Cell2ECG: A virtual laboratory to simulate cardiac electrograms

Dietrich Romberg Biomedical Engineering

Anhalt University of Applied Sciences Köthen, Germany

John W. Dyer Electrical and Computer Engineering

University of Oklahoma Norman, OK, USA

Edward J. Berbari Biomedical Engineering

Indiana University Purdue University Indianapolis, IN, USA

Abstract—Engineering approaches to indentify and reverse electrical instabilities of the injured heart have lead to innovative diagnostic and therapeutic solutions. Therefore, curricula of Biomedical Engineering (BME) courses include cardiac electrophysiology. This paper describes a virtual laboratory designed for the study of the generation of the electrocardiogram (ECG) based on cellular electrophysiology. In detail, the virtual lab includes (1) an equivalent source model for cardiac cells, (2) a formulation of the mathematical relationship between the transmembrane potential and the transmembrane current, (3) the determination of resulting extracellular potentials, and (4) the calculation of the ECG by a weighted summation of transmembrane currents. Since these tasks require specific presumption and knowledge concerning cardiac geometry, a geometrical model was constructed using anatomical stylized segments of the left ventricle. Propagation parameters control the activation sequence as well as velocity and direction for anisotropic conduction. In order to verify the developed models and algorithms simulation results were compared against experimentally obtained data under various physiological conditions. Although there is considerable scatter in the measurements, the comparison indicates that a definite relationship exists between measured and computed waveforms. Simulations interactively show physiological and pathophysiological changes in the ECGs for various user setting of the cell function. In conclusion, the interactive laboratory enables the user to study the relationship between the electric activity of cardiac cells and the resulting extracellular potentials including the ECGs on the body surface. Students increase their knowledge of cardiac electrophysiology, applied electrical circuit theory and the understanding of differential equations as well as numerical methods for solving them.

Keywords—Biomedical Engineering, virtual laboratory, simulation, heart model, electrocardiogram

I. INTRODUCTION Engineering approaches to indentify and reverse electrical

instabilities of the injured heart have lead to innovative diagnostic and therapeutic solutions. Therefore, curricula of

BME courses include cardiac electrophysiology to provide a framework for understanding the genesis of the electrical signals themselves. In contrast to other engineering disciplines, experimental approaches in BME require noninvasive and invasive procedures on living systems. Thus, computer modeling is an essential tool for describing and analyzing biological systems [1]. One of the major challenges to teaching cardiac electrophysiology is gaining an understanding of the link between the electrocardiogram (ECG) and cellular electrophysiology. This paper describes a virtual laboratory, called Cell2ECG, and related projects, designed to simulate the genesis of cardiac electrical potentials under various physiological and pathophysiological conditions. The interdisciplinary approach of Cell2ECG includes cardiac electrophysiology as well as electrical circuit theory, mathematics and software development.

II. CONCEPT The prediction of the ECG on the body surface requires

knowledge of (1) cellular basics of electrical activation in the heart; (2) the sequence of these activations; (3) geometry and conductivity of the heart; and (4) a biophysical model describing current flow in the heart and current flow from the heart, through the intervening tissues, to the body surface. Therefore, Cell2ECG specifically incorporates

• a geometrical model of the heart, • a biophysical model of an equivalent source generator, • a simulation tool for action potentials (APs), and • a simulation tool for specifying the propagation of

electrical excitation in the heart. In addition, a database of experimental data [2] is included in order to validate the results of the simulation. Based on these modules, the students are required to complete three experimental projects to estimate (1) a transmembrane current, (2) an extracellular potential, and (3) an ECG.

In contrast to more sophisticated simulations [3-5], the simple mathematical theory of the model allows [6] the

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students to implement a numerical solution for these projects. The simulation tools are based on LabVIEW/Mathscript, since it provides access to powerful programming and visualization functions and the students have had previous experience in this environment. The modular structure of the tools allows it to be easily modified or extended to other projects that are related to cardiac electrophysiology. Therefore, Cell2ECG should encourage active learning, group discussion, and problem solving skills.

III. MODELLS AND SIMULATION TOOLS

A. The Heart Model The model is focused on the left ventricle (LV) since the

effects of the right ventricle are not essential for understanding the genesis of the ECG. According to a modification of the Solomon-Selvester model [7], the LV is divided into three levels (apical, middle and basal) and four sections (quadrants) resulting in 12 segments (Fig. 1). Each of these 12 segments consists of endocardial, mid-myocardial, and epicardial layers of identical cells.

Fig. 1: Model of the left ventricle (LV). The LV is divided into quadrant-segments in the apical, middle, and basal level in accordance with the International Society of Computerized Electrocardiology (ISCE) nomenclature of myocardial wall segments [7]. All 12 segments are assemblies of endocardial, mid-myocardial, and epicardial layers.

