i

omne ignotum pro magnifico

from Vesalius, 1545 from Arterial Model ,1992

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Recent developments in non-invasive techniques of measurement of arterial pressure waveforms have increased the need for better understanding of the human arterial pressure pulse. This thesis examines the information carried by the propagating arterial pressure pulse in human systemic arteries, particularly in the peripheral upper limb arteries, in relation to properties and transmission characteristics of the arterial vasculature.

Simulations were performed in a mathematical model of the whole arterial tree where arterial segments are represented as uniform transmission lines and arranged as a binary tree structure. Experiments involved measurement and analysis of propagating pressure waves (a) in a single elastic tube (b) in aortic, carotid, brachial, radial, femoral and dorsalis pedis arteries of human subjects during: (i) altered heart rates, (ii) external compression of the vascular beds of the hand, (iii) vasodilatation with nitroglycerine and nitroprusside, (iv) ingestion of glucose.

Findings from model simulations and studies in human subjects are: (i) pressure waves in the central and peripheral upper limb arteries are different in the time domain and the relationship in the frequency domain (transfer function) is relatively constant; (ii) local reflections in the peripheral terminations modulate the initial oscillations seen in the peripheral arterial pressure pulse in early systole; (iii) reflected waves originating from the lower aortic trunk modulate the secondary oscillations seen in the arterial pressure pulse; (iv) under normal conditions wave reflection in the hand is high and relatively constant; (v) vasodilatation does not produce substantial reductions in reflected waves from the hand; (vi) ingestion of glucose significantly reduces the intensity of reflected waves from the vasculature of the lower trunk segment but does not alter reflections from the upper and lower limb terminations explaining the paradoxical disparity of wave shapes in central and peripheral upper limb arteries after vasodilatation.

The above findings resulted in the design and development of a computer based clinical tool to reconstruct the calibrated central aortic pressure waveform from pressure wave recordings in the radial artery. This application complements the current methods of non-invasive measurement of blood pressure by providing a comprehensive analysis of the arterial pressure waveform.

i

DECLARATION

I hereby declare that this submission is my own work and that, to the best of my

knowledge and belief, it contains no material previously published or written by another

person nor material which to a substantial extent has been accepted for the award of any

other degree or diploma of a university or other institute of higher learning, except

where due acknowledgment is made in the text.

Mustafa Karamanoglu

December, 1992

Sydney

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SIMULATION, MEASUREMENT AND ANALYSIS OF THE PROPAGATING PRESSURE PULSE IN THE HUMAN

ARTERIAL SYSTEM

Mustafa Karamanoglu

(BE, MBiomedE)

A Thesis Presented for the Degree of

DOCTOR OF PHILOSOPHY

at the

University of New South Wales

December, 1992

Sydney

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ACKNOWLEDGMENTS

I would like to thank my supervisor Dr. Albert Avolio for his patience in shaping this

work. During the entire investigation, he was at enough distance to let free thought

flourish, yet at times he was very close to scrutinise each and every statement that I

made. I will also remember Prof. Michael O'Rourke with his always youthful

enthusiasm to research. This work would never have started, let alone be finalised,

without his knowledge, his active participation and the research environment that he

created and maintained. I thank Dr. Raymond Kelly for his help in during my initial

acquaintance with the field. Thanks to my colleague and friend Dr. David Gallagher,

who over many years shared the joy and the stress of exploring the unknown. We

performed most of the experiments together. His calm nature and frankness will always

be with me. Sincere thanks to Dr. Gary Dobson who has contributed to theoretical

discussions. Special thanks to Miss Lina L. Lee, who spent many long hours in

recruiting the subjects and collecting the data presented in this thesis. Her contribution

to the analysis and presentation is most appreciated. I would also like to thank Ms

Georgina McPhee who typed the thesis and patiently dealt with the confusions that I

created. I also thank Mr. Jason Young on his help in correcting the manuscript. Finally

I would like to express my gratitude to many who helped me to develop the necessary

curiosity and persistence in realising this work.

My greatest thanks to my always understanding wife Sahure who recognised the

consequences of undertaking such a task.

ii

ABSTRACT

Recent developments in devices for non-invasive recording of arterial pressure contour such as arterial tonometry and photoplethysmography have highlighted the inadequacies of conventional methods of arterial pressure measurement and increased the need for better understanding of the human arterial pressure pulse. The arterial pressure waveform is a complex quantity containing much more information than the two singular points - systolic and diastolic pressures - provided by the widely used sphygmomanometric techniques. The arterial pulse contour undergoes marked alterations during its travel in the arterial system under normal conditions which is modified by aging, level of mean blood pressure and vasoactive state. This thesis examines the information carried by the propagating arterial pressure waveform in human systemic arteries in relation to vascular properties and transmission characteristics of the arterial tree. Since the arterial pulse is often measured in the upper limb, the thesis concentrates particularly on the relationship between the central aortic and the peripheral upper limb pulse.

Propagated pressure waves were simulated in a mathematical model of the whole arterial tree consisting of 142 uniform transmission line segments arranged in a binary tree structure. This model allowed detailed investigation of the effect of reflected waves originating from different regions in the arterial system on the central and peripheral pulse. Wave transmission in the upper limb during simulated changes in upper limb vascular properties were investigated and transmission parameters were estimated from experimental results. Experiments involved: (a) pulse propagation in a single elastic tube to test the model implementation (b) measurement of the aortic, carotid, brachial, radial, femoral and dorsalis pedis arterial waveforms and their relationship during: (i) altered heart rate in subjects instrumented with implantable pacemakers, (ii) alterations in regional wave reflection in the hand caused by graded compression of the palm, (iii) vasodilatation with nitroglycerine and nitroprusside, (iv) altered lower aortic trunk wave reflection at the region of splanchnic circulation caused by glucose ingestion.

Results from experimental investigations and model simulations have produced the following findings in the human arterial system: (i) pressure waves in the central and peripheral upper limb arteries are different in the time domain and the relationship in the frequency domain (transfer function) is relatively constant; (ii) reflected waves from upper and lower limbs are mainly confined within respective segments and they form

iii

the initial peak of the pressure waveforms in these segments; (iii) the vasoactive state of arterial beds of the trunk segment modulates global reflected waves which form the secondary oscillations seen in the central aortic and upper limb pulses; (iv) under normal conditions the intensity of reflected waves originating from the vasculature in the hand is high and close to that obtained by total occlusion of this bed. This finding explains the relative stability of the upper limb transfer function under these conditions; (v) Vasodilating agents such as nitroglycerine and sodium nitroprusside do not alter reflections from the palm vasculature significantly but decrease reflections from the trunk vasculature. This explains the paradoxical disparity of wave shapes in central and peripheral upper limb arteries after the administration of these drugs; (vi) ingestion of glucose causes significant reduction in the intensity of wave reflection from the vasculature of the lower trunk segment as evidenced by the arterial pressure wave contours and by analysis of transfer functions of the upper limb, trunk and lower limb segments. These changes are similar to those observed after administration of vasodilators.

The above findings resulted in the design and development of a computer based clinical tool where sphygmomanometric measurement of arterial pressure are augmented by recording of the radial artery waveform from which the calibrated central aortic pressure waveform is reconstructed. This application complements the current methods of non-invasive measurement of blood pressure by providing a comprehensive analysis of the arterial pressure waveform.

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TABLE OF CONTENTS

ACKNOWLEDGEMENTS i ABSTRACT ii TABLE OF CONTENTS iv LIST OF SYMBOLS x

INTRODUCTION 1

SECTION I BACKGROUND

CHAPTER 1 HISTORY 7

CHAPTER 2 THEORETICAL FOUNDATIONS 13

2.1 MATHEMATICAL MODELS OF THE ARTERIAL SYSTEM 14 2.1.1 Steady Flow 15 2.1.2 Oscillatory Flow 17

Windkessel Approach 17 One Element Windkessels 17 Modified Windkessels 19

Womersley's Approach 20 Discussion of the Validity of the Womersley Approach 27 Validation of Womersley's Approach 30

Other Approaches 31 Linear Models 31 Non-linear Models 31

(i) Method of Characteristics 31 (ii) Wavelet Analysis 36

2.2 ANALOGUES OF THE ARTERIAL SYSTEM 37 2.2.1 Lumped Analogues 38

Zero Order Analogues 38 First Order Analogues 38 Second Order Analogues 39 High Order Analogues 40

2.2.2 Uniform Transmission Line Analogues 41

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2.2.3 Non-Uniform Transmission Line Analogues 44 2.2.4 Implications of Transmission Line Concepts 47

Transmission Direction 47 Impedance 47

Longitudinal Impedance 47 Transverse Impedance 48 Characteristic Impedance 49 Input Impedance 51

Propagation Coefficient 51 Transmission Ratio 52

2.2.5 Application of Transmission Line Concepts in Haemodynamics 53 Interpretation of Impedance 53 Interpretation of the Transfer Function 58

2.3 APPLICATION OF TUBULAR MODELS OF THE ARTERIAL SYSTEM 64

2.3.1 Single Tube Models 64 2.3.2 Two Tube Models 65

Tubes in Parallel 65 Tubes in Series 66

2.3.3 Three or More Tube Models 66 Random Branching Models 67 Anatomically Arranged Branches 67

Tree Structure 67 Mesh Structure 68

CHAPTER 3 MEASUREMENT OF ARTERIAL BLOOD PRESSURE 70

3.1 INVASIVE MEASUREMENTS 71 3.2 NON-INVASIVE MEASUREMENTS 73 3.2.1 Peak Systolic Value Measurement 74

Auscultatory Method 74 3.2.2 Contour Measurement 75

Sphygmography 75 Plethysmography 76 Applanation Tonometry 77

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SECTION II MODELLING AND SIMULATION

CHAPTER 4 MODEL CONSTRUCTION 83

4.1 REPRESENTATION OF THE ARTERIAL TREE 84 4.1.1 Parent Daughter Relations 85

Zero Order Relation 85 First Order Relation 88 Second Order Relation 88

4.1.2 Computational Algorithm 89 4.2 ANATOMICAL DATA 91

CHAPTER 5 A BINARY TREE MODEL OF THE HUMAN ARTERIAL SYSTEM 96

5.1 PRESSURE WAVE PROPAGATION IN THE HUMAN ARTERIAL MODEL 98

5.1.1 Pressure Contour Maps 98 5.1.2 Introduction of Reflected Waves 101 5.1.3 Tracking Of Reflected Waves 102 5.1.4 Wave Velocities and Reflection Coefficients 103 5.1.5 Input to the Model 104 5.1.6 Input Impedance 104 5.1.7 Pressure Wave Transmission 106 5.1.8 Reflections From Compartments 107 5.2. PRESSURE WAVE PROPAGATION IN THE UPPER LIMB 110 5.2.1 Effect of Change in Wall Elastance (E) 112 5.2.2 Effect of Change in Reflection Coefficient (Γ) 113 5.2.3 Effect of Changes in Time Constant (τ) 114 5.2.4 Effect of Change in Wall Viscosity (Θo) 116 5.3 DISCUSSION 117

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CHAPTER 6 MODEL VALIDATION AND PARAMETER ESTIMATION IN A SINGLE ELASTIC TUBE 125

6.1 SINGLE ELASTIC TUBE EXPERIMENT 126 6.1.1 Estimation of Tube Elastance 128

Calculation of Elastance from Stress-Strain Relation by Longitudinal Stretch 128

Calculation of Elastance from Pressure-Volume Relations 129 Calculation of Elastance from Wave Velocity: 129

6.2 ESTIMATION OF MODEL PARAMETERS 130 6.3 RESULTS 132 6.3.1 Tube Elastance 132 6.3.2 Convergence of the Model 133 6.3.3 Predicted Pressure Waves 134 6.3.4 Transfer Function 136

Modulus 136 Phase Velocity 139

6.3.5 Reflection Coefficient 139 6.4 DISCUSSION 140

SECTION III EXPERIMENTAL STUDIES

CHAPTER 7 DETERMINATION OF WAVE PROPAGATION CHARACTERISTICS IN THE HUMAN UPPER LIMB 145

7.1 METHODS 147 7.1.1 Subjects 147 7.1.2 Measurements 147

Calibration of Waves 148 7.1.3 Data Analysis 148

Pressure Wave Amplification 148 Pressure Wave Shapes 148 Transfer Functions 149 Parameters Obtained from the Transfer Function 149

Resonant frequency 150 Reflection coefficient 150

Reduced Model Of the Upper limb Arterial System 151 7.1.4 Statistics 152 7.2 RESULTS 152 7.3 DISCUSSION 158

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CHAPTER 8 QUANTIFICATION OF THE INTENSITY OF REFLECTED WAVES IN THE UPPER LIMB 161

8.1 METHODS 162 8.1.1 Subjects 162 8.1.2 Procedure and Data Acquisition 163 8.1.3 Estimation of Intensity of Reflected Waves 166

(i) Estimation from Phase Velocities 167 (ii) Estimation from Pulse Pressures 167 (iii) Estimation from the Model 167

8.1.4 Statistics 168 8.2 RESULTS 168 8.3 DISCUSSION 177

CHAPTER 9 EFFECTS OF VASODILATORS ON UPPER LIMB WAVE PROPAGATION PROPERTIES 185

9.1 METHODS 186 9.1.1 Nitroglycerine (NTG) 186 9.1.2 Sodium Nitroprusside (SNP) 188 9.1.3 Statistics 188 9.2 RESULTS 189 9.2.1 Nitroglycerine 189 9.2.1 Sodium Nitroprusside 191 9.3 DISCUSSION 194

CHAPTER 10 EFFECT OF THE SPLANCHNIC CIRCULATION ON THE FORMATION OF THE ARTERIAL PULSE 196

10.1 METHODS 197 10.1.1 Subjects 197 10.1.2 Protocol 198 10.1.3 Measurements and Data Analysis 198 10.1.4 Statistics 200 10.2 RESULTS 200 10.3 DISCUSSION 205 10.3.1 Reduced Model of the Arterial System 209

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SECTION IV APPLICATIONS

CHAPTER 11 THE SYNTHESIS OF THE CENTRAL PRESSURE WAVEFORM FROM THE PERIPHERAL PULSE IN THE UPPER LIMB 217

11.1 METHODS 219 11.1.1 Synthesis in the Frequency Domain 219 11.1.2 Synthesis in the Time Domain 221 11.2 RESULTS 222 11.2.1 Frequency Domain 222 11.2.2 Time Domain 226 11.3 DISCUSSION 228

CHAPTER 12 A SYSTEM FOR ON-LINE ANALYSIS OF BLOOD PRESSURE WAVEFORMS 233

12.1 IMPLEMENTATION 235 12.1.1 Data Acquisition 235 12.1.2 Signal Conditioning 237 12.1.3 Feature extraction 238 12.1.4 Calculated Parameters 241 12.1.5 Database Engine 244 12.2 THE TRIAL OF THE SYSTEM IN A CLINICAL ENVIRONMENT 247

SECTION V SUMMARY AND CONCLUSIONS

REFERENCES 256 APPENDIX I 283

x

LIST OF SYMBOLS

<, j Phase angle

α Womersley's non-dimensional number

C Capacitance

c, co Wave velocity, Inviscid wave velocity

D Diameter

E, Ec Elastance, Complex Elastance

ε Strain

F10 the expression ( )

( )230

23

2312

jJjjJαα

α

f Frequency

Γ Reflection coefficient

γ Propagation constant

H Weighted volume of the wall substance taking into account external inertial loading of pe

H(ω) Frequency (ω) dependent transfer function

h(t) Impulse response function of H(ω)

h Vessel wall thickness

η Viscosity coefficient of the vessel wall

I Current

J0, J1 Zeroth and first order Bessel functions of the first kind with complex arguments

K Spring coefficient of the tethered tissue

L Inductance

r, l Radial and axial distance

M Mass of the vessel wall element

M Modulus of complex quantity

µ Viscosity of the blood

N Vessel's tethering coefficient

xi

P Pressure

Q Flow, Charge

θ Phase angle of Ec

R Internal radius

R Resistance

ρ, ρ0 Vessel wall and blood density

r, z, θ Radial, axial and tangential coordinates in cylindrical axes

S Vessel wall area

σ, σc Poisson's ratio, Complex Poisson's ratio

τ Time constant of the Windkessel

t Time

u, v, w Radial, tangential and axial component of the instantaneous flow velocity

V Voltage, Volume

ω, ωn Angular, Resonant frequency

ξ, ζ Radial and Axial displacement of the vessel wall

Z Impedance

Z0, Zin Characteristic and Input Impedance

Zl, Zr, ZT Longitudinal, Transverse and Terminal Impedance

Ψ Efflux of fluid through the walls per unit length

1

INTRODUCTION

Although pulsatile haemodynamics forms a relatively small branch of cardiovascular

physiology, investigators from many different disciplines have confronted the many

challenges and complexities of the field. From disciplines of mathematics, physics and

engineering investigators studied the principles of the field of haemodynamics, with the

aim to gain basic understanding leading to improved clinical diagnosis and treatment of

cardiovascular disease. Although many important discoveries were made, clinical

applications resulting from basic investigations in pulsatile haemodynamics have been

scarce. Practicing clinicians have found them either too complex to understand or too

difficult to apply. Inevitably, it has proved difficult for experimental and theoretical

concepts to emerge from the confines of the laboratories.

