejects not onlythe signals from other locations but also the signal from the location r if the orientation ofthe source at r differs from 7]. Then, the magnitude of the source moment is estimated usinga simple linear operation ,

M

S{r, t ) = w T(r, 7])b(t ) = L wm(r , 7])bm(t ),m=1

(7.22)

where the estimate of the source magnitude is denoted s(r , t ). When using the scalar-typebeamformer in Eq. (7.22), we need to first determine the beamformer orientation 7] toestimate the source activity at a specific location r . However, this 7] is generally unknown,although several techniques have been developed to obtain the optimum estimate of thesource orientation (Sekihara and Scholz , 1996; Mosher et al., 1992).

The vector spatial filter uses a weight matrix W(r ) that contains three weight vectorswAr), wy(r ), and wz(r ), which respectively estimate the x, y , and z components of thesource moment. That is, the source current vector is estimated from

S(r , t ) = W T(r )b(t ) = [w Ar), wy(r), wz(r )fb(t), (7.23)

where s (r , t) is the estimate of the source current vector. The vector spatial filter estimatethe source orientation as well as the source magnitude.

The application of a spatial filter weight artificially focuses the sensitivity of a sensorarray on a specific location r , and this location r is a controllable parameter. Therefore, in


(7.24)

a post-processing procedure, we can scan the focused region over a region of interest toreconstruct the entire source distribution.

7.4.2 RESOLUTION KERNEL

The major problem with spatial filter techniques is how to derive a weight vector withdesirable properties.To develop such weight vectors, we need a criterion which characterizeshow appropriately the weight has been designed. The resolution kernel can play this role.Combining Eqs. (7.1) and (7.23), we obtain the relationship,

s (r, t ) = ! WT(r )L(r ' )s(r ' , t )dr' = ! lR(r, r ' )s(r ' , t )dr' ,

where,

lR(r,r ') = WT(r)L(r ') . (7.25)

This lR(r, r ') is called the resolution kernel, which expres ses the relationship between theoriginal and estimated source distributions (Menendez et al, 1997; Menendez et al., 1996;Lutkenhoner and Menendez, 1997). Therefore, the resoluti on kernel can provide a measureof the appropriateness of the filter weight. In other words, the weight must be chosen so thatthe resolution kernel has a desirable shape, which generally satisfies the three propert ies:(i) peak at the source location, (ii) a small main-lobe width, and (iii) a low side-lobeampl itude .

The most important property among them is that the kernel should be peaked at thesource locat ion. Only in this case, the recon structed source distribution can be interpretedas a smoothed version of the true source distribution . However, if this condition is not met,the reconstructed source distribution should contain systematic bias and may be totallydifferent from the true source distribution. The kernel should also have a small main-lobewidth, so that the reconstruction results have high spatial resolution. When the kernel hasa lower side-lobe amplitude, the results have less systematic noise and artifacts.

7.4.3 NON-ADAPTIVE SPATIAL FILTER

Minimum norm spatialfilter

There are generally two types of spatial filter techniques. One is a non-adaptive methodin which the filter weight is independent of the measurements. The other is an adaptivemethod in which the filter weight depends on the measurements. The primary interest ofthis chapter is the application of the adaptive spatial filter techniqu e to the neuromagneticsource reconstruct ion. However, before proceeding to describe the adaptive spatial filter,we briefly describe the non-adaptive spatial filter in order to clarify the difference betweenthese two types of spatial filter methods.

The best-known non-adaptive spatial filter is the minimum-norm estimate (Hamalainenand Ilmoniemi , 1984; Hamalainen and Ilmoniemi , 1994; Wang et al. , 1992; Graumann,1991). The filter weight can be obtained by the following minimization:

min! IIlR(r, r' ) - 8(r - r ') 1I 2dr' , (7.26)


where b(r) is the three-dimensional delta function. By making the resolution kernel closeto the delta-function, the weight is obtained and it is expressed as

The estimated current density is then expressed as

(7.28)

The matrix G is often referred to as the gram matrix. The p and q element of G is given bycalculating the overlap between the lead fields of the pth and qth sensors,

o.. = f Ip(r)l~(r)dr. (7.29)

Unfortunately, in biomagnetic instruments, the overlaps between the adjacent sensorlead fields are very large, as depicted in Fig. 7.1(a). As a result, Gp •q has a more-or-lesssimilar value for various pairs of p and q. Consequently, the matrix G is generally verypoorly conditioned. This fact greatly affects the performance of this non-adaptive spatialfilter method because it requires calculation of the inverse of G, a process which is veryerroneous if G is nearly singular.

This gram matrix G is usually numerically calculated by introducing pixel gridsthroughout the source space. Let us denote the locations of the pixel grid pointsr 1, r 2, ... , r N, and the composite lead-field matrix for the entire pixel grids L N:

Then, the matrix G is calculated from

(7.30)

(a) (b)

FIGURE 7.1. Schematic views of the sensor lead field. (a) Biomagnetic instrument and (b) X-ray computedtomography.


However, to avoid the numerical instability when inverting it, the following regularizedversion is usually used:

(7.31)

where y is the regularization parameter. The final solution is expressed as

(7.32)

The regularization, however, inevitably introduces considerable amounts of smearing intothe reconstruction results. Besides, the solution obtained using the minimum-norm spatialfilter suffers a geometric bias in that the current estimates are forced to be closer to thesensor array than their actual locations.

It should, however, emphasized that the poor performance of the minimum-norm spatialfilter is not because the method itself has a serious defect but because a mismatch existsbetween the method and the biomagnetic instruments. This can be understood by consideringa situation for other imaging modalities such as the X-ray computed tomography (CT). Asis shown in Fig. 7.1(b), the overlaps between the lead fields of different sensors are verysmall for the X-ray CT. As a result, the matrix G is close to the identity matrix and thenon-adaptive spatial filter method works quite well. Indeed, the minimum-norm spatial filtertechnique is considered identical to the filtered-backprojection algorithm (Herman, 1980)used for image reconstruction from projections in commercial X-ray CT systems. In thenext subsection, we briefly describe investigations into ways of improving the performanceof the minimum-norm-based spatial filter.

Least-squares-based interpretation ofthe minimum-normmethods

The minimum-norm spatial filter is commonly derived by minimizing the leastsquares-based cost function. Actually, this least-squares-based interpretation is much morepopular than the spatial-filter-based interpretation described in Section 7.4.3. Namely, thesolution in Eq. (7.32) minimizes the cost function,

(7.33)

where SN is a source vector whose elements consist of the current estimate at the pixel points,i.e.'sN = [S(rj, t), ... ,S(rN, t)]T. In Eq. (7.33), the first term on the right-hand side is theleast-squares error term and the second term is the total sum of the current norm. Therefore,the optimum solution minimizes the total current norm as well as the least-squares error.This is why the method is often referred to as the minimum-norm estimate.

The trick to improving the performance of the minimum-norm method is to use a moregeneral form of the cost function, expressed as

(7.34)

where iJ! represents some kind of weighting applied to the solution vector SN, and Yrepresents the weighting applied to the residual of the least-squares term. The solution


derived by minimizing this cost function is expressed as

225

(7.35)

In this solution, the gram matrix becomes G= LNP-IL~ + yy-I. The inclusion of thematrices P and Y gives a greater degree of freedom in regularizing G, and by choosingappropriate forms for these matrices, the numerical instability can be improved withoutintroducing unwanted side effects such as image blur.

In general, the matrix P is derived from a desired property of the solution. One widelyused example is the minimum weighted-norm constraint in which we use P = p L whosenon-diagonal elements are zero, and diagonal terms are given by

pL _ 13k+l,3k+l - IlfY(rd[[2'

1and pL = (7 36)

3k+2,3k+2 IJlZ(rk) 11 2 •

This weight pL can reduce the geometric bias of the minimum-norm solution to some extent, compensating for the variation in the lead-field norm. Low resolution electromagnetictomography (LORETA) (Pascual-Marqui and Michel, 1994; Wagner et al., 1996) is anotherpopular application of this particular type of P. It seeks the maximally smooth solution byusing P = pLpR, where pR is the Laplacian smoothing matrix.

Bayesian-type estimation methods determine P based on prior knowledge of the neuralcurrent distribution (Schmidt et al., 1999; Baillet and Gamero, 1997). Determination of Pby fMRI has been proposed (Liu et al., 1998; Dale et al., 2000). The matrix Y is generallydetermined from the noise properties. When the measurements contain non-white noise andwe know the noise covariance matrix, Y is usually set to the inverse of the noise covariancematrix. The determination of the optimum forms for the matrices p and r has been an activeresearch topic, and many investigations have been performed in this direction. However,we will not digress into the details of these investigations. Instead, in the following sectionwe describe different approaches known as the adaptive spatial filter, which does not usethe gram matrix of the lead field.

7.4.4 NOISE GAIN AND WEIGHT NORMALIZATION

The spatial filter weights determine the gain for the noise in the reconstructed results.In the scalar spatial filter techniques, the output noise power due to the noise input, Pn, isgiven by

(7.37)

where R; is the noise covariance matrix. When the noise is uncorrelated white Gaussiannoise, the output noise power is equal to

(7.38)

where 0'2 = (n(t)nT (t)) is the power of the input noise. Therefore, the norm of the filterweight vector IIw(r)[!2 is called the noise power gain or the white noise gain. In vector


spatial filter techniques, the output noise power is expressed as

Pn = tr{W T(r)(n(t)nT(t)) W T(r)} = tr{W T(r)Rn W T(r)}.

When the input noise is uncorrelated white Gaussian noise, this reduces to

(7.39)

(7.40)

Here, the square sum of the norm of the filter weights tr{WT(r)W(r)} = IlwxCr)/I2 +Ilwy(r)f + /Iwz(r)11

2is the noise power gain. A minimum-norm spatial filter with weightnormalization has also been proposed (Dale et al., 2000) The output of this spatial filter isexpressed as

~ WT(r)b(t)s(r, t) = T .

tr{W (r)W(r)}(7.41)

Because the weight norm is the noise gain, the output of this spatial filter is interpreted asbeing equal to the SNR of the minimum-norm filter outputs.

7.5 ADAPTIVE SPATIAL FILTER TECHNIQUES

7.5.1 SCALAR MINIMUM-VARIANCE-BASED BEAMFORMER TECHNIQUES

The adaptive spatial filter techniques use a weight vector that depends on measurement. The best-known adaptive spatial filter technique is probably the minimum-variancebeamformer. The term "beamformer" has been customarily used in the signal-processingcommunity with the same meaning as "spatial filter". In this method, the spatial filter weightsare obtained by solving the constrained optimization problem,

and consequently we get

subject to (7.42)

(7.43)

where I(r) is defined as I(r) = L(r )17.The idea behind the above optimization is that the filter weight is designed to minimize

the total output signal power while maintaining the signal from the pointing location r.Therefore, ideally, this weight only passes the signal from a source at the location r withthe orientation 17, and suppresses the signals from sources at other locations or orientations.One difficulty arises when applying it to actual MEGfEEG source localization problems.That is, when we use the spherically symmetric homogeneous conductor model to calculateI(r), the beamformer output has erroneously large values near the center of the sphere. Thisis because III(r) II becomes very small when r approaches the center of the sphere.


A variant of the minimum-variance beamformer, proposed by Borgiotti and Kaplan(Borgiotti and Kaplan, 1979), uses the optimization,

subject to (7.44)

The resultant weight vector is expressed as

(7.45)

Because w T (r )w(r) represents the noise power gain, the output of the above bearnformerdirectly corresponds to the power of the source activity normalized by the power of the outputnoise. This Borgiotti-Kaplan beamformer is known to provide a spatial resolution higherthan that of the minimum-variance beamformer (Borgiotti and Kaplan, 1979). Moreover,it can easily be seen that the output of the beamformer in Eq. (7.45) does not depend onIII(r) II. Thus, III(r) 11-related artifacts are avoided.

Another more serious problem with the adaptive beamformer techniques described sofar is that they are very sensitive to errors in the forward modeling or errors in estimatingthe data covariance matrix. Since such errors are nearly inevitable in neuromagnetic measurements, these techniques generally provide noisy spatio-temporal reconstruction results,as demonstrated in Section 7.7.

One technique has been developed to overcome such poor performance (Cox et al.,1987; Carlson, 1988). The technique, referred to as diagonal loading, uses the regularizedinverse of the measurement covariance matrix, instead of its direct matrix inverse. Althoughthis technique has been applied to the MEG source localization problem (Robinson and Vrba,1999; Gross and Ioannides, 1999; Gross et al., 2001), it is known that the regularizationleads to a trade-off between the spatial resolution and the SNR of the beamformer output.

7.5.2 EXTENSION TO EIGENSPACE-PROJECTION BEAMFORMER

We here describe the eigenspace-projection beamformer (van Veen, 1988; Feldmanand Griffiths, 1991), which is tolerant of the above-mentioned errors and provides improvedoutput SNR without sacrificing the spatial resolution in practical low-rank signal situations.Using Eqs. (7.43) and (7.17), and defining a = l/[IT(r)Rbl/(r)], we rewrite the weightvector for the minimum-variance beamformer as

(7.46)

where

In Eq. (7.46), the second term on the right-hand side, arNI(r), should ideally be equalto zero because the lead-field matrix I (r) is orthogonal to EN at the source locations asindicated by Eq. (7.21). Various factors, however, prevent this term from being zero, and a


non-zeroaTNL(r) seriouslydegradesSNR as explainedin the nextsection.Therefore,theeigenspace-based beamformeruses only the first term of Eq. (7.46) to calculate its weightvector w(r); i.e.,

_ Ts/(r )w(r ) = aTs/(r ) = T 1 •

I (r )RJ; I (r )(7.47)

Note that w(r) is equal to the projection of w(r) onto the signal subspace of Rb • Namely,the following relationship holds (Feldman and Griffiths, 1991;Yu and Yeh, 1995):

w(r) = EsEJw(r). (7.48)

Therefore, the extension to an eigenspace-projection beamformeris attained by projectingthe weight vectors onto the signal subspaceof the measurement covariance matrix.

7.5.3 COMPARISON BETWEEN MINIMUM-VARIANCE AND EIGENSPACEBEAMFORMER TECHNIQUES

Althoughthe minimum-variancebeamformerideallyhas exactlythesameSNR as thatof theeigenspace-based beamformer, theSNRof theeigenspacebeamformeris significantlyhigher in practicalapplications. The reason for this high SNRcan be understoodas follows.Let us assume that a single source with a moment magnitudeequal to set) exists at r. Weassume that the estimated lead field, l (r ), is slightly different from the true lead fieldI (r) .The estimateof set),s et), is derivedbys(t ) = w T(r )b(t ) = w T(r)/(r)s(t ) and the averagepowerof the estimatedsource momentFir ), p." is expressed as

(7.49)

where P, is the averagepower of set) definedby P, = (s(t) 2). For the minimum-variancebeamformer, this Ps is expressedas

~ 2 [i T -1 T]2 2 ~ T T 2r, = a Ps I (r )Rb I (r) = a Ps[1 (r )rsl (r)] , (7.50)

where we use the orthogonality relationship IT(r)EN = O. For the eigenspace-projectionbeamformer, it is also expressed as

~ 2 ~ T T 2r. = a Ps[1 (r )TsI (r) ] . (7.51)

The average noise power Pn is obtained using Eqs. (7.38) and (7.46) and, for theminimum-variance beamformer, it is expressedas

(7.52)


For the eigenspace-projection beamformer, Pn is expressedas

229

(7.53)

Thus, the output SNR of the minimum-variance beamformer, SNR(MV), is expressed as(Chang and Yeh, 1992;Chang and Yeh, 1993)

(7.54)

The SNR for the eigenspace-based beamformer, SNR(ES). is thus

(7.55)

The only difference between Eqs. (7.54) and (7.55) is the presence of the second termI T(r)r~I(r ) in the denominatorof the right-handside ofEq. (7.54). It is readily apparentthatSNR(MV)andSN~ES) are equalifwecanusean accuratenoise subspaceestimateandanaccuratelead-field vector, because the termi

T(r )r~I(r) isexactlyequal tozero in thiscase.

It is, however, generally difficultto attain the relationship,I T(r)r~I(r) = O. One obviousreasonfor thisdifficultyis that whencalculatingr~ inpractice,pstead of usi'¥ Rb , the~ampIecovariancematrix Rb must be used; Rb is calculated from Rb = 1/ k L k=l b(tk )b T (tk)where K is the number of time points.

~T ~

Anotherfactorthat is specifictoMEGandcausesl (r )r~1(r ) tohavea non-zerovalueis that it is almost impossible to use a perfectly accurate lead-field vector. This is becausethe conductivity distribution in the brain is usually approximated by using some kind ofconductor model-such as the spherically homogeneous conductor model-to calculatethe lead field matrix. Although this error may be reduced to a certain extent by using arealistic head model, the error cannot be perfectly avoided. Let us define the overall errorin estimatingI(r) as e. Assumingthat 11/(r) [1 2»11 e /12, we can rewriteEq. (7.54) as

(7.56)

Note that, in the denominatorof the right-hand side of this equation, the norm of the matrixe T r~e has an order of magnitudeproportionalto /I e 11

2 /A~, whereANrepresentsone ofthe noise-level eigenvalues of Ri : The eigenvalue AN is usually significantly smaller thanthe signal-level eigenvalues. Therefore, Equation (7.56) indicates that even when the error" e " is verysmall, the terme Tr~g may not be negligiblysmallcompared to the first termin the denominator. Thus, in practice theeigenspace-projection beamformerattainsan SNRsignificantly higher than that of the minimum-variancebeamformer.

230

7.5.4 VECTOR-TYPE ADAPTIVE SPATIAL FILTER

K. Sekihara and S. S. Nagarajan

The scalar beamfonner techniques described in the preceding subsections requiredetermination of the beamformer orientation "l to calculate l(r). We, here, describe theextension to vector-type adaptive spatial filter techniques, which does not require the predetermination of "l.

Problem ofvirtual source correlation

A naive way of extending to the vector beamformer is to simply use the scalar beamformer weight vector obtained with "l = f ~ to estimate the source in the ~ direction (where

~ = x, y,orz).Letustrytoestimate~(t)byusing~(t) = wT(r, f~)b(t),where uif (r , f~)is obtained using Eq. (7.43). The use of such weight vectors, however, generally gives erroneous results, and the cause of this estimation failure can be explained as follows. Let usassume that a single source with its orientation equal to "l = [1]x, 1]y, 1];f exists at r. Itsactivation time course is assumed to be s(t). Then, we can express the measured magneticfield as bet) = 1]xs(t)r(r) + 1]ys(t)fY(r) + 1]zs(t)[Z(r). This can be interpreted as showingthat the magnetic field is generated from three perfectly correlated sources located at thesame location r , with moments equal to 1]xs(t)fx' 1]ys(t)fyo and 1]zs(t)fz.

Let us, for example, consider the case of estimating the x component of the sourcemoment. The estimated moment, s,,(t), is expressed as

s,,(t) = w]' (r )b(t) = [1]x w T(r, fx)r (r) + 1]vw T(r, f,)fY (r) + 1];w T(r, fz)F(r )]s(t).

(7.57)

Since the weight w T(r, fx) is obtained by imposing the constraint w T(r, fx)r(r) = 1,we have

(7.58)

In this equation, there is no guarantee that the relationships w T (r , fx)fY (r) = 0 andw T(r , f x)F(r) = 0 hold. Instead, w Ttr , f x)fY(r) and w T(r , f x)[Z(r) generally have fairlylarge negative values, resulting in considerable errors.

A vector-extended minimum-variance beamformer

The above analysis also suggests how we can avoid such errors. Equation (7.57)indicates that the weight should be derived with the multiple constraints,

w]'r(r) = 1, (7.59)

That is, we impose the null constraints on the directions orthogonal to the one to be estimated. We here omit the notation (r) for the weight expression unless this omission causesambiguity. Similarly, to derive Wy and lL!:, the following constraints should be imposed,

w;r(r) = 0,

wIrer) = 0,

TW y fY(r) = 1,

wzTfY(r) = 0,

and

and

w T[Z(r) = 0y ,

w;[Z(r) = 1.

(7.60)

(7.61)


The minimum-variance beamformer with such multiple linear constraints, referred to asthe linearly constrained minimum-variance beamformer (Frost, 1972), is known to have thefollowing solution:

(7.62)

It is clear from the discussion above that, when estimating one of the three orthogonalcomponents of the source moment, we need to suppress the other two components. Bydoing this, we can avoid the errors caused by the perfectly correlated virtual sources, so thebeamformer can detect the source moment projected in three orthogonal directions. Notethat the set of weight vectors in Eq. (7.62) has been previously reported (van Dronge1enet al., 1996; van Veen et al., 1997; Spencer et al., 1992). In these reports, however, thenecessity for imposing the null constraints was not fully explained.

Vector-extended Borgiotti-Kaplan beamformer

The extension of the Borgiotti-Kaplan beamformer in Eq. (7.45) is performed in thesimilar manner. The weight vectors are obtained by using the following constrained minimizations,

. TR subject to T w;fY(r) = 0, and w;"fZ(r) = 0, (7.63)mmwx bWx Wx Wx = 1,Wx

. TR subject to Tx T and w TIZ(r) = 0 (7.64)mmw y bWy wv 1 (r) = 0, w y w y = 1,w,

y ,

T subject to w/"r(r) = 0, w!fY(r) = 0, and T (7.65)min Wz Rbw z Wz Wz = 1.W;:: , ,.

We first derive the expression for w .r- Let us introduce a scalar constant ~ such thatw;r(r) = ~ where ~ can be determined from the relationship w;Wx = 1. Then, the constrained optimization problem in Eq. (7.63) becomes

(7.66)

The solution of this optimization problem is known to have the form

Then, we have

T 2 TWx Wx = ~ f x nf x'

where

n = [L T(r)R{;1 L(r)rlL T(r)R{;2L(r)[L T(r)R/;1 L(r)r

l.

(7.67)

(7.68)


Thus , we get ( = l/jI; nIxfrom the relationship w;W x= 1. Using exactly the same

derivation, the weights wyand W z can be derived, and a set of the weights is expressed as

RbI L(r )[L T (r)Rbl L(r )r

lI~

w~= .jI[nh

(7.69)

Extensionto eigenspace-projection vector beamformer

The extension to the eigenspace-projection vector beamformer is attained by using

(7.70)

The projection onto the signal subspace , however, cannot preserve the null constraintsimposed on the orthogonal components.This can be understood by considering, for example ,the case of wx' The null constraints in this case should be w.~fY(r ) = 0 and w~F(r ) = O.However, let us consider

w;JY(r) = (EsEJ wxfJY(r) = w;EsEJF(r),

w.~fZ(r ) = (EsE J wxfZZ(r) = w; EsE JfZ(r). (7.71)

Because P(r) and fZ(r) are not necessarily in the signal subspace, we generally haveEsEJP(r ) i= P(r) and EsEJZZ(r) i= fZ(r) , and therefore w~P(r) i= 0 and w~ZZ(r) i= O.It can, however, be shown that the eigenspace-projection beamformer in Eq. (7.70) canstill detect the three orthogonal components of the source moment even though the nullconstraints are not preserved (Sekihara et aI., 2001).