B. The Source Model For each segment of the heart model, an endocardial, mid-

myocardial, and epicardial layer was approximated by a two-dimensional sheet of electrically coupled elements. Each element is modeled by a 2D-repetitive network of length ∆x and ∆y with the current source Im (transmembrane current) as described by the core-conductor approximation [8]. The network topology allows parameters governing propagation to be addressed easily and avoids computationally burdensome modeling. For the sake of mathematical simplicity and without losing the conceptual basis, Fig. 2 illustrates the one-dimensional case.

Fig. 2: Equivalent circuit of an element. Potential, resistance, and current are

expressed per unit length ∆x. Currents Ii and Ie are driven by the potential gradient Vm= Ve-Vi.

Vi, Ve: intracellular and extracellular potential, respectively Ri, Re: the intracellular and extracellular potential, respectively. Ii, Ie: intracellular and extracellular current, respectively Vm: transmembrane potential Im: transmembrane current According to Fig. 2, the intracellular and extracellular current, Ii and Ie, respectively, are driven by potential gradients, so that,

Re · Ie = ⁄ (1)

Re · Ii = ⁄ (2)

where Ri and Re stand for the intracellular and extracellular resistance, respectively, and Vi and Ve for the intracellular and extracellular potential, respectively. By definition,

Vm = Ve - Vi (3)

Taking the derivative on both sides of (1) and (2) produces,

Re · Ie = / (4)

Ri · Ii = / (5)

Since the total current must be preserved per unit length, the change in Ii and Ie must precisely equal the transmembrane current, Im (application of Kirchhoff’s current law). In mathematical terms,

Im = ⁄ (6)

Im = − ⁄ (7)

Replacement of (6) and (7) in (4) and (5), respectively, yields,

Re · Im = / (8)

Ri · Im = − / (9)

Summation of (8) and (9) and recalling (3) leads to,

Im = 1/(Re+Ri) · / (10)

basal

apical

middle

epicardial

endocardial

mid-myocardial

Including the extracellular current flow in the y direction, superposition of both results in,

Im = 1/(Re+Ri) · [ / + / ] (11)

Thus, the transmembrane current density is directly proportional to the second spatial derivative of the transmembrane potential.

C. Simulation of APs The transmembrane potential remains constant (resting

membrane potential) without exterior influences. However, as the result of a local stimulus, an action potential (AP) is generated by the cell. In order to simulate the transmembrane current in the equivalent source model an estimation of the transmembrane potential, in particular for the AP, is required (11).

As proposed by [9] APs were composed using a parameterized AP profile where separate components control the initial upstroke (phase 0), the immediate fast repolarization and the AP plateau (phase 1 and 2, respectively) as well as the repolarization part (phase 3). In this way, APs were assigned individually to the cells of specific layers. The software enables the student to specify or modify the AP shape interactively, including the timing of depolarization and repolarization (Fig. 3).

Fig. 3: Menu for interactive parametrization of APs. The scaling factors

andtime constants relate to the different phases of the AP.

Differences between epicardial, mid-myocardial and endocardial cells are primary related to the AP duration as illustrated in Fig. 4.

Fig. 4: Simulated APs assigned to epicardial (top) , mid-myocardial (middle) and endocardial (bottom) layers. The AP durations, given at 90 % repolarization (APD90) are 350 ms at the endocardium, 380 ms at the mid-myocardium, and 340 ms at the epicardium. The scale of amplitude for APs is arbitrary but normalized for all APs.

D. Simulation of Activation Sequence The activation time (AT) represents delay between the start

of myocardium excitation and activation of each individual element, and controls the activation sequence. The excitation starts at the endocardial layer and spreads through the mid-myocardium and the epicardium. Timing of activation may be changed interactively using a provided graphical user interface (GUI) as depited in Fig. 5. Fig. 5: GUI for interactive selection of activation times. The schematic

drawing of a cross section of the heart represents the pathway of myocardial excitation starting at the endocardial layer and spreading through the mid-myocardium and the epicardium. Numbers indicate the activation time.

IV. PROJECT APPLICATIONS

A. Framwork The subsequent projects are devoted to studying and

understanding basic problems in cardiac electrophysiology related to (i) the estimation of transmembrane currents, (ii) the characterization of extracellular potentials, and (iii) the interpretation of the ECG on the body surface. The application of Cell2ECG requires knowledge of the physiology and anatomy of the heart, the understanding of cellular electrophysiology and a mathematical background for differential equations as well as numerical methods for solving them. After an introduction to the concept of Cell2ECG, students are required to reproduce the mathematical description of the equivalent source model as developed in (1)-(11). In contrast to other approaches [3-6], the numerical implementation of the source model is an essential step in the learning process. In order to verify the models, simulation results are compared against experimentally obtained data under various physiological conditions as provided by the database. For each project, a report is delivered consisting of derivations, physiological interpretation, figures, tables, programming code, results and discussion.