Recent advances in technology and results of large scale epidemiological studies are

producing considerable changes in many traditional and largely conservative attitudes.

It is now possible to obtain important basic quantities non-invasively by echo doppler

imaging (Nichols and O'Rourke 1990) and tonometry (Kelly et al 1989b). Due to the

developments made in computer technology it is also possible to perform complex

calculations in real time. Recent studies also highlighted the importance of the arterial

pulse wave contour in cardiovascular medicine and increased the awareness of the effect

of pulsatile load on the heart and on the arteries. Systolic pressure is now considered a

major risk factor for cardiovascular disease (Kannel et al 1981) and the shape of the

central aortic pulse is an important indicator of left ventricular-vascular coupling

(Takazawa 1987). A current trend in clinical cardiology is to develop techniques to

measure and establish background work to interpret the arterial pressure pulse.

2

Not surprisingly, the pulse in the human arterial system has been investigated

extensively in the past. In these early studies, specific attention was directed towards

the pressure wave in the central aorta and the lower limbs while the pressure was often

recorded in the upper limb. The pressure pulse contour in the upper limb is not only

considerably different from the central pulse but also undergoes different directional

changes under certain circumstances. Compared to the central pulse, the pulse in the

upper limb is affected differently by aging, vasodilatation and physical manoeuvres

leading to difficulties in interpretation. Although much has been published in

documenting the morphology of the upper limb pulses under these circumstances (Kelly

et al 1989a, Simkus and Fitchett 1990), there has been little systematic investigation on

the characteristics of pressure wave propagation in the upper limb.

This thesis aims to extend the understanding of what arterial parameters are relevant in

the formation of arterial pressure pulse. It attempts to understand the information

content of propagating arterial pressure in human systemic arteries by examining the

vascular properties and transmission characteristics of the arterial tree. It also aims to

characterise the properties of the arterial system by extraction of specific relevant

features from the pressure waveform obtained from different arteries. It particularly

concentrates on the pressure pulses in the central aorta and in the peripheral upper limb

arteries.

In Section I, the background work to the broad field of haemodynamics and wave

propagation phenomena will be discussed. The historical account of events leading to

present knowledge is followed by a theoretical survey. In this section, various concepts

of pulsatile haemodynamics will be discussed. Since these concepts will form the

foundations of the modelling section, the underlying assumptions will be studied in

detail. In the remainder of this section the current methods on measuring the pressure

waveform will be reviewed.

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Section II is devoted to the simulation of pressure wave propagation by computer

implementation of theoretical modelling concepts mentioned in Section I.

The first chapter of this section, Chapter 4, describes the construction of a mathematical

model of the pressure/flow relations in a viscoelastic tube. Using methods originally

developed by Taylor (1966a, b) and Avolio (1976) a binary tree model of the arterial

system is constructed which finds extensive use in Section III. Experiments performed

in Section III are simulated by the model to estimate the properties of the arterial

vasculature.

Chapter 5 describes the application of the binary tree model to the human arterial system

where a new approach in exploring pressure wave propagation is developed. The

pressure contour maps developed in this chapter provide new insights into the origins of

reflected waves. The important physical parameters in the upper limb wave propagation

are also described in this chapter. The sensitivity of transfer functions of the upper limb

model on propagation parameters are explored by altering physical quantities.

The third of these chapters, Chapter 6, is devoted to the verification of the model and the

development of parameter estimation techniques based on this model. Using a bench

top experiment, the theory and the integrity of model algorithm are tested.

Section III presents chapters describing the clinical studies. In Chapter 7 the presence of

upper limb transfer function and its clinical implications were investigated by

conducting experiments on eighteen human subjects. Results of this study confirmed

predictions made by the model.

Chapter 8 describes experiments directed towards investigating the effect of reflected

waves on the formation of pressure waves in the upper limb. The effect of reflected

waves originating from the palm were investigated by graded compression of the hand

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in thirty three human subjects. It was found that the reflection indices correlate well

with the systolic pressure and could even be reduced by increasing the compliance of the

peripheral hand vessels.

Chapter 9 describes findings of the analysis of changes in central and upper limb pulses

in response to vasodilatation as reported by other investigators using techniques

developed in earlier chapters. It was found that the effect of vasodilator agents on the

propagation properties of the upper limb vasculature is negligible and does not explain

the reported differences.

The outcome of experiments in Chapter 9 and model predictions in Chapter 5 suggested

studies to explore vasodilatation of the splanchnic circulation. In Chapter 10, the effects

of glucose ingestion on the pressure waveforms along the arterial tree is explored in ten

human subjects. Results explained the reported discrepancies of pressure waveforms in

different arteries in response to vasodilators. These findings were confirmed by results

from the arterial model.

The general findings described above established that the propagation properties of the

upper limb arterial system are relatively stable and do not account for observed changes

in a wide variety of conditions. Section IV investigates the possibility of synthesis of

the central pressure wave from the peripheral pulse in the upper limb using the finding

of a relatively constant transfer function. In Chapter 11, methods to synthesise central

pressure waveforms from peripheral recordings are described. These methods are then

tested using published records of central and peripheral pulses. Chapter 12 describes a

practical application resulting from this investigation. In this chapter, an on-line system

is described to synthesise the central aortic pulse from peripheral pulses in the upper

limb. It also describes the inner workings of the system together with algorithms

developed to extract features and analyse the pulses from different recording sites in the

5

upper limb. The system described in this chapter is designed and implemented as a

clinical tool to be used in the clinical environment.

This investigation has highlighted the importance of quantification of wave propagation

characteristics in the arterial system especially in the upper limb vasculature. Systems

that measure the various features of the arterial pulse do not take wave propagation into

account. Information obtained using these techniques can be limiting in both content

and interpretation. Results of the present study suggest means of overcoming these

inadequacies leading to improved clinical applications through measurement and

analysis of the arterial pressure waveform.

6

SECTION I

BACKGROUND

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HISTORY The beating of the heart, the presence of the arterial pulse and the movement of the chest

are the three vital signs that have been traditionally associated with the presence of life.

A large part of the history of medicine covers the attempts to reconcile these signals of

life and volumes of monographs summarise this historical treatment (McDonald 1960,

Fishman and Richards 1964; Schwartz et al 1981; Milnor 1989; Nichols and O'Rourke

1990; O'Rourke et al 1992). In this chapter, the arterial pressure wave and its

distribution along the arterial tree will be addressed with respect to historical

development.

Ancient Greeks had an inaccurate notion of the arterial pressure pulse. They believed

that the arteries (air ducts) actively contracted and dilated, like bellows, to pump vital

spirits to and from the heart. Prominent amongst these thinkers was Praxagoras who

Chapter 1

8

lived in 400 BC (Harris 1981). The first opposing view came in 300 BC by Erasistratus

of Cos, who on pure mechanical grounds, argued the role of the left ventricle and

proposed the propagation of the wave from the heart (Schechter et al 1969). This point

was ignored by Galen (129-199 AD) who by his works, influenced medicine for the next

fourteen centuries. The only improvement of the Galenic theory was to allow blood to

run in the arteries rather than just the spirits. Jean Furnel (1497-1558) and Realdo

Columbus (1510-1559) both argued that the pulse travelled away from the heart

(Webster 1981). This was taken up by William Harvey (1578-1637) in his influential

work "De motu cordis" in 1628. Harvey, regarded as the father of modern

cardiovascular medicine for his work in establishing the modern views on circulation,

failed to point out the propagation delay in wave transmission (Franklin 1979). John

Floyer (1649-1734) was first to demonstrate the effect of the elastic wall on removing

the intermittency of the pump on his cow gut experiments. Twenty-five years earlier

than Hales, he quoted "The force of water injected (by a pump) protruded the gut, and

the annular fibres by their natural restitution promoted the motion of the water, and

kept the stream from any interruption though the injection was made by intervals"

(Schechter et al 1969; Taylor 1991). Stephen Hales (1677-1761) best known for

measuring blood pressure in a horse, arrived at the same conclusion through a different

route. He reasoned that although there were pulsations in the arteries, none was

observable in the veins. In his acclaimed book "Haemostatics' (Hales 1773) he also

mentioned the sites of resistance on the arterial system, undoubtedly capillaries as

described by Malpighi in 1620 and the estimates of cardiac output by post mortem

measurements of chambers of the heart. This book was translated into German and

formed the important foundation for German physiologists in the late 19th and early

20th century. Cardiac output was measured in vivo by the German physiologist Fick in

1870, a hundred years later, while the amount of resistance was measured by J.P.

Poiseuille (1797-1869) by assuming a laminar viscous flow. Poiseuille was the first to

introduce mercury as a fluid to measure the height of the blood pressure column. This

mercury manometer was improved upon by Carl Ludwig in 1842 whose pupils

9

dominated physiology throughout the latter part of the 19th century and middle of 20th

century (Milnor 1989).

The introduction of calculus to physics by Isaac Newton led to the proliferation of

scientists applying this tool to different natural phenomena. Leonhard Euler (1707-

1783), not only contributed to expansion of techniques of calculus, but also was the first

to develop the equation of motion in liquids. His theory was hailed as the complete

theory of fluid flow by others but did not include the viscosity effect "defectus lubricus"

as first mentioned by Newton. These equations remained insoluble until 1958, when

Lambert tackled it with the method of characteristics (Skalak 1972). The effect of

viscosity is taken into account by Navier in 1822. This formulation was later extended

and corrected by Stokes to be known as the Navier-Stokes equation (Milnor 1989).

The arterial wall properties were first investigated by Spallanzani in 1773, who fitted

rings of fixed diameters around the aorta to detect dilation of the vessel by pressure

waves. He was the first to show that the pressure wave is propagated through the wall

rather than the blood (Fishman and Richards 1964). The same idea was also taken up by

Thomas Young (1773-1829) (also known for the theory of colour-vision) when he

investigated the propagation speed of pressure waves in an artery. Young (1808)

concluded the ongoing debate of contraction of arterial wall muscle and the formation of

the pressure pulse. He examined the speed of a free falling body in a compressible tube-

fluid combination (Noordergraaf 1969). The resulting equations related the pressure-

diameter relationship (elasticity) to the wave speed - later rediscovered by Bramwell-

Hill and Otto Frank (McDonald 1974). Wave equations and their implications became

very topical in the early 19th Century, thus when the Weber brothers published the

monograph "Wellenlehre" in 1825 they laid the fundamental principles of wave

propagation and wave reflection. The later work done independently by Moens in 1878

and its analysis by Korteweg and Resall (McDonald 1974) led to the well-known wave

speed equation.

10

While these advances in wave travel were being made, the Chinese views of the pulse

were spreading (Osler 1921). These views which first arrived in Europe in the 17th

Century stipulated that the disturbances in the function of any organ could be detected

by changes in the pulse waves in the radial artery (Fung 1984). The detection of the

pulse from superficial arteries (specifically radial and carotid) and interpretation of the

pulse reached its peak in the 19th Century with the continuous addition of numerous

versions of sphygmographs (sphygmo - Greek: beat, throb) to the arsenal of the

practitioner who only possessed the pulse watch, thermometer and stethoscope at the

time (O'Rourke et al 1992). Etienne Marey (1830-1904) advanced the kymography

technique introduced by Ludwig by incorporating it into instruments of pressure

measurement (Snellen 1980). He also developed the first intra-cardiac pressure

measuring catheter (Geddes 1970) which he used in a horse.

In the late 19th Century, the tide of research was turning more towards cardiac

properties. Mackenzie in the preface to his monograph 'The Study of the Pulse' stated

that "It is now fashion to decry the value of the sphygmograph" (Mackenzie 1902). It is

therefore not surprising that the works done by Mahomed (1874, 1892), Broadbent

(1890) and others were forgotten for almost one hundred years (O'Rourke et al 1992).

Instead blood pressure began to be assessed by non-invasive occlusion methods under

clinical conditions. Works of Riva-Rocci (1896) and Korotkoff (Korotkoff 1905;

Geddes 1966) laid the foundations of the sphygmomanometer. Although the

consequences of wave propagation were not properly addressed, this technique gained

wide acceptance and still dominates clinical practice (O'Rourke et al 1992).

Since the study of the heart involves invasive techniques, researchers needed better

equipment to measure pressures inside the cavities. The development of fluid-filled

manometers by Rollesten and Hurtle (Geddes 1970) and many others attempted to fulfil

this demand. However, there were problems in the fidelity of the manometers and there

was no standardisation. Otto Frank (1865-1944), the last pupil of Ludwig, called for

11

physical principles to be followed in the design of these manometers. He solved the

differential equation of motion of a single degree of freedom system (Frank 1903). The

solutions are quite easy to follow and still in use today (Nichols and O'Rourke 1990).

However, preoccupied with manometer theory he went on to apply the same equations

to the arterial system while borrowing from Hales' concept of an elastic reservoir (Frank

1905). The "Windkessel" theory, as we know it today, views the arterial system as an

overdamped catheter to explain the aortic pressure waveform. It is important to note

that although he must have been aware of the wave equations and transmission lines (the

transatlantic telegraph cable was already laid down in 1866), he failed to employ them

in the basic equations. Nevertheless, Frank and his followers such as Wiggers (1928),

Hamilton (1944), Wetterer (1954) and Remington and Wood (1956) dominated the area

of haemodynamics for the next fifty years (O'Rourke 1965).