7.6 NUMERICAL EXPERIMENTS: RESOLUTION KERNELCOMPARISON BETWEEN ADAPTIVE AND NON-ADAPTIVE SPATIALFILTERS

7.6.1 RESOLUTION KERNEL FOR THE MINIMUM-NORM SPATIAL FILTER

We compare the resolution kernels for the minimum-norm and the minimum-variancespatial filter techniques. These two methods are typical and basic spatial filter techniquesin each category. In these numerical experiments, we use the coil configuration of the 148channel Magnes 2500™ neuromagnetometer (4D Neuroimaging, San Diego). The sensorcoils are arranged on a helmet-shaped surface whose sensor locations are shown in Fig. 7.2.The coordinate origin is chosen as the center of the sensor array. The z direction is definedas the direction perpendicular to the plane of the detector coil located at this center. The xdirection is defined as that from the posterior to the anterior, and the y direction is definedas that from the left to the right hemisphere. The values of the spatial coordinates (x, y , z)


o

-5

1- 10N

-15

-2020

......,. .• • •..,~ -',1-.,... . .'.'., . .. .

: ~. . . .. . . :.. ...- ..e. •• I • " . ..

• IIt '. . : . . . • • _ e.... .. . ..~... . :. ..... . . ...-.'.

. : ,... .. ..,- . . ~..••• • • •.........- ~:. . • ' e "••• '.

'. :

-20

y (em) -20 20 x (em)

FIGURE 7.2. ThesensorlocationsandthecoordinatesystemusedforplottingtheresolutionkernelsinSection7.6.The filledspots indicate the locationsof the 148sensors, and the hatched rectangleshows the plane of x = 0 onwhich the resolutionkernelswere plotted.

are expressed in centimeters. The coordinate system is also shown in Fig. 7.2. The originof the spherically symmetric homogeneous conductor was set to (0, 0, -11).

To plot the resolution kernel, we assume a vertical plane x = 0 located below the centerof the sensor array (Fig. 7.2). The power ofthe resolution kernel II lR 11 2 was calculated usingEqs. (7.25), (7.27), and (7.31) for the minimum-norm method. A point source was assumedto exist at (0,0, -6), i.e., r' in Eq. (7.25) was set to (0,0, -6). The kernel was plottedwithin a region defined as -5 :s y :s 5 and -9 :s z :s -Ion the vertical plane of x = O.The resulting resolution kernels are shown in Fig. 7.3. Here, the results in Fig.7.3(a) showthe kernel obtained from the original minimum-norm method. It is well known that theoriginal minimum-norm method suffers from a strong geometric bias toward the sensors.The results in Fig. 7.3(a) confirm this fact. The kernels of the minimum-norm method withthe lead-field normalization are shown in Fig.7 .3(b) and (c). Here, cpL in Eq. (7.36) was usedand the regularization parameter y was set at O.OOOlA l for (b) and O.OOlA l t for (c). Theseresults show that the lead-field normalization significantly improves the performance of theminimum-norm method. However, the resolution is still significantly low, particularly inthe depth direction. Moreover, the peak of the kernel is located a few centimeters shallowerthan the assumed location; the depth difference depends on the choice of the regularizationparameter. The results in Fig.7.3(d) show the kernel from the minimum-norm method withthe normalized weight (Eq. (7.41)). The main lobe is significantly sharper than those in thelead-field normalization cases of (b) and (c). However, the peak is located 2cm deeper thanits original position.

t This A1 is the largesteigenvalueof the gram matrixL J; LN.

-9'-------~~-~--~--'

- I

" ~

--1

~ -5

. ()

-7

-8

. <)

(b) y (e m )

- I

-2

-3

. -1

c -5

. (>

·7

-8

. <)

(d)

2oy (em)

(c) y (em)

-l -2(a)

-2

-3

--1

-6

-7

-8

~ -5

FIGURE 7.3. Results of plotting the resolution kernels for minimum-norm-based spatial filter techniques.(a) Results for the original minimum-norm method. (b) Results for the minimum-nom, method with the normalized lead field. The regularization parameter y was set to O.OOO])cj. (c) Results for the minimum-normmethod with the normalized lead field. The regularization parameter y was set to 0.00 lAI. (d) Results for theminimum-norm method with weight normalization. The parameter y was set to 0.000 IAI for the results in(a) and (d). Here, Al is the largest eigenvalue of the gram matrix L~LN.

7.6.2 RESOLUTION KERNEL FOR THE MINIMUM-VARIANCE ADAPTIVESPATIAL FILTER

Next, the resolution kernel for the minimum-variance vector beamformer techniquewas plotted. The kernel was calculated using Eqs. (7.25) and (7.62). Here, the covariancematrix was calculated from Rb = (a 21+ PJ(rl)[T(r l)) wherer ' is setto the sourcelocationequal to (0,0, -6). The calculated resolution kernels for the four SNR values of Jp"ja 2

are shown in Fig. 7.4. First of all, the kernel is peaked exactly at the target source location(0,0, -6). The kernel's sharpness depends on the SNR value. This is one characteristic ofthe adaptive spatial filter methods. We encounter the SNR between 8 and 2 in most actualmeasurements. In such an SNR range, the kernel sharpness obtained with the minimumvariance spatial filter is significantly higher than that with the non-adaptive minimum-normmethod. These results clearly demonstrate that the minimum-variance filter can providemore accurate reconstruction results with significantly higher spatial resolution than theminimum-norm spatial filter.


- I -1

-2 -2

-3 -3

-4 -4

~ -5 ~ -5'" -6 -6

-7 -7

-8 ·8

-9 -9-4 -2 0 2 4 -4 -2 0 2

(a) y(em) (b) y Icrn)

. J -1

-2 -2

-3 -3

-4 -4

.s -5 , Q -5

" '"-6 -6

-7 ·7

-8 -8

-9 -9-4 -2 0 4 -4 -2 0 2 4

(c) y(em) (d ) y (em)

FIGURE 7.4. Results ofplotting the resolution kernels for the minimum-variance adaptive spatial filter technique.The plots are for SNR values of (a) 8, (b) 4, (c) 2, and (d) I.

7.7 NUMERICAL EXPERIMENTS: EVALUATION OF ADAPTIVEBEAMFORMER PERFORMANCE

7.7.1 DATA GENERATION AND RECONSTRUCTION CONDITION

We next conducted a series of numerical experiments to test the performance of theadaptive beamformer technique. Here, we assume the 37-sensor array of Magnes" neuromagnetometer in which the sensor coils are arranged in a uniform, concentric array ona spherical surface with a radius of 12.2 em. The sensors are configured as first-order axial gradiometers with a baseline of 5 em. Three signal sources were assumed to exist ona plane defined as x = O. The source configuration is schematically shown in Fig. 7.5(a)and their time courses are shown Fig. 7.5(b). The magnetic field was generated at a l-msinterval from 0 to 400 ms. Gaussian noise was added to the generated magnetic field, andthe SNR, defined as the ratio of the Frobenius norm of the signal-magnetic-field data matrixto that of the noise matrix, was set to 16. The generated magnetic field is shown also inFig.7.5(b).

The covariance matrix Rb was calculated using the data obtained with this time windowbetween 0 and 400 ms. We express the source-moment vector using the two tangentialcomponents ((), ¢), and the radial component is assumed to be zero. The reconstruction was

236

-5

..--Eu

';:;' -10

-15

(a)

K. Sekiha ra and S. S. Nagarajan

37-channel sensor array

three sources

5

sphere for the forward modeling

FIGURE 7.5. (a) The coordinate system and the source configuration used in the numerical experiments inSection 7.7. The cross section atx = °is shown. The square shows the reconstruction region for the experimentalresults shown in Figs. 7.6-7.9. (b) Time courses of the three sources assumed in the numerical experiments. Timecourses from the first to the third sources are shown from the top to the third row. respectively. The three verticalbroken lines indicate the time instants 220, 268, and 300 ms, at which the source-moment magnitude is displayed.The bottom row showsthe generated magnetic field.

performed by using

sq,(r, t ) = wI (r)b(t) and so(r, t) = w[ (r)b(t) . (7.72)

The reconstruction region was defined as the area between -4 :::: y :::: 4 and -8 :::: z :::: - 3on the plane x = 0, and the reconstruction interval was 1 mm in the y and z directions.

Once sq,(r, t) and so(r , t) were obtained, cp, an angle representing the mean sourcedirection in the ¢ - eplane was calculated using

cp = arctan ((so(t?) ) if

(so(t )) > 0(Sq, (t )2) (sq,(t)) - ,

~ = -an:tan ((;,(1)') )

if(so(t)) < 0 (7.73)

(sq,(t)2) (sq, (t )) ,

where (.) indicates the average over the time window with which Rb was calculated. Then ,the time course expressed in the mean source direction , slI(r , t ), and that in its orthogonal

220 268 300

"0~

.~:;E0c

- I1

"0~

. ~:; 0E0c

-I1

"0~

.!:::; 0E0c

100 200 300 400late ncy (ms)

<:J"'C

2·cc:I)«:E 0<:J

'S«:

"E0

(b )

100 200latency (ms)

FIGURE 7.5. (cont .)

300 400

direction, S1-(r , t) were given by

slI(r, t ) =s¢(r, t ) cos q; +so(r, t ) sin q; ,

h(r, t ) = so(r, t ) cos q; - ;;P(r, t) sin q;. (7.74)

In the following experiments, we used slI(r, t ) and h(r. t) when displaying the time courseof a source activity.

To display the results of the spatio-temporal reconstruction, three time points at 220,268, and 300 ms were selected . The amplitude of the second source happened to be zeroat 220 ms, and all the sources had non-zero amplitudes at 268 ms, while only the second


source had a non-zero amplitude at 300 ms. The snapshots of the source magnitude dis

tribution [S(r , t )1= s ~ (r , t ) +s~(r , t ) at these three time points, and the time averaged

reconstruction J(s (r , t )2) are presented in the following experiments.

7.7.2 RESULTS FROM MINIMUM-VARIANCE VECTOR BEAMFORMER

The results of the spatio-temporal reconstruction obtained using the minimum-variancevector beamformer in Eq. (7.62) are shown in Fig. 7.6(a). The estimated time courses atthe pixels nearest to the three source locations are shown in Fig. 7.6(b). These results showthat the reconstruction at each instant in time was fairly noisy : the snapshot at 220 msshowed some influence from the second source, and the snapshot at 300 ms contained theactivities of the first and third sources. The time-averaged reconstruction, however, resolvedthree active sources. In Fig. 7.6(b), only slI(r, t) shows the source activity time courses, butS.L(r , t) contains no significant activities. This confirms the fact that the source orientationis fixed during the observation period.

We next tested the minimum-variance beamformer with the regularized inverse(Rb + y I)-I instead of R"b 1. The regularization parameter was set to 0.003).. It . The results in Fig. 7.7(a) show that a considerable amount of blur was introduced. The estimated time courses are shown in Fig. 7.7(b). This figure shows that the SNR of thebeamformer output increased considerably in this case, although each time course showssome influence from neighboring sources. The results demonstrate that the regularization leads to a trade-off between the spatial resolution and the SNR of the beamformeroutput.

7.7.3 RESULTS FROM THE VECTOR-EXTENDED BORGIOTT/-KAPLANBEAMFORMER

The reconstruction results from weight vectors obtained using Eq. (7.69) are shownin Fig. 7.8. The weight is equivalent to the vector-extended Borgiotti-Kaplan (B-K) beamformer without the eigenspace projection. Comparison between the time-averaged reconstruction in Fig. 7.6(a) and in Fig. 7.8(a) confirms that the Borgiotti-Kaplan-type beamformer has a spatial resolution much higher than the minimum-variance beamformer. Thespatio-temporal reconstruction, however, is very noisy in both cases.

7.7.4 RESULTS FROM THE EIGENSPACE PROJECTED VECTOR-EXTENDEDBORGIOTTI- KAPLAN BEAMFORMER

We then applied the eigenspace-projected B-K beamformer obtained using Eqs. (7.69)and (7.70) to the same computer-generated data set. The reconstructed source distribution sare shown in Fig. 7.9(a), and the estimated time courses are shown in Fig. 7.9(b). Comparisonbetween Figs. 7.8 and 7.9 confirms that the eigenspace projection can improve the SNRwith almost no sacrifice in spatial resolution. Comparing the results in Fig. 7.9 with theminimum-variance results in Fig. 7.6, we can clearly see that the eigenspace-projected B-Kbeamformer technique significantly improved both spatial resolution and output SNR.

:j: This AI is the largest eigenvalue of Rh.

Neuromagnetic Source Reconstruction and InverseModeling 239

-I

E.... 6

8

-I

Q, 6

82

(a)

~~:-

. ~

"~

( 0)

100 200latency (m s)

300 -100

2

FIGURE 7.6. (a) Results of the spatio-temporal reconstruction obtained using the minimum-variance-basedvector beamformer in Eq. (7.62). The upper-left, upper-right, and lower-left maps show the snapshots of thesource-moment magnitude at 220 ms, 268 ms, and 300 ms, respectively. The lower-right map shows the timeaveraged reconstruction. (b) Estimated time courses from the first to the third sources are shown from the top tothe bottom, respectively. The two time courses in each panel correspond to slI(r, t) and h(r, I). The three verticalbroken lines indicate the time instants at 220, 268, and 300 ms.

240

E~N 6

4


E~N 6 .

(a)

2 0 2 4 2 0 2 4

Y(em) y (cm)

u:>c:;>u 0

.~;;11

(b)

100 200

latency (ms)

300 400

FIGURE 7.7. (a) Results of the spatio-temporal reconstruction obtained using the minimum-variance-basedvector beamformer in Eq. (7.62) together with the regularized inverse (Rb + YI)-I. The parameter y was set to0.003Al' where Al is the largest eigenvalue of Rs; (b) Estimated time courses from the first to the third sourcesare shown from the top to the bottom, respectively.

Neuromagnetic SourceReconstruction and Inverse Modeling 241

4

E2-N 6

4

E2N 6

(a)

2 oy (em)

2 4

~.

~.

2 0 2 4

y (em)

4003001001"----- -'-- -'--L.-_L.-...L- -"

o(b)

FIGURE 7.8. (a) Results of the spatio-temporal reconstruction with the vector-extended Borgiotti-Kaplan-typebeamformer (Eq. (7.69)). (b) Estimated time courses from the first to the third sources are shown from the top tothe bottom, respectively.


400300200

latency (rns)

100

(b)

4

E -,-=-" 6

8

4

'2Q

" 6

82 0 2 4 2 0 2 4

(a) y (ern) y (ern)

"-=g:".e;;"E

FIGURE 7.9. (a) Results of the spatio-temporal reconstruction obtained using the eigenspace-projected BorgiottiKaplan vector beamformer technique (Eqs. (7.69) and (7.70». (b) Estimated time courses from the first to the thirdsources are shown from the top to the bottom, respectively.

z

243

FIGURE 7.10. The x, y, and zcoordinates used to express the reconstructi on results in Section 7.8. The coordinateorigin is defined as the midpoint between the left and right pre-auricular points. The axis directed away from theorigin toward the left pre-auricular point is defined as the +y axis, and that from the origin to the nasion asthe +x axis. The +z axis is defined as the axis perpendicular to both these axes and is directed from the origin tothe vertex.

7.8 APPLICATION OF ADAPTIVE SPATIAL FILTER TECHNIQUETO MEG DATA

This section describes the application of the adaptive spatial filter technique to actualMEG data. The MEG data sets were collected using the 37-channel Magnes" neuromagnetometer. The first data set is an auditory and somatosensory combined response, whichcontains two major source activities. We show that the adaptive spatial filter technique canreconstruct these two sources and retrieve their time courses. The second data set is thesomatosensory response with very high SNR achieved by averaging 10000 trials. With thisdata set, we show that the adaptive technique can separate cortical activities only 0.7-cmapart . Throughout this section, we use the head coordinates shown in Fig. 7.10 to expressthe reconstruction results.

7.8.1 APPLICATION TO AUDITORY-SOMATOSENSORY COMBINED RESPONSE

The evoked response was measured by simultaneously presenting an auditory stimulusand a somatosensory stimulus to a male subject. The auditory stimulus was a 200 ms puretone pulse with 1 kHz frequency presented to the subject's right ear, and the somatosensorystimulus was a 30 ms tactile pulse delivered to the distal segment of the right index finger.These two stimuli started at the same time. The sensor array was placed above the subject'sleft hemisphere with the position adjusted to optimally record the Nlm auditory evokedfield. A total of 256 epochs were measured, and the response averaged over all the epochsis shown in the upper part of Fig . 7.11.

The adaptive vector beamformer technique was applied to localize sources from thisdata set. The covariance matrix Rb was calculated with the time window between 0 msand 300 ms. We calculated, by using Eqs. (7.69) and (7.70), the eigenspace-projectedBorgiotti-Kaplan weight matrix containing the two weight vectors [w </> , we], and estimatedthe source magnitude vector Flr, t ) using Eq. (7.72). The signal subspace dimension Q was


§ 200

.~c

0'".5."v

2000,::

50 0 50 100 150 200 250 300 350latency (ms)

10®E 5 Q 5 65ms~

0

5 5

10®,E 5 E 5 38ms~ ~

0

5 5

10

(~}= 5 ~ 5 94msE-O

55 0 5 10 5 0 5 5 0 5 10

(em) (em) (em)

FIGURE 7.11. Results of the spatio-temporal reconstruction from the auditory-somatosensory combined response shown in the upper trace of this figure. The auditory-somatosensory combined response was measured bysimultaneously applying an auditory stimulus and a somatosensory stimulus. A total of256 epochs were averaged.The contour maps show reconstructed source magnitude distributions at three different latencies (65, 138, and194 ms). The reconstruction grid spacing was set to 5 mm. The maximum-intensity projections onto the axial (leftcolumn), coronal (middle column), and sagittal (right column) directions are shown. The letters Land R indicatethe left and right hemispheres. The circles depicting a human head show the projections of the sphere used for theforward modeling.

set to two because the eigenvalue spectrum of Rb showed two distinctly large eigenvalues.The maximum-intensity projections of the reconstructed moment magnitude Us(r, t) 11

2 ontothe axial, coronal, and sagittal planes are shown in Fig. 7.11. The source magnitude at threelatencies, (65, 138, and 194 ms), is shown in this figure. The source magnitude map at138 ms contains a source activity presumably in the primary somatosensory cortex. Thesource magnitude map at 194 ms shows a source activity in the primary auditory cortex.The map at 65 ms contains both of these activities.

The time courses of points in the primary somatosensory and auditory cortices areshown in Figs. 7.l2(a) and (b), respectively. The coordinates of these cortices were determined from the maximum points in the source magnitude maps at 138 ms and 194 ms.In Fig. 7.12(a) the P50 peak, which is known to represent the activity of the primary


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somatosensory cortex, is observed at a latency of about 50 ms. In Fig. 7.12(b), the auditoryNlm peak is observed at a latency of about 100 ms.

7.8.2 APPLICATION TO SOMATOSENSORY RESPONSE: HIGH-RESOLUTIONIMAGING EXPERIMENTS

Electrical stimuli with 0.2 ms duration were delivered to the right posterior tibialnerve at the ankle with a repetition rate of 4 Hz. The MEG recordings were taken from thevertex centering at Cz of the intemationall0-20 system. An epoch of 60 ms duration wasdigitized at a 4000 Hz sampling frequency and 10000 epochs were averaged. The upper partof Fig. 7.13 shows the MEG signals, recorded over the foot somatosensory region in the lefthemisphere. The eigenspace-projected Borgiotti-Kaplan beamformer was applied to thisMEG recording. The covariance matrix Rb was calculated with a time window between20 and 45 ms containing 100 time samples. The maximum-intensity projections of thereconstructed source magnitude IIs(r, t) 11 2 onto the axial, coronal, and sagittal planes areshown in Fig. 7.13.

The source magnitude maps revealed initial activation in the anterior part of the S1foot area at 33.1 ms, followed by co-activation of the posterior part of S1 cortex at 36.2 ms.The posterior activation became dominant at 37.2 ms and the initial anterior activationcompletely disappeared at 39.1 ms. Fig. 7.14 shows the source magnitude map at 36.9 msoverlaid, with proper thresholding, onto the subject's MRI. Here, the anterior source wasprobably in area 3b and the posterior source was in an area near the marginal sulcus. The

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FIGURE 7.13. Results of the spatio-temporal reconstruction from the somatosensory response shown in the uppertrace of this figure. The somatosensory response was measured using the right posterior tibial nerve stimulation.The contour maps show reconstructed source magnitude distributions at four different latencies. The reconstructiongrid spacing was set to I mm. The maximum-intensity projections onto the axial (left column), coronal (middlecolumn), and sagittal (right column) directions are shown. The letters Land R indicate the left and right hemispheres.The circles depicting a human head show the projections of the sphere used for the forward modeling.

separation of the two sources was approximately 7 mm, demonstrating the high-resolutionimaging capability of the adaptive spatial filter techniques. Details of this investigationhave been reported (Hashimoto et al., 2001a), and the results of applying the adaptivebeamformer technique to the response from the median nerve stimulation have also beenreported (Hashimoto et al., 200Ib).

ACKNOWLEDGMENTS

The author would like to thank Dr. D. Poeppel, Dr. A. Marantz, and Dr. T. Roberts forproviding the auditory data. We are also grateful to Dr. I. Hashimoto, and Dr. K. Sakuma for


FIGURE 7.14. The source magnitude reconstruction results at a latency of 36.9 ms. The source magnitude mapwas properly thresholded and overlaid onto the sagittal cross section of the subject's MRI. The colors represent therelative intensity of the source magnitude; the relationship between the colors and relative intensities is indicatedby the color bar. The anterior source was probably in area 3b and the posterior source was in an area near themarginal sulcus. The separation of the two sources was approximately 7 mm in this case. See the attached CD forcolor figure.

providing the somatosensory data and for useful discussion regarding the interpretation ofthe reconstructed results. This work has been supported by Grants-in-Aid from the KayamoriFoundation of Informational Science Advancement; Grants-in-Aid from the Suzuki Foundation; and Grants-in-Aid from the Ministry of Education, Science, Culture and Sports inJapan (C13680948). This work has also been supported by the Whitaker Foundation, andby National Institute of Health. (P4IRRI2553-03 and ROI-DC004855-0IAI).