B. Estimation of the Transmembrane Current Using (11), transmembrane current in an endocardial, mid-

myocardial, and epicardial layer must to be determined. Specific aims include the investigation of the influence of the intracellular and extracellular resistance. A typical example of a simulated and an experimentally obtained transmembrane current is shown in Fig. 6.

Fig. 6: A comparison of simulated (left) and experimentally obtained (right) transmembrane currents. Differences may be caused in part by the contributions of distant sources (far field effects). Scales arbitrary but normalized for both traces.

C. Estimation of Extracellular Potentials The extracellular potential V(x,y) for a transmembrane

current source, Im, is related to the distance r between the current source and the observation point [8]. In the two-dimensional case, the superposition principle for multiple linear current sources leads to

V(x,y) = ∑ | | (12)

where Re is the extracellular resistivity.

Besides assigning an AP, activation times need to be defined for each element. Specific aims include the

investigation of the effect of the intracellular and extracellular resistance as well as the influence of the number of elements used for the simulation. Fig. 7 provides an example of a simulated and an experimentally obtained extracellular potential.

Fig. 7: A comparison of a simulated (left) and an experimentally obtained

(right) extracellular potential. Significant differences may be due in part to the fact that the wavefront in the model is assumed to be planar and propagating with uniform conduction velocity. Scales arbitrary but normalized for both traces.

D. Simulation of the ECG Extending the approach of Project 2, the simulation of

extracellular potential will include current sources from all segments of the LV. For this purpose, APs and activation times are assigned and a representative current source is calculated for the center element of an endocardial, mid-myocardial,and epicardial layer in each segment. Assuming a homogeneous volume conductor, the ECG is obtained by the weighted sum of these representative current sources with reference to the distance between current source and observation point (12). Specific aims include the investigation of the relevance of the AP shape as well as the activation sequence. As Fig. 8 reveals, simulation leads to similar waveform morphologies observed in the experimentally obtained data.

Fig. 8: A comparison of a simulated (left) and an experimentally obtained

(right) ECG. The discrepancies were accounted for by variations in the AP shape and the activation sequence. Scales are arbitrary but normalized for both traces.

E. Effects of Cardiac Ischemia In order to simulate the effects of epicardial ischemia on the

ECG, the epicardial AP was modified as shown in the upper panel of Fig. 9. The severity of the ischemia can be modeled by an increase in the resting potential and a shortening of the duration of the AP plateau in the affected layer of a specific wall segment. The corresponding elevation of the middle part of the simulated ECG (lower panel) is related to the changes in the plateau of the epicardial AP. Similar effect can be demonstrated by modifying the endocardial AP.

Fig. 9: Upper panel: Simulations of epicardial APs from normal (dotted line) and ischemic myocardium (solid line). Lower pabel: Simulated ECG corresponding to the ischemic epicardial AP. For reference see Fig. 8.

V. DISCUSSION Cell2ECG provides a concept for an interactive simulation

program that enables the user to study the relationship between the electric current sources of the heart and the resulting electrocardiographic signals on the body surface. A primary goal of the projects is for the student to gain proficiency in developing a mathematical description as well as a numerical implementation of the equivalent source model.

Comparison testing of Cell2ECG shows simulated and experimentally obtained data reveal quite similar waveforms. Furthermore, discrepancies were accounted for in the simple heart model due to the assumption of an infinite homogeneous volume conductor. A more accurate approach would have been to calculate the ECG from each point in the heart model. However, this would dramatically increase the complexity of the model and require a significant increase in the number of computations. This reduces the effectiveness of the simulation as a learning tool. At the designed level of complexity, it is hoped that the proposed projects will cultivate the student’s interest in the analysis of physiological systems as well as in computational modeling, encouraging further study of more in-depth models.

In summary, Cell2ECG represents a simulation platform for electrical activities of cardiac cells, which provides a tool to examine the mechanisms underlying the genesis of the ECG under various physiological and pathophysiological conditions. It aims to serve as an educational tool and as a research tool as well, such as analysis of slow conduction in the infract border zone, simulating ECGs under ischemic circumstances, or during ectopic activation. The modular structure of Cell2ECG permits easy modification or expansion by adding more function to the defined menus, for example adding visualization of the activation sequence synchronously with the ECG waveform generation.

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