It was Aperia (1940), one of the followers of Frank, who pointed out the inconsistencies

in the entire assumptions made by Frank. The Windkessel theory could not explain the

different wave shapes in different arteries without resorting to the wave equations.

Aperia tried to reconcile these concerns but his premature death prevented him doing so

(McDonald 1974).

Although the first complete set of analyses of wave propagation was made by Witzig in

his PhD thesis in 1914, this was not widely circulated and thus remained obscure

(Milnor 1989). The treatise of transmission equations was starting to have an impact on

physiology when McDonald and Womersley collaborated in 1952 to publish their

elaborate one on arterial fluid dynamics. Almost simultaneously with Hodgkin and

Huxley, who used telegraph equations for description of action potential propagation in

nerve cells (Hodgkin and Huxley 1952), Womersley linearised and solved the Navier-

Stokes equations with well-documented assumptions (Womersley 1955a). The

mathematics of this theory required heavy computation that had hindered earlier

attempts. Having access to computer facilities that were scarce at the time, helped him

12

to generate tables necessary for the theory (Dinnar 1981). When the question of

reflected waves appeared in 1957 (Taylor 1957a) the solutions he proposed were easily

modified to include these effects (Womersley 1958). The results of this theory and its

predictions were published in textbook form by DA. McDonald in 1960. The book

'Blood Flow in Arteries' became the standard reference in the field of arterial

haemodynamics (McDonald 1960).

The success of this linearised theory and its applications dominated research in

haemodynamics in the 1960s and 1970s. The arterial impedance (Patel et al 1963,

O'Rourke and Taylor 1966, 1967a) and transmission of the pulse (Taylor 1966b), led to

a widespread continuation of research into wall properties (Learoyd and Taylor 1966;

Gow and Taylor 1968, Gow 1972) and wave reflection phenomena (O'Rourke 1967b,

Westerhof et al 1972). During this period several models of the arterial system were

proposed (de Pater and Van den Berg 1963; Taylor 1966a, 1966b; Westerhof 1969;

O'Rourke 1967b; Anliker et al 1978; Avolio 1980). By the 1980's, the influence of

reflected waves on arterial waveforms (Murgo et al 1980a, 1980b; Fuji et al 1987, Kelly

et al 1989a) and on aortic impedance (O'Rourke 1982; O'Rourke et al 1987) on left

ventricular function (Laskey et al 1985) was investigated. Arterial haemodynamics has

never experienced such consistent and fruitful collaboration between theory and

practice.

13

THEORETICAL FOUNDATIONS The conclusion drawn from the recorded pressure waveform depends on the observer's

conception of the arterial vasculature, that is, it will depend on the assumed circulatory

"model".

The "model" is as vague a term as the phenomenon it tries to describe. There appears to

be little agreement upon the definition, its use and its interpretation. It was previously

referred to as "something simple by scientists to help them understand something

complicated" (Fitzhugh 1969), or "an effective visualisation of a functional relation"

(Kenner 1978), or as "a set of mathematical relations between relevant quantities" (Arts

1978). In this respect there are different ways to approach modelling. One can define

important parameters for the system under investigation and observe only these

parameters to construct a set of rules. Alternatively, one can define sets of rules of

Chapter 2

14

importance and eliminate the rules according to the subsequent observations. A final

approach would be to permute all the available rules and all the available sets of

observations and determine the ones that fulfil a pre-defined optimum criterion. This

optimum criterion clearly dictates the rules and observations to be made, yet it can not

change the experimental data. It is equivalent to saying that interpretation of data in

different ways will not create more data points. Thus, a model is a tool, an abstraction

technique, to transform real world observations to a known domain of few rules, to

interpolate and/or extrapolate the real world in an abstract predefined optimum. The

model, therefore, is not meant to replace the real system. Rather it is to gain better

understanding of the real system given certain constraints.

2.1 MATHEMATICAL MODELS OF THE ARTERIAL SYSTEM

Previous investigators attempted to model the arterial system to understand the complex

interaction between the heart and the arterial tree (Yin 1987a), to understand the

interaction between the flow and pressure waves at each point in the tree (Avolio 1980)

and to explain differences in the pressure and flow shapes due to altered vascular

properties (Westerhof and Noordergraaf 1970). To assert the existence of a steady state

condition, the nervous control of the heart and the vasculature is either ignored or is

assumed negligible in most attempts. In these models, it has always been acknowledged

that the heart is a force generator in the circulation that keeps blood in motion. This

motion of blood ensures a steady concentration gradient of nutrients and metabolic by

products to exist at the cellular level. In gross terms, the general model of the

circulation includes a power generator, whose energy is both taken up by the vessel

walls as elastic energy and dissipated by the viscous forces in the wall and in the blood

during circulation. Thus it is not surprising that the first attempts to model the

15

circulation concentrated on this dissipative force as a first order approximation to the

circulation.

2.1.1 Steady Flow

Probably the first observation leading to this model was done by Hales. Using a time

clock and the gut of an animal he observed a reduction of flow velocity in the region

close to the capillaries, accounting for the presence of resistance at around the capillary

level (Schechter et al 1969). This work was taken up by Poiseuille who could not

document any difference in blood pressure along the large arteries but between the veins

and the arteries indicating the small arteries beyond 2 millimetres. When he studied the

water flow in small tubes of 30µm to 140µm internal diameter, he noted the following

relation (Milnor 1989).

( )

lPPKD

Q 214 −

= (2.1)

Where Q, P1-P2 and l are the volume flow, pressure drop and length of the tube

respectively. D is the tube internal diameter, and K is a constant. He found that K is

independent of everything else but temperature. Subsequent developments indicated

that K was related to viscosity. Independent theoretical studies by Wieden in 1856 and

Hagenbach in 1860, found (Nichols and O'Rourke 1990) that

K =

πµ128 (2.2)

Consider an axisymmetric flow of an incompressible fluid of viscosity µ across a

pressure gradient (P1-P2)/l in a tube of radius R and length l. Then, the expression for

velocity may be obtained using the Navier-Stokes equations in cylindrical coordinates

16

∂∂

∂∂

∂µ∂

∂µ∂

2

2w

rw

r rPz

r wt

+ + = (2.3)

Since ∂w/∂t=0 the equation reduces to

∂∂

∂∂

∂µ∂

2

2w

rw

r rPz

+ = − (2.4)

or

z

Pdr

rdwrdrd

µ∂∂

−=

(2.5)

Integrating twice yields

( ) 212 CrLnCr

zPw ++−=

µ∂∂ (2.6)

w should be finite at the axis and at the radius R. This yields C1 0= , and

C Pz

R22

4= −

∂µ∂

. Substituting for C1 and C2

( )22

4Rr

zPw −−=

µ∂∂

(2.7)

Volume flow then becomes

( )µ

π∂∂π

82

4

0

RzPdrrwQ

R

−== ∫ (2.8)

substituting ( )lPP 21 − for − ∂

∂Pz

yields equation 2.1

17

The assumptions and validity of these assumptions in the derivation of this formulation

are quite important and discussed elsewhere (Caro et al 1978; Milnor 1989; Nichols and

O'Rourke 1990). The term 8µl/πR4 can be thought of as the resistance term to bolus

flow by analogy with electrical circuits where the pressure drop along the tube is the

voltage and the input bolus flow becomes the current. Analogous to electrical networks,

any arterial bed in the systemic circulation could be modelled as a parallel and series

combination of resistances. Since the element lengths are fixed for anatomical reasons,

the resistance of the entire network becomes proportional to the diameter of individual

elements. The most useful and probably the most used rule of resistance is the total

peripheral resistance (TPR)derived as

TPR MeanArterial essure MeanVenous essureCardiacOutput

=−Pr Pr

(2.9)

2.1.2 Oscillatory Flow

Windkessel Approach

Apart from the slowing down of flow, Hales also observed the diminishing of pulsations

from the arterial to the venous side. Thus he likened the arterial system to a

contemporary fire engine with reservoirs, which operated like elastic chambers. These

ideas were later taken up by German physiologists, particularly by Otto Frank, from the

German translation of the book 'Haemostatics' by Hales. In German translation the

elastic reservoir was called Windkessel or "air kettle" and was attached to this

description by Otto Frank.

One Element Windkessels

According to Frank, the left ventricular ejection would be taken up during systole by the

arterial wall, which is to be passed to the blood to drive it against a resistance, namely

18

the peripheral resistance (Nichols and O'Rourke 1990; Milnor 1989; Skalak 1972).

Although Frank was aware of concepts of wave phenomena (Frank 1899) which was

evident from his work on manometers (Frank 1903), he totally ignored the forces

associated with blood inertia and blood viscosity as well as wall viscosity. Thus the

resultant differential equation contained only two terms. Assuming an elastic chamber

where rate of pressure change dP is related to volume change dV as

( )PfdVdP

= (2.10)

and a discharge from the resistance element, R, of the form

dVdt

PR

= − (2.11)

then the equation 2.10 and 2.11 lead to

( ) Rdt

PdP

Pf−=

1 (2.12)

In special case where f(P) is a constant of the form f(P)=1/C then

C dPP

dtR

= − (2.13)

integration and assuming P=Po at t=0 yields

P P e t RC= −0 (2.14)

This equation and its solution is basically an RC network in electrical engineering.

Overall, it is an improvement on a one element (resistive) model of the circulation. It

attempts to describe the systolic, diastolic and pulsatile pressures and the shape of the

19

central aortic pressure waveform. The model found great enthusiasm amongst the

German school (Wetterer 1954) and in the USA (Wiggers 1928; Remington et al 1948;

Remington and Wood 1956) and dominated the physiological research until the 1950s.

The major drawback of the model is that it does not allow for wave transmission since

the chamber is filled instantaneously, and by implication there is no wave reflection. It

also neglects the inertial forces. Noticing these flaws Aperia (1940) tried to reconcile

wave transmission with the elastic chamber theory.

Remington tried to explain the differences in peripheral arterial wave shapes due to

resonances and thus incorporated second order systems (Remington and O'Brien 1970;

Remington 1974). The system is described by two parameters, R and C, where one

should be calculated independently since the other could be determined from the

diastolic decay of the arterial pulse. This led to a major effort in calculating the

effective Windkessel capacitance. Although the model did not consider the propagation

of waves, there were attempts to relate wave velocity to the capacitance, C (O'Rourke et

al 1968). The model also had problems describing the pressure flow relations during

early part of the systole. Although it implied that the pressure rise was proportional to

the inflow, the experimental results suggested that pressure itself was proportional to

inflow. Furthermore, it not only directly related the diastolic pressure and the peripheral

resistance but also systolic pressure to the contractility of the heart and to the wall

stiffness (O'Rourke et al 1992). The model, although under close scrutiny, finds use in

predicting stroke volume from pressure waveform (Skalak 1978). Due to shortcomings

of two-element Windkessels, three and more element Windkessels were devised.

Modified Windkessels

Given the inherent limitations of the original Windkessel theory, it is often modified to

take into account blood inertia and the transmission delay in filling the chamber, leading

to incorporation of inductance and series resistance to the original RC parallel network

20

(Westerhof et al 1977; Milnor 1989). The influence of non-linear properties of the

arterial wall was also incorporated into a model by assuming non-linear pressure-

volume relations in quadratic or higher forms (Cope 1965). The assumption of

combined parabolic and linear forms yielded deviations of 5-10% of the actual values,

when stroke volume was estimated from pressure waveform (Cope 1965). This led to

investigation by McDonald (1974) of the estimation of stroke volume by pulse contour

methods. These Windkessel equations appear to give satisfactory results when the

effect of wave travel becomes small due to increased wave speed.

Versions of modified Windkessels and especially second order systems have been

modelled (Guier 1981). They are also used to explain the differences in wave contours

in different arteries (Wiggers 1928; Warner 1957; Remington 1974). These models all

suffer from infinite wave speed and lack of wave reflection (McDonald 1974).

Because modified Windkessels are easy to implement as hydraulic analogues, they enjoy

widespread use in testing cardiac valves, intra-aortic balloon pump (Niederer and Schilt

1988) and modelling of the function of the heart (Sunawaga et al 1983). However, they

ignore the effects of wave reflection therefore their use is limited. Modified

Windkessels are also used to lump arterial beds with few model parameters in modelling

studies of the systemic circulation (Liu et al 1989, Burattini et al 1989b).

Womersley's Approach

Previous models described above take into account neither the nature of blood flow in

the arteries, (although "the most prominent feature of blood flow in arteries is that it is

pulsatile" (McDonald 1960)), nor the effect of wave propagation. Amongst the many

previous attempts (Witzig 1914; Crandal 1927; Iberal 1950; Morgan and Kiely 1984),

Womersley's treatise of the subject is particularly significant (Womersley 1955a, 1955b;

1957a, 1957b, 1958). The model he described is the most realistic compromise due to

21

close collaboration with DA McDonald and MG Taylor who supplied the necessary

information about the validity of the assumption. In his work, he documented every

assumption and the final work was simplified for immediate testing. Thus, the model he

devised is still the best recognised, although there were many other attempts to

supersede it (Noordergraaf 1968; Cox 1969).

Womersley assumed a homogenous, Newtonian, incompressible, viscous fluid in a long

and straight cylindrical tube with thin, isotropic walls and longitudinally tethering while

externally constrained. The flow is considered to be laminar and axisymmetric with no

slip at the wall. The pressure gradient is assumed to be oscillatory.