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8

MULTIMODAL IMAGING FROMNEUROELECTROMAGNETIC AND

FUNCTIONAL MAGNETICRESONANCE RECORDINGS

Fabio Babiloni and Febo CincottiDipartimento di Fisiologia Umana e Farmacologia, Universita di Roma "La Sapienza", Roma, Italy

8.1 INTRODUCTION

Human neocortical processes involve temporal and spatial scales spanning several orders ofmagnitude, from the rapidly shifting somatosensory processes characterized by a temporalscale ofmilliseconds and a spatial scales of few square millimeters to the memory processes,involving time periods of seconds and spatial scale of square centimeters. Information aboutthe brain activity can be obtained by measuring different physical variables arising from thebrain processes, such as the increase in consumption ofoxygen by the neural tissues or a variation of the electric potential over the scalp surface. All these variables are connected in director indirect way to the neural ongoing processes, and each variable has its own spatial andtemporal resolution. The different neuroimaging techniques are then confined to the spatiotemporal resolution offered by the monitored variables. For instance, it is known from physiology that the temporal resolution of the hemodynamic deoxyhemoglobin increase/decreaselies in the range of 1-2 seconds, while its spatial resolution is generally observable with thecurrent imaging techniques at few mm scale. Today, no neuroimaging method allows a spatial resolution on a mm scale and a temporal resolution on a msec scale. Hence, it is of interestto study the possibility to integrate the information offered by the different physiologicalvariables in a unique mathematical context. This operation is called the "multimodal integration" of variable X and Y, when the X variable has typically particular appealing spatialresolution property (mm scale) and the Y variable has particular attractive temporal properties (on a ms scale). Nevertheless, the issue of several temporal and spatial domains is

Corresponding author: Dr. Fabio Babiloni, Dipartimento di Fisiologia Umana e Farmacologia, Universita diRoma "La Sapienza", P.le A. Mow 5, 00185 Roma, Italy, Tel: +39-06-49910317, Fax: +39-06-49910917, Email:fabio.babiloniteuniromal.it

251

252 F.Babiioni and F. Cincotti

critical in the study of the brain functions, since different properties could become observable, depending on the spatio-temporal scales at which the brain processes are measured.

Electroencephalography (EEG) and magnetoencephalography (MEG) are two interesting techniques that present a high temporal resolution, on the millisecond scale, adequateto follow brain activity. Unlikely, both techniques have a relatively modest spatial resolution,beyond the centimeter. In spite ofa lack of spatial resolution, neural sources can be localizedfrom EEG or MEG data by making a priori hypotheses on their number and extension. Amore detailed description of the techniques involved in the high resolution EEG and MEGrecordings and imaging can be found in Chapter 7 and Chapter 8. Here, we briefly recallthat the so called high resolution EEG methods included: (i) subject's multi-compartmenthead model (scalp, skull, dura mater, cortex) constructed from magnetic resonance images, the sampling of EEG potentials from 64-128 electrodes; (ii) the computation ofthe surface Laplacian from the scalp potential recordings and/or the use of multi-dipolesource model for characterizing the neural active sources. However, the spatial resolutionof the EEG/MEG techniques is fundamentally limited by the inter-sensor distances and bythe fundamental laws of electromagnetism (Nunez, 1981). On the other hand, the use ofaprioriinformation from other neuroimaging techniques like functional magnetic resonanceimaging (fMRI) with high spatial resolution could improve the localization of sources fromEEG/MEG data.

This chapter deals with the multimodal integration of electrical, magnetic and hemodynamic data to locate neural sources responsible for the recorded EEG/MEG activity. The rationale of the multimodal approach based on fMRI, MEG and EEG data to locate brain activity is that neural activity generating EEG potentials or MEG fields increases glucose and oxygen demands (Magistretti et aI., 1999). This results in an increase in the local hemodynamicresponse that can be measured by fMRI (Grinvald et al., 1986; Puce et al., 1997). On thewhole, such a correlation between electrical and hemodynamic concomitants provides thebasis for a spatial correspondence between fMRI responses and EEG/MEG source activity.The chapter is organized as follows: first, a brief introduction on the principles at the basis ofthe fMRI recordings will be presented; then a recall of the principal techniques used for EEGand MEG for locating neural sources will be presented, with special emphasis on corticalimaging and linear distributed solutions. This last technique will be employed to show boththe mathematical principle and the practical applications of the multimodal integration ofEEG, MEG and fMRI for the localization of sources responsible for intentional movements.

8.2 GENERALITIES ON FUNCTIONAL MAGNETICRESONANCE IMAGING

A brain imaging method, known as fMRI, has gained favor among neuroscientists overthe last few years. Functional MRI reflects oxygen consumption and, as oxygen consumption is tied to processing or neural activation, can give a map of functional activity. Whenneurons fire, they consume oxygen and this causes the local oxygen levels to briefly decreaseand then actually increase above the resting level as nearby capillaries dilate to let more oxygenated blood flow into the active area. The most used acquisition paradigm is the so-calledBlood Oxygen Level Dependence (BOLD), in which the fMRI scanner works by imagingblood oxygenation. The BOLD paradigm relies on the brain mechanisms, which overcompensate for oxygen usage (activation causes an influx of oxygenated blood in excess of that

Multimodal Imaging from Neuroelectromagnetic

A B

253

FI GURE 8.1. Physiologic principle at the base of the generation of fMRI signals . A) Neurons increase theirfiring rates, increasing also the oxygen consumption. B) Hemodynamic response in a second scale increases thediameter of the vessel close to the activated neurons. The induced increase in blood flow overco mes the need foroxygen supply. As a consequence, the percentage of deoxyhemoglobin in the blood flow decreases in the vesse lwith respect to the figure A). See the attached CD for color figure.

used and therefore the local oxyhemoglobinconcentration increases). Oxygen is carried tothe brain in the hemoglobin molecules of blood red cells. Fig. 8.1 shows the physiologicprinciple at the base of the generation of tMRI signals. In this figure it is shown how thehemodynamic responses elicited by an increased neuronal activity (A) produces a decreaseof the deoxyhemoglobincontent of the blood flow in the same neuronal district after fewseconds (B). The magnetic properties of hemoglobin differ when it is saturated with oxygen compared to when it has given up oxygen. Technically, deoxygenated hemoglobin is"paramagnetic" and therefore has a short T2 relaxation time. As the ratio of oxygenatedto deoxygenated hemoglobin increases, so does the signal recorded by the MRI. Deoxyhemoglobin increases the rate of depolarizationof hydrogen nuclei creating the MR signalthus decreases the intensityof the T2 image. The bottom line is that the intensity of imagesincreases with the increaseof brain activation.The problem is that at the standard intensityused for the staticmagnetic field (1.5 Tesla) this increase is small (usually less than 2%) andeasily obscured by noise and different artifacts. By increasing the static field of the fMRIscanner, the signal to noise ratio increases to more convenient values. Static field values of3 Tesla are now commonly used for research on humans, while recently fMRI scanner at 7Teslawas employed to map hemodynamic responses in the human brain (Bonmassar et al.,2001). At such high field value, the possibility to detect the initial increase of deoxyhemoglobin (the initial "dip") increase. The interest in the detection of the dip is based on thefact that this hemodynamic response happens on timescale of 500 ms (as revealedby hemodynamic optic measures; Malonek and Grinvald, 1996), compared to 1-2 seconds neededfor the response of the vascular systemto theoxygendemand. Furthermore, in the lattercasethe response has a temporal extension well beyond the activation occurred (10 seconds). Asa last point, the spatial distribution of the initial dip (as described by using the optical dyes;

254 F. Babiloni and F. Cincotti

Malonek and Grinvald, 1996) is sharper than those related to the vascular response of theoxygenated hemoglobin. Recently, with high field strength MR scanners at 7 or even 9.4Tesla (on animals), a resolution down to cortical column level has been achieved (Kim et al.,2000). However, at the standard field intensity commonly used in fMRI studies (1.5 or 3Tesla), the identification of such initial transient increase of deoxyhemoglobin is controversial. Compared to positron emitted tomography (PET) or single photon emitted tomography(SPECT), fMRI does not require the injection of radio-labeled substances, and its imageshave a higher resolution (reviewed in Rosen et al., 1998). PET, however, is still the most informative technique for directly imaging metabolic processes and neurotransmitter turnover.

8.2.1 BLOCK-DESIGN AND EVENT-RELATED fMRI

Though dynamic fMRI experiments were early recognized to be fundamentally different from previous hemodynamically based functional imaging methods (like, for instance,Positron Emitted Tomography ; PET), early studies in fMRI typically used experimentalparadigms that could have been easily performed by using previous nuclear technologies.Spec ifically, most experiments were performed by using extended periods of "on" versus"off" activations, in a way called block designs paradigm . Such paradigm had been usedin dozens of functional studies of sensory and higher cortic al function using PET andsingle photon emission computed tomography (SPECT) for more than a decade. Nevertheless, although such block designs are a necessity when imaging hemodynamics by usingtechn iques that require a quasi-equilibrium physiological state for periods up to I min,they were clearl y not required for fMRI experiments, where activity was detectable withinseconds from stimulus onset. Movement away from block designs was gradual and aidedby a number of studies exploring fMRI signal responses to brief stimulus events (as long as2 seconds or less; Blamire et al., 1992). A detectable signal change in fMRI was shown tobe produ ced by 2 s or shorter stimulus (Blamire et al., 1992; Bandett ini, 1993). Moreover,it was also shown that visual stimulation as brief as 34 msec in duration could elicit small,but clea rly detectable, signal changes (Savoy et al., 1995). All these data suggest that tMRIis sensitive to transient phenomena and can provide at least some degree of quantitativeinformation on the underlying neuronal behavior. Together, these results thus suggest thatit should be possible to interpret transient fMRI signal changes in ways directly analogousto electrophysiologic evoked potenti als. A first step in this direction was made by Daleand Buckner (reviewed in Rosen et al., 1998) who showed that visual stimuli lateralizedto one hemifield could be detected within intermixed trial paradigms. By using methodssimilar to those applied in the field of evoked response potential research, the trials wereselectively averaged to reveal the predicted pattern of contralateral visual cortex activation.Taken together with the above observations, these collective data demonstrate convincinglythat fMRI is capable of detecting changes related to single-task events and brief epochsof stimulation. Hence, the paradigm in which the fMRI information was collected on atrial-by-trial basis is called "event-related fMRI" .

8.3 INVERSE TECHNIQUES

The ultimate goal of any EEG, MEG and fMRI recordin gs is to produce informationabout the brain activity of a subject during a particular sensorimotor or cognitive task.

Multimodal Imaging from Neuroelectromagnetic 255

The mathematical procedures that allow us to recover information about the activity of theneural sources from the non-invasive EEG/MEG recordings are called inverse techniques.The inverse techniques have been systematically treated and reviewed, with application toECG (Chapter 4), and MEG (Chapter 7). Mathematical models for the head as volumeconductor and for the neural sources are employed by linear and non-linear minimizationprocedures to localize putative sources of EEG data. Several studies have indicated theadequacy of the equivalent current dipole as a model for the cortical sources (Nunez, 1981,1995), while the importance of realistic geometry head volume conductor models for thelocalization of cortical activity has been stressed more recently (Gevins, 1989; Gevinset aI., 1991, 1999, Nunez, 1995). Results of previous intracranial EEG studies have ledsupport to the idea that high resolution EEG techniques (including head/source models andproper regularization inverse procedures) might model with an acceptable approximationthe strengths and extension of cortical sources of surface EEG data, at least in certainconditions (Le and Gevins, 1993; Gevins et aI., 1994; He et aI., 2002). We briefly presenta survey on the principal inverse techniques, with a particular emphasis on the so-called"distributed" solutions, which we will use to demonstrate the multimodal integration ofEEG/MEG and fMRI.

8.3.1 ACQUISITION OF VOLUME CONDUCTOR GEOMETRY

A key point of high-resolution EEG and MEG technologies is the availability of theaccurate model of the head as a volume conductor by using anatomical MRIs. These imagesare obtained by using the MRI facilities largely available in all research and clinical institutions worldwide. Reference landmarks such as nasion, inion, vertex, and preauricular pointsmay be labeled using vitamin E pills as markers. Tl-weighted MR images are typically usedsince they present maximal contrast between the structures of interest.

Contouring algorithms allow the segmentation of the principal tissues (scalp, skull,dura mater) from the MR images (Dale et aI., 1999). Separate surfaces of scalp, skull, duramater and cortical envelopes are extracted for each experimental subject, yielding a closedtriangulated mesh. This procedure produces an initial description of the anatomical structurethat uses several hundred thousands points--quite too much for subsequent mathematicalprocedures. These structures are thus down-sampled and triangulated to produce scalp, skulland dura mater geometrical models with about 1000-1300 triangles for each surface. Thesetriangulations were found adequate to model the spatial structures of these head tissues.

A different number of triangles are used in the modeling of the cortical surface, sinceits envelope is more convoluted than the scalp, skull and dura mater structures. A numberof triangles variable from 5000 to 6000 may be used to model the cortical envelope for thepurpose of following the spatial shape of the cerebral cortex. In order to allow coregistrationwith other geometrical information, the coordinates of the triangulated structures are referredto an orthogonal coordinate system (x, y, z) based on the positions of nasion and pre-auricularpoints extracted from the MR images. For instance, the midpoint of the line connectingthe pre-auricular points can be set as the origin of the coordinate system and, with they axis going through the right pre-auricular point, the x axis lying on the plane determinedby nasion and pre-auricular points (directed anteriorly) and the z axis normal to this plane(directed upward). Once the model of scalp surface has been generated, the integration of theelectrodes' positions are accomplished by using the information about the sensor locations

256 F.Babiioni and F. Cincotti

FIGURE 8.2. Realistic MRI-constructed head of a human subject. Electrode positions (128) are shown on theMRI-constructed scalp surface and on the underlying cortex surface.

produced by the 3-D digitizer. The sensors positions on the scalp model are determined byusing a non-linear fitting technique. Fig. 8.2 shows the results of the integration betweenthe EEG scalp electrodes position and a realistic head model.

8.3.2 DIPOLELOCALIZATION TECHNIQUES

Dipole localization techniques produced estimates of the position and moment of oneor several equivalent current dipoles localized in a head model from the non-invasive EEGand/or MEG recordings. From the position of the localized current dipoles in the head model,inferences about the neural sources in the real brain are inferred. So far, two approaches todipole localization have become popular in neuroscience, and both rely on the solution ofnon-linear minimization algorithms.

The first approach is the so-called "moving dipole" method (Cohen et al., 1990).Dipoles are found at a succession of discrete times, with no a priori assumption aboutthe relation of the localized dipoles at different time instants. In general, it is difficult tolocate more than two dipoles for each potential/magnetic field recorded due to the numericalinstability of the inverse procedure. However, also with this limitation, this procedure isvery popular and allows locating sources mainly in the primary sensory cortical areas.Fig. 8.3 shows the localization of a current dipole (the red arrow) indicating the restrictedcortical areas responsible for the generation of the characteristic magnetic field distributionrecorded by the magnetic sensors 20 ms after a stimulus delivery at right wrist in humans.


FIGURE 8.3. Localization of the equivalent current dipole (the red arrow) indicating the restricted cortical areasresponsible for the generation of the characteristic magnetic field distribution occurring over the magnetic sensors20 ms after a stimulus delivered at the right wrist of a subject (N201P20). The position of the dipole is integratedinto a realistic head model built by segmenting sequential magnetic resonance images of the subject. See theattached CD for color figure.

The position of the localized dipole was then integrated into a realistic head model builtaccording to the procedures described above.

The second approach to the dipole localization combines both the spatial and temporalproperties of scalp potentials/fields, to increase the ratio of available data to degrees offreedom for the minimization procedures. This results in an increase of the number ofdipoles that may be reliably localized from EEG and MEG recordings. Different constraintsare applied to find the best inverse solutions, for example setting the position of the dipolesand estimating the time/series of the dipole moments, or determining the orientation of thedipoles and setting their positions. This last approach is called multiple source analysis(MSA; Scherg and von Cramon, 1984; Ebersole 1997, 1999; Scherg et aI., 1999).

8.3.3 CORTICAL IMAGING

The possibility to model the complex head geometry with the finite element techniqueallowed Alan Gevins and colleagues to derive a method, they called deblurring, that estimates potential distribution on the dura mater surface by using non invasive EEG recordings(Le and Gevins, 1993; Gevins et aI., 1994). This method still uses non-linear minimizationtechniques but did not use any explicit model of the neural sources. In fact, by just applyingPoisson's equation, Gevins and co-workers were able to move back from the scalp potentialdistribution to the dura mater potential distribution. This method was also validated by

258

3) Generated corticalf

potentials from theestimated dipolestrenghts

F. Babiloni and F. Cincott i

i~ · II) EEG signa s

2) Estimated corticalstrenghts from EEGsiznals

FIGUR E 8.4. Possible representation of the cortical imaging technique. The acquired EEG scalp potentials ( I) areused to estimate the cortical dipole strengths at the dipo le layer level, here represented with the realistic corticalsurface . The estimated cortical dipole strengths (2) are then used to generate the potential distribution over adura mater surface (3) using the basic laws of electromagnetism. Such distribution can be generated at any othermodeled head structure.

using epicortical recording s, and the deblurred dura mater potential distributions showed aclear improvement with respect to the examination of the raw potential distribution s overthe scalp. It is worth of note that the mathematical model supporting the deblurring methodis not suitable to accommodate fMRI or PET information. In fact, mathematical framework that allow integration between electromagnetic and metabolic modalities, require thesources of currents in the brain to be explicitly modeled.

Another technique useful to recover improved images of cortical distributions fromEEG scalp recording s is known as cortical imaging . In this technique, an explicit modelof the neural sources, i.e. the current dipole, is used. In general, a layer of current dipolessimulates the cortical surface, and the retrieved dipole strengths are then used to generatepotential distributions over a surface of the head model simulating the dura mater. Fig. 8.4shows the idea at the base of the cortical imaging technique. A three-shell (scalp, inner andouter surface of the skull) realistic head volume conductor is represented, together withthe cortical dipole layer. It has been proven that even the use of homogeneous sphericalvolume conductor for the head and a realistic cortical surface for the dipole layer providedmore focused and detailed information than the raw scalp potential s (Sidman et al., 1992,Srebro et aI., 1993, Srebro and Oguz, 1997). However, it must be noted that the conductivityratio between skull and scalp is far than 1 as assumed in homogeneous models. The valueadopted for such ratio by all the researchers in the field in these last 30 years is 1:80(Rush and Driscoll, 1968), or even I:15 as stated more recently (Ooste ndorp et al., 2000) .According to this observation, several researchers (He et aI., 1996; Babiloni et aI., 1997;He, 1999; He et al., 1999) developed cortical imaging techniques that took into accountthe inhomogeneity of the head as volume conductor by using realistic head models andboundary element mathematics. By regularization, dura mater potentials obtained both


from simulation and real recordings presented improved spatial characteristics with respectto the use of raw scalp potentials. It is worth of note that the mathematical framework of thecortical imaging technique allows in principle the integration of tMRI priors. In fact, sincethe cortical imaging method is a linear inverse technique (He, 1999), all the multimodalintegration we will present in the following paragraph for the EEG/MEG and tMRI data forthe distributed linear inverse solution can be in theory applied also for the cortical imaging.

Beside to the use of Green's second identity, another approach to the imaging of thecortical potential distribution from non-invasive recordings was made by using the sphericalharmonic functions (Nunez et al., 1994; Edlinger et al., 1998). However, the mathematicalframework developed for the estimation of cortical potentials do not allow easily the integration of tMRI information. In fact, since this method recovers the "deblurred" corticalpotential distribution but not the cortical current strengths, it is difficult to integrate theinformation about the activation of patches of cortical tissues obtained by tMRI.

8.3.4 DISTRIBUTED LINEAR INVERSE ESTIMATION

As seen before, when the EEG activity is mainly generated by circumscribed corticalsources (i.e. short-latency evoked potentials/magnetic fields), the location and strength ofthese sources can be reliably estimated by the dipole localization technique (Scherg et al.,1984, Salmelin et al., 1995). In contrast, when EEG activity is generated by extended corticalsources (i.e. event-related potentials/magnetic fields), the underlying cortical sources canbe described by using a distributed source model with spherical or realistic head models(Grave de Peralta et al., 1997; Pascual-Marqui, 1995; Dale and Sereno, 1993). With thisapproach, typically thousands of equivalent current dipoles covering the cortical surfacemodeled and located at the triangle center were used, and their strength was estimatedby using linear and non linear inverse procedures (Dale and Sereno, 1993; Uutela et al.,1999). Taking into account the measurement noise n, supposed to be normally distributed,an estimate of the dipole source configuration that generated a measured potential b can beobtained by solving the linear system:

Ax-l n e b (8.1)

where A is a m x n matrix with number of rows equal to the number of sensors and numberof columns equal to the number of modeled sources. We denote with A j the potentialdistribution over the m sensors due to each unitary j-th cortical dipole. The collection ofall the m-dimensional vectors A j , (j = 1, ... , n) describes how each dipole generates thepotential distribution over the head model, and this collection is called the lead field matrixA This is a strongly under-determined linear system, in which the number of unknowns,dimension of the vector x, is greater than the number of measurements b of about oneorder of magnitude. In this case from the linear algebra we know that infinite solutions forthe x dipole strength vector are available, explaining in the same way the data vector b.Furthermore, the linear system is ill-conditioned as results of the substantial equivalenceof several columns of the electromagnetic lead field matrix A In fact, we know that eachcolumn of the lead field matrix arose from the potential distribution generated by the dipolarsources that are located in similar positions and have orientations along the cortical modelused. Regularization of the inverse problem consists in attenuating the oscillatory modes

generated by vectors that are associated with the smallest singular values of the lead fieldmatrix A, introducing supplementary and aprioriinformation on the sources to be estimated.In the following, we characterize with the term "source space" the vector space in whichthe "best" current strength solution x will be found. "Data space" is the vector space inwhich the vector b of the measured data is considered. The electrical lead field matrix Aand the data vector b must be referenced consistently. Before we proceed to the derivationof a possible solution for the problem drawn in (8.1) we recall few definitions of algebrauseful for the following. A more complete introduction to the theory of vector spaces isout of the scope of this chapter, and the interested readers could refer to related textbooks(Spiegel, 1978; Rao and Mitra, 1977). In a vector space provided with a definition of a innerproduct (-,.), it is possible to associate a value or modulus to a vector b by using the notation(b, b) = [b]. The notion of length of a vector can be generalized even in a vector space inwhich the space axes are not orthogonal. Any symmetric positive definite matrix M is saida metric for the vector space furnished with the inner product (.,.) and the squared modulusof a vector b in a space equipped with the norm M is described by

(8.2)

With these recalls in mind, we now face the problem to derive a general solution of theproblem described in Eq. 8.1 under the assumption of the existence of two distinct metricsNand M for the source and the data space, respectively. Since the system is undetermined,infinite solutions exist. However, we are looking for a particular vector solution ethathas the following properties: 1) it has the minimum residual in fitting the data vector bunder the norm M in the data space 2) it has the minimum strength in the source spaceunder the norm N. To take into account these properties, we have to solve the problemutilizing the Lagrange multiplier A and minimizing the following functional that expressthe desired properties for the sources x (Tikhonov and Arsenin, 1977; Dale and Sereno,1993; Menke, 1989; Grave de Peralta and Gonzalez Andino, 1998; Liu, 2000) :

 = (IIAx - bll~ +)..1 IIxlI~) (8.3)

The solution of the variational problem depends on adequacy of the data and source spacemetrics. Under the hypothesis ofM and N positive definite , the solution of Eq. 3 is given bytaking the derivatives of the functional and setting it to zero. After few straightforwardcomputations the solution is

(8.4)

where G is called the pseudoinverse matrix, or the inverse operator, that maps the measureddata b onto the source space e.Note that the requirements of positive definite matrices forthe metric Nand M allow to consider their inverses . Last equation stated that the inverseoperator G depends on the matrices M and N that describe the norm of the measurementsand the source space, respectively. The metric M, characterizing the idea of closeness inthe data space, can be particularized by taking into account the sensors noise level byusing the Mahalanobis distance (Grave de Peralta and Gonzalez Andino, 1998). If noa priori information is available for the solution of linear inverse problem, the matrices M

MultimodaI Imaging from Neuroelectromagnetic 261

and N are set to the identity, and the minimum norm estimation is obtained (Hamalainenand Ilmoniemi, 1984). However, it was recognized that in this particular application thesolutions obtained with the minimum norm constraints are biased toward those sourcesthat are located nearest to the sensors. In fact, there is a dependence of the distance on thelaw of potential (and magnetic field) generation and this dependence tends to increase theactivity of the more superficial sources while depresses the activity of the sources far fromthe sensors. The solution to this bias was obtained by taking into account a compensationfactor for each dipole that equalizes the "visibility" of the dipole from the sensors. Suchtechnique, called column norm normalization by Lawson and Hanson in 1974, was usedin the linear inverse problem by Pascual-Marqui, 1985 and then adopted largely by thescientists in this field. With the column norm normalization the inverse of the resultingsource metric is

(8.5)

in which (N- I)ii is the i-th element of the inverse of the diagonal matrix N and II Ai II isthe L2 norm of the i-th column of the lead field matrix A. In this way, dipoles close tothe sensors, and hence with a large II Ai II, will be depressed in the solution of the inverseproblem, since their activations are not convenient from the point of view of the functionalcost. The use of this definition of matrix N in the source estimation is known as weightedminimum norm solution (Pascual-Marqui, 1995; Grave de Peralta et aI., 1997).