The equations of motion in cylindrical coordinates for the type of fluid given above in

Navier-Stokes form are

++−=− 2

2

2

2

0 zw

rrw

rw

tw

zP

∂∂

∂∂

∂∂µ

∂∂ρ

∂∂ (2.15)

−++−=− 22

2

2

2

0 ru

zu

rru

ru

tu

zP

∂∂

∂∂

∂∂µ

∂∂ρ

∂∂ (2.16)

The tangential velocity component v is omitted. The convective non-linear terms

u∂w/∂r, w∂w/∂z, u∂u/∂r, and w∂u/∂z are assumed to be negligible and also dropped

from the equations. Continuity equation gives

∂∂

∂∂

ur

ur

wz

+ + = 0 (2.17)

After non-dimensionalising r by setting y r R= −1 and assuming a travelling pressure

wave of the form

22

( )cztjeAP −= ω1 (2.18)

where ω and c are the angular frequency and wave velocity respectively. The solutions

of equations 2.15 and 2.16 using 2.17 and 2.18 become

( )( )

( )cztjec

AjJ

yjJCw −

+= ω

ραα

0

123

0

230

1 (2.19)

( )( )

( )cztjec

yAjJjyjJC

crju −

+= ω

ραααω

0

123

023

231

12

2 (2.20)

where α is the non-dimensional parameter giving the ratio of inertial to viscous forces as

µωρα 022 R= (2.21)

J0 and J1 are zero and first order Bessel functions with complex arguments. C1 is

integration constant. If one defines

( )( ) 101023

023

231

102 ε

ααα M

jJjjJF == (2.22)

where M10 is the modulus and ε10 is the phase of F10, then the fluid motion at the

boundary becomes

( )cztjy e

cACw −

=

+= ω

ρ0

111 (2.23)

( )cztjy e

cyACF

cRju −

=

+= ω

ρω

0

11101 2

(2.24)

23

The equation of the wall motion is obtained by assuming

( )EjEEc ∆+= ω1 (2.25)

( )σωσσ ∆+= jc 1 (2.26)

for complex elastic modulus, Ec ,and complex Poisson ratio,σc , of the wall. Defining ρ

as the wall density, ξ and ζ as the radial and axial displacements of the wall which are

travelling sinusoidal waves as

( )cztjeD −= ωξ 1 (2.27)

( )cztjeE −= ωζ 1 (2.28)

The descriptive equations for the radial wall motion become

( ) 01 222

2

=

+−

+−RzRH

hEH

Pt

c

c

c ξ∂∂ζσ

σρρ∂ξ∂ (2.29)

where the first term is the inertial, the second is the transmural pressure and the third is

the viscoelastic component of the wall movement in radial direction. The axial

component is also

( ) 01 2

2

22

2

=

+

−−

++−

= zzRHhE

zu

rw

HHK

tc

c

c

Rr ∂ζ∂

∂∂ξσ

σρ∂∂

∂∂

ρµ

ρζ

∂ζ∂ (2.30)

The terms in the equation from left to right are inertial component, spring force of the

external constraint, the drag force from the fluid motion, and viscoelastic deformation of

the wall substance in axial direction. K represents the tethering tissues spring

24

coefficient and is related to undamped natural frequency of wall element in the z

direction as

2HMK ρ= (2.31)

The wall and the fluid velocity should couple at the boundary, ie

( )cztjeDjdtdu −== ωωξ

1 (2.32)

( )cztjeEjdtdw −== ωωζ

1 (2.33)

Substituting equations 2.32 and 2.33 to equations 2.23 and 2.24 at the boundary (y=1)

yields

c

ACEj0

111 ρ

ω += (2.34)

+=

cACF

cRD

0

11101 2 ρ

(2.35)

Equations 2.29, 2.30, 2.34 and 2.35 form a set of simultaneous equations with four

unknowns: A1, B1, C1 and D1 or if A1 is given C1, D1, E1 and c , the complex wave

velocity. Inspection of 2.29, 2.30, 2.34 and 2.35 indicates that the solution contains only

the known constraints plus the complex wave velocity. Solving for c gives

( )χσρ 20 11

cchE

Rc

−= (2.36)

or

25

( )χσ 2

0

12

11c

cEE

cc−= (2.37)

or written as real and imaginary parts

( ) ( )10

20

11

FjYX

cc c

−

−=−=

σ (2.38)

where c0 of the pulse wave velocity in an infinitesimally thick tube filled with inviscid

fluid and equal to the one given by Moens-Korteweg equation

c EhR0

02=

ρ (2.39)

then X cω 0 and Y cω 0 now become the phase shift and the attenuation coefficient

respectively. The χ is given as

( )

( )2

22

11

c

c HGGσ

σχ

−

′−−±= (2.40)

where

G

Fkc

c=−

−+

′+ −

5 41 2

1 410

σσ

(2.41)

′ =

+ ′−

−H kF

1 21

110 (2.42)

−= 2

0

' 1ωρρ

ρHK

RHk (2.43)

On the other hand, inserting equations 2.34, 2.35 into equation 2.27 gives

26

c

NAC

01

1

ρ= (2.44)

where N is defined as

( ) c

c

c FFN

σσ

σχ 221

22

1010 −−

−−

= (2.45)

The mean velocity across the lumen is obtained by integrating equation 2.19 from y=0

to y=1 which yields

( ) ( )cztjeNFc

Aw −+= ω

ρ 100

1 1 (2.46)

This final equation does not take into account the wave reflection, yet it can be included

if pressure, P, is given as superposition of forward and backward travelling pressure

waves, ie

( ) ( )cztjcztj eAeAP +− += ωω21 (2.47)

then using equation 2.47, equation 2.46 becomes

( ) ( )

( ) ( ) ( )cztjcztj eNFc

AeNFc

Aw +− +−

++= ωω

ρρ 100

210

0

1 11 (2.48)

or,

( ) ( ) ( )[ ]

−+= +− cztjcztj eAeA

cjNF

jw ωωω

ωρ 21100

11 (2.49)

the terms in curly bracket is in fact − ∂ ∂P z , which can be rewritten as

27

− = ′∂∂

ωPz

A e j t (2.50)

then equation 2.49 becomes

( ) tjeNFjAw ω

ωρ 100

1+′

= (2.51)

The instantaneous average blood velocity across the lumen is proportional to the

pressure gradient and the term ( )100 1 NFj +ωρ becomes the longitudinal impedance of

the tube, Zz. The characteristic impedance of the tube is related to longitudinal

impedance by wave velocity (McDonald 1974) as

zZj

cZω

=0 (2.52)

expansion gives

( )10

00 1 NF

cZ+

=ρ (2.53)

and inserting equation 2.37 in equation 2.53 and considering a strongly tethered tube, ie.

N= -1, gives

210

00 11

c

FcZ

σρ

−−

= (2.54)

Discussion of the Validity of the Womersley Approach

Womersley introduced several assumptions in derivation of the solution to fluid flow.

These have been discussed previously (McDonald 1960, Fry and Greenfield 1964;

Milnor 1989, Nichols and O'Rourke 1990). The important points to be related to these

assumptions are the following:

28

(i) Flow is laminar. This condition is fulfilled for most of the cardiac cycle.

However, small bursts of turbulence might occur in the ventricular outflow tracts. Non-

laminar flow increases the viscous dissipation, hence pressure gradient might be higher

for a given flow. The curvature of the ascending aorta introduces rotational flow which

might introduce some error.

(ii) Newtonian flow. In arteries larger than 500 microns the shear rate is high leading

to an asymptotic value of viscosity of blood. However, below this diameter because of

the Fahraeus-Lindquist effect, anomalous viscosity will be apparent (Nichols and

O'Rourke 1990).

(iii) Uniform cylindrical tube. Although arteries are uniform between branching,

there exists a tapering which might lead to convective acceleration (Fry and Greenfield

1964).

(iv) Entrance effects. The entrance effects are shown to exist in the ascending aorta

and major bifurcations leading to an unestablished flow.

(v) Reflected waves. The existence of reflected waves is shown not to affect the

longitudinal impedance provided that forward and backward propagation coefficients

are equal.

(vi) Linearisation of equations. During linearisation, Womersley assumed that the

terms given by u∂w/∂r, w∂w/∂z, u∂u/∂r, and w∂u/∂z are small with respect to the linear

terms, considering no convective acceleration in radial and axial direction. He also

assumed that axial flow velocity to pulse wave velocity ratio, w/c, and radial flow

velocity to pulse wave velocity, u/c, were small. During the cardiac cycle, c is about

500 cm/sec, while the peak flow is 100 cm/sec, which lasts only briefly. The non-linear

pressure-diameter relation of the vessel wall also introduces errors in the boundary. He

29

also neglected the terms ( ) αω ⟨⟨2cR implying that at all times the radius, R, is much

smaller than the wave length. Although the non-linearities described above impose

errors, experiments (Dick et al 1968) showed that they are small and negligible with

respect to the measurement errors. It is thus justifiable that a closed form solution

obtained by linearisation can be used in vivo without sacrifying the ease of calculation.

(vii) Thin tube. It is known that the wall thickness to diameter ratio increases towards

to the periphery. This assumption is used for the Moens-Korteweg equation of pulse

wave velocity for inviscid fluid. This increase, however, in the order of 10-18% causes

10% of errors in wave velocity estimations.

(viii) Isotropic tube. Although it has been assumed isotropic, the vessel wall is far

from being so. Although this point does not introduce major errors it has to be taken in

account in interpretation of wall stresses calculated using these equations.

(ix) Incompressible wall. The compressibility effects the Poisson's ratio, s, making it

different from 0.5. Yet, it is been reported that the wall is normally incompressible

(Carew et al 1968).

(x) Newtonian wall viscosity. Although the viscosity of the wall in arteries is far

from constant, the phase angles in the complex elastic moduli are in the order of 10°-

15°. Being small and not changing with pressure (Gow 1972), the errors introduced by

this assumption are negligible.

(xi) Wall and fluid densities are equal. Arterial wall density is 2% higher than that

of blood.

Although none of the above assumptions are correct in vivo, the errors that could be

introduced are normally less than the measurement errors. Yet, there are consistent

30

differences in the estimation of pressure gradient and derived longitudinal resistance,

which is consistently underestimated (Fry and Greenfield 1964) and differences in the

attenuation of harmonics are greater.

Validation of Womersley's Approach

During validation of any theory, the only and most valuable form of action is to compare

theoretical predictions with experimental findings. In an elaborate theory such as

Womersley's, one then has to measure all the boundary conditions (physical properties

of the wall), fluid properties (viscosity and density) and pressure and flow

simultaneously. One has to measure these variables as precisely as possible to be

confident about the accuracy of predictions. However, even with current

instrumentation the limits of measurement error are not less than about 5%, thus the

theory can only be confirmed within the observable limits.

In the last four decades it has been repeatedly shown (McDonald 1955; Bergel et al

1958; Greenfield 1966; Milnor 1978) that flow wave contour estimated from the

pressure gradient is satisfactory. However, there is growing evidence that the resistance

term and the attenuation predicted by the theory is different from the experimental

findings (Fry and Greenfield 1964, Milnor 1989). These differences have also shown to

exist in every linear theory (Cox 1969). Thus it can be speculated that this is the result

of linearisation of the Navier-Stokes equations, by eliminating the convective

acceleration terms. It normally does not cause gross errors under normal conditions.

However, when one includes the effect of tapering, this effect can cause discrepancies as

much as 200 % (Ling and Atabek 1972). The other effect that has to be taken into

account is the non-linear wall properties which causes steepening of the wave front as it

moves distally (Rockwell 1969).

31

Other Approaches

Linear Models

Womersley was not the only investigator who linearised the Navier-Stokes equations

(see Witzig 1914; Crandall 1927; Jager et al 1965; Whirlow and Rouleau 1965; Atabek

1968; and many others). Witzig (1914) was the first to linearise the Navier-Stokes

formulation for a thin elastic tube. In 1927, Crandall published the expressions for the

velocity profile, average velocity and fluid impedance for an oscillatory flow in a thin

rigid tube. In 1950, Iberal published the analysis of the same flow in a one-dimensional

radial-elasticity equation with restrained wall. After Womersley's solution, there has

been an explosion of linear theories for blood flow in arteries. The detailed analysis of

all these models with respect to Womersley solutions is described in a paper by Cox

(1969) and subsequently by Milnor (1989). It was shown that in the limiting case all

these models predict the same relations for fluid flow and wave propagation. The

individual differences constituted less than 5%. Addition of wall viscosity predicts an

increase in phase velocity and attenuation. The effects of wall compressibility make the

predictions of thin and thick walled models converge as Poisson's ratio approaches zero

and propagation constants in thick and thin walled models become similar as h/R falls

below 0.1.

Non-linear Models

The existence of tapering and non-linear wall elastance as seen above, causes

discrepancies between theory of linear models and experimental findings. Although

small and negligible, advances in computer capabilities often encouraged investigators

to explore the addition of these terms on the predicted flows.

(i) Method of Characteristics

32

It was known by Euler that the closed form solutions to non-linear differential equations

were not available using known analytical techniques, when he developed the first

theoretical analysis of inviscid flow in a tube in 1775. In developing this analysis he

became aware that the boundary was non-linear, ie wave velocity was dependent on

pressure. The problem was not tackled until 1958 when digital computers began

making an impact on mathematical solutions to non-linear equations. It was Lambert

who used this opportunity to suggest a method of characteristics which could be used to

solve the problem.

The major drawback of the approach is that flow is one-dimensional and requires

directional terms to be supplied by another theory or experimentation. One-

dimensionality leads to the characterisation of flow and pressure in the axial direction in

time but lacks the description of radial distribution of the velocities. In that sense, the

technique is cruder than using linearised solutions.

The basic equation of motion is in radial coordinates

fzP

zww

tw

=++∂∂

ρ∂∂

∂∂

0

1 (2.55)

where w is the axial velocity and f the net effect of shear stresses due to viscosity of

fluid at the boundary of the vessel. The continuity equation is (Rockwell 1969, Skalak

1972)

( ) 0=Ψ++z

SwtS

∂∂

∂∂ (2.56)

S, the vessel lumen area is considered to be a function of pressure and distance.

S S P z= ( , ) (2.57)

33

In equation 2.56, Ψ denotes the efflux of fluid through the walls per unit length. This is

a conventional way to include branching. The equation of motion and the equation of

continuity can be combined to the form

21 LLL λ+= (2.58)

where L1 is the continuity and L2 motion equations. After eliminating dependent

variable S by considering

zP

PS

tS

z ∂∂

∂∂

∂∂

= (2.59)

and

Pz z

SzP

PS

zS

+

=

∂∂

∂∂

∂∂

∂∂ (2.60)

equation 2.58 leads to

( ) 01

0

=Ψ+

+−

+

+

+

++= λ

∂∂λ

∂∂

∂∂

∂∂λρ∂

∂λ∂∂

∂∂λ

P

z

z PSwf

tP

zP

PS

wPS

tw

zwwL (2.61)

Taking derivatives of P and w with respect to t leads to

tP

dtdz

zP

dtdP

∂∂

∂∂

+= (2.62)

and

34

tw

dtdz

zw

dtdw

∂∂

∂∂

+= (2.63)

Inspecting equations inside the curly brackets in equation 2.61 it is evident that the first

bracket will be dw/dt if

λ+= wdtdz (2.64)

and the second brackets will be dP/dt if

zPS

wdtdz

+=

∂∂λρ0

1 (2.65)

then combining equation 2.64 and 2.65 one obtains

zPS

ww

+=+

∂∂λρ

λ

0

1 (2.64)

which is

cSP

z mm =

=0ρ

∂∂

λ (2.65)

where c is the pulse wave velocity. Then the four sets of equations obtained

dzdt

w c= m (2.66)

and

35

Ψ

=± mmm

tS

zSw

Scf

dtdP

cdtdw

∂∂

∂∂

ρ0

1 (2.67)

After writing the equations on a finite difference scheme using a mesh ratio of

Θ ∆ ∆= t z and satisfying the condition of stability and convergence by

( )cwz +⟩∆ (2.68)

At all times the intermediate values can be obtained with respect to the generated mesh

using boundary conditions and the known relations between P, z and S and functions

for f, Ψ (Corey et al 1975).

Porenta et al (1986) considered ∂ ∂P z = 0 at the branch level of an arterial bed which

was represented as a modified Windkessel. The branch flow becomes

012

1

22 =

+−+− ββ

RR

RP

dtdCR

dtdPC (2.69)

where C is the compliance, R1 and R2 are the series and parallel resistances to the

compliant element, β is the difference between the flow before and after the branching

point in the main arterial stem. The frictional force was written as f FV ni i i= −2 where

Fi, Vi and ni are the constants in i= +z, -z directions. The solution of these equations in

such a leaking tube displayed the progressive increase in pulse amplitude and the

steepening of the pressure wave. It also displayed a good agreement in calculated mean

pressure.

The models of this kind also employed viscoelastic wall (Anliker et al 1978), branches

(Porenta et al 1986) and obstructions (Anliker et al 1978; Porenta et al 1986). The

predicted wave shapes are close to the ones obtained in vivo.

36

Extensive use of computers in the field of biomedical research increases the potential

application of these models. Since the computing time was quite expensive and this

technique did not produce enough improvements to the modelling, it coined the term

"steam-hammer for crushing peanuts". The technique also needs measurement of

diameter change and/or viscous forces which are difficult to achieve.

(ii) Wavelet Analysis

An alternative approach has been suggested by Parker and Jones (1990) using wavelets

to describe the energy per unit area by constituting equations of

zz dPdPdP −+ += (2.70)

zz dwdwdw −+ += (2.71)

indicating forward, +z, and backward, -z, waves. It was assumed that if the viscous

losses are ignored in large arteries the integral

∫±=±

P

Pz cdPwR

0 0ρ (2.72)

is constant along the characteristic paths dzdt

w c= ± . Thus, the energy of the wavelet

becomes

220 zz dRdRcdPdw −+ −= ρ (2.73)

leading to the separation of expansion of wavelets from other factors. Real time

separation of reflected waves by this technique is its big promise.

37

2.2 ANALOGUES OF THE ARTERIAL SYSTEM

"An analogue implies a recognised relationship of consistent mutual similarity between

the equations and structures of two or more fields of knowledge." (Attinger 1964). To

facilitate the transfer of knowledge between two systems, the parameters and structural

elements of the similar kind are identified and related. For this purpose, often the

differential equations describing two different systems are inspected and related. The

most common of these systems is the electrical circuit theory, since it is easy to

implement and to confirm results with the other fields.

Therefore, the electrical analogies are of practical importance not only for didactic

purposes but also for the establishment of an elegant, clear and relatively simple

treatment. The analogues of fluid systems in electrical circuitry terms are well

established (Schonfield 1953), as given in Table 2.1.

Symbol Electrical

Symbol Mechanical

Q Charge x Displacement I Current dx

dt

Velocity

V Voltage F Force L Inductance M Mass R Resistance R Frictional Resistance C Capacitance K Spring Constant

Table 2.1 Analogies indicated by differential equations for the electrical system and the mechanical systems.

38

2.2.1 Lumped Analogues

Zero Order Analogues

The simplest analogy is given in terms of electricity is Ohm's law to the one of

peripheral resistance:

( )tIzC

zI

∂∂

∂∂

=− (2.74)

where V is the voltage, corresponding to pressure gradient and I is the current,

corresponding to flow. Since the differential equation describing this equation is zero

order, they are called the zero order approximation. A borrowed term from electrical

engineering called impedance relates the voltage to current. The impedance has a

frequency dependent nature in describing this relation. Basically it is written as

( ) ( )( )ωωω

IVZ = (2.75)

thus the impedance of zero order system is

( ) pRZ =ω (2.76)

First Order Analogues

This analogue represents the Rp and C elements in parallel, where Rp is the Poiseuille

resistance and C is the compliance of the bed, both given as per unit length

RRp =8

4µ

π (2.77)

39

+

+

=

hRE

hRR

C21

13 2ρ (2.78)

The impedance of the analogue is given as

( )CRj

RZ

p

p

ωω

+=

1 (2.79)

which is also called the impedance of the Windkessel (Nichols and O'Rourke 1990).

There are other variations of this circuitry, one called Westkessel (Nichols and O'Rourke

1990) described as

( )CRj

RRZ

p

pc ω

ω+

+=1

(2.80)

where Rc is the characteristic resistance, added in series, after Westerhof et al (1977).

These analogues as they imply, lump the behaviour of the arterial bed. Therefore, they

can describe the overall behaviour of this lumped system rather than its components

(Cevenini et al 1987).

Second Order Analogues

Another lumped analogue is the second order system where the inertia of the blood is

also taken into account. The inductance, L, thus is in series with the Windkessel.

( )CRj

RLjZ

p

p

ωωω

++=

1 (2.81)

Where inductance per unit length, L, is given as

40

LR

=ρ

π 2 (2.82)

The equation given above also describes the action of the fluid filled manometer. It is

not surprising therefore that the Windkessel as first described by Frank (1899), is also

used in conjunction with manometer theory which is also a derivation of Frank (1903).

The second order system given above also describes a resonance phenomenon-at

frequency LCn /1=ω . This notion is taken up by followers of Frank to explain the

different shapes of the pulse in different arterial beds (Warner 1957; Remington and

O'Brien 1970). This concept of resonance is still in fashion in different circles (Watt

and Burrus 1976; Wang et al 1991).

The models described above lack the description of flow shapes in different arteries.

However, they can be quite useful in lumping an arterial bed downstream without

having to know individual geometry of branching and vessel characteristics. This point,

when taken in gross terms, had been used in the past to calculate stroke volume from

pressure pulses (Frank 1930; Cope 1965; see McDonald (1974) for historical review).

This is useful in limited cases because it does not account for wave reflection and

transmission. Also, satisfactory results have been obtained in modelling the arterial

system as transmission lines and where distal arterial beds are terminated with complex

loads (Liu et al 1989; Burattini et al 1989b).

High Order Analogues

The arterial system could be better modelled as a composition of several viscoelastic

tubes representing vessel properties as a function of space. This type of dispersion of

physical properties along the arterial tree generates parameters of wave travel. It also

solves many of the problems associated with lumping such as infinite wave speeds, no

conception of reflections and no change in wave shape. The arterial segments can be

represented with various analogues. If arterial segments are chosen significantly small

41

enough, the model constructed with these analogues approximates the real system. In

early representation of arterial elements by electrical analogues (de Pater and Van den

Berg 1963, Jager et al 1965; Westerhof et al 1969), modified Windkessels with inertial

terms had been used to represent the artery under investigation. By taking data from a

wide variety of sources, early investigators were able to construct an entire arterial

system by adding elements representing individual arteries. The values that were used

were represented by equations 2.79 to 2.82 which converged to give encouraging results.

However, there were considerable discrepancies with the experimental data, probably

due to lumping of the individual segments and ignoring frequency dependence of

elements. The other problem with this approach was that the model elements could

hardly be changed; each element contained four passive elements and there were 121

elements in the entire representation.

2.2.2 Uniform Transmission Line Analogues

If it was possible to reduce the arterial segment size represented by each element to a

finite minimum, then a lumped analogue for that segment approaches a uniform

electrical transmission line. The arterial system represented by these elements then

would represent the continuum of varying physical properties reflected on the electrical

analogue. This approach coupled with the increased computing power led to the

exploitation of transmission line equations and other properties derived from it, as

analogues of fluid flow in arteries (Taylor 1957a, 1957b).

Linearisation of Navier-Stokes equations together with continuity equations reduces the

partial differential equations describing the fluid flow in a tube to the telegraphist

equation, with parameters Rz, Rr, L and C per unit length (Nichols and O'Rourke 1990)

42

tILIR

zV

z ∂∂

∂∂

+=− (2.83)

and

− = +∂∂

∂∂

Iz

VR

CItr

(2.84)

Where Rz is the longitudinal resistance, L is the fluid inertance, C is the arterial

compliance, Rr is the radial resistive component of the vessel wall and V(z, t) and I(z, t)

is the pressure and flow. Since steady state oscillations are assumed to be present in the

system, one can write current, I, and voltage, V, as

( ) ( ) tjezItzI ω=, (2.85)

( ) ( ) tjezVtzV ω=, (2.86)

Then from equation 2.83 and 2.84 one obtains

( ) ( )zIz

zI 22

2

γ∂

∂= (2.87)

where the propagation coefficient, γ , and characteristic impedance, Z0 , are defined as

( )( )CRj

LjRCjr

z

ωω

ωγ+

+=

12 (2.88)

( )( )( )Cj

CRjLjRZ rz

ωωω ++

=1

0 (2.89)

The solution of equation 2.87 then becomes

zz BeAezI γγ += −)( (2.90)

43

where A and B are arbitrary constants. If the boundary conditions are given as

0)0( II = , at z=0 (2.91)

V l I l ZT( ) ( )= , at z=l (2.92)

where ZT is the terminal impedance. If, reflection coefficient, Γ , is defined as

0

0

ZZZZ

T

T

+−

=Γ (2.93)

one can now write

V z I Z e ee

z z l

l( )( )

=−

+

− −

−0 0

2

21

γ γ

γΓ

Γ (2.94)

Using equations 2.88 and 2.89 together with 2.52 and defining

( )jYXcj

−=ωγ (2.95)

One obtains analogues that are also defined in Womersley equations(Taylor 1959b)

( )

+−==

1000 1

1ImReNF

ZRz ρωµγ (2.96)

( )

+==

1000 1

1ReIm1NF

ZL ργω (2.97)

( )( )( ) ( )

+−==

1020

20 1

1ImReNFjYX

cZRr ρωγ (2.98)

44

( ) ( )( ) ( )

+−=−=

1020

20 1

1ReIm1NFjYX

cZC

ργω (2.99)

Thus, there is a one-to-one correspondence between Womersley solutions and the

electrical uniform transmission lines. Thus under the assumptions of linearisation one

can utilise existing notions of transmission lines such as propagation constant,

definitions of impedances and reflection coefficient. These terms in turn, help to

understand the reasons why there are reflections and delays in the arterial system and the

implications of wave propagation in the arteries. Backed by solid theoretical and

experimental evidence this analogue is the most elaborate. It can be used as the

fundamental tool in investigation of the problems with pressure measurements. See

chapters (6.2 through 6.4).

2.2.3 Non-Uniform Transmission Line Analogues

It is well known that the arterial system is not composed of uniform tubes, it undergoes

both geometric and elastic taper with subsequent increase in wall stiffness and wave

velocity. Although it is possible to model this system as a series of tubes of increasing

stiffness, general analytic expressions using uniform tubes are not available. Taylor

(1965) investigated wave travel and reflection in non-uniform tubes, where losses are

ignored. Replacing the C and L in equations 2.85 and 2.86 by C(z) and L(z) and

ignoring radial and axial losses one can obtain

( )tIzC

zI

∂∂

∂∂

=− (2.100)

( )tIzL

zV

∂∂

∂∂

=− (2.101)

Writing I, and V in complex form

45

I I jI= +1 2 (2.102)

21 jVVV += (2.103)

where dependence on z is not indicated. With boundary conditions of

( ) ( ) TZlIlV 11 = (2.104)

( ) ( ) TZlIlV 22 = (2.105)

( ) ( ) ( )000 21 jVVV += (2.106)

where ZT is the terminal impedance. The reflection coefficient at distance z becomes

)()(

)()(

211

)()()(

)(2

11

000

000

lZZzZzlZ

lcj

lZZzZzlZ

lcj

Tlz

Tlz

+

+

−

−

=Γ

=

=

∂∂

ω

∂∂

ω (2.107)

where c(z) is the nominal wave velocity and Z0(z) is nominal characteristic impedance

defined as

)()(

1)(zCzL

zc = (2.108)

)()(

1)(0 zCzLzZ = (2.109)

The voltage at point z then becomes:

∂∂

∂∂

∂∂

ω2

2

2

21 0V

z L zL z

zVz c z

V− − + − =( )

( )( )

(2.110)

46

where the solution of equation 2.110 becomes

∫−

=

z

zcdzj

eZ

zZVzV 0

)(

0

0

)0()()0()(

ω

(2.111)

By considering a wave velocity of the form )(2)0()( zCosczc π−= and inductance of

the form L(z)=1, Taylor (1965) was able to show that non-uniformity enabled the load

to be decoupled from the input, with a smoother impedance spectrum than the one with

the uniform line. Furthermore, the wave amplitude was dictated by equation 2.111.

This kind of dependence of wave amplitude to characteristic impedance ratios partly

explains the progressive increase in pulse pressure from central to peripheral arteries. In

contrast to the uniform line, the transmission ratio was always higher than unity, even

for higher harmonics. This agrees with experimental findings.

A non-uniform tube was analysed by Fich et al (1966), who considered spatial

distribution of radius of the form 2)0()( kzeRzR −= . This was then converted for

expressions of C(z) and L(z), which was than subsequently employed in obtaining

closed form solutions for spatial distribution of pressure and flow along this tube.

Another tube where an exponential function was assigned to the inductance L(z) term is

also proposed (Einav et al 1988). The solutions were similar to the one proposed by

Fich et al (1966).

The non-uniform analogy does not provide a closed form solution, unless spatial

distribution of physical properties is given in analytical form. Thus, it does not give

expressions for spatial distribution of wave velocity and characteristic impedance.

47

2.2.4 Implications of Transmission Line Concepts

Transmission Direction

The concept of transmission, certainly involves wave travel in given directions, namely

upstream and downstream. All the concepts given below assume that the waves are

travelling downstream which can be affirmed by looking at the wave and/or its

components at two arbitrary sites. If the time delay between the two waves is negative,

ie. the wave at the second site arrives later, then the first one is called the downstream

direction, +z, indicating the sources of these waves are on the upstream direction, -z,

and vice versa.

Impedance

Impedance can be generally defined as the opposition to flow (or current) as dictated by

the downstream properties of the system. It can be subdivided into 4 categories:

longitudinal, transverse, characteristic and input impedances.

Longitudinal Impedance

This is the opposition of fluid in the direction of pressure gradient. It can be formulated

as

( )102

0

1 NFRj

zPZz +

=−=π

ωρ∂∂ (2.112)

Since ∂ ∂P z is a function of the spatial direction, it is a point description. For an

oscillatory flow this definition is not effected by the properties of vasculature at the

terminations that might introduce reflected waves. It can be described as electrical

analogue

48

zzz LjRZ ω+= (2.113)

where Rz is the longitudinal resistance describing the losses during the movement of

fluid along the z direction. Lz becomes the hydraulic inductance, related to the inertial

forces opposing to the fluid flow. It can be shown (Fry and Greenfield 1964) that those

terms can be confined with 2.112 after normalisation to the resistive term at zero

frequency

( )

+=

10

2

1Re

NFRz

α (2.114)

( )

+=

101Im

NFjLz (2.115)

It has been shown (Fry and Greenfield 1964, Ling and Atabek 1972) that although Lz

measured experimentally agrees well with the theory, Rz derived this way can be quite

different. This is probably due to the convective terms in the Navier-Stokes equations

which were ignored during the derivation of the equation 2.112.

Transverse Impedance

This form of impedance takes into account the vessel wall properties and continuity

equations in radial directions. In short form it is written as:

zQ

PZr ∂∂−= (2.116)

where zQ ∂∂− denotes the movement of the wall. If the wall element is thought of as

a series resistive element Rr with a capacitive one C the transverse impedance in

analogue form becomes

49

Z Rj Cr r= +

1ω

(2.117)

where Rr denotes the viscous losses due to movement of the wall, while C is the elastic

properties of the wall. It can be shown that the wall element's capacitance can be

written (McDonald 1974) as:

EhRC

22π= (2.118)

indicating a dependence on the wave velocity c0. The value of Rr cannot be deduced in

an analytical way, but it can be described by taking into account interaction between the

fluid and the wall (Taylor 1957b) as in equations 2.98 and 2.99.

Characteristic Impedance

If the transmission line is not terminated, the relation between pressure and flow

becomes the characteristic impedance.