The described mathematical framework is able to accommodate the information coming from EEG, MEG and tMRI data, as we will demonstrate in the following paragraphs.

8.4 MULTIMODAL INTEGRATION OF EEG, MEG AND FMRI DATA

Before we describe how it can be possible to implement methods that fuse data fromall modalities, some remarks are necessary about the neural sources that mayor may notbe retrieved by multimodal EEG-MEG-tMRI integration. In the following paragraphs, wewill present possible techniques for multimodal integration of EEG, MEG and tMRI databy using a particularization of the metrics of the data and the source space, in the context ofthe distributed linear inverse problem. In particular, we will show that the metric of the dataspace M can be characterized to take into account the EEG and MEG data. Furthermore,we will demonstrate how the source metric N can be particularized by taking into accountthe information from the hemodynamic responses of the brain voxels.

8.4.1 VISIBLE AND INVISIBLE SOURCES

Any neuroimaging technique has its own visible and invisible sources. The visiblesources for a particular neuroimaging technique are those neuronal pools whose spatiotemporal activity can be at least in part detected. In contrast, invisible sources are thoseneural assemblies that produce a pattern of the spatio-temporal activity not detectable bythe analyzed neuroimaging technique. In the case ofEEG (or MEG) technique, it is clear thatthe visible sources are generally located at the cortical level, since the cortical assembliesare close to the recording sensors, and the morphology of the cortical layers allows the


generation of open (rather than closed) electromagnetic fields. On the other hand, it is oftenpoorly understood that the invisible sources for the EEG (or MEG) are all those corticalassemblies that do not fire synchronously together. In fact, in a dipole layer composed by Mcoherent sources and N incoherent ones, the potentials due to individual coherent sources arecombined by linear superposition, while the combination of the incoherent sources is onlydue to statistical fluctuations. The ratio between the contributions of coherent to incoherentsource can be expressed by M 1v'N(Nunez, 1995). Hence, if N is very large, say about 10million of incoherent neurons that fire continuously, and M is a small percentage of suchneurons (say 1%; about 100,000 neurons) that instead fire synchronously, we obtain that thepotential measured at the scalp level will be determined by 1051M ,with a net result ofabout 30. Hence, only 1% of the active sources produce a potential larger than the other 99%by a factor of 30 just because of the synchronicity property. This means that a cortical patchmay generate an EEG signal with no modification of its metabolic consumption, simplyby increasing the firing coherence of a small percentage of neurons. As a consequence ofthat, neuroimaging techniques based on imaging of the metabolic/hemodynamic request ofthe neural assemblies may detect no activity change with respect to the baseline condition.However, there are other situations in which the visible sources for metabolic techniquessuch as fMRI and PET can be invisible for EEG or MEG techniques. Stellate cells areneurons present in the human cerebral cortex, and represent 15% of the neural populationof the neocortex (Braitenberg and Schuz, 1991). These cells occupy a spherical volumewithin the cortex, thus generating essentially a closed-field electromagnetic pattern. Sucha field cannot be recorded at the scalp level by electrical or magnetic sensors, although theactual firing rate of such stellate neurons is rather high with respect to the other corticalneurons. This means that these neuronal populations present high metabolism requirementsthat can be detected by the fMRI or PET techniques, while at the same time they are"invisible sources" for the EEG and MEG techniques. Other example of invisible sourcesfor the EEG and MEG techniques are represented by the neural assemblies located at thethalamic level, since they are also arranged in such a way to produce closed electromagneticfield, while having high metabolic requirements.

8.4.2 EXPERIMENTAL DESIGN AND CO-REGISTRATION ISSUES

8.4.2.a Experimental design

Experimental setups that take into account both the electrical and the hemodynamicresponses as dependent variables have to be designed with particular attention. There aretwo main important classes of setups that can be considered in a study of this type, dependingwhether simultaneous EEG and fMRI measurements or just separate EEG/MEG and fMRIrecordings are scheduled. In the first case, many issues related to the co-registration ofthe head can be easily overcome. However, in both cases the differences between thehemodynamic and electric behavior have to be taken into account. In fact, considerationabout the signal to noise ratio (SNR) can limit the use of similar paradigms for EEGand fMRI recordings. For instance, EEG/MEG response to very brief stimuli (such asSomatosensory Evoked Potentials; SEPs, i.e. short electrical shock) can be recorded with ahigh SNR, while the hemodynamic responses decrease its SNR by decreasing the stimulationlength. Furthermore, it has also been demonstrated that while EEG amplitudes decrease


by increasing the stimulation rate, the opposite it is true for the hemodynamic responseamplitudes (Wikstrom et al., 1996; Kampe et al., 2000). Experimental design for eitherseparate or simultaneous collection of electrophysiological and hemodynamic variables canbe easier when event-related fMRI technique is used, in contrast to the block-design fMRI.In these experimental paradigms, the availability of the time behavior of the hemodynamicresponse can be useful to design similar stimulation setup for both modalities.

S.4.2.b Co-registration

In the multimodal integration ofEEGIMEG data with the fMRI, a common geometricalframework has to be derived in order to locate appropriately the voxels whose EEG responsesis high and voxels whose hemodynamic response is increased/decreased during the taskperformance. The issue of deriving a common geometrical framework for the data obtainedby different imaging modalities is called the "co-registration" problem (van den Elsenet al., 1993). Several techniques can be used to produce an optimal match between therealistic head reconstruction obtained in the high resolution EEGIMEG by the MRIs ofthe experimental subject and the fMRI image coordinates. The first body of techniquesis based on the presence of landmarks on the both images used for the co-registration.Corresponding landmarks have to be determined in both modalities (Fuchs et al., 1995).A second body of techniques is based instead on the matching of surfaces belong to thesame head structure, as obtained by the different image modalities. In these techniques aprerequisite is the segmentation of the structures whose surfaces have to be matched (Wagnerand Fuchs, 2001). With the volume-based registration technique no additional informationas landmarks or surface detection is necessary (Wells et al., 1997). In the case in which themultimodal EEG and fMRI is performed simultaneously, the setup of a common geometricalframework becomes simpler. In this case registration can be performed based on a scannercoordinate system. As additional advantage, simultaneous measurement of EEG and fMRIalso allows an accurate co-registration of the electrode positions, a problem that in the othercases have to be solved by using non-linear minimization techniques.

8.4.3 INTEGRATION OF EEG AND MEG DATA

As mentioned before, electroencephalography (EEG) and magneto-encephalography(MEG) are useful tools for the study of brain dynamics and functional cortical connectivitydue to their high temporal resolution (in the range of milliseconds). While the EEG reflectsthe activity of neural generators oriented both tangentially or radially with respect to thesurfaces of electrodes, the MEG is more sensitive to cortical generators oriented tangentiallyto the surfaces of sensors. However, the recorded EEG is a distorted copy of the corticalpotential distribution due to the poor conductivity of the skull, while the MEG is insensitiveto the different head tissues conductivities. In this framework an important question arises,namely the importance of using one (EEG, MEG) or both modalities (EEG and MEG)for increasing the accuracy of the estimated neural activity. Simulation studies aimed atintegrating data from MEG and EEG sensors with phantoms demonstrated an improvementof spatial accuracy of the reconstruction methods when MEG and EEG data are fusedtogether (Phillips et al., 1997, Baillet et al., 1997, 1999). These simulation studies suggest thepossibility to practically integrate data from both EEG and MEG modalities in the solution


of some neurophysiological problems also in the case of distributed source problem. It wasalso demonstrated that the use of combined EEG and MEG data increase the stability andthe accuracy of the source activity estimate from primary sensory cortical areas of man withrespect to using modalities separately (Stock et aI., 1987; Fuchs et aI., 1998). In this context,the question if the use ofcombined EEG and MEG measurements lead to a better estimate ofthe distributed cortical activity with respect to the use of EEG or MEG separately, has beenrecently addressed (Babiloni et aI., 2001). Here, we would like to expand in the followingparagraph concepts about the possibility to integrate EEG and MEG data in the context ofdistributed linear inverse solutions.

At a first look, the attempt to integrate the EEG and MEG data in order to increase thequality of the source reconstruction fails when we consider that the units of the potential andmagnetic field differ. How can one fuse such data together? In order to combine the differentmeasures of electric and magnetic data, both have to be converted to a common basis. Thisconversion was performed by normalizing the measured signals to their individual noiseamplitudes, yielding unit-free measures for both electric and magnetic modalities. Suchnormalization procedure was accomplished by using the covariance matrix of the electricand magnetic noise as metric in the data space for the solution of the linear inverse problem.The estimation of the noise covariance matrices requires the recording of several singlesweeps of EEG and MEG data, and the possibility to determine a segment of the recordeddata in which no task-related activity is present. Then, on all the sweeps recorded and forthe time period of interest, the maximum likelihood estimate for the covariance matrices ofthe electrical N, and magnetic Nm noise matrices have to be computed. With the use of thesematrices we can produce the block covariance matrix of the electromagnetic measurement,by posing S = [ N, Nm ] , i.e horizontally adjoining the two matrices. The forward solutionspecifying the potential scalp field due to an arbitrary dipole source configuration is solvedon the basis of the linear system

(8.6)

where (i) E is the electric lead field matrix obtained by the boundary element technique forthe realistic MRI-constructed head model; (ii) B is the magnetic lead field matrix obtainedfor the same head model; (iii) x is the array of the unknown cortical dipole strengths; (iv) vis the array of the recorded potential values; and (v) m is the array of magnetic field values.The lead field matrix E and the array v were referenced consistently. In order to scale EEGand MEG, the rows of the lead field matrix E and B were first normalized by the norm ofrows (Phillips et aI., 1997). This scaling was equally applied on the electrical and magneticmeasurements arrays, v and m.

As noted before, the inverse operator G is expressed in terms of the matrices M andN that regulates the metric in the measurement and source space, respectively. Here, M isnow equal to the inverse of the covariance matrix S of the noise of the normalized EEGand MEG sensors, while N is the matrix that regulates how each EEG or MEG sensor isinfluenced by dipoles located at different depths of the source model. The covariance matrixS was derived from the normalized EEG (v) and MEG (m) data by maximum likelihoodestimation as described before. The matrix N is a diagonal matrix in which the i-th elementis equal to the norm of the i-th column of the normalized lead field matrix A.

Multimodallmaging from Neuroelectromagnetic 265

RIGHT MOVEMENT

EEG

-t .{l.5 0 -Hl.5 +1

TIME~

ME G

-t -ns 0 -Hl.5 +1T1ME<-ec)

FIGURE 8.5. Disposition of the electric (up) and magnetic (bottom) sensors for the recording of EEG and MEGdata related to unilateral voluntary fingermovements (separate recording sessions). Averaged MEGand EEG timeseries (waveforms) recorded from two selected magnetic (Ml and M2) andelectric (El and E2) sensors are shownon the right of the figure. These sensors overlay the primary sensorimotor cortex contralateral to the movement.See the attached CD for color figure.

In the following , the above methodology is applied to the EEG and MEG data relatedto the preparation and the execution of the voluntary movement of the right index finger.Fig. 8.5 presents the disposition of the electric (up) and magnetic (bottom) sensors for therecording of EEG and MEG data (separate recording sessions). Averaged MEG and EEGtime series (waveforms) recorded from two selected magnetic (M I and M2) and electric(E I and E2) sensors in a normal healthy subject executing the movement are shown on theright of Fig. 8.5. These sensors overlay the primary sensorimotor cortex contralateral tothe movement, which is known to be active both during preparation and execution of themovement. Fig. 8.6 shows linear inverse estimates from EEG, MEG, and combined EEGMEG data recorded from the subjec t about 110 ms after the onset ofEMG activity associatedwith self-paced right finger movement. Raw EEG and MEG distributions present large anddistant negative and positive maxima preponderant in the side contralateral to the movement,the electric field being tilted of 90· with respect to the magnetic field. In contrast, linearinverse estimates were characterized by circumscribed zone of negativity and positivity inboth sides. Linear inverse estimate of EEG data (Movement-Related Response 1; MRR 1)shows negative maxima in the mesial-frontal and contralater al frontal areas , and a zoneof minor negativity in the ipsilateral frontal area. In addition, there is a reversed parietal


FIGURE 8.6. Amplitude gray scale 3-D maps showing linear inverse estimates of electroencephalographic (EEG),magnetoencephalographic (MEG), and combined EEG-MEG data recorded (128-50 channels, respectively) froma subject about 110 ms after the onset of the electromyographic response accompanying a voluntary brisk rightmiddle finger extension. Percent gray scale is normalized with reference to the maximum amplitude calculatedfor each map. Maximum negativity (-100%) is coded in white and maximum positivity (+ 100%) in black.

positivity in both sides. Compared to this distribution, linear inverse estimate of MEG data(Movement Evoked Field l; MEF 1) present closer bilateral frontal negativity and parietalpositivity as well as no negativity in the mesial-frontal area. The linear inverse estimate ofthe combined EEG-MEG data show a high spatial resolution content integrating features oflinear inverse estimate of EEG and MEG data considered separately. This result does notdepend on the increase of the total number of sensors (EEG electrodes + MEG coils) sinceit was also reproducible by comparing purely EEG data and data obtained substituting someof the EEG channels with MEG channels (thus preserving the total number of channels inthe two data-sets, Babiloni et al., 2001).

It is worth of note that the presented results are obtained with separate EEG and MEGrecordings. An important source of variance in the linear inverse source analysis of combinedEEG-MEG data might be caused by the non-simultaneous recording of these data sets, whenattentional, learning and emotional variables are unpaired across the recording blocks.Regarding the experiments presented above, it must be stressed that movement-relatedpotentials/fields are very stable across experimental sessions performed in different days.Furthermore, the participant subjects were preliminary trained to stabilize a simple motor


performance in which learning, attentional attentive and emotional concomitants seem to benegligible. Finally, the possibility of combining MEG and EEG data recorded in differentexperimental sessions offers the opportunity of scientific and clinical cooperation betweencenters that own only one of the imaging systems. In contrast, the use of simultaneousmultimodal EEG-MEG recordings is nowadays possible only in very few laboratories.

In conclusion, the present results for the multimodal integration of EEG and MEG suggest that the linear inverse source estimates of the combined EEG-MEG data improve withrespect to those of EEG or MEG data considered separately. The methodological approachincludes MR-based realistic head and cortical source modeling, high spatial sampling ofreal EEG and MEG data, and regularized linear inverse estimate mathematics (weightedminimum norm source estimate). The application of this technology supports the hypothesis that in humans the preparation and execution of one of the simple unilateral volitionaldigit acts is subserved by a distributed cortical system including SMA and contralaterallypreponderant Ml and S1.

8.4.4 FUNCTIONAL HEMODYNAMIC COUPLING AND INVERSE ESTIMATIONOF SOURCE ACTIVITY

Here we describe how it is possible to take into account the information from thehemodynamic coupling between the cortical areas in the estimation of the time-varyingsource strengths by solving the electromagnetic inverse problem.

8.4.4.a Multimodal integration ofEEG/MEG andfMRI data with dipolelocalization techniques

As described previously, the dipole localization techniques locate neural activity byusing few current equivalent dipoles from non-invasive EEG and/or MEG recordings. Theuse of spatial information from hemodynamic methods as a constraint to the electromagneticinverse problem necessitates the assumption that the brain areas that appear active withdifferent methods are to some extent the same. Dipole localization techniques could use thetMRI information essentially in two way: 1) by setting the foci of the tMRI hotspots inthe brain as initial locations of the equivalent current dipoles localization procedure. Sincethe minimization procedure is non linear, it is hence dependent by the initial position of thecurrent dipole(s); 2) by setting the position of the current dipoles at the tMRI hotspots,allowing to the localization procedure to rotate freely only the direction of the dipoles tofit the EEG or MEG data. Methods I) and 2) can also be combined. In order to take intoaccount possible electrical sources that were not detected as active fMRI spots, some currentdipoles that are not constrained to fMRI spots' positions may be added to the source model.

Recently, a simulation study suggested that EEG dipole fits could benefit from fMRIconstraints (Wagner and Fuchs, 2001) . A reconstructed dipole in the vicinity of each fMRIhotspot yields the corresponding source time course . However, spatially unconstraineddipoles are then necessary to account for remaining simulated EEG activity, to appropriatelylocate also sources that are invisible to the fMRI hotspots . Moving from the simulationto the application studies, the spatial correspondence between EEG/MEG and tMRI hasbeen mostly investigated in the context of motor and somatosensory evoked activity. Inthe localization of the primary sensory and motor areas using equivalent current dipole

268 F. Babiloni and F.Cincotti

modeling, typical spatial differences between the dipole location and the center of tMRIactivation have been between 10 to 16 millimeters (Sanders et al., 1996; Beistener et al.,1997). In light of the current limited knowledge, the localization results seem to agreereasonably well when locating responses from primary sensory areas. Extending comparisonto more complex cortical networks has indicated mostly converging activation patterns(Ahlfors et al., 1999; Korvenoja et al., 1999). In the last study of Korvenoja and co-workers,five subjects participated in both a somatosensory evoked field (SEFs) recording with aNeuromag 122 and a tMRI recording. The goal was to maximize the response amplitudein both imaging modalities, in order to achieve the best possible signal-to-noise ratio. TheSEFs were first modeled independently of tMRI results. Thereafter a multi-dipole modelwas constructed by placing equivalent current dipoles (ECDs) to tMRI activation centroids.IfMEG data indicated activity in some of the eight source areas but no tMRI activation wasseen, then ECD location from independent MEG data analysis was used. The eight dipoleswere spatially fixed, but were allowed to change their orientation and amplitude to explainthe data (rotating dipoles). The ECD orientations were remarkably stable over the wholeanalysis period (0 to 400 ms post stimulus). The time-courses of activation in the modelwere found to agree with data, which have been obtained with invasive electrophysiologicalmethods (Allison et al., 1989; 1996). In another study performed by the same researchgroup, the temporal dynamics of visual motion areas was analyzed by combining tMRI andMEG data (Ahlfors et al., 1999). Also in that case, results indicated that while completeoverlap of activation patterns determined independently from MEG and tMRI did not existon individual level, the activation patterns did converge at the group level. In a study onthe sources of movement-related magnetic fields (Baillet et al., 2001), the locations of thecontralateral and SMA sources found with MEG dipole localizations were found ratherclose to the maximum of the closest tMRI clusters (12 mm and 4 mm, respectively).Cortical remapping of the focal parametric source model was performed and the corticalclusters corresponding to the contralateral and central sources indicated a good match inlocation with the tMRI regions. Other studies have combined the analysis of hemodynamicand electrophysiological data that were collected separately, by using both the Positronemission tomography (PET) and EEG (Heinze et al., 1994; Snyder et al., 1995; Heinzeet al., 1998) or the tMRI and EEGIMEG (Belliveau 1993; Morioka et al., 1995; Georgeet al., 1995; Menon et al., 1997; Opitz et al., 1999). Example of integration of hemodynamicor metabolic techniques and invasive cortical recordings can be found in the study of Luckand colleagues with tMRI and EEG invasive recordings (Luck, 1999) or in that of Lamusuoand co-workers, that integrates PET, MEG and invasive recordings (Lamusuo et aI., 1999).

Simultaneous acquisition of EEG and tMRI data is necessary when the activity ofinterest cannot be easily reproduced, as it happens in the case of the epilepsy studies. Inthis case the epileptiform activity between seizures could be produced by different corticalgenerators. Simultaneous EEG/tMRI recordings (Huang-Hellinger et al., 1995; Warachet aI., 1996; Seeck et aI., 1998; Krakow et al., 1999) have been used to measure suchactivity, by hypothesizing that epileptiform discharges are likely to produce neural activitymeasurable by tMRI (Ives et aI., 1993). These studies recorded interleaved EEG and tMRIto monitor for the presence of interictal activity without, however, localizing the EEGactivity. In a recent study of comparison of spike-triggered tMRI activation hotspots andEEG dipole model in six epileptic patients, an average of 3 em of mismatch between thetMRI hotspots and EEG localized current dipoles was found (Lemieux et al., 2001). In this


FIGURE 8.7. Comparisons between foci of interictal epileptic activity as obtained by MEG dipole localizationtechnique, over threshold fMRI clusters and site of brain lesion in a patient evaluated before surgery. See theattached CD for color figure.

case the authors concluded that the combination of EEG and tMRI techniques offers thepossibility of advancing the study of the generators of epileptiform electrical activity. Suchresults are in line with those found by the group of Romani and co-workers, in which theyfound some centimeters of displacement between tMRI hotspots, localized dipoles on thebase of MEG recordings and brain l~sions before surgical operation in epileptic patients.Fig. 8.7 presents such displacements between the various hotspots and the brain lesion.

There are occasions where disagreement in spatial activation patterns could exist forthe integration of EEG/MEG and tMRI data. It is not clear, for example, whether veryshort-lasting synchronous firing, which can be detected in EEG and MEG, will producea detectable hemodynamic change. Event-related synchronization and resynchronizationare phenomena that possibly remain undetected by observing hemodynamic changes. Forexample, in the study by Ahlfors et al. (1999) MEG indicated activity over the frontal cortexbilaterally while tMRI did not demonstrate any activity in similar areas. This is a typicalexample of the co-existence of visible and invisible sources in a same behavioral task forthe MEG and tMRI techniques.