0

00 Q

PZ = (2.119)

Where P0 and Q0 denotes pressure and flow when there is no load at the end of the line

respectively. It indicates the interaction between the wall properties and the fluid

motion and can also be written as

Z Z Zz r0 = (2.120)

The expression of the electrical analogue is given by equation 2.89. Noting its relation

to propagation coefficient and longitudinal impedance, characteristic impedance is

50

ωjcZZ z=0 (2.121)

Considering a viscoelastic wall the elastance can be rewritten in complex form (Bergel

1961a, 1961b; Gow 1972)

ωηjEEc += (2.122)

where η is the wall viscosity coefficient of the wall. The phase of Ec then becomes

= −

ETan ωηθ 1 (2.123)

The phase angle is shown to approach a constant with higher frequencies. The

exponential form which is used to model this behaviour is given as (Taylor 1966a)

( )ωθθ ke−−= 10 (2.124)

where k is found the be around 2 (Gow and Taylor 1968, Imura et al 1990).

Then using equation 2.54; where it was shown that characteristic impedance is directly

proportional to wave velocity and complex elastance, the characteristic impedance per

unit area for strongly tethered tube becomes

2

1022

000 1

11

θ

σπ

ρ jeFR

cZ −

−−= (2.125)

None of these equations take reflected waves into account, so they are only valid for

propagation of forward or backward waves.

51

Input Impedance

This is the impedance seen at the source (or input) of the transmission line, whether the

line is loaded (terminated with a load) or not. It is expressed as

( ) ( )( )ωωω

QPZin = (2.126)

Where P(ω) and Q(ω) represent the frequency components of pressure and flow

respectively. This equation, when displayed as modulus and phase, presents the

characteristics of a transmission line as a function of frequency. It expresses the load

presented to the source, lumping all the properties of the line at the point of source and

beyond. When there is no load at the termination, this equation reduces to the one of

characteristic impedance. Thus the relation between characteristic and input impedance

is given as

l

l

in eeZZ γ

γ

2

2

0 11

−

−

Γ−Γ+

= (2.127)

where Γ is the coefficient of reflection given in Equation 2.93. By definition Γ has a

value of 1 for closed and -1 for open end termination.

Propagation Coefficient

As waves propagate, their amplitude becomes attenuated. Also due to finite wave speed

the propagation occurs in a finite time (see equation 2.87). The two events can be

described by propagation coefficient in the form of

jba +=γ (2.128)

52

where a is the attenuation constant (nepers/cm), describing the losses in the line due to

wall and fluid viscosity. The phase constant, b, is the time delay introduced by the

interaction of the forces between the wall and the fluid and denoted by radians/cm.

Another form of writing this expression is

γω σ

=−

+c NF0

2

10

11

(2.129)

Transmission Ratio

Any measured wave any point along the line, Pp , can be represented as combination of

forward Pf and backward Pb travelling waves

bfp PPP += (2.130)

If at distance l from the origin O a reflecting site L exists with a complex reflection

coefficient Γ , then the forward pressure wave at this point becomes P efl−γ and the

reflected pressure wave will be ΓP efl−γ . Using equation 2.130, the measured pressure

wave at point L will be

lf

lfL ePePP γγ −− Γ+= (2.131)

If reflected wave travels upstream to site O, it will attenuate by e l−γ amount. Again

using 2.130 the pressure wave at O will be

llffo eePPP γγ −−Γ+= (2.132)

Then, the relation of PL with respect to Po will be

PP

e ee

L

o

l l

l=+

+

− −

−

γ γ

γΓ

Γ1 2 (2.133)

53

Which is called the transmission ratio or transfer function between sites O and L.

Obviously, for flow waves the Γ has to be denoted as -Γ for energy considerations.

2.2.5 Application of Transmission Line Concepts in Haemodynamics

The arterial system is a network of tubes, not a single chamber. Therefore, transmission

line theory on the properties of fluid flow is more appropriate. This approach was

extensively used by Taylor (1957a) who first considered the electrical transmission line

as an analogue of the arterial system. By assuming the arterial system to be in steady

state oscillation and linear, since the pulsations are small and wave lengths are long,

Taylor was able to apply telegraph equations to the problem of fluid flow in arteries.

Although Porjé (1946) was the first to use Fourier analysis in this system, it was Taylor,

being a close associate of McDonald and Womersley, who pointed out the physical

meaning of this analysis.

In the treatment of the problem, he used electrical analogues to represent the fluid flow

and considered concepts such as wave propagation and wave reflection, input

impedance, terminal impedance and characteristic impedance. Subsequent works by

Gabe (1965), Patel et al (1965), O'Rourke and Taylor (1966) and Milnor et al (1966)

highlighted these concepts as useful in describing the characteristics of certain arterial

beds.

Interpretation of Impedance

Input impedance of an arterial bed can be calculated by making measurements of

pressure and flow at a site proximal to an arterial bed. If the site is the ascending aorta

then the impedance will be the systemic impedance. After calculation of the frequency

54

components of the pressure, P , and of the flow, Q , by discrete Fourier Transform

(DFT) or by correlation techniques (Taylor 1966c), one can then obtain

( ) ( )( ) ( ) ( )ωϕωωωω jeM

QPZ == (2.134)

where modulus, ( )ωM , and the phase, ( )ωϕ , of impedance are expressed as

( ) ( )( )ωω

ωQP

M = (2.135)

( ) ( ) ( )ωωωϕ QP ∠−∠= (2.136)

The input impedance modulus when plotted against frequency displays the peripheral

resistance as the zeroth harmonic. It then descends steeply from this value to essentially

a minimum as dictated by the reflections present in the system. Modulus then oscillates

around a mean value, (characteristic impedance), to approximate it asymptotically. The

phase on the other hand starts from a negative value, slowly approaches zero, often

crosses it, then oscillates around zero mostly being positive (Nichols and O'Rourke

1990). The spectrum obtained in this way will obviously be an estimate of the actual

spectrum. Since it is sampled at the fundamental rate, heart rate, the continuous

spectrum can be estimated by interpolating the harmonics by linear analysis or by pacing

the heart at a range of frequencies (Taylor 1966c). It is often suggested (Nichols and

O'Rourke 1990) that the characteristic impedance, Zo, be estimated by averaging the

values after the first minimum or above 2 hz. The complex reflection coefficient can be

estimated by

0

0

ZZZZ

in

in

+−

=Γ (2.137)

55

Although there are no propagating waves, thus no wave reflections at zero frequency,

zero intercept of the reflection coefficient can be calculated as analogous to ZT in

equation 2.93. This takes into account the low frequency components of the impedance

spectrum but excludes the often used peripheral resistance (O'Rourke and Taylor 1966)

or the oscillations in the frequency spectrum (Nichols and O'Rourke 1990) both of

which can change independently by certain drugs (Yaginuma et al 1986).

The minima of the impedance spectrum can be used to estimate the distance to the

reflection site. When a single tube is being considered there will be one termination and

the distance can be estimated using quarter wavelength formula:

min

0

44 fc

l ==λ (2.138)

where co is the assumed wave velocity for this tube. This equation for a single tube

assumes that the second minimum will be at triple of this minimal frequency. If it is not

the case, ie a second minimum is found at frequency other than this, it suggests a second

tube parallel to the first one that is also been terminated by another load. The distance

to this reflection site can also be estimated using equation 2.138 (O'Rourke 1967b;

Nichols and O'Rourke 1990). Note that the wave speed remains the same in both cases.

This kind of impedance spectrum was mostly found in the ascending aorta of different

species, including man (Avolio 1976a, Nichols and O'Rourke 1990), leading to the

reduced model of systemic arteries as an eccentric T-Tube, where the long arm supplies

to the lower part of the body but the short is to the upper part (O'Rourke and Taylor

1967a). This view is later taken up by Liu et al (1989) and Burattini et al (1989b) who

extended the model and applied it to estimate vascular parameters. The influence of

wall viscosity, blood viscosity, reflection coefficient and wave speed has been studied

extensively using models (Taylor 1966a, Avolio 1980, Liu et al 1989) and in vivo

56

(O'Rourke et al 1968). It was shown that viscosity of the blood and of the wall

influences the oscillations in the impedance spectra.

The effects of various vasoconstrictor drugs on input impedance had been investigated

in the past using epinephrine, angiotensin or norepinephrine (O'Rourke et al 1992).

Data show little difference from the control, apart from the effects of increased wave

velocity brought by increased mean arterial pressure. This indicates the presence of

significant vasoconstrictive state under control conditions. The estimated reflection

coefficient obtained using peripheral resistances is 0.8 for control and 0.98 under

vasoconstrictor therapy.

The effects of vasodilator therapy are profound, however. Acetylcholine (O'Rourke and

Taylor 1966), nitroprusside (Pepine et al 1979; Gundel et al 1981; Yin et al 1983,

Merrillon et al 1984; Laskey et al 1987, Chang et al 1990) and nitroglycerine (Westling

et al 1984, Yaginuma et al 1986; Latson et al 1988, Fitchett et al 1988) profoundly

effects the lower harmonics of the aortic input impedance and thus the reflection

coefficient from 0.8 to 0.5 or lower.

The effect of various physical manoeuvres; exercise (Laskey and Kussmaul 1987;

Murgo et al 1980a) and Valsalva manoeuvres (Murgo et al 1981) have been investigated

and have shown to change the impedance spectrum. These alterations are brought by

changes in the magnitude of reflected waves or wave velocity or the peripheral

resistance.

The systemic input impedance has also been used as a unique way of expressing the left

ventricular load. When expressed as steady and pulsatile load, it can differentiate

between the efficiencies of different arterial systems. It is also related to the oxygen

demand of the left ventricle for a given supply (Weber and Janicki 1977). The external

power output of the heart, Wn , is defined through impedance as

57

nn

N

nnn CosZQQPW ϕ∑

=

+=1

200 2

1 (2.139)

where the first term is the zeroth and second term is the higher terms of the Fourier

components of the pressure and flow and impedance respectively and ϕn is the phase of

the impedance. This equation does not include power dissipation due to kinetic energy

of the blood since this component is relatively small. The term P Q0 0 thus represents

the steady power while the other term is the pulsatile component. The ratio of pulsatile

to total power is used as an index of "arterial efficiency" with respect to

ventricular/vascular coupling. This ratio is around 10% under normal conditions

(Nichols and O'Rourke 1990).

Since the basic characteristic shape of the flow pulse does not alter from individual to

individual, even from species to species (Nichols and O'Rourke 1990), it was suggested

that the alterations in the pressure waveform are mainly due to changes in the impedance

spectrum. Furthermore, since in experimental animals and in young healthy humans

pulse wave velocity is similar in corresponding arteries, pressure wave patterns are

largely determined by length of arterial segments and by body patterns (O'Rourke 1982).

Murgo et al (1980b) investigated the possibility of classifying the arterial pulse due to

ascending aortic input impedance spectrum. It was concluded that the second shoulder

appearing in central pressure waveform was due to an oscillatory impedance spectrum

brought by either earlier or increased wave reflection. This study was later supported

with the finding that, in older subjects with increased characteristic impedance (Avolio

et al 1983b, 1985a), the systolic pressure augmentation also increases (Kelly et al

1989a, 1989b, 1990a) due to early wave reflection (O'Rourke 1988).

The normalisation of impedance spectrum facilitates the comparison from different sets

of data from different arterial beds or from different species. For this purpose the units

used in describing the modulus is often normalised (Nichols and O'Rourke 1990). It

was found not surprisingly that when modulus is described as a proportion of the

58

characteristic impedance, it normalises the impedances obtained from different species.

Using equation 2.127 and expressing input impedance as a proportion of the

characteristic impedance leads to

l

lin

ee

ZZ

γ

γ

2

2

0 11

−

−

Γ−Γ+

= (2.140)

This normalisation allows one to express input impedance as a function of reflection

coefficient, propagation coefficient and the length. Further normalisation is possible if

one combines equation (2.129) with equation (2.138) to obtain

10

2

minmin

0

10

2

0 11

2411

NFff

fc

NFcl

+−

=+−

=σπσωγ (2.141)

leading γl to be proportional to f fmin . Thus, if the abscissa is normalised by fmin, the

impedance spectra for both modulus and phase can be expressed as a unique identity,

just a function of ( 101 NF+ ) and thus α and the complex reflection coefficient Γ . This

enables comparison of experiments as a function of α and Γ .

Interpretation of the Transfer Function

Transfer function (transmission ration) between two sites, proximal and distal, can be

calculated by taking two simultaneous pressure measurements from these sites and

subjecting them to Fourier analysis either by discrete Fourier transform (DFT) or

spectral techniques (Nichols and O'Rourke 1990). The obtained moduli, ( )ωdM and

( )ωpM , and phases, ( )ωϕd and ( )ωϕ p , of the proximal and distal pressures are then

related as

( ) ( )

( ) ( ) ( ) ( )ωϕωϕ

ωϕ

ωωω j

jp

jd

DP eMeMeMH

p

d

==− (2.142)

59

to obtain the transfer function HP D− . Here, while ( )ωM describes the modulus or the

amplification ( )ωϕ describes the phase delay between these two signals.

The modulus of the transfer function when plotted against frequency, displays

successive minima and maxima depending on the intensity of reflected waves (Figure

2.1, Top).

Figure 2.1 A typical transfer function between ascending aortic and the radial arterial pressure pulse. Modulus of the transfer function starts from unity at zero hertz and exceeds unity at higher frequencies to reach a peak due to presence of reflected waves. The phase delay presents itself as time difference between the ascending aortic and radial pressure waves. Phase of transfer function starts from zero radians at zero frequency indicating no delay. With increase in frequency, phase becomes negative indicating a progressive phase delay. Note the oscillations in the phase values which settle to a constant value dictated by wave speed (see text for more explanation).

The component at zero frequency displays the ratio of segment resistance to that of the

resistance distal to the measurement site and it is often very close to unity (Nichols and

60

O'Rourke 1990). Since there is an elastic tapering (Taylor 1964) as well as reflections

in the arterial system under normal circumstances, the modulus reaches a value above

unity with increase in frequency. The position of this peak on the abscissa can be

estimated using the quarter wave length formula (equation 2.138). The modulus then

gradually decreases to a value below unity.

Because of this prominent peak, it was often assumed that the segments of arteries

behave like a resonating second order systems (Warner 1957; Remington and O'Brien

1970) drawing an analogy between this and manometers. Utilising this similarity,

Warner used four pole networks to simulate the effects of pulse transmission in arteries

(1957). Although, this manometer analogy was proved successful to an extent, it is not

complete due to discrepancy in attenuation at higher frequencies. The second order

system analogues always attenuate at 40 dB per octave above "resonant frequency"

(Oppenheim et al 1983) while lesser attenuation was observed in experimental settings

(Lasance et al 1976; Karamanoglu et al 1992). The values obtained approached close to

the theoretical predictions (Taylor 1965) brought by presence of elastic tapering in the

arteries (Taylor 1969).

Phase of the transfer function starts from zero and remains negative, being linearly

dependent on the frequency (Figure 2.1, Bottom). This behaviour had been treated more

carefully than modulus and shown to contain information about reflected waves. Porjé

analysed the phase relation between the two pressure signals to obtain a reliable estimate

of the pulse wave velocity (Porjé 1946). For this purpose, he used the following

equation to calculate phase velocity

( )ωϕωlcph −= (2.143)

61

where cph ( )ω is expressed in cm/sec and ( )ωϕ is the phase of transfer function. To his

surprise, he found that the lower harmonics were travelling at speeds higher than the

high frequency components. Porjé correctly identified the cause of this phenomenon-as

wave reflection (Cox 1971). In solving the telegraphist equations, Taylor derived the

formula relating the wave speed to reflections and later confirmed his findings in

experiments in rubber tubes (Taylor 1957a, 1957b). The expression was of the form

( )[ ] ( )[ ]( )[ ] ( )[ ]βα

βα

−−−−−

−−−−−=

′

bzl2Sinbaazl2Sinh

bzl2Cosazl2Coshcc

0

(2.144)

where a, b are the real and imaginary parts of the complex propagation coefficient; α

and β are the elements of the reflection coefficients which is of the form

( )βα je +=Γ 2 (2.145)

The expression 2.144 is a point description and transforms to

( )[ ] ( )[ ]( )[ ] ( )[ ]βα

βα

−−−−−

−−−−−−

=′

bzl2Sinbaazl2Sinh

bzl2Cosazl2Coshln)za(z2

1cc

120

(2.146)

when actual measurements are done. In this form z1 and z2 are the proximal and distal

measurement sites respectively. These equations point out that wave velocity is

dependent on the distance to the reflecting site and to the magnitude and phase of the

reflection coefficient.

The phase velocity is also related to impedance by (McDonald 1974) by

10

20

MRcZin π

ρ= (2.147)

62

where c is the wave velocity derived using equation 2.143. Since radius, R , and M10

are relatively constant, α does not change considerably since it approaches its limit

value very quickly, there is a linear relationship between the modulus of the impedance

spectrum and the phase velocity (McDonald 1974). It was shown that the phase velocity

obtained this way was closely correlated with impedance under a wide variety of

conditions (Latham et al 1985; Milnor 1989), including valve disease.

Although viscosity of the wall and the blood as well as the cross-sectional area have

been shown to be very influential on the transmission of the pressure wave, these factors

have little influence on the input impedance (Taylor 1966b).

The relation between the modulus and phase of the transfer function is complementary

as they both contain similar information. For example, the peaks of moduli correspond

to the slower wave speed as estimated from the phase velocity. The positions of the

prominent peaks are also related to the asymptotic value obtained by the phase velocity

by equation 2.143.

The normalisation of the transfer function for comparative purposes can be obtained by

scaling the frequency axis by the quarter wave length formula using equation 2.143.

The phase velocity axis can also be scaled using the asymptotic value as proposed in the

impedance section. This leads to the direct comparison of the impedance spectrum with

the phase velocity spectrum. The reflection coefficient from phase velocity spectra can

be obtained by using formulation given by Taylor (1957a)

( )azlecccc

22

0

0 −

+′−′

≅Γ (2.148)

63

where the assumptions are that reflections are positive and that the point z2 is close to

the termination with respect to z1. A much easier formulation can be obtained by

assuming l z− ≅2 0

0

0

cccc

+′−′

≅Γ (2.149)

where c' is the phase velocity estimate at around zero hertz. If this value is not

available, it can be obtained by linear interpolation. This expression of reflection

coefficient and the equation 2.137 is obvious.

The wave travel phenomenon is most apparent when one considers the delay introduced

by transmission. Therefore, most of the early investigations of wave travel were

devoted to accurate measurement of wave velocity. In order to measure wave velocity, a

point is defined on the wave and time needed for this point to travel some distance was

measured. The most probable candidate for such an identifiable point was obviously the

foot of the wave, and it turned out to be the most valid. There are problems, however, in

the accurate description of the foot of the wave. It was defined as

(i) the maximum point of the first derivative,

(ii) the minimum pressure point before the upstroke,

(iii) the intersection of the tangents drawn from the diastolic point and from the

systolic point,

(iv) maximum point of second derivative etc.

Since the upstroke of the wave is less contaminated by the reflected waves that often

interfere with measurements (Nichols and O'Rourke 1990), the consistency of these

descriptions around the upstroke is not surprising. Detecting a point around the

64

upstroke involves often differentiation, or multiplication of the wave by ω, to enhance

the high frequency components of the wave. Thus, the ideal candidate of a pre-defined

point becomes the part of the wave where high frequency components are rich or where

phase velocities are constant. These predictions have been verified by measurements of

pulse wave velocities which are made by the techniques described above (i-iv) (Chiu et

al 1991).

2.3 APPLICATION OF TUBULAR MODELS OF THE ARTERIAL SYSTEM

2.3.1 Single Tube Models

The simplest application of transmission line concept not surprisingly involves a single

tube. This approximation was first used by Taylor to validate predictions of Womersley

theory (Taylor 1957a, b). It was later utilised in modelling of the arterial system. The

maxima and minima of harmonic components of the pressure waveform are already

studied in the arterial tree of the dog (McDonald 1974). These points are then used in

estimation of the reflection site by drawing analogies to nodes and antinodes present in a

single tube. Similarly, the minima and maxima of the impedance modulus at various

arterial segments and beds were also investigated through modelling (Taylor 1966b) and

experimentation (O'Rourke and Taylor 1967a). Although this analogue was initially

useful, the theoretical predictions of the input impedance spectrum for single tube

deviated from in vivo measurements. It was further shown that the model parameters

needed to fit for a given impedance spectrum might not have any physiological meaning

(Patel et al 1963, Sipkema and Westerhof 1975, Campbell et al 1989). However, this

model still finds use in assessing gross functional parameters from the impedance

65

spectra obtained in various ways, yet the attempts are directed to more sophisticated

models (Nichols and O'Rourke 1990).

2.3.2 Two Tube Models

The presence of two minima in the input impedance spectrum that are separated by

frequency doubling rather than tripling, led to models incorporating two tubes in parallel

(T-Tube) or in series.

Tubes in Parallel

The T-tube model introduced by O'Rourke and Taylor (1967a, b) attempts to reconcile

anatomical structure, body size and distribution of body mass with the ascending aortic

impedance spectra. According to this model, the heart, being eccentrically placed, is

connected to two parallel circulations, one to the head and the upper limbs and the other

perfusing the lower body and the lower limbs (O'Rourke 1967b). This hypothesis is

supported by the occlusion of the descending aorta and the brachiocephalic artery of the

dog. When the former is occluded the ascending aortic input impedance was found

similar to that of the brachiocephalic arteries. In contrast, when the brachiocephalic

artery was occluded it was similar to that of the descending aortas. Furthermore, flow

pulses in the brachiocephalic artery and descending aorta showed reciprocal oscillations,

indicating waves travelling to and from these arteries (Mills et al 1970). This kind of

interpretation coupled with aortic impedance data implies that reflections from these

two functionally separate circulations cancel each other and present a more uniform load

to the heart. However, the model had two weaknesses. Firstly, the aortic input

impedance spectra was very oscillatory . Although smaller than the single tube, this

oscillatory behaviour could not be dampened out with physiological viscous forces

unless complex loads at the terminations were introduced (Sipkema and Westerhof

66

1975). The second and probably the most important drawback is that the model

parameters cannot be estimated without recordings made in the ascending aorta (Liu et

al 1989; Burattini 1989a; Burattini et al 1989b). In this respect, this model can only be

thought of as a framework.

Tubes in Series

Two tubes in series were first proposed by Wetterer (1968) to explain the ascending

aortic impedance spectrum. According to this model, a tube representing the thoracic

aorta is terminated by another tube representing the femoral bed. The model was

revived in a study done by Latham et al (1985), where the model found use in

explaining the wave shapes along the aorta. In this model, the positions of the junction

of these two tubes are calculated to be at or below the renal arteries while the second

tube terminates at around the aortic bifurcation.

2.3.3 Three or More Tube Models

Increasing the number of tubes has been shown to yield better results even for a few

extra branches (McIlroy et al 1986). The models involving more branches obviously

exhibit an improved description of the behaviour of the arterial system, since they

acknowledge the presence of multiple reflection sites in the arterial system. In that

respect, the models with more elements explain different shapes of pressure waves in

different arteries, but they introduce extra difficulties associated with estimating the

physical parameters of individual tubes. These models are, however, superior in taking

into account the arrangement or morphology of the connections that might be important

in deciding the outcome.

67

Random Branching Models

These models are implemented in assessing the impedance and transmission

characteristics due to variation of various parameters. Taylor (1966a, b) was the first to

implement this kind of model in a digital computer, which later became a template for

subsequent investigations. In this model, it was shown that other than the non-

uniformity of the arteries, the scattered terminations reduced the reflected waves. As

Womersley (1955a) predicted, the terminations of random lengths behave like a single

unit because the wave length becomes comparable to the segment length, especially at

higher frequencies. The generated phase differences led to cancellation of reflected

waves at higher frequencies. The model was successful in predicting realistic input

impedance spectra, representative of an arterial bed. It was concluded that the

architecture of the arterial system has an important role in determining the characteristic

shape of aortic impedance.

Anatomically Arranged Branches

Tree Structure

When Taylor's random branching model (1966a, 1966b) was modified to allow for the

architecture of the arterial system, it gave representative values for ascending aortic

impedance (Avolio 1980) and for wave transmission (Salotto et al 1986). Both being

derivations of Taylor's original model, they viewed the arterial system as a continuously

branching tree of parallel elements. This simplified the solution, since the

implementation on a digital computer was relatively easy but tedious. It required the

input of many physical parameters for individual elements and a specification of their

interconnection in a matrix form.

In these models, parameters such as vessel diameters and lengths, wall properties and

input wave shapes, had to be given externally and were obtained from the literature

68

(Noordergraaf 1969; Westerhof et al 1969) and from Roentgenographic recordings

(Avolio 1976a). When applied to different species (Avolio et al 1976b, 1983, 1984a)

and for intra-aortic balloon pumps (Avolio 1976a), the models predicted the

experimentally observed impedance spectra. This does seem surprising because these

models do not take into account branching angles and inlet lengths. Peripheral

resistance values measured in vivo could not be estimated accurately probably due to

finite number of elements in the models. In an Avolio model there were 128 branches.

For simulation purposes peripheral resistance values were obtained from known values

of cardiac output and mean pressure. The models were then used to simulate ageing by

changes in pulse wave velocity values. Being multi-branched, the models predicted

realistic pressure shapes along the arterial tree. This was achieved for large arteries

(Avolio 1980) and in small arteries (Salotto et al 1986). It seems that the computational

procedure, as described by Taylor (1966a, b), is valid for estimating the vascular

parameters. This procedure, thus is useful in investigating the regional pressure and flow

patterns in different parts of the body.

Mesh Structure

The computational technique for tree structures cannot be directly applied to model

loops and grafts. To overcome this problem, an alternative computational method was

suggested involving 2-Port networks (Helal et al 1990). In this method the equations of

pressure and flow transfer functions are rewritten.

Considering the pressure, Pu, and flow, Qu, waves were input to the two-port, the

output Pressure, Pd, and flow, Qd, is related to input as

Γ+

Γ+=

−

1

ll

dueePP

γγ

(2.150)

and

69

d

dT Q

PZ = (2.151)

combining 2.154 and 2.93 yields

Γ =−+

P Z QP Z Q

d d

d d

0

0 (2.152)

Using 2.152 and 2.150 gives

( ) ( )lSinhZQlCoshPP ddu γγ 0+= (2.153)

and replacing Γ with -Γ for flow relations gives

( ) ( )lCoshQlSinhZPQ d

du γγ +=

0

(2.154)

Equations 2.153 and 2.154 represent a set of linear equations with unknowns Pd and

Qd. Then the solution of proximal waveforms can be calculated for the downstream

properties along a given path (Helal et al 1990). This kind of representation eliminates

one of the steps normally needed (backward iteration for impedance and forward for

transmission ratios) in Taylor's computational method. The model based on this

computational scheme predicted pressure patterns in the Circle of Willis in line with in

vivo recordings. The arterial models involving species that have a confluence (rather

than branching) of large vessels, such as reptiles seem best suited for this

scheme(Avolio et al 1982, 1983a).

70

MEASUREMENT OF ARTERIAL BLOOD PRESSURE Arterial blood pressure is one of the most fundamental physical quantities related to

haemodynamics. Because of the perceived need to quantify pressure levels throughout

the cardiovascular system, measurement techniques have been developed which, at

times have obscured the true nature of this pulsatile phenomenon. In quantifying blood

pressure oscillations, these techniques emphasised the mean value, peak values or both.

Blood pressure is, however, a composite wave that undergoes significant changes in its

contour while travelling within the arterial system. Thus the term "blood pressure",

when used alone, does not express this wave nature.

Chapter 3

71

The measurement techniques are divided into direct registration of pressure wave by

invasive means and non-invasive methods where the pressure wave is obtained by

methods employing external transducers.

3.1 INVASIVE MEASUREMENTS

The first invasive measurement of blood pressure was done by Hales in 1733. By

inserting a glass tube into the femoral artery of a horse he observed the blood column to

reach "eight feet three inches perpendicular above the level of the left ventricle". He

further observed that "when it was at its full height (the blood column), it would rise and

fall at and after each pulse two, three or four inches" (Geddes 1970). The former

observation then described as the mean pressure and the latter as the pulsatile pressure.

Since the proportion of the latter to the former was quite small, it was ignored. The

same phenomenon-was observed in Poiseuille's experiments in 1828, where he replaced

the blood column with a mercury column, hence the token of millimetres of mercury

(mmHg) as the unit of blood pressure. This observation led to the belief that the major

pressure in the arteries is the mean component and the pulsatile component can be

ignored, pressure oscillations were of course again attenuated by the inertia of the

mercury column. Quantification of the pulsatile component was first proposed by Carl

Ludwig, who attached a smoked drum to the mercury manometer using a cantilever

mechanism in 1847. Using this graphic recorder (kymograph), Ludwig was convinced

that the true systolic and diastolic pressure values were different from the recorded ones,

and he attributed this discrepancy to the mass of the system he employed. Later

attempts were directed to employing mechanical systems to transmit the pressure wave

to the kymograph rapidly. The most remarkable of these was the one developed by

Marey (1860), in which the instrument was so refined that it was possible to record the

cardiac pressures (Geddes 1970). The introduction of fluid to transmit pressures, since

72

fluid is incompressible and so it is relatively easy to build devices around it, was first

suggested by Rolleston in 1887. The theoretical problems related with this mode of

transmission were treated by Otto Frank in 1903 who proposed an index or figure of

merit (Güte) of the manometers by solving second order differential equations

describing the operation of manometers (Wiggers 1928). Figure of merit stated that the

manometers should be designed with high stiffness to mass ratio to accomplish a high

frequency response.

The implementation of Frank's recommendations on manometers increased the

resolution necessary to distinguish the features of the pressure waveform (Wiggers

1928). It further accelerated the investigation on the formation of the pressure pulse

(Hamilton 1944; Wood et al 1951; Remington and Wood 1956). The introduction of

mechano-electric transducers of capacitance type by Schutz in 1937, of inactive

transducer type Wetterer in 1943 and of strain-gauge type by Lambert in 1947, not only

increased the frequency characteristics but also improved the amplification and

recording stages of the measurement system (Geddes 1970).