Taken together, the results above indicate that while it may not be possible to simplyrestrict the source model solutions to areas where tMRI shows activation, it still seems tobe a valuable aid in the validation of the source model. Converging lines of evidence frommultiple methods will increase the likelihood of correct solution. The ultimate way to validate the inverse solution would be the invasive recordings. However, it is worth of note that

fMRI priors used in conjunction with the dipole localization techniques require a significantmanual intervention, since one must decide in which regions of the fMRI activation a dipoleis to be "placed". This could provide at least a difficulty in the replication of findings between different researchers, since the dependence of the results by experimenter's choice. Inthis respect, it is of interest to note that the combined fMRI, EEG and MEG procedure thatutilizes the mathematical framework of distributed linear inverse solutions do not requiressuch "manual" intervention. Such procedure will be presented in the following paragraphs.

8.4.4.b Multimodal integration ofEEG/MEG andfMRI data with distributed model byusing diagonal source metric

Here, we present two characterizations of the source metric N that can provide the basis for the inclusion of the information about the statistical hemodynamic activation of i-thcortical voxel into the linear inverse estimation of the cortical source activity. In the fMRIanalysis, several methods to quantify the brain hemodynamic response to a particular taskhave been developed. However, in the following we analyze the case in which a particularfMRI quantification technique has been used, called Percent Change (PC) technique. Thismeasure quantifies the percentage increase of the fMRI signal during the task performancewith respect the rest state (Kim et aI., 1993). The visualization of the voxels' distributionin the brain space that is statistically increased during the task condition with respect to therest is called the PC map. The difference between the mean rest- and movement-relatedsignal intensity is generally calculated voxel-by-voxeI. The rest-related fMRI signal intensity is obtained by averaging the pre-movement and recovery fMRI. Bonferroni-correctedStudent's t-test is also used to minimize alpha inflation effects due to multiple statisticalvoxel-by-voxel comparisons (Type I error; p < 0.05). The introduction of fMRI priors intothe linear inverse estimation produces a bias in the estimation of the current density strengthof the modeled cortical dipoles. Statistically significantly activated fMRI voxels, which arereturned by the percentage change approach (Kim et aI., 1993), are weighted to account forthe EEG measured potentials.

In fact, a reasonable hypothesis is that there is a positive correlation between localelectric or magnetic activity and local hemodynamic response over time. This correlationcan be expressed as a decrease of the cost in the functional of Eq. 8.3 for the sources Xj inwhich fMRI activation can be observed. This increases the probability for those particularsources Xj to be present in the solution of the electromagnetic problem. Such thoughts canbe formalized by particularizing the source metric N, to take into account the informationcoming from the fMRI. The inverse of the resulting metric is then proposed as follows(Babiloni et aI., 2000):

(8.7)

in which (N- 1)ii and IIAi II has the same meaning described above. g(Ui) is a function of thestatistically significant percentage increase of the fMRI signal assigned to the i-th dipole ofthe modeled source space. This function is expressed as

U·g(uJ2=1+(K-1)

1, K2:1, Ui2:0maxto.)

(8.8)


where a ; is the percentage increase of the tMRI signal during the task state for the i-th voxeland the factor K tunes tMRI constraints in the source space. Fixing K = 1 let us disregardtMRI priors , thus returning to a purely electrical solution; a value for K » 1 allows only thesources associated with tMRI active voxels to participate in the solution . It was shown thata value for K in the order of 10 (90% of constraints for the tMRI information) is useful toavoid mislocalization due to over constrained solutions (Liu et aI., 1998; Dale et aI., 2000;Liu, 2000). In the following the estimation of the cortical activity obtained with this metricwill be denoted as diag-tMRI, since the previous definition of the source metric N resultsin a matrix in which the off-diagonal elements are zero.

8.4.4.c Multimodal integration ofEEG/MEG and fMRIdata with distributed modelby using full source metric

In the previous paragraphs, we observed that incorporating a priori information foreach cortical voxel about geometrical orientation with respect to the cortical surface andabout hemodynamic response we obtain an estimate of the cortical activity that improvesthe reconstruction generated without such constraints. However, it must be noted thatall the formulations presented in literature on the integration of EEG, MEG with tMRIdid not take into account information about the functional coupling of the neural sources.In fact, these formulation s only use the information about the presence or absence of a particular source located at the voxel level in the set of those whose hemodynamic responseshave been elicited by the considered task. However, the theoretical possib ility to includethis source of information in the linear inverse problem was already mentioned in previousarticles (Dale and Sereno , 1993; Liu et aI., 1998; Dale et aI., 2000; Liu, 2000).

In this paragraph we present an extension of the linear inverse problem aimed totaking also into account information about the functional coupling of the cortical sources,as provided experimentally by the hemodynamic responses returned by the event-relatedtMRI. In particular, we estimate the hemodynamic correlation of the neural sources byusing the cross-correlation technique on the hemodynamic waveforms obtained during theperformance of the task under the tMRI scanner. These correlation values are then used asadditional a priori constraints in the solution of the electromagnetic linear inverse problemtogether with the cortical orientation constraints and the presence of statistically significantactivation of the hemodynamic response. We take advantage of the off-diagonal elements ofthe matrix N to insert the information about the functional coupling of the cortical sources.In particular we set the generic (i, j) entry of the inverse of matrix N as in the following

(8.9)

where IIA.; II and g(a ;) have the same meaning described above and cor rij is the degreeof functional coupling between source i and source j during the particular task analyzed .Information on coupling is revealed by the correlation of their hemodynamic responsesobtained by the event-related tMRI data. In the following the estimation of the corticalactivity obtained with this metric will be denoted as corr-tMRI. It is of interest that inthe case of uncorrelated sources tcorn , == 0, i i= j; corn, == 1), the corr-tMRI formulationleads back to the diag-tMRI one. Fig. 8.8 summarizes the different approaches pursued here

272

i-th source

F. Babiloni and F.Cincotti

lO3r----------,.--. .....~ 102

~.J::, 101 . .. .: './.~:-.~ ....! ;: .-.•... .y...... .. .... • sac;,C . -.." .~ 100 . • .... - 'W .

00

123 • 56 7 aTime[sl

N ij -l = 11 A.;112 - 1· IIA' j I12 - I . g (a ; ) . g (a j ) . Corr (5i,5j)

{

(51(/),5/1)) , corr - fMRl

CDrI' (Si , 5j) =8 ij , diag - fMRI

FIGURE 8.8. Upper part: Estimate of the hemodynamic couplin g between two generic cortical sources (i-th andj-th) as obtained by the computat ion of the cross-correlati on between the waveforms of the IMRI responses. Thesewaveforms (Sr, Sj) are obtained during a simple voluntary movement (right middle finger extension). Lower part:mathem atical formulation of the inverse of the source metric N to be used in the solution of the linear inverseproblem . Corr(Sj, Sj) is the zero-lag correlation between the two hemodynamic waveforms Sj, and Sj, and <lij isthe Kroneker symbol.

in order to insert the hemodynamic constraints in the solution of the linear inverse problemfor the estimation of the cortical sources of the recorded EEG in a unique mathematicalformulation.

8.4.4.d Application ofthe multimodal EEG-fMRI integration techniques to theestimation ofsources ofself-paced movements

In this section we will provide a practical example of the application of multimodalintegration techniques of EEG, MEG and fMRI (as theoretically described in the previoussections) to the problem of detection of neural sources subserving unilateral self-pacedmovements in humans. The high resolution EEG recordings (128 scalp electrodes) wereperformed on normal healthy subjects by using the facilities available at the laboratoryof the Department of Human Physiology, University of Rome "La Sapienza". Realistichead models were used, each one provided with a cortical surface reconstruction tessellatedwith 3,000 current dipoles . Separate block design and event-related fMRI recordings ofthe same subject s were performed by using the facilities of the Istituto Tecnologie Avanzate Biomediche (ITAB) of Chiety, Italy, leaded by prof. Gian Luca Romani. Distributedlinear inverse solutions by using hemodynamic constraints were obtained according to themethodology presented above.

An example of multimodal integration between EEG and fMRI related to a simple voluntary movement task by using only the hemodynamic information relative to the strength


FIGURE 8.9. Amplitude gray scale 3-D maps showing linear inverse estimates from high resolution electroencephalographic (HREEG) and combined functional magnetic resonance image (fMRI)-HREEG data computedfrom a subject about 20 ms after the onset of the electromyographic activity associated with self-paced right middlefinger movements (motor potential peak, MPp). Percent gray scale of HREEG and combined fMRI-HREEG datais normalized with reference to the maximum amplitude calculated for each map. Maximum negativity (-I00% )is coded in white and maximum positivity (+ I00%) in black.

of fMRI data (according to Eq. 8.7) is provided in Fig. 8.9. This figure shows amplitudegray scale maps of linear source inverse estimates from EEG and combined fMRI-EEGdata, computed about 20 ms after the onset of the electromyographic response to voluntaryright finger movements (Motor Potential peak; MPp). fMRI data indicate maximum activated voxels clustered in bilateral primary motor (Ml), primary somatosensory (Sl), andsupplementary motor (SMA) areas, the fMRI signal intensity being much more higher onthe contralateral (left) side. The linear inverse estimate of neural activity for the HREEGand combined fMRI-HREEG data were mapped over the cortical compartment ofa realisticMRI-constructed subject's head model. The MPp map presents maximum responses in thecontralateral Ml and Sl and in the modeled SMA. With respect to the HREEG solutions(left), the fMRI-HREEG solutions present more circumscribed Ml, S1, and SMA responses.In addition, the contralateral Ml and Sl responses have similar intensity and are spatiallydissociated.

An example of the multimodal integration between EEG and fMRI data by using bothblock and event related experimental designs is depicted in Fig. 8.10. In this figure, the upperrow illustrates the topographic map of readiness potential distribution recorded at the scalpabout 200 ms before a right middle finger extension for another subject analyzed. Note theextension of the maximum of the negative scalp potential distribution, roughly overlying


Right finger movement

Readiness Potential tMRI Response

no-tMRI

Linear Inverse solutions

diag-tMRI corr-tMRI

-100% +100%

FIGURE 8.10. Top left: scalp potential distribution recorded about 200 ms before the movement onset (128recording channels) in a separate session. This distribution is representative of the so-called readiness potential.Percent color scale in which maximum negativity is coded in red and maximum positivity is coded in black. Topright: cortical tMRI response related to the movement. Only the tMRI voxels close to the cortical surface contributeto source weighting. The intensity of yellow codes the percentage of the increase of the tMRI response. Bottomrow: Cortical distributions of the current density estimated with a linear inverse approach from the readinesspotential shown in the top row. Linear inverse estimates are obtained with no tMRI constraints (left, no-tMRI)and two kinds oftMRI constraints, one based on the strengths of the cortical tMRI responses (center, diag-tMRI)and the other on the correlation between tMRI responsive cortical areas (right, corr-tMRI). Percent color scale:maximum negativity is coded in red and maximum positivity is coded in black. See the attached CD for colorfigure.

frontal and centro-parietal areas contralateral to the movement. The percent values of thetMRI response during the movement in a separate experimental session are also illustrated.The maximum values of the tMRI responses are located in the voxels roughly corresponding to the primary somatosensory and motor areas (hand representation) contralateral tothe movement. In fact, during the self-paced unilateral finger extension, somatosensoryreafference inputs from finger joints as well as cutaneous nerves are directed to the primarysomatosensory area, while centrifugal commands from the primary motor area are directed


toward the spinal cord via the pyramidal system.The lower row of Fig. 8.10 illustrates thecortical distribution of the current density estimated with linear inverseapproach from thepotentialdistributionof the upperrow. Linearinverseprocedureusedno-tMRIconstraintaswell as two types offMRI constraints, i.e. one based on block-design (diag-fMRI, Eq. 8.7)and the other on event-related design (corr-fMRI, Eq. 8.9). The cortical distributions arerepresented on the realistic subject's head volume conductor model. Linear inverse solutions obtainedwith the fMRI priors (diag- and corr-fMRI)present more localized spots ofactivations with respect to thoseobtainedwith the no fMRIpriors.Remarkably, the spotsofactivation are localizedin the hand regionof the primary somatosensory (post-central) andmotor (pre-central) areas contralateralto the movement. In addition, spots of minor activation wereobservedin the fronto-central medialareas (includingsupplementary motor area)and in the primary somatosensory and motor areas of the ipsilateral hemisphere. Similarresultswereobtainedin the othermaincomponentsof the movement-relatedpotentials(i.e.motorpotentialsand movement-evoked potential).

8.5 DISCUSSION

Thischapterrevieweda mathematical framework for the integration ofEEG, MEGandfMRIdata. Besides, advantages anddrawbacks relatedto the localizationtechniquesin conjunctionwith fMRIand PET data havealsobeen reviewed. In general, there is a rather largeconsensus about the need and utility of the multimodal integration of metabolic, hemodynamicand neuroelectrical data.Resultsreviewedin literatureas wellas thosepresentedheresuggesta real improvement in the spatialdetailsof theestimatedneuralsourcesby performingmultimodal integration. However, whilefor themultimodalintegration ofEEG andMEGdata a precise electromagnetic theoryexists, a clear mathematical and physiologiclink betweenmetabolic demandsand firing ratesof the neuronsis still lacking.It is outof doubtthatwhenthislinkis furtherclarified, themodelingof the interaction betweenhemodynamic andneural firing rate can be further refined. This will lead us to a more proper characterizationof the issues of visible and invisiblesource that at the momentrepresent the major concernabout the applicability of the multimodal integration techniques (Nunez et aI., 2000).

The results for the multimodal integrationof EEG/MEG and fMRI presented in thischapter are in line with those regarding the coupling between cortical electrical activityand hemodynamic measure as indicated by a direct comparison of maps obtained usingvoltage-sensitive dyes (which reflectdepolarization of neuronal membranes in superficialcorticallayers) andmapsderivedfromintrinsicopticalsignals(whichreflectchangesin lightabsorptiondue to changes in blood volumeand oxygenconsumption, Shohamet aI., 1999).Furthermore, previous studies on animals have also shown a strong correlation betweenlocal field potentials,spikingactivity, and voltage-sensitive dye signals(Arieli et aI., 1996).Finally, studies in humans comparing the localization of functional activity by invasiveelectrical recordings and fMRI have provided evidence of a correlation between the localelectrophysiological andhemodynamic responses(Puceet aI., 1997). It is worthof note thatrecentlya studyaimedat investigating this link has been produced(Logothetis et aI., 2001).In this study, intracortical recordingsof neural signalsand simultaneousfMRI signalswereacquired in monkeys. The comparisons were made between the local field potentials, themulti-unit spiking activity and BOLD signals in the visual cortex. The study supports the

276 F.Babiloni and F.Cincotti

link between the local field potentials and BOLD mechanism, which is at the base of theprocedure of the multimodal integration of EEGIMEG with fMRI described above. Thismay suggest that the local fMRI responses can be reliablyused to bias the estimationof theelectrical activity in the regions showing a prominent hemodynamicresponse.

It may be argued that combined EEG-fMRI responses could be less reliable for themodeling of cortical activation in the case of a spatial mismatch between electrical andhemodynamicresponses.However, previousstudies havesuggestedthat by using the fMRIdata as a partial constraint in the liner inverse procedure, it is possible to obtain accuratesourceestimatesof electricalactivityevenin thepresenceof somespatialmismatchbetweenthe generators ofEEG data and the fMRI signals (Liu et al., 1998; Liu, 2000).Furthermore,it is questionable whether the level of bias for the hemodynamic constraints in the linearinverse estimation can be the same with the diag-fMRI and corr-fMRI approaches. Thisissue seems to deserve a specific simulation study, using the literature indexes capable ofassessing the quality of the linear inversesolutions(PascualMarqui, 1995;Gravede Peraltaet al., 1996;Grave de Peralta and Gonzalez Andino, 1998,Babiloni et al., 2001).

The multimodalintegrationof fMRI, MEGand EEG data constitutesan unsurpassablenon-invasive technologyfor the analysisof humanhigher brain functions at a high temporaland a good spatial resolution.

ACKNOWLEDGMENTS

The Authors express their gratitude to the following colleagues that have participatedin the researches described above: dr. Claudio Babiloni, dr. Filippo Carducci, prof. GianLuca Romani, dr. Cosimo Del Gratta, dr. VittorioPizzella, prof. Paolo Rossini.

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9

THE ELECTRICAL CONDUCTIVITYOF LIVING TISSUE: A PARAMETER IN

THE BIOELECTRICAL INVERSEPROBLEM

Maria J. Peters, Jeroen G. Stinstra, and Ibolya LevelesFacultyof AppliedPhysics, LowTemperature Division, University of Twente

9.1 INTRODUCTION

Electrically active cells within the human body generate currents in the tissues surroundingthese cells. These currents are called volume currents. The volume currents in turn give riseto potential differences between electrodes attached to the body. When these electrodes areattached to the torso, electrical potential differences generated by the heart are recorded.The recording of these electrical potential differences as a function of time is called anelectrocardiogram (ECG). ECG measurements can be used to compute the generators withinthe heart. This is called the solution of the ECG inverse problem. This solution may be ofinterest for diagnostic purposes. For instance, it can be used to localize an extra conductingpathway between atria and ventricles. This pathway can then subsequently be removed byradio-frequent ablation through a catheter. When the active cells are situated within thebrain and the electrodes are attached to the scalp, the recording of the potential differencemeasured between two electrodes as a function of time is called an electroencephalogram(EEG). The EEG inverse problem can, for example, be used to localize an epileptic focusas part of the presurgical evaluation. The frequencies involved in electrocardiograms andelectroencephalograms are in the range of I-1000Hz. Therefore, the Maxwell equationscan be used in a quasi-static approximation, implicating that capacitive and inductive effectsand wave phenomena are ignored as argued by Plonsey and Heppner (1967).

To solve the inverse problem a model is needed of the source and the surroundingtissues, i.e. the volume conductor. Customarily, the source is modeled by a current dipoleor a current dipole layer and the volume conductor is described by a compartment model,

Corresponding author: Prof. Dr. M. J. Peters, Faculty of Applied Physics, University of Twente, P.O. Box 217,7500 AE Enschede, The Netherlands, Tel. 31534893138, Fax 31534891099, E-mail m.j.peterstetn.utwente.nl

281

282 M. J. Peters, J. G. Stinstra, and I. Leveles

where all compartments are considered to be homogeneous. The head may have a scalp, askull, a cerebrospinal fluid and a brain compartment. The torso compartment model mayinclude the ventricular cavities, the lungs and the surrounding homogeneous medium. Theshape may be a rough approximation of the real geometry, or the surfaces of the variouscompartments may have a realistic shape that is obtained from magnetic resonance images.An electrical conductivity is assigned to each compartment. If the inverse solution is used tolocalize the sources of the measured potentials, then only the ratio between the conductivitiesassigned to the various compartments is of importance. If the inverse solution is also usedto estimate the strengths of the sources, then the absolute values of these conductivities areof importance.

In general, the conductivities of the various human tissues are among other thingsdependent on the blood content and temperature, they are a function of the frequencyand strength of the applied current, they show an inter-individual variability, and theyare inhomogeneous and anisotropic (Robillard and Poussart, 1977; Rosell et aZ., 1988;Law, 1993). Moreover, the conductivity may be dependent on the health of the subject,for instance, edema will change the conductivity, so does the presence of scar tissue ortumors. The conductivity is called inhomogeneous when the conductivity differs fromplace to place. The conductivity is called anisotropic when the conductivity is different indifferent directions. For low current densities, the current density is linear with the appliedelectric field, in other words the law of Ohm is valid in this case. The averaged Ohmicconductivity that is assigned to a compartment is called the effective conductivity. Theeffective conductivity of an inhomogeneous tissue is the conductivity of a hypotheticalhomogeneous medium, which mimics the potential distribution that is found outside theinhomogeneous tissue. For instance, in case of EEG, the effective conductivities assigned tothe various tissues in the head have to give approximately the same potential distribution atthe scalp as the real inhomogeneous tissues . The problem addressed in the present chapteris: Which value should be assigned to a certain compartment?

9.1.1 SCOPE OF THISCHAPTER

In order to estimate the effective conductivity of a certain tissue, two approaches arepossible. First, the conductivity of a tissue can be measured in vitro or in vivo by applyinga potential difference by means of a set electrodes and measuring the resulting current.A second approach is applying knowledge of the chemical composition and biologicalmorphology of the tissue in order to compute the effective conductivity. As measuredvalues of the effective conductivities of tissue reported in the literature vary widely, thesecond approach may be of help to restrict the uncertainties in the conductivities involvedin the volume conduction problem . In section 9.1.2, the concept that the conductivity is nota straightforward property of the material is discussed. The effective conductivity from anexperimental point of view will be discussed in section 9.1.3.

In section 9.2 the tissue is modeled as a suspension ofcells in an aqueous surrounding. Ifonly one type of cells is present and the suspension is a dilute one, the effective conductivityis expressed by Maxwell's mixture equation. This equation will be discussed in section 9.2.3.If more than one type of cell is present and the cells are densely packed or clustered, theeffective conductivity is expressed by Archie's law. This will be the subject of section 9.2.4.The applicability of Maxwell's mixture equation and Archie's law will be illustrated forvarious tissues, such as blood, fat, liver, and skeletal muscles .

The Electrical Conductivity of Living Tissue 283

In section 9.3, it is taken into account that most tissues have a layered structure, eachlayer having a different conductivity. The effective conductivity of an entire compartmentwill be discussed in section 9.4. The conductivity of a composite medium, like humantissue, cannot have any value but will be within certain limits. section 9.5 of this chapter isdedicated to these limits.

9.1.2 AMBIGUITY OF THE EFFECTIVE CONDUCTIVITY

Electrical currents in the human body are due to the movements of ions. As a resultof an electric force acting on an ion that is dissolved in fluid, the ion will develop a meandrift velocity between collisions. Despite the fact that the average spacing between cellsmay be no more than 20 nm, the mean free path of an ion in the extracellular space isonly about 0.01 nm. This represents the distance between collisions with other molecules.Almost all these collisions take place with water. An ion rarely encounters a cell membraneand behaves most of the time as though it were in a continuum. The drift velocity in Ohmicconductors is directly proportional to the electric field. The current density reads:

(9.1)

where (T is the conductivity and E the electric field.The electrical conductivity cannot be determined unambiguously, because it depends

on the direction of the currents, the extent of the current source, and the positions of themeasuring electrodes. This is illustrated in Fig. 9.1 where two different situations are compared. In the first situation, a uniform current is applied between two parallel flat electrodes.In the second situation the current is generated by a point source. The expressions for theconductivity are different for the two situations showing that the effective conductivity isnot merely dependent on the material, but also on the configuration given by the orientationand location of the source and the measurement points.