Since all these systems use incompressible fluid as a medium for wave transmission, the

equations of transmission line apply to the description of operation of fluid filled

catheters. This is especially important when transmission time cannot be neglected,

otherwise it is quite possible to use the system with a single degree of freedom as first

described by Frank. In the latter the behaviour of the system is related to the

parameters: damping coefficient, β , and natural frequency, ωn , (McDonald 1974)

ERl

n 20

πρ

ω = (3.1)

73

El

R πρµβ 0

3

4= (3.2)

and the amplitude at frequency Ω = ω ωn to its true value is

( ) 2222 41

1

Ω+Ω−=

βA (3.3)

and the phase is

Ω−Ω

= −2

1

12βϕ Tan (3.4)

Expressions for resonant frequency 3.1 and damping coefficient 3.2 can be combined

with Moens-Korteweg equation to yield similar expressions

lc

n 42 0π

ω = (3.5)

20

16Rc

lµβ = (3.6)

Obviously, these equations are only valid when the termination of the measuring tube is

totally closed (Gabe 1972), otherwise the system contains more than one degree of

freedom and equations 3.1 through 3.6 do not hold.

3.2 NON-INVASIVE MEASUREMENTS

The most common non-invasive methods of blood pressure measurement give either

peak values and/or mean value of the pressure wave or contour of the pressure wave

74

(uncalibrated or with some form of indirect calibration) or both. All of these techniques

involve the arterial wall as a prime transmitting medium and thus prone to errors due to

wide variations in vascular, muscular and subcutaneous tissue properties of different

individuals.

3.2.1 Peak Systolic Value Measurement

In these measurement methods the information content of the pressure wave is assumed

to be contained only in the peak values and/or mean value of the waveform. They often

include compression of an artery and surrounding tissue with a cuff filled with air. The

pressure inside the cuff is then raised and subsequently decreased. During this cycle,

blood flow distal to the cuff or oscillations in the cuff or the sounds generated by the

non-linear oscillations of the wall and of the fluid are monitored. These signals are then

related to the pressure inside the cuff. The cuff pressure at the onset of these signals is

often used as the systolic, diastolic and mean pressures. The validation of the methods

is done by comparing the pressures obtained by the technique and the measurements

done invasively. Detailed histories of different methods can be found in Geddes (1970).

However, because of its importance in clinical medicine the auscultatory methods will

be briefly described here.

Auscultatory Method

The sounds generated by the non-linear behaviour of the collapsible tubes (Conrad

1969) were first described by Korotkoff in 1905. This was followed by extensive

research into the phenomenon-(Rodbard et al 1953; Ur and Gordon 1970) and its

validation (Van Bergen et al 1954, Borow et al 1982). Since its introduction,

sphygmomanometry has remained virtually unchanged. The American Heart

Association proposed five revisions for measurement methods in the last forty years.

75

These were concerns over the size of the cuff and whether the phase IV or phase V of

the Korotkoff sounds represented the diastolic pressure. Research has also been

performed on the establishing the relation between values obtained manually and by

microprocessor controlled devices. Yet, neither the pressure wave amplification in the

upper limb nor the generations of sounds that depends on arterial elasticity, heart rate,

distance to the central artery is addressed in these recommendations. Furthermore, the

explanation of the variation of the differences between invasive and non-invasive blood

pressure measurement in different individuals is still lacking.

3.2.2 Contour Measurement

Sphygmography

Since the arterial pulse had been described in many ways qualitatively, (in ancient

Sanskrit medicine there were 100 different descriptions), the graphic display of the

arterial pulse had created a large interest in the 19th Century. The devices developed in

that era were all mechanical and non-invasive. They were required to be applied

externally and be pressed firmly on an artery, often the radial. Etienne Jules Marey was

first to record arterial pulse accurately in man (Schechter 1969). The sphygmographs he

developed were taken up by Mahomed (1874), Broadbent (1890) and Mackenzie (1902).

They made changes in the original design leading to improvements in amplification and

conditioning of the obtained pressure pulses (Geddes 1970). However, because of the

inherent problems associated with these instruments such as high frequency attenuation

and high inertia brought by air or fluid filled capsules, they could not record the pressure

waveform with sufficient fidelity (O'Rourke et al 1992). Instead, these instruments find

use in detecting the presence of pulse and its rhythm.

76

Plethysmography

Plethysmography is another technique to register the pressure wave contour. The

change in vessel dimension caused by pressure pulse is detected by strain gauges, photo-

electric sensors or electrical impedance electrodes. Assuming a linear relation between

blood pressure and vessel dimension then yields a pressure contour. Although widely

used because of ease and relative low cost, the non-linear relation between the pressure

and diameter of the vessel often invalidates the technique. The pulse waves recorded

are not pressure waves.

A technique introduced by Penaz (1973) initially described by Wiederhelm (O'Rourke et

al 1992) tries to overcome this difficulty introduced by non-linearity of the stress strain

relation of the wall. It relies on the fact that, when the arterial wall is unloaded at zero

transmural pressure, the wall becomes totally compliant and thus exhibits the maximum

strain per unit stress. When the wall is clamped at this maximum strain level by

adjusting the external pressure at all times, it is possible to obtain the pressure in the

artery (Yamakoshi et al 1980). In effect, this procedure is not much different from

calculation of instantaneous pressure using a linear pressure volume relation. However,

in this technique, the non-linear pressure-volume curve of the vessel is estimated first

and subsequently used to calculate instantaneous pressure by measured instantaneous

volume.

The instrument first designed by Penaz and subsequently developed by Wesseling et al

(1982) is marketed by OHMEDA as a device known as "FINAPRES". This device first

"learns" the pressure-volume curve of the digital artery of the middle finger. It measures

the photoplethysmographic volume of the finger while increasing the pressure in steps

within a small cuff surrounding the finger. After detecting the maximum volume

oscillations in the photoplethysmographic recordings, it then uses a servo driven

pressure pump to clamp finger arterial volume at this maximum value. The pressure in

77

the cuff, needed to achieve this, is then considered to be the intra-arterial pressure in the

finger artery. The technique claims to yield calibrated pressure waveforms, but it

definitely fails when the compliance curve cannot be estimated or changes actively

during measurement (Wesseling 1985).

Applanation Tonometry

The applanation tonometry technique relies on the elimination of vessel-wall curvature

with a known force, thus transferring the normal stresses to the applanating element.

The theoretical basis for the technique has solid foundations (O'Rourke et al 1992)

based on a curved beam theory (Drzewiecki and Noordergraaf 1980). The intra-arterial

pressure waveform has been shown to be registered faithfully provided that curvature

changes introduced by the tonometer are small with respect to the radius and

axisymmetric (Drzewiecki et al 1984). A probe has been designed based on these

model predictions (Hayward 1989). The probe has a Millar micromanometer in its tip

that operates on the principle of piezo resistivity. Application of force on to the active

element on this tip, bends this cantilever element which changes its resistance. The

element forms one arm of a Wheatstone bridge, where this unbalanced resistance is

detected and generated voltage is amplified. The design of the tonometer as a pencil

probe, facilitates the applanation of an artery that can be readily palpated. The pressure

sensitive area on the probe is 0.5 mm2 and sits on an area of the probe tip of 133 mm2.

This construction allows most of the arteries being applanated and the force being

detected quite accurately which is then converted to pressure. The validation of this

tonometer (Kelly et al 1989b) displayed a very high accuracy and fidelity compared to

the intra-arterial pressure waveform. The frequency response of the system was more

than 1 kHz and had a sensitivity of 0.2 volts/mmHg when used with its own

preamplifier boxes (Millar TCB-500).

78

SECTION II

MODELLING AND SIMULATION

The in vivo estimation of important parameters in the arterial system is often hampered

by the type of instrumentation and methodology used in experiments. The noise present

in the experimental set up often blunts the responses, causing repeat of experimentation

and increases the cost. The introduction of digital computers into the laboratory

environment, however, changed this entirely. The simulation techniques now available

enable experimenters to perform the experiments with a reduction in time spent on real

experiments. Various modifications to the computer model can be done repeatedly in

considerably shorter time. The experimental findings can then be used to confirm the

predictions made by the model, enabling the adjustments on the model parameters. All

these assertions are valid, provided that the model is close to the simulated experimental

phenomena. However, the model fails in reproducing events if it does not use the

physical constraints and formulations of real life situations. Deviation from reality

causes models to converge to an arbitrary solution, if any, very slowly. The strict

79

adherence to the reality, on the other extreme, increases computation time significantly

with a concomitant requirement for exact duplication of boundary conditions. This

approach then approximates the system under investigation, nullifying the benefit

gained from modelling.

The properties of the arterial system are traditionally described by pressure-flow

relations in the frequency domain. However, pressure wave propagation parameters can

also be used to describe the same arterial properties. Quantification of pressure-pressure

relations in the frequency domain has an undisputed advantage over the pressure-flow

relations. Instead of being restricted by the technique of flow measurements, one can

use pressure measurements to quantify the gross properties of the arterial system. The

pressure wave recordings can also be made non-invasively. With the advent of non-

invasive pressure wave registration with high-fidelity tonometers (Kelly et al 1989b),

limitation of measurement technique is becoming less restricted. Tonometry is best

suited for arteries that can be applanated against a rigid structure such as the carotid

artery, radial artery, femoral artery and dorsalis pedis artery. These sites can also be

used to obtain Doppler-derived flow profiles, thus enabling determination of pressure-

flow relations. However, since the flow measurement techniques are not as accurate as

those of pressure (Nichols and O'Rourke 1990), this technique is used less frequently.

This fact imposes certain limitations in reality. Firstly, the number of measurements or

known variables often exceed the number of equations that wholly describe pressure-

flow relations. Secondly, the global nature of wave propagation phenomena in the

arterial system deviates from local properties. For example, any change in arterial

properties in the lower body can change the pressure waves recorded in an upper limb

artery (Karamanoglu et al 1990). The introduction of modelling to the system

overcomes these two problems, given that the model accurately represents the

underlying physical phenomena. Using this model, one can then estimate spatial and

temporal distribution of pressure and flow waves, solving the second problem. The first

80

problem is however more complex to solve. One needs to explore all the possible

solutions of the equations by successively assuming different values for one or more

equation. This technique is commonly known also as parameter estimation methods.

Using a heuristic approach, then the model can be optimised for some parameter sets,

which can then be assumed to be the system parameters set.

In this thesis, modelling of the arterial system is limited to the following investigations:

i) To reproduce documented changes in the arterial pulse due to ageing,

vasodilatation and vasoconstriction in humans.

ii) To explain the differences between central and peripheral, upper limb pressure

pulses under wide variety of conditions in humans.

iii) To suggest ways of analysing pressure waveforms under clinical conditions, for

diagnostic and therapeutic purposes in humans.

iv) To suggest ways to monitor central haemodynamic events from peripheral

recordings in humans.

The above limitations assert that the model implemented has to take into account the

distributed nature of the arterial system. The emphasis in this model has to be the

contribution of physical quantities on the formation of the pulse. It should also take into

account the changes induced by ageing, vasoactive drugs and the anatomical differences

between individuals. The effect of local as opposed to general properties of the vessels,

has to be treated separately.

The model that satisfies these criteria was implemented in a computer first by Taylor

(1966a, 1966b) as random branching networks. This work was subsequently extended

81

to the mammalian arterial system by Avolio (1980) who obtained reasonable agreement

with real measurements. Although both were satisfactory, these investigators did not

extend their simulations to the problems of pressure waveform genesis and propagation

in clinical situations. Taylor's model suffered from the fact that the parameters selected

bore no resemblance to the ones observed in the real human arterial system. For

example, although the progressive increase in the stiffness of the arteries was present,

the extent and variation were not taken into account; also no attempt was made to

simulate anatomical branching patterns. The Avolio model was considerably enhanced,

since it took into account the anatomical distribution of arterial segments and utilised

reasonable physical properties for the segments. However, it also suffered from the lack

of realistic alteration of these properties due to ageing and vasoactivation. These data

became available recently (Ho 1982; Avolio et al 1983b; Latham et al 1985). Both

models also suffered from their applicability to the clinical environment. Being

extensively dependent upon computer resources, they were unfortunately tied to

mainframe hardware. This meant that they could only be used in circumstances where

access to a mainframe was possible and this imposed severe limitations. For example,

the limited number of peripheral terminations due to computer storage limitations

imposed restrictions on how far along the arterial tree vascular impedance could be

faithfully determined. This however did not influence calculation of input impedance

and transmission along the aortic trunk.

Since computers are now regarded as another instrument in experimental physiology

thanks to recent developments in microprocessor design and implementation, they are in

abundance. This fact and the others described above, necessitated the implementation

of these modelling algorithms in a much more widely used computer platform while

taking into account the vital data exchange methods between the existing laboratory

equipment. It also required that the implementation be done in a higher level language

that is supported by contemporary hardware and software. Finally, the model has to be

designed and implemented in a user friendly way (difficult-to-use systems often

82

discourage use) and enabling the end user, most likely the physician, to be the ultimate

judge.

83

MODEL CONSTRUCTION Several models of the human circulation have been suggested previously. These models

were either linear or non-linear, lumped or distributed, implemented in hardware or in

software. The model of Avolio (1980) was the best to be chosen to represent the human

arterial system for this investigation. This model is chosen because it is not only a

distributed model, representing the architecture of the human arterial tree with its

morphology and physical properties of the arterial segments, but also because it is based

on linear theory, allowing one to use techniques such as spectral analysis and

transmission line theory on pressure and flow waves. This advantage brought by linear

approximation outweighs any disadvantage due to restrictions of this model, since not

only the non-linear behaviour of the elastic properties is small within the physiological

range of frequencies and pressures but also the tapering, which introduces convective

non-linear terms in the Navier-Stokes equations, is considered to be negligible due to

side branches (Chapter 2).

Chapter 4

84

The basic unit in the Avolio model is an arterial segment that was represented by a

viscoelastic, thin and tethered tube of radius r, length l and wall thickness h. The

solutions of the linearised Navier-Stokes equations for fluid flow in this tube are as

described by Womersley. Description of these equations, assumptions made during

derivation and limitations of the theory verified by experimentation are already

described in detail (see Chapter 2). Furthermore, the uniform lossless transmission line

analogue is assumed to hold for representation of this tube. The concepts and

implications that are derived from this analogue have also been discussed previously

(Chapter 2).

4.1 REPRESENTATION OF THE ARTERIAL TREE

Any vascular tree, excluding anastomoses and grafts, can be represented as a binary tree

(B-tree) since branches are likely to bifurcate (Zamir 1978). B-tree's are finite element

sets which empty or contain a single element called the root of the tree and whose

remaining elements are partitioned into two diagonal subsets, each of which is itself a

binary tree (Figure 4.1). These two subsets are called the left and right subtrees of the

original tree. Each element of a binary tree is called a node of the tree. The B-Tree in

Figure 4.1 consists of nine nodes with A as the root. Its left subtree is rooted at B and

right subtree is rooted at C. Here, node A is the parent of nodes B and C. B and C is in

turn daughters of node A and parents of D, E and F respectively. The nodes D, G, H

and I have no descendants and hence are called the leaves.

For the arterial bed displayed in Figure 4.1, the coding technique is such that, a unique

key is assigned to each element for the path it takes to reach the node. This is achieved

by assigning the root of the entire tree, A in this case, code M, short for Main. If one

85

E

I J

F

H

D

G

CB

A

Figure 4.1 A representative binary tree composed of nine nodes and four generations.

then travels to the Left a letter L is appended to the previous code and for Right a letter

R is appended. Thus, for example, node E in this scheme of coding becomes MLR,

which designates a departure from main, turn left then right. Obviously this coding

technique does not take into account the functionality of the branching but