If a medium is piece-wise homogeneous and isotropic then the proper boundary conditions are that the normal component of the current density is continuous and the normal

co nductivity betweenflat electrodesa=VA / IL

source

co nductivity of apoint so urcea=V 4rcr / I

p

FIGURE 9.1. This figure illustrates that the conductivity measured in a homogeneous medium of infinite extentdepends on the electrodes used.


component of the dielectric displacement makes a step that is equal to the surface chargedensity. Hence, all interfaces (including the outerfaces of the cells) will have a continuousdistribution of surface charge representing genuine accumulations of charge. If the materialis non-uniform or if the conductivity is anisotropic, we get accumulations of charge withinthe material as well as on the interfaces.

The microscopic electrical conductivity is the conductivity that characterizes a part oftissue that is comparable in size with the dimensions of the cells. The macroscopic effectiveconductivity characterizes a part of the tissue that is large compared to the dimensions of thecells. Several levels of inhomogeneities can be distinguished. We will restrict ourselves tothree levels of inhomogeneities, the microscopic level with typical dimensions of microns,the millimeter level and the macroscopic level, e.g. compartments with dimensions of severalcentimeters. This is illustrated in Fig. 9.2. In the microscopic point of view, forms anddimensions ofcells and the interstitial fluid are taken into account. Near a cell, the electricalfield may change in direction or amplitude due to the charge density on the surface and thepresence of counterions near the surface. For macroscopic purposes one has to consider thefield averaged over regions large enough to contain many thousands of cells or fibers sothat microscopic fluctuations are smoothed over, the 'graininess' of the material is blurredby distance. At a millimeter level, the layered structure or columnar structure of an organ istaken into account. For instance, the skin is composed of three layers, namely the epidermis(the outer non-sensitive and non-vascular layer of the skin that overlies the dermis), thedermis and subcutis. The epidermis is composed of stratum corneum, stratum lucidum,stratum granulosum and stratum germinativum. These layers differ in composition andmorphology and consequently in conductivity. The conductivity of the various layers isaveraged and the material acts as a continuum, the averaged conductivity being the effectiveconductivity.

9.1.3 MEASURING THE EFFECTIVE CONDUCTIVITY

The uncertainty in measured conductivity values is high, because the measurements arevery complicated. The reasons why they are so complicated will be shortly discussed in thissection. Usually, the Ohmic low-frequency conductivity of a piece of tissue is determinedfrom the current-voltage relation using a two- or four points method. The currents usedfor the measurements have to be low (about lmA) in order not to trigger an activation ofcells. The field near one cell is very much influenced by the presence of the cell. In orderto have an effective conductivity thousands of cells have to be present within the piece oftissue measured. Therefore the effective conductivity has to be measured with electrodeswith dimensions of millimeters, these electrodes have to be millimeters apart. The sourcesin the brain and heart of EEG and ECG are due to at least 105 cells that are active insynchrony else the EEG or ECG would not be measurable. Consequently, the conditionthat the source has dimensions of millimeters is met if brain or heart activity is used forconductivity measurements.

If the measurements are carried outin vitro (i.e. outside the body), the accuracy may below because tissue properties change after death. The conductivity will initially drop aftercirculatory arrest due to emptying blood vessels and drainage of fluids. For instance, it wasfound that the conductivity of frog muscle at 10Hz decreased a factor two after 2.5 hours ofdeath and that of chicken muscle decreased 70 percent in the first 60 minutes (Zheng et al.,


FIGURE 9.2. Illustration of the three levels of inhomogeneities discussed in this chapter. at the left) the head thatis usually modeled by three compartments; in the middle) the layered gray matter; at the right) a suspension ofcells.

1984). However, the conductivity of dog muscle increased immediately after death (Zhenget al., 1984). This may be caused by a change in the osmotic pressure causing some cellsto swell and burst. Gielen (1983) measured in vivo the low-frequency conductivity of themuscles ofa rabbit that were prepared free , the blood supply was unimpaired. He found afterfinishing his experiments that the outermost muscle layer was damaged. The cross sectionsof the fibers were much larger and rounder than normal. This damage was related to anincrease in conductivity. This phenomenon was blamed to osmotic proces ses. The increase


will grow in time (Schwan, 1985), because cell membranes after death allow currents to passmore easily. In other words, all in vitro conductivity measurements should be completedwithin a very limited time.

If the measurements are carried out in vivo, commonly animal tissue instead of humantissue is used. However, it is not clear whether the animal tissue has the same electricalproperties as human tissue although sometimes they seem to be comparable. Moreover, thein vivo measurements depend on the surrounding tissues. When electrodes are implanted inliving tissue, the currents applied by these electrodes will not be confined to the tissue thatis between the electrodes, but will spread out through all surrounding tissues. Hence, it isdifficult to estimate which part of the current will flow through the tissue of interest. Someinvestigators try to avoid this problem by measuring the conductivity with two electrodesspaced closely together. However, this raises the question whether on such a scale the inhomogeneities in the structure of the tissue do not disable the measurement of an effectivemacroscopic conductivity because the electric field within the tissue near cells has a complicated pattern. If one happens to be near a cell, the field may be small or point in a totallydifferent direction. Another problem experienced is a relatively large extra capacity betweenelectrode and tissue at low frequencies. Moreover, the electrode-electrolyte interface canproduce large errors that depend on the pressure between electrode and organ tissue.

Thence, it is not surprising that the measured low-frequen cy conductivities reported inthe literature vary over a wide range. As example some values found for the low-frequencyconductivity of skeletal muscle tissue at 37°C are given in table 9. 1. No attempt to givea complete overview has been made. The conductivity in muscle tissue is anisotropic, thecondu ctivity along the fibers a"is higher than the conductivity perpendicular to the fibersal . Five degrees misalignment from true parallel or perpendicul ar orientation during themeasurement would result in an 18 percent overestimate of al and a 0.4 underestimate of a h

(Epstein and Forster, 1983). Consequently, misalignment errors will be smallest in a h andthe anisotropy factor will be easily underestimated. Out of theoretical studies, there mightdevelop insight into the nature of volume conduction that would permit a proper choicefrom the values that are reported .

TABLE 9.1. Som e values fo und in the literature of the conductivity of skeletal

muscle at 37°C in the frequency range of 0-1000 Hz

anisotropyspecies ah(S/m) a\(S/m) factor ahla[ referenc e

cow 0.41 0.15 2.7 Burger and van Dongen (1961)

rabbi t 0.8 0.06 13dog 0.67 0.04 17 Rush et al. (1963)dog 0.43 0.2 1 2.0 Burger and van Milaan (1943)frog(2I CC) 0.09 0.05 1.8 Hart et al. (1999)dog 0.70 0.06 II Zheng et al. (1984)rabbit 0.75 0.04 17monkey 0.8 1 0.06 13dog 0.52 0.08 6.5 Epstein and Foster (1983)rabbit 0.5 0.08 6.3 Gielen et at. (1984)

rat 0.5 0.Q7 6.1 McCrae and Esrick (1993)

The Electrical Conductivity of Living Tissue

TABLE9.2. Examples of the temperaturedependenceof tissues

287

material conductivity (S/m)

bile (cow-pig) 1.661.28

amniotic fluid (sheep) 1.542.04

Urine (cow-pig) 3.332.56

skeletal muscle (rat)parallel 0.42

0.66perpendicular 0.08

0.20

9.1.4 TEMPERATURE DEPENDENCE

temperature e C)

37202537.53720

36.744.5 (after 30 minutes)

36.644.5 (after 30 minutes)

The electrical conduction in living tissue depend s on the temperature. Me Rae andEsrick (1993) heated freshly excised rat skeletal muscle from 39.5 to 50°C. Above 44°Chealthy tissue will be damaged irreversibly. Initially. a rapid increase of the low-frequenc yconductivity (with about a factor two) was observed followed by a much slower increaseduring which the low-frequency conductivity graduall y approached the high-frequencyvalues.The initial rapid change was associated with microscopically observed fiber roundingand shrinkage in the radial direction and increasing edema. The subsequent slow changewas associated with disintegration of the tissue. After the cell membranes are destroyed.the conductivity is no longer frequency dependent and the low-frequency conductivity hasthe same value previously measured at high frequencies. The temperature dependency of theconductivity of tissue is also caused by the temperature-dependent regulation of the vesseldiameter (vasodilatation). In other words. the blood supply is temperature dependent. Thetemperature dependen ce of the conductivity is illustrated in Fig. 9.3 and table 9.2.

9.1.5 FREQUENCY DEPENDENCE

The low-frequency conductivity of most tissues like heart muscle . skin. liver. lung. fat.and uterus is not strongly dependent on frequency. although measurements show that theconductivity increases with frequency. also at low frequencie s.Commonly. this phenomenonis not taken into account and the conductivity is presumed to be frequency independent forfrequencies lower than 1000Hz (e.g., Schwan and Foster. 1980). Nevertheless. the conductivity does increase with the frequency regardless the kind of tissue. Nicholson (1965)measured the conductivity of cerebral white matter of a cat. He found an enhancement ofthe conductivity in the direction normal to the fibers from 0.11Sim to 0.13S/m. when thefrequency changed from 20 to 200Hz. No enhancement was found for the conductivityparallel to the fibers. Comparison between the experimental data presented by Gabriel et al.(1996a

) . and corresponding data previously reported in the literature show good agreement(Stuchly and Stuchly, 1980). The conductivity of the gray matter as measured by Gabrielet al. (1996b) in the frequency range from 10Hz to 20GHz is given in Fig. 9.4.

288

E 1 .2Cf)

>-...:>...c:::l"0C0o

36

M. J. Peters,J.G. Stinstra,and I. Leveles

42 4340 4 1

39 00 '~ ,

\e~

FIGURE9.3. Variation of the conductivity with the temperature and time for skeletal muscle, based om measurements reported by Gersing (1998).

0.1.------r---r-----,----.-----.---,.---.,.--~--~-~

0.08

,.......§V>......,0 0.06:~'0:=

"Cc00

0.04

0.02

01L-_ ....L..-_ ---l__....L._----l__....L.__L---_-L__L-_--L_---.J

20 40 60

frequency (Hz)

80 100

FIGURE 9.4. The frequency-dependent conductivity of gray matter based on the parameters given by Gabrielet al. (l996b) .


9.1.5.1 Impact ofthe frequency dependence on the EEG

The frequency dependence of the conductivity may influence the EEG. In order toestimate the impact on EEG, the effects of a frequency-dependent conductivity were simulated. The volume conductor model consisted of three concentric spheres representingthe brain, skull and scalp, with radii of 87, 92 and 100mm. This model is often used todescribe the head as a volume conductor. The source was modeled as a current dipole.Simulations were carried out for three current dipoles, i.e. a central dipole, and a radialand a tangential dipole at a radius of 80mm. As observation point, a point at the surfacewas taken where the potential, found for the lowest frequency, was maximal. The transferfunction was calculated. The transfer function is the relationship between the strength ofthe current dipole and the potential in a point at the outermost surface as a function of thefrequency. To enable a comparison between the different cases, all transfer functions werenormalized. If there is no frequency dependence, the transfer function would have the valueone for all frequencies.

Two cases were studied. First, the gray matter conductivity was taken from Fig. 9.4in which the conductivity increases by a factor four; the scalp conductivity was 0.33S/mand that of the skull 0.0042S/m. Second, the conductivity of all compartments increased20% in the interval from 1Hz to 100Hz. At 1Hz the conductivity of the brain and scalpwas 0.33S/m and that of the skull 0.0042S/m. The results of these simulations are givenin Fig. 9.5. As shown the volume conductor acts as a low-pass filter. The potential maydrop by approximately a factor two when the frequency is increased from 1 to 100Hz. Thetransfer function depends on the depth of the source.

9.2 MODELS OF HUMAN TISSUE

Tissues are composed of cells. The interstitial space between the cells contains fluid.So, the effective conductivity ofa tissue depends on the conductivity of the cells, the volumefraction occupied by the cells, and the conductivity of the extracellular medium.

9.2.1 COMPOSITES OF HUMAN TISSUE

This section starts with a brief description of tissue at a cellular level. Next, the conductivity of a cell and that of the extracellular fluid will be discussed.

9.2.1.1 Cells

All human cells stem from the round-shaped fertilized egg cell. There is no typical cellshape. Cells come in all shapes: cubes (cells lining sweat ducts), spheres (white blood cellsof the immune system), Bismarck doughnuts (red blood cells), columnar cells, balloon-likecells (cells lining the urinary bladder), needle shaped ellipsoids or rods (skeletal musclecells) and pancakes (cells on the surface of the skin) as illustrated in Fig. 9.6.

Cells vary also considerably in size, and function. For instance, the diameter of a redblood cell is 7.5f.1m, the diameter of a human egg cell is 140fJ..m, a smooth muscle cell hasa length of 20 to 500f.1m, while a skeletal muscle cell may have a length of 30cm. All cells

10090807040 50 60frequency (Hz)

3020

central dipo leradial dipole at z = 80mmtangential dipole at z = 80mrnall dipole when all conductivities increase by 20%

10

"- '- "- .- ._ "- "- .- 0_. _ .- 0_ .- 0_ .- 0_ .- 0_ .. _.0.8

0.9

c.2U 0.7e<2...~ 0.6tilt:

'".'="0 0.5<:J

'"~c 0.4::0c

0.3

0.2

0.1

0

FIGURE 9.5. Transfer functions for EEG for three dipoles using a three-sphere model of the head. Two casesof frequency-dependent behaviour are considered. In the first case the conductivity of the brain compartment ischose n according to Fig. 9.4. In the second case the conductiv ity of all three compartments increases linearly.

Fat cells

•

?FNerve cell

blood cells

o

0 -muscle cells

skin cells

r----~--r=' .----

Zygote

FIG URE 9.6. Cells vary considerably in shape.

The ElectricalConductivity of LivingTissue

TABLE9.3. Values ofacellas foundin theliterature(adaptedfromKotniket al., 1997)

291

cell values

extracellular mediumconductivity ae

membraneconductivity amcytoplasmicconductivity a i

cell radius RAverage membranethickness: t = 5 nm

lower limit

5.0 x IO-4Sm- 1

1.0 x IO-8Sm- 1

2.0 x IO- 2Sm- 1

1 11m

higher limit

2.0 Sm- I

1.2 x IO- 6Sm-1

1.0Sm- I

10011m

are surrounded by a membrane 5 to lOnm thick, visible only with the electron microscope.The electrical properties of the cell components are different. The highest and lowest valuesreported in the literature for the electrical properties of biological cells are given in table 9.3.

The intracellular fluid accounts for about 70 percent of the inner cell volume. Themembranes maintain the integrity of the cell, breaking down after death. Membranes arehighly selective permeability barriers and consist mainly of lipids and proteins. Lipid bilayermembranes have a very low permeability for ions and most polar molecules. An exceptionis water that easily crosses such membranes, and creates a balance in the whole organism.An ion such as Na+ crosses membranes very slowly because the removal of its shell of watermolecules is highly unfavorable energetically. Membranes contain specific channels andpumps that regulate the molecular and ionic composition of the intercellular compartment.A nerve impulse, or action potential is mediated by transient changes in Na+ and K+permeability. An action potential is generated when the membrane potential is depolarizedbeyond a critical threshold value. When the transmembrane potential is not exceeding thethreshold value, the relationship between the potential and the current is approximatelylinear, so it obeys Ohm's law. For low frequencies, this is the case when the current densityis smaller than 0.5J.LNcm2•

9.2.1.2 Volume fraction occupied by cells

One can analyse blood and determine the hematocrit content (the volume fractionoccupied by the red blood cells) by measuring the electrical conductivity of blood. Theelectrical conductivity is a function of the hematocrit. This function is given in Fig. 9.7.

9.2.1.3 The extracellularfluid

Mammals have a water content of 65 to 70 percent. In the early fetal period of humans,approximately 95 percent of the fetus is water. The proportion of total body weight that iswater decreases throughout the fetal period to reach 75 percent at term. The water content ofvarious tissues ofadults is given in table 9.4. Biological tissues are inhomogeneous materialswith discrete domains: the extracellular and the intracellular space. Although the cells are ofmicroscopic size, still they are much larger than the ions in the extracellular space so that theextracellular fluid can be considered as a continuum. The extracellular and intracellular fluidsare electrically neutral, however, the ion concentrations are quite different in each. Thus,the electrical properties of the intracellular fluid and the extracellular fluid are different.


TABLE 9.4. Water content of various organs (Pehtigand Kell, 1987; Foster and Schwan, 1986)

tissue type volume fraction of water

gray matter (brain)white matter (brain)skeletal muscleFatLiverSpleen

0.840.740.7950.090.7950.795

X 10.5

16.--- - -.- - - ,.-- - --.-- - - ,...-- - --,- - - ....,

14

12

E 10....~

~'s 813:::J

"Cc:8 6

.4

,,~

" " .... ", ... ... .... .., ... ... ..

706030 40 50percentage hematocrit

202l....-----l.-----1 -L-__----L ....L..-__---I

10

FIGURE 9.7. The conductiv ity of blood as a function of the percentage of red blood cells (i.e. the hematocri t).The dashed line is the measured curve , the solid line is calculated using Maxwell's mixture equation, the line withpoints and dashes is calculated using Archie's law.

9.2.2 CONDUCTIVITIES OF COMPOSITES OF HUMAN TISSUE

A composite cell includes a nucleus, cytoplasm, and a cell membrane. The nucleusis enclosed by a thin cell membrane. The cytoplasm is a mass of fluid that surrounds thenucleus and is encircled by the plasma membrane. Within the cytoplasm are specializedstructures called cytoplasmic organelles. In other words, the structure of a cell is such thatthe conductivity cannot be expected to be homogeneous. The effective conductivity is the


E

FIGURE 9.8. Model used to calculate the effective conductivity of a spherical cell.

conductivity ascribed to an entire cell. In the next section this effective conductivity isestimated using a simple model for the cell.

9.2.2.1 Effectiveconductivity ofa sphericalcell

The effective conductivity of a cell at low frequencies and for low current densitieswill be estimated for a spherical model of a cell. In the model, a sphere of radius rj andconductivity (Tj is surrounded by a thin concentric shell of thickness t and conductivity (Tm

representing the membrane. The cell is suspended in a medium with conductivity (Te. Themodel is depicted in Fig. 9.8. Typically, tis Snm for a cell and its radius R = r, + tis 10/-Lm,thence t « R.

The law of conservation of charge in quasistatic approximation reads:

(9.2)

Inserting J= (TEand E= - VV yields for a homogeneous region where a is constantLaplace's equation

(9.3)

We assume that a homogeneous electric field E is applied along the z-axis. The electricalpotential outside the two-layered particle can be calculated by solving Laplace's equationwith the proper boundary conditions. The boundary conditions are that the potential andthe normal component of the current density are continuous across the boundary. On theother hand we can calculate the potential outside a homogeneous sphere of conductivity(Teff suspended in a medium of conductivity (Te. The effective conductivity of the cell isthe conductivity of a uniform sphere that gives the same electrical potential outside thecell as the two-layered one. Equating the two solutions for the potential and neglectingthe higher order terms of tIR in both the denominator and numerator as t « R, yields

294

I I I I I \ ,. ,

, ..

M. J. Peters, J. G. Stinstra, and I. Leveles

FIGURE 9.9. Thecurrentdensity around a sphericalcell.The applied fieldwas initially uniform.The intracellularcurrent density is too small to be depicted.

(Takashima, 1989)

(9.4)

Inserting the values given in table 9.3 shows that cells can be described as non-conductingparticles because the effective conductivity of a cell is about 10- 5 times that of the surrounding fluid. In good approximation, at low frequencies the currents flow around the cellsrather than through them as shown in Fig. 9.9. Thence, at low frequencies, the conductivityis dominated by the conductivity of the extracellular space .

9.2.2.2 Effective conductivity ofa cylindrical cell

Similar calculations can be carried out to deter mine the effective conductivity of acylindrical cell. We assume that a current is applied perpendicular to the axis. The effectiveconductivity reads

(9.5)

leading to the conclusion that in this case the cylindrical cell like the spherical one can beconsidered as a non-conducting particle.

However, if the field is applied parallel to the axis it is a different situation. The cellhas a resistance for a current parallel with the axis. The resistance of an element of material


of length L and cross section A is

R = L/(aA)

295

(9.6)

The cylindrical shell of thickness t, that represents the membrane, is connected in parallelwith the inner cylinder of radius (r - t). The part of the membrane at the top and that at thebottom of the cylinder are connected in series with the intracellular fluid of the cell. So theresistance reads

1

Reff

Jl' (r - t)2

L - 2t 2t--+-

aj am

(9.7)

where the effective conductivity is defined as the conductivity of an element of homogeneousmaterial with the same dimensions, that has the same resistance as the inhomogeneouselement. In good approximation expression (9.7) leads to the expression

(9.8)

Expression (9.8) is only true when the cells are intact, i.e. the membranes at the bottom andtop of the cylinder are not damaged.

9.2.2.3 Conductivity ofextracellularfluid

Since the intracellular space is not playing a role, as the effective conductivity of acell is in general zero for low frequencies, the intracellular fluid is not considered herefurther. Materials with the highest conductivity are fluids with a low concentration ofcells like urine, amniotic fluid, cerebrospinal fluid and plasma, which will be discussedbelow.

Using the ionic concentrations of the different cations and anions in the extracellularfluid, the conductivity can be approached by

(9.9)

where Ai are the ionic conductivities at 37°C and Cj are the molar concentrations of thedifferent ions. In table 9.5 the ionic conductivities of various ions at a temperature of 37°Care given as well as the ionic concentrations in the interstitial fluid, the cerebrospinal fluidand the blood plasma. Using these values, the conductivity of these fluids can be estimated.In table 9.6 the computed values are compared with the values cited in literature. From thesevalues it becomes apparent that the computed values are about 15 to 25 percent higher thanthe measured values. This overestimation may be explained by the presence of proteins inthe actual solution and the presence of counterions surrounding its cells.

The extracellular concentration of ions such as K+ undergoes frequent small fluctuations, particularly after meals or bouts of exercise. An exception is the brain; if the brain


TABLE9.5. The ionic concentration mmoIll of differentfluids encounteredin the body and the specific conductivity of each ion in a dilute solution at37°C.The valueswereobtainedfromAseyev(1998). "Thesevalueswereonlyavailable for 25°Cand have beencorrectedby 2 percent/tC

Ion blood plasma interstitial fluid cerebrospinal fluid Ai 1O-4Sm2 jmol

Na+ 142 145 153 63.9K+ 4 4 2 95.7Mg2+ 2 I 3 142.4Ca2+ 5 4 3 154.9Cl- 102 116 123 95.4HC03 26 29 55*

PO~- 2 2 296*protein 17.0 0.0Other 6.0 6.7

TABLE9.6. Calculated and measured values of the conductivityof body fluids at 37°C. The measuredvalues are taken from literature: "Geddes and Baker (1967), "Schwan and Takashima (1993),cBaumann et al. (1997)

conductivity of body fluids interstitial cerebrospinal(at 37°C in S/m) blood plasma fluid fluid

computed 2.08 2.22 2.12measured 1.58" 2.0b 1.79c

were exposed to such fluctuations the result might be uncontrolled nervous activity, becauseK+ ions influence the threshold for the firing of nerve cells. The conductivity of electrolytesis dependent on the temperature. The conductivity of the amniotic fluid was measured by DeLuca et al. (1996) at a temperature of 20°C. The mayor contributors to the conductivity arethe Natand CI- ions. The ionic conductivity of the former increases by 2.1 percent per °Cand that of the latter by 1.9 percent per "C. An overall increase of 17 x 2 = 34 percent forthe amniotic fluid heated from 20 to 37°C is thus expected. The same rate of increase wasalso obtained by Baumann et al. (1997), who measured the conductivity of cerebrospinalfluid at 25 and 37°C.

9.2.3 MAXWELL'S MIXTURE EQUATION

Spherical particles suspended in a solvent are considered to be a relevant model ofbiological tissues. Maxwell (1891) derived an equation for the effective conductivity ofdilute suspensions in aqueous media of spherical particles (that do not have a permanentelectric moment). The derivation of this equation is given in short below. Next, Maxwell'sequation is extended for ellipsoidal particles, as many biological cells are better describedby ellipsoids instead of spheres. The applicability of Maxwell's equation will be illustratedfor blood.


FIGURE 9.10. A spherical particle of conductivity apart surrounded by a solvent with conductivity a e placed in

an original uniform field E.

9.2.3.1 A dilute solution ofspheres

First, let us consider a spherical particle of radius R and conductivity apart embedded ina homogeneous solvent with conductivity a; when a uniform electric field Eo is applied. Anexpression for the potential outside the sphere is found by solving the Laplace equation withthe proper boundary conditions using spherical co-ordinates. The origin of the co-ordinates(r, e, rp) is at the center of the sphere . This yields for the potential V outside the sphere ofradius R:

(ae - apart R3

)V(r, e) = - 1 + 3 Eorcose

2ae + apart r(9.10)

Within the sphere, the field is parallel and uniform. Outside the sphere, the field is theoriginal field plus the field of a dipole at the center of the sphere (see Fig. 9.10).

Second, the suspension is modeled by N small spherical particles of radius R andconductivity apart that are surrounded by a large spherical boundary of radius R' as depictedin Fig. 9.11. Two assumptions are made. First, the volume fraction occupied by the cells isassumed to be low, the average distance between them being larger than their dimensionsand as a consequence the spheres do not influence each other although they act as dipoles .The potential is the sum of the potential due to N small homogeneously distributed particles,yielding:

(ae - apart R3

)V(r, e) = - 1 + N , 3 Eorcose.

2ae -r-apart r(9.11)

Instead of the microscopic point of view, we can look at the sphere from a macroscopicpoint of view. We have a spherical shaped medium consisting of an aqueous solution ofspherical particles. The effective conductivity of this sphere is per definition the conductivitythat gives a potential outside of the sphere that is expressed by (9.10). Hence, the potential at


p

FIGURE 9.11. The model used for the derivation of the Maxwell mixture equation.

a distance r > R' from the center of the sphere describing the solution reads:

(ae - aeff R/3 )

V(r, B) = - 1 + -3 EorcosB2ae + aeff r

(9.12)

where aeff is the effective conductivity of the large sphere containing N particles.Formulas (9.11) and (9.12) are equivalent expressions for the potential outside the

sphere with radius R'. Noting that N R3/ R'3 = P is the volume fraction occupied by theparticles that are suspended in the large spherical boundary we obtain by equating expressions (9.11) and (9.12) the so-called Maxwell's mixture equation:

(9.13)

As argued before at low frequencies and low current densities, the currents will in generalbe only in the extracellular space, in that case one can insert apart = O.

A spherical model is a good approximation for many colloidal particles, includingbiological cells. However, biological cells often have a complex geometry. Many cells arebetter described by ellipsoids; a, b and c are the semi-axes of the ellipsoid. Maxwell'smixture equation has been derived for ellipsoidal particles that are orientated in parallel bySillars (1937). When p is very low and the field is applied along the a-axis, it reads:

[_ -,-p_(a-,-part_-_a_e_)_]

aeff = ae 1+a« + (apart - ae)La

(9.14)

where La is the depolarization factor of the ellipsoid in the direction of the a-axis, (Boyle,1985).

abc ['JO ds

La = 2 10 (a2 + s)J(a2 + s)(b2 + s)(c2 + s);La + L, + L, = 1 (9.15)


0 .9

0 .8

0.7

0.6

~ 0.5

0.4

0.3

0.2

0 .1

00 2 3 4 5 6 7

alb

299

FIGURE 9.12. Thedepolarization factor in thedirectionof thea-axisforanellipsoidof revolution withsemi-axesa, b, and c, whereb = c.

There are three cases of considerably interest for us: spheres, needle-shaped ellipsoidsand disk-like particles. For a needle-shaped ellipsoid or a particle with the shape of a longcylinder with a = b « c, La,L, and L, tend to 1/2, 1/2 and 0, respectively. For a sphere witha = b = c, the depolarization factor La = 1/3. For a disk-shaped ellipsoid with a = b » c,La, t., and t, tend to 0, 0 and 1 respectively (Fricke, 1953). In Fig. 9.12 the depolarizationfactor La is depicted for a spheroid with semi-axes a, b = c.

Maxwell's mixture equation has been experimentally tested by Cole et al. (1969) forvariously shaped objects. The particles could be spheres, cubes, or cylinders arranged ina cubic or hexagonal array or randomly distributed plates. They found that the Maxwellequation was surprisingly accurate within one percent at concentrations from 30 percent orless to 90 percent.

9.2.3.1.a. Blood

To illustrate the applicability of Maxwell's mixture equation, the conductivity of bloodis discussed. Blood of normal subjects consists predominantly of red blood cells (erythrocytes) in plasma. The mature erythrocyte is a cell surrounded by a deformable membranewell adapted to the need to transverse narrow capillaries. The red cells are biconcave disks,each with a diameter of about Sum, a thickness of 2fJ..m at its edge, and a volume of about94fJ..m3 . In normal adults, the red cells occupy on the average about 48 percent of the volumeof blood of males and about 42 percent in the blood of females. The percentage of the volumeof blood made up by erythrocytes is defined as the hematocrit. Erythrocytes are essentially


TABLE 9.7. Parametervaluesusedtoestimate theconductivityperpendicularto the fibers and thatparallel to the fibers. ("Kobayashi and Yonemura, 1967)

parameter symbol minimum maximum

effective intracellular conductivity a j 0.55 S/m 0.80 S/mextracellular conductivity a. 2.0 S/m 2.4 S/mmembrane conductivity a m 1.0 X 1O-8S/m 1.2 X 1O- 6S/m

volume fraction occupied by fibers p 0.85 0.9"fiber length L 5 mm 30cmmembrane thickness t 0.1 lim I lim

non-conducting in fields with frequencies up to 100kHz (Trautman and Newbower, 1983).A typical value for the low-frequency conductivity of human blood at body temperature andnormal hematocrit is in the range 0.43-Q.76S/m (Geddes and Baker, 1967). The orientationof the ellipsoids with respect to the electric field depends on the blood flow. There is aperiodic noise component in continuous measurements of conductivity, which is presumedto result from the cyclic reorientations due to the flow. The magnitude of the change withflow is of the order of three percent of the baseline conductivity. Since the shape of redblood cells of most species is non-spherical and is subject to changes in aggregation andorientation with flow, variations in the effective conductivity are understandable.

Geddes and Sadler (1973) measured the conductivity for human , canine, bovine andequine blood at 25kHz and 37°C, having a hematocrit range extending from 0 to 70 percent.For human blood they found an exponential fit between the measured data and an exponentialcurve for p = 53.2eo.022H with a correlation coefficient of 0.99. Blood can be modeled as adilute solution of non-conducting spherical particles within a conducting medium . In Fig. 9.7both the experimental measurements and the results using Maxwell 's mixture equation forspherical particles are displayed. From this figure one can conclude that the description ofMaxwell is rather accurate. With increasing concentration of red blood cells (p is larger)the effective conductivity is reduced. The relationship between p and the conductivity canbe used to estimate the hematocrit content (Ulgen and Sezdi , 1998).

9.2.4 ARCHIE'S LAW

Measurements of the conductivity of the cortex do not fit so well with the theoryof Maxwell, they fit better with Archie's law (Archie, 1942; Nicholson and Rice, 1986).Moreover, Maxwell's theory can only be applied if there is only one type of cell presentand most human tissues are constituted of various types of cells . Archie's law that willbe discussed in the next section is applicable if there are various types of densely -packednon-conducting cells suspended in a conducting solution.

To derive Archie's law the following assumptions are made . a) Tissue is modeledby cells of various shapes that are suspended in an aqueous conducting medium; b) Theeffective conductivity is calculated for a region containing thousands of particles; c) Theparticles are assumed to be homogeneou sly distributed; d) Currents are Ohmic and areonly present in the extracellular space; e) The solvent is homogeneous and isotropic ; f) Anelectric field is applied which is initially uniform; g) The particles do not have a permanentelectric dipole moment.


9.2.4.1 Ellipsoidal particles withthe sameorientation

The derivation of an equation for the effective conductivity for higher values of p isgiven by Hanai (1960). It starts with equation (9.14). Subsequently, the number of particlesis increased. Due to this increase, the effective conductivity aeff increases with a fraction~aeff' In the next stage the previous mixture acts as a host, and so on. An expression for theeffective conductivity of a densely packed suspension of ellipsoidal particles is obtained byintegration yielding:

( )

L'aeff - apart a e----'----- - =l-pa e - apart aeff

Inserting in this equation apart = 0 yields Archie's law:

(9.16)

(9.17)

where ae is the conductivity of the fluid surrounding the non-conducting cells, p is the volumefraction occupied by the cells, and m is the so-called cementation factor that depends on theshape and orientation of the particles, but not on their sizes. According to Archie's law, theeffective conductivity is proportional to the extracellular conductivity o.: The value of mdepends on the shape of the cells. For instance, its value is 3/2 for spherical particles, m is2 for long cylinders with the axis perpendicular to the external field, and it is 1 for cylinderswith the axis parallel to the field. Archie's law also holds for ellipsoidal particles. If the fieldis applied along the a-axis of the ellipsoids and the ellipsoids have the same orientation,

(9.18)

9.2.4.1.a. Fat

Since fat tissue consists of about 9 percent water (Foster and Schwan, 1989) andthe interior of the cells is almost completely filled with fat, the interstitial fluid occupiesabout 9 percent of the volume (upper limit). Histology shows that fat cells are sphericalshaped particles. According to equation (9.17) an upper limit for the conductivity is 1.9 x(0.09)3/2 ~ 0.05 S/m.

9.2.4.1.b. Skeletal muscles

A skeletal muscle fiber represents a single cell of muscle. Each skeletal muscle fiber is athin elongated cylinder with rounded ends that are attached to connective tissues associatedwith the muscle. Just beneath its cell membrane, the cytoplasm of the fiber contains manysmall nuclei and mitochondria. The cytoplasm also contains numerous threadlike myofibrilsthat lie parallel to the cylinder axis. The diameter may vary within muscles, between musclesin the same animals, and between species. Muscle fibers increase in diameter from birth tomaturity and also in response to exercise. The length of a fiber may vary between millimetersand several tens of centimeters. The amount of connective tissue relative to muscle fibers ismuch greater in some muscles than in others and may range from 3 to 30 percent. Connectivetissue is composed of collagen fibers, reticular fibers, elastic fibers and several varieties ofcells, such as fat cells. There is an increase in elastic tissue with aging. Apart from the

fibers in the tongue, the fibers have no branches. In long muscles , as in the longest musclein the human body the sartorius (52cm long) , the fibers are arranged in parallel. Coveringthe surface of each muscle fiber is a thin membrane of about O.I/-Lm thickness. The modelused to describe muscle tissue consists of homogeneously distributed cylinders. So far asconnective tissue, blood vessels, and nerve tissue cannot be described by parallel cylinders,their influence will be neglected.

The effective conductivity ofone fiber when the field is applied parallel to the axis is according to formula (9.9) dependent on the length ofthe fiber. Inserting the values ofTable 9.7in formula (9.8) yields 0.25 x 1O-5S/m (short fiber) < apart < 15 x 1O- 2S/m (long fiber) .When the fiber is damaged (forinstance, Mc Rae and Esrick (1993) trimmed the fibers) thenaj may be as high as the effective intracellular conductivity. Currents parallel to the fiberswill be in both domains, the intracellular and extracellular space are connected in parallel.The effective conductivity of the muscle tissue will be given by (1 - p) aextracell + paeff.

When the current is applied perpendicular to the fibers, the cell can be described as anon-conducting cylinder. The effective conductivity perpendicular to the fibers at accordingto equation (9.17) will be within the limits 2.0 x (0.1)2 S aI,eff ::: 2.4 x (0.15)2. Thus theeffective conductivity perpendicular to the fibers will be in the range of 0.02-0.05S/m.

9.2.4.1.c. Cardiac tissue

Cardiac cells can be described as cylinders with a diameter of about 15/-Lm and a lengthof about 100/-Lm. The currents perpendicular to the fibers will circumvent the fibers. Theeffective conductivity in the direction perpendicular to the fibers can be calculated usingexpression (9.17) with m = 2. The volume fraction occupied by the fibers reported byClerc (1976) is p = 0.7. The conductivity of the extracellular space is assumed to be that ofRinger (p = 69Qcm) yielding a conductivity of I.4S/m. So the conductivity perpendicularto the fibers is estimated to be 1.4 x (0.3)2 = O.13S/m. Rush et ai. (1963 ) measured theconductivity of cardiac tissue and found for the conductivity perpendicular to the axis0.18S/m.

Currents parallel to the fibers will be present both in the extracellular and in theintracellular space. Cardiac cells are joined at their ends by intercalated disks and eachcell is connected to its neighbors by gap junctions passing through these disks. Cardiacmuscle fibers can be modeled by cylinders that are interconnected by junctions as depictedin Fig. 9.13. All cylinders are homogeneously distributed and are arranged in parallel. Thecurrent is applied parallel to the axes, such that the volume between the electrodes used toapply the current contains 104 fibers or more. Each cylinder has a length of lOO/-Lm anda radius of 7.5/-Lm. All parameters are chosen conform the parameters chosen by Plonseyand Barr (1986) from literature. These values were measured in mammalian ventriculartissue at a temperature of about 20°e. The resistance of a junction is Rj = 0.85 x lO6Q.The resistivity of intracellular fluid is 282Qcm (Chapman and Frye, 1978). Consequently,the resistance of a cylinder with a length of lO-2cm and a diameter of 15/-Lm of intracellularfluid is

The junction and the intracellular space are connected in series , so the resistance of a single


Current

Cross section

,,

ae=2 S/~,,

,

Electricalscheme

p =0.7

1.p =0.3

FIGURE 9.13. Model of cardiac tissue.

cylindrical cell is

Rcell = R, + Rj = (0.85 + 1.60) x 106Q .

A homogeneous cylinder with the same dimensions will have the same resistance Rcell,when an effective conductivity aeff is ascribed to the entire cylinder.

aeff = Ij(RcellO), where 1= 10-4m, O = n(7.5 x 10-6)2, yielding aeff = 0.23Sjm.

The two composites of the cardiac tissue, namely the intra- and extracellular medium areconnected in parallel. Thus, the effective conductivity along the fibers for cardiac tissue is

(1 - p)aextracell + paeff = 0.3 x 1.4 + 0.7 x 0.23 = 0.58Sjm.

Rush et al. (1963) found for the conductivity parallel to the axis of cardiac fibers a value ofO.4S/m.

9.2.4.2 Randomlyorientatedellipsoidal particles

The architecture of human tissue may be such that fibers and cells are oriented alongeach other. For instance, skeletal muscle fibers are more or less parallel. However, theassumption that the cells or fibers have the same orientation is often not plausible. Forexample, the red blood cells in blood outside of the body can roughly be described by oblate

304

E


4 r-- --r-- - r-- --r-- - ..---- ---,-- - ....--- ---"T- -----,

3 .5

3

2.5

FIGURE 9.14. The cementat ion factor for a solution of randomly orientated spheroids with semi-axes a. b. andc, where b = c.

spheroids (ellipsoids with a < b = c) that are randomly oriented . Boned and Peyrelasse(1983) derived an expression for a solution of randomly orientated ellipsoidal particles.Every orientation of the particle has the same probability and therefore the Hanai procedureis performed such that infinitesimal amounts of particles are added such that 1/3 of theellipsoids have their a-axes in the direction of the electric field, 1/3 their b-axes and 1/3their c-axes. For non-conducting particles again Archie's law is found with a cementationfactor

(9.19)

The low-frequency conductivity of suspensions of ellipsoidal particles and that of doubletshaped (budding yeast cells) or biconcave particles (erythrocytes) differs only a few percentas can be concluded from simulation studies using as numerical method the boundaryelement method (Sekine, 2000).

In Fig. 9.14 the cementation factor is depicted for an ellipsoid of revolution with semiaxes a, b and c, where b =c. As can be seen m is minimal for La = 1/3, the depolarizationfactor for a sphere, in that case CTeff is maximal.

9.2.4.2.a. Blood

Archie's law can be applicated for blood. Fitting the measured data for human bloodof Geddes and Sadler (1973) to Archie's law leads to a fit for m = 1.46 (see Fig. 9.7). Forblood of different animals, a slightly different cementation factor is found .


9.2.4.3 Cells ofdifferent shape

Human tissue can often be described by particles of different shapes that are immersedin an aqueous medium. The cortex, for instance , has pyramidal cells and glial cells . Thesizes of the particles do not play a role in the theory derived. A different shaped ellipsoidalparticle is one that has a different ratio of the radii along the three axes. In order to obtain anexpression for a suspension of particles of different shapes, we applied the Hanai procedureby adding successively infinitesimal numbers of particles in proportion to their relativevolume fractions , leading again to Archie 's law

n

. h "PiWIt m = L..., mj-i= 1 p

I(C I)and m, = - ,,--3 L...,1-L-

J=a J

(9.20)

where n is the number of types of non-conducting ellipsoids, Pi the volume fraction ofa particle of type i, p the sum of all volume fractions occupied by particles and Lj thedepolarization factor in the direction of one of the axes (Peters et al., 2001).

9.2.4.3.a. Gray matte r

As an example , the effective conductivity of the superficial cortex is estimated, usingparameters found in the literature. The principal neuron types in superficial cortex arepyramidal cells in layer II and stellate cells in layer III. Axons and dendrites from deeperlayers and axons from subcortical layers occupy a substantial portion of the tissue. The glialcells of superficial cortex are mostly star-shaped cells, i.e. astrocytes and a lesser fraction ofoligondendrocytes. The latter resemble astrocytes, but are smaller and have fewer processes.A branching astrocyte process has a length of about 40 microns and an average diameterof about one micron. The shape of the glial cell body in vitro is an intermediate between aflat disk and a sphere. The pyramidal cells have a cell body and dendrites. The cell bodyhas a roughly pyramidal shape. The branching properties of dendrite s of neurons show amean branching angle of about 60 degrees. The diameter of motoneurons is reported as7 to 14f.lm for the proximal segment with decreasing values of about 5, 3, and 2f.lm forsuccessive branches. The fact that solutes of high molecular weight readily pass betweencerebrospinal fluid (CSF) and interstitial fluid supports the supposition that no significantconcentration gradients exist between the two compartments in the steady state. Hence theCSF reflects the ionic composition of the interstitial fluid and the conductivity of the CSFwill be identical to that of the interstitial fluid i.e.1.8S/m (Baum ann et al., 1997).

The gray matter is composed of glial cells that are modeled as spheres, occupying avolume fraction of 38 percent and pyramidal cells that are modeled as cylinders, occupyinga volume fraction of 46 percent (Havstad, 1967). Both neurons and glial cells are describedby non-conducting particles, i.e. a particle = O. The effective conductivity is computed usingequation (9.20) yielding a cff = 0.097S/m. A measurement that used a uniform current todetermine the conductivity of gray matter has been carried out by Ludt and Hermann (1973),who reported a value of 0.10S/m for the rabbit. However, the measurements were carriedout in vitro 15 minutes after death and at room temperature. As found by van Harreveldand Ochs (1956), the conductivity drops after circulatory arrest by 30 to 35 percentdue to the emptying of blood vessels and drainage of fluid. Thus Ludt and Hermann's


measurements indicate that the effective conductivity of cortical tissue in vivo will be about0.15S/m. Archie's law led to a value of about 0.10S/m. However, the values used for thecalculation may be not optimal. The volume fraction occupied by the extracellular fluidis in the range 17 to 28 percent (van Harreveld et aI., 1965; Nicholson and Rice, 1986).The interstitial space and the vascular volume together constitute the extracellular space.The vascular space has a volume of about 1 to 3 percent. So the value of (l - p) in practicewill be somewhat higher than the value handled by us in the computation of the effectiveconductivity.

9.2.4.4 Clustered cells

When spherical particles are clustered in a chain or in a close-packed lattice, theconductivity is such that these clusters behave as randomly orientated ellipsoids and thusArchie's law will be obeyed (Grandqvist and Hunderi, 1978).

9.2.4.4.a. Blood

Blood with a high hematocrit content may aggregate. Pftitzner (1987) measured theconductivity of blood samples containing red blood cells of varying diameter between100Hz and 100kHz. The conductivity was essentially independent of the diameter of thecells and the frequency. He found that at a hematocrit of60 percent or more that the dielectricconstant shows a distinct decrease. This was explained by the aggregation of blood cells.An aggregation in a single cell chain, for instance, would lead to effective depolarizationfactors La = 0.13; Lb = L, = 0.435, leading according relation (9.19) to m = 1.6. In suchcase, the effective conductivity at 80 percent hematocrit would practically be the same aswithout aggregation (2 percent difference).

9.2.4.4.b. Liver

The liver consists of different types of cells. These can be divided in hepatocytes andnon-hepatocytes (Raicu et al., 1998a) . But since the volume fraction of non-hepatocytes(0.06) is relatively small to that of the hepatocytes (0.72) a model is used consisting ofone cell-type. The hepatic cells are clustered forming plates of one layer in thickness (seeFig. 9.15). The close-packed clusters of cells are assumed to act as oblate spheroids with axes

FIGURE 9.15. A schematic drawing of a part of a lobule of the liver.


Epiderm is

Dermis

Fat

FIGURE 9.16. A schematic drawing of the layered human skin.

a, b, c, where a = b » c. Choosing the effective depolarization factors La, Lb, and L, tobe 0.1,0.1 and O.S, respectively and applying Archie's law with p = 0.72 and as = 0.65S/m(the conductivity of blood) results in O'eff = 0.03S/m. This finding is in accordance withfindings of Gabriel et al. (1996a) who reported values in the range of 0.02-o.03S/m.Choosing La, Lb, and L, to be 0.2, 0.2 and 0.6 leads to an effective conductivity ofO'eff = O.OSS/m, which is more in accordance with the values given by Geddes and Baker(1967) and Raicu et al. (199Sb) .

9.3 LAYERED STRUCTURES

Many tissues are organized in layers, such as the gastrointestinal tract, the retina, orthe gray matter. An example of a layered structure is shown in Fig. 9.16, where the scalp isdepicted. At an interface current lines will change direction. Corning from a medium withhigher conductivity to a medium with lower conductivity, current lines will bend in thedirection of the normal. Coming from a medium with lower conductivity to a medium withhigher conductivity, current lines will bend from the normal. As a consequence, currentstend to cross the skull in a direction perpendicular to its surface. In other words , the layersof the skull are approximately traversed in series. Currents in the scalp tend to be parallelto its surface and the layers of the scalp can be approximated by a parallel connection ofresistances. The simulations discussed in sections 3a and 3b show that this is indeed thecase .

9.3.1 THE SCALP

The outer layer of the scalp is the epidermis of about 0.2mm thickness. This layer is nonsensitive and non-vascular and overlies the dermis. The epidermis is a poorly conductive


layer; this is ascribed to the dead nature of one of the layers within the epidermis, thestratum corneum. The conductivity of the epidermis is approximately 0.026S/m (Sernrovet al., 1997). The dermis is connective tissue with a thickness of 2rnrn and its conductivityis estimated to be 0.22S/m (Yamanoto and Yamanoto, 1976). The basis of the dermis is asupporting matrix with a remarkable capacity for holding water. The dermis has a very richblood supply. At the side of the head, subcutaneous fat is found of about 3mm thickness.Sernrov et al. (1997) used a conductivity value of 0.08S/m for this layer.

To assess the effective conductivity of the scalp, a simulation is performed with aspherical volume conductor. Two models are used. The first model consists of five shellsrepresenting the brain, skull, fat layer (3mm), dermis (2rnrn) and the epidermis (0.2rnrn).The conductivities of the three layers are mentioned above. The radii and conductivity of thebrain and skull are 78 and 83mm, and 0.33 and 0.0042S/m, respectively. The other modelhas 3 shells: the brain, skull and the skin. The radii and conductivities of the brain and skullare equal to those of the first model. The thickness of the skin layer is chosen to be the sumof the thickness of the fat layer, dermis and epidermis. The effective conductivity of theentire skin layer is calculated (assuming that the currents are crossing the scalp such thatthe layers are approximately connected in parallel) by means of

(9.21)

where d, and a i denote the thickness and conductivity of layer i. This yields a value of0.13S/m for the effective conductivity of the skin. A current dipole on the z-axis is used assource. The potential is calculated at the outer sphere by means of an analytical expression(Burik, 1999). The potential using the three-shell model is compared with the potentialusing the other five-shell model. The differences are expressed by the relative differencemeasure (RDM) defined as

RDM= (9.22)

where Vi is the calculated value of the potential for the three-shell model, Vc,i is thecalculated value of the potential for the five-shell model and N is the number of pointswere the values are calculated, The calculations are repeated varying the eccentricities andorientations of the dipole, and the ratio of the thickness of the three layers of the skin. Someresults are shown in Fig. 9.17. It can be seen that equation (9.21) can be used to calculatethe effective conductivity of the scalp when the thickness of the various layers is known.If these thicknesses are not known then a = 0.13S/m will be an appropriate choice for theeffective conductivity of the scalp.

9.3.2 THE SKULL

Another example of a layered structure is the skull that can also be subdivided intothree layers. The upper and lower layers are structures made out of bone, which are badconductors. The middle layer is a relatively good conductor, because it is spongy and


6,95E-03

:i!: 6,80E-03

c~

6,65E-Q3

-- radial dipole

.. . .. annential

6,50E-Q3 -+--...,....----r-- ----,-- ---,----,- - ,----.,--...,.......- -'0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 o.s

eccentricity

FIGURE 9.17. The RDM between the scalp potential calculated using an effective conductivity and the onecalculated for a skin composed of 3 layers, namely the epidermis, the dermis and the fat.

contains blood. The total thickness varies across the skull between 4 and 11mm, the mean isaround 6mm. The skull has a conductivity in the direction perpendicular to the surface of afactor 10 smaller than in the direction parallel to the surface. Usually, a value of 0.0042S/mis assigned to the conductivity of the entire skull, although recent measurements of theconductivity of the skull using bone that was temporarily removed during epilepsy surgeryled to values that were a factor ten higher (Hoekema et al., 2001).

Simulations are performed with a spherical volume conductor. The volume conductorconsisting of the brain, skull and scalp is modelled by either a three-shell model in which aneffective conductivity is assigned to the skull or a five-shell model. The radii and conductivityof the brain and scalp are 78 and 89.2 mm, and 0.33 and O.l3S/m, respectively. The skullis described by three layers of conductivity of 0.0029,0.029 and 0.0029S/m, respectively.These values are chosen such that for a mean skull thickness of 6mm consisting of layersof 2mm each, the effective conductivity is 0.0042S/m.

The other model has 3 shells: the brain, skull and the scalp. The radii and conductivitiesof the brain and scalp are equal to those of the first model. The thickness of the skull isvaried between 2 and 10mm. The ratio between the thicknesses of the three layers describing the skull is varied as well. A current dipole is taken as the source, whose position andorientation are varied. As the conductivity of the skull is much lower than that of the surrounding tissues one may expect that the currents will cross the skull perpendicularly. Thisimplies that the three layers of the skull are crossed in series. If the effective conductivity isgiven by

(9.23)

then the RDM is well below 2 percent. Hence, we may conclude that the effective


conductivity of the skull is approximated quite well by expression (9.23). When realisticallyshaped models are used in the inverse solution, both the thickness and the conductivityshould vary.

9.3.3 A LAYER OF SKELETAL MUSCLE

In the ECG inverse problem the muscle layer may be considered as one of the compartments. Some simulation studies suggest that this anisotropic layer is the only thoracicinhomogeneity with a significant effect on the relationship between epicardial and torso potentials (Stanley et al., 1991). The muscles underneath the skin of the torso are essentiallydirected parallel to the body surface but otherwise almost uniformly distributed over allangles(Rush, 1967). This implies that the conductivity parallel to the body surface differsfrom that perpendicular to it. Half of the fibers that are parallel to the body surface areperpendicular to the other half. The conductivity parallel to the body surface will be aboutam = (at + ah) /2. The conductivity perpendicular to the body surface is al. We choose aco-ordinate system such that ax = ay = am and az = aj.

Expression (9.2) reads in this case:

(9.24)

If the muscle layer is modeled as layer of infinite extent with thickness t, then the properboundary conditions are that at the surface the normal component of the current density iscontinuous and the potential is continuous.

Equation (9.24) can be solved by a co-ordinate transformation from (x,y,z) to (x' ,y' ,z'),where

x' = x;y' = y and z' = zJam/al (9.25)

The scale transformation is such that the currents and the potential are chosen to be invariant.In other words they are the same at corresponding points of the primed and unprimed system.The components of the current density being a current divided by a surface transform as:

and '1 •Jz = Jz (9.26)

The electric field being the gradient of the potential transforms as:

As a consequence, in the primed system:

n2lv, = 0 and J71 ~aaE~v = = yUmUj

(9.27)

(9.28)

Hence, after transformation we have to solve Laplace's equation with the proper boundarycondition (i.e. the z-component of the current density is continuous), which is invariantunder the co-ordinate transformation. The primed system is isotropic with conductivity


-Jamal. In other words, the potential of the anisotropic muscle layer is equivalent to thepotential in a homogeneous isotropic medium with an effective conductivity -Jamal, if thethickness is enhanced with a factor -Jam/a,.

Inserting aj = 0.05S/m, ah = 0.5S/m for skeletal muscle and taking a layer thicknesst = lcm yields for the effective conductivity of the muscle layer aeff ;:::, O.IIS/m and aneffective thickness of the muscle layer of 5cm.

Stanley et at. (1986) showed that the agreement between calculated and measuredtorso potentials significantly improved if the anisotropic nature of the muscle layer wastaken into account. Their results were based on a canine study.

9.4 COMPARTMENTS

Customarily, the volume conductor is described by a compartment model. Each compartment is considered to be homogeneous. The number of compartments may vary. Tosolve the inverse problem for EEG, the head is often described by three compartments,the brain, skull and scalp. Usually, the conductivity assigned to the brain and to the scalpcompartment are identical. The geometry of these compartments may be a sphere coveredby two spherical shells or the compartments may have a realistic shape. Especially in thecase that a three-sphere model of the head is used with standard radii of the spheres, theinterfaces between the compartments will not coincide with the actual interfaces betweenthe different tissues. Consequently, a compartment will consist of more than one type oftissue. For instance, the muscles that are present in the occipital and orbito-frontal areas ofthe scalp may be partly assigned to the scalp compartment and partly to the skull compartment. The cerebrospinal fluid may be assigned mainly to the brain compartment, but alsopartly to the skull compartment. Consequently, the effective conductivities do not representthe actual conductivities of brain matter, skull bone or skin. The effective conductivities arethose values that minimize the differences between the actual EEG and the calculated EEG.

The effective conductivity of a compartment can be estimated in three ways. First, implanted electrodes can be used to act as a source. Measurements of the potential or magneticfield distribution generated by the source can be used to estimate the source. The actualsource can be compared with the calculated one. By varying the effective conductivitiesused in the models the differences between the actual source and the calculated one can beminimized. The conductivities that give the smallest differences are taken as the effectiveones. Second, the effective conductivities can be determined by fitting evoked magnetic fieldmeasurements with those of the electrical potential in case that the potentials and magneticfields are due to the same source. The third method is based on impedance tomography.

9.4.1 USING IMPLANTED ELECTRODES

Sometimes electrodes are inserted in the brain through trephine holes during presurgical evaluation of epileptic patients (Veelen et al., 1990). Normally, these electrodes areused for the measurement of the potential, especially during long term seizure monitoring.However, when a current is applied to a pair of electrodes, it will act as an artificial source. Itslocation and orientation is known from X-ray or magnetic resonance images. The artificialsource can be localized from potential measurements on the scalp. The true and calculated


location should coincide. This can be established by varying parameters of the model usedin the inverse solution, such as the ratio between the conductivities in a three-compartmentmodel of the head. Homma et al. (1994) used a realistically shaped model and found thatthe inverse solution was the best when the ratio between the conductivities was 1 : 1/80 : 1.

9.4.2 COMBINING MEASUREMENTS OF THE POTENTIAL AND THEMAGNETIC FlEW

Cohen and Cuffin (1983) measured magnetic fields and electrical potentials that wereevoked by the same stimulus. Both types of measurements were used to localize a dipolewithin a standard three-sphere model of the head. The locations coincided when usingthe ratio 1 : 1/80 : 1 for the equivalent conductivity of the brain, the skull, and the scalpcompartment. Goncalves et at. (2001) repeated these measurements in four subjects. Theyreported values for the conductivity ratio between scalp and skull that varied between 43and 85. Part of the observed variability may be ascribed to errors in the volume conductormodel, numerical errors, or errors in the measurement. The effect of these errors on thepotential distribution will differ from that on the magnetic field distribution. Part will bedue to the differences between the heads of the four individuals. In each head the distributionof inhomogeneities within compartments will be different.

9.4.3 ESTIMATION OF THE EQUIVALENT CONDUCTIVITY USINGIMPEDANCE TOMOGRAPHY

When a current is applied to the scalp surface, a potential distribution develops acrossthe head. The relation between the applied currents and the resulting potential depends onthe conductivity distribution within the head. The distribution of the internal conductivitycan be estimated from measurements of this scalp potential. This type of research is knownas electrical impedance tomography. Goncalves et al. (2001) used electrical impedancetomography to estimate the ratio between the effective conductivity of the skull and thebrain compartment for five subjects under the condition that the brain has the same effective conductivity as the scalp. They used both a spherical and a realistically-shapedthree-compartment model. For the spherical model they found for the ratio of the brain andscalp conductivities values that varied between 31 and 124. They ascribed the observed variability to geometrical errors. Differences between the actual head geometry and the threecompartment model are compensated by adjusting the values of the electrical conductivityof the compartments. These errors will be much smaller in case realistically-shaped models are used. And indeed for their realistically-shaped models the spread is decreased. Inthe latter case the ratio of the effective conductivity of brain and skull varied between 17and 65. They suggest that electrical impedance tomography should be a part of the EEGinverse problem in order to take the individual differences in effective conductivities intoaccount. Oostendorp et al. (2000) used this method to estimate the effective conductivityof the skull and scalp compartment for two subjects. Fitting their potential measurementsto the potentials computed by means of the boundary element method yielded a skull conductivity of O.013S/m and a brain conductivity of 0.20S/m. The value found for the ratio ofbrain-to-skull conductivity was 15 : 1.

Impedance tomography can also be used to estimate the effective conductivities usedin a compartment model of the torso. Eynboglu et al. (1994) used this method to estimatethe effective conductivities of the heart, lungs and body in a dog. However, the accuracy ofthe estimated effective conductivities was only about 40 percent, because the method is notsensitive for changes in conductivities in the various compartments.

9.5 UPPER AND LOWER BOUNDS

In most cases, it is very difficult to actually calculate the effective conductivity of acomposite material because oflack of detailed information on the micro-geometric structure.Based on information available, lower and upper bounds for the effective conductivity canbe given. The range between these bounds decreases with increasing knowledge about theconstituents of the composite medium.

The conductivity of a composite medium cannot be higher than that of the best conducting phase and cannot not be lower than the worst conducting phase. Thus, for a materialconsisting of n phases, denoted by the index i = 1,2,3, ... , n, with (Jj increasing with i,the limits are

(9.29)

Stricter bounds are found when apart from the conductivities also the volume fractionsPi of all phases are known. These bounds read (Hashin and Shtrikman, 1962)

(9.30)

The upper bound is, for instance, attained for a two-phase composite medium consistingof a suspension of needle-shaped particles with the applied field in the direction of theirprincipal axes. The lower bound is, for instance, attained for a suspension of thin disks thatare stacked and the field is applied perpendicular to the disks.

In case it is known that the distribution of the constituents is homogeneous and themedium is isotropic, more rigorous limits are applicable (Hashin and Shtrikman, 1962). Inthis case, the upper and lower bound for a two-phase material is given by:

(9.31)p]

---+~(J\ - (J2 3(J2

P2(J\ + --;---=----- < (Jeff < (J2 + --;--------+~ - -(J2 - (J\ 3(J]

If the composite constitutes of homogeneously distributed and randomly orientatedparticles within a conducting medium, and the particles are non-conducting spheroids(L, = Lc) , the upper and lower bounds can be obtained from Archie's law. The depolarization factors are restrained to La + L, + L, = 1 and La > 0, L, > 0, L, > O.

According to expression (9.19), the cementation factor m in Archie's law reads in thiscase:

(9.32)

314

The maximum effective conductivity is found when


a~ff m am- = O"solv(l - p) In(l - p)- = 0a~ a~

yielding an upper bound for La = L, = L, = 1/3

3/2O"crr .:::: O"solv(l - p)

(9.33)

(9.34)

In other words , the upper bound is attained when the particles are spheres. Apparently, sucha suspension has the least impact on the flow of currents through the material. The lowerbound O"cff = 0 is attained when the particles are thin disks (La = 1, L, = L, = 0). Oftenthe ratio between the axes of the spheroidal particles is not known, but it is known that a >b = c. In that case the lower bound is found for cylinders, where La = 0, L, = L, = 1/2,leading to

O"solv(l - P)5/3 .:::: O"cff (9.35)

In summary, for elongated, homogeneously distributed and randomly orientated nonconducting spheroids in suspen sion the effective conductivity is limited by the followingbounds

O"solv(l - P)5/3 .:::: O"cff .:::: O"solv(l _ P)3/2 (9.36)

In Fig. 9.18 the various upper and lower bounds are plotted for a two-phase compositemedium consisting of a non-conducting phase embedded in a conducting medium as afunction of the volume fraction occupied by the non-conducting phase.

9.5.1 WHITE MATTER

White matter appears white because it contains fiber groups that possess myelin sheathsthat consist of layers of membranes. These membranes, composed largely of a lipoproteincalled myelin, have a higher proportion of lipid than other surface membranes. The oligodendrocytes (i.e. star-shaped cells) are arranged in rows parallel to the myelinated nervefibers, with long processes running in the same direction. The white matter can be modeledas a suspension of elongated particles that may have any direct ion. The extracellular spacemeasured by van Harreveld and Ochs (1956) in the cerebellum of mice varied between 18.1and 25.5 percent with a mean of 23.6 percent. The conductivity of the interstitial fluid isthat of the cerebrospinal fluid, i.e.l .8S/m. Thus according to equation (9.36), the effectiveconductivity will be between the limits O.lOS/m < O"cff < 0.23S/m.

9.5.2 THE FETUS

In order to simulate the fetal electrocardiogram, a single compartment may describethe fetus. As no measured values for the conductivity of the human fetus are available,its conductivity is estimated. In order to be able to use the theory presented above, the


1a

x 0 21 ---- - ,- - - - -..-....,...----r-----,------,

0.2

0.11b,2b,3b,4b

0.8

:f O.7[~ 0.6"0c:80.5Ql>'U 0.4~Ql 0.3

0.2 0.4 0.6 0.8volumefraction particles(p)

FIGURE 9.18. Upper and lower bounds for a material consisting of two composites where one phase is nonconducting as a function of the volume fraction occupied by the non-conducting phase. The conductivity of theconducting phase is oz.(I a) Upper bound if the only information that is available is the value of (f 2 ; the lower bound (l b) coincides withthe x-axis.(2a) upper bound and (2b) lower bound if the only information that is available is the value of (f2 and the volumefraction p.(3a) upper bound, (3b) lower bound if the phases are homogeneously distributed and (f 2 and p are known.(4a) upper bound and (4b) lower bound if the non-conducting phase consists of spheroidal, homogeneouslydistributed particles and the values of (f2 and p are known.(Sa) upper bound and (5b) lower bound if the non-conducting phase consists of elongated particles that arehomogeneously distributed and randomly orientated and the values of oz and p are known.

fetus is assumed to be a homogeneous conductor. It is assumed that the cells in the fetusare homogeneously distributed and randomly orientated and have a shape somewhere between a sphere and a cylinder. Looking at the histology of the fetus most tissues consist ofelongated spheroids or spheres. Disc-like cells are less commonly encountered. Based onthese assumptions the conductiv ity of the fetus can be estimated to be between the limitsoAI - p)5/3 :s afetus :s ae(1 - p)3/2. The volume fraction of the extracellular space at theend of gestation is about 40 percent of the total body volume (Brace, 1998; Costarino andBrans, 1998). The extracellular space includes besides the interstitial fluid the fluids in thebody cavities like the cerebrospinal fluid and the blood plasma . The blood plasma is about18 percent of the total extracellular water content. As the fetus is considered as one singleentity, there is no objection in taking all the extracellular fluid in account in estimatingthe conductivity, as all contribute to the conductivity. In compari son the extracellular fluidfraction in an adult is about 20 percent. Thence, the fetus is at least a factor two more conducting than the maternal abdomen. Assuming the conductivity of the extracellular spacein both fetus and adult to becomparable at a value of 2S/m, equation (9.36) predicts a range


of 1.9 X (004)5/3 ~ OAIS/m::::: afetus ::::: 1.9 X (0.6)3/2 ~ 0.88S/m. Assuming that the volume fraction will be somewhere between 40 and 60 percent and will approach 40 percentat the end of gestation, a value of O.SS/m is a reasonable choice for the conductivity of thefetus in the third trimester of pregnancy. This value is used for the solution of the inverseproblem for fetal ECG and leads to reasonable results (Stinstra, 2001).

9.6 DISCUSSION

The accuracy of measurements will be limited because the measurements are verycomplicated. The accuracy of the computation is limited because the cells vary in shape, theyare not homogeneously distributed, blood supply plays a role, etcetera. Since the model usedto describe a tissue in this chapter is a simplification the results are only an approximation.However, the results are useful in clarifying the relation between the conductivity and thestructure of the tissue. The results can be used to predict the effects of changes due to,for instance, temperature, illnesses or age. Anyhow, it makes no sense to use values of theeffective conductivity that suggest an accuracy higher than ten percent by giving the valueswith too many digits.

The effective electrical conductivity is a macroscopic parameter that represents theelectrical conductivity of the tissue averaged in space over many cells. Many of the tissuesin the body such as lung, liver, fat, and blood have cells structures that macroscopically showno preferred direction. Even the heart, which is muscular, has its muscle strands wound insuch a complicated fashion that, overall, no preferred direction can be readily discerned.Baynham and Knisley (1999) measured the effective epicardial resistance of rabbit ventriclesand found that in contrast to isolated fibers the ventricular epicardium exhibits an isotropiceffective resistance due to transmural rotation of fibers. Only skeletal muscle cells have adefinite preferred direction when many cells are averaged (Rush et al., 1984). Most cellshave an elongated shape. Thence, the bounds given in section 9.5 can be used to estimatethe effective conductivity. These bounds are not so far apart, so they will help to restrict theuncertainties in the effective conductivity to be used in the bioelectrical inverse problem.An exception form long skeletal muscle and heart tissue, as the conductivity parallel to thefibers will take place both in the extracellular and the intracellular space.

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Action potential 2, 89, 120Activation time imaging 167Adaptive spatial filter 226Anisotropic bidoma in 50Archie's law 300

Beamformer 226, 230, 231Bidomain model 124Bidomain myocardi um 46Biot-Savart law 216Block-design fMRI 254Body surface isopotential map 107Body surface Laplacian 144, 183Body surface Laplacian mapping 192Body surface potential map 126BOLD 252BSLM 192BSPM 126

Cable theory 17Cardiac action potential 7, 8, 9Cardiac arrhythmia 27Cardiac tissue 302Cell networks 23Cells 289Computer heart model 62Conductivity tensor 51Co-registration 262, 263Cortical imaging 257Current dipole density 46

DAD 14,25Defibrillation 72Dipole distrib ution imaging 164Dipole distribution 163, 164Dipole localization 256, 267Dipole source imaging 163, 165Dipole source 44

INDEX

Drug integration 35

EAD 14,25ECG 183EEG 183,263Effective conductivity 47, 283Electrical conductivity 28 1Electrocardiographic tomographic imaging 16 1,

168,175Endocardial potential imaging 129, 147Endocardial potential 127, 128Epicardial potential imaging 129, 138Epicardial potential 57Equivalent conductivi ty 312Equivalent current density 49Equivalent dipole 54Equivalent moving dipole 163Event-related fMRI 254Extracellular electrogram 20Extracellular fluid 29 1, 295

Fat 301Fiber orientation 52, 86Finite difference laplacian 186Finite difference method 58Finite element method 58Finite volume method 60FitzHugh-Nagumo model 100fMRI 252Forward problem 43, 53Functional magnetic resonance imaging

252

Genetic integration 35Global Laplacia n estimate 188Gradiometer 2 15Gray matter 305Green's function 54, 125

321

modeling bio electrical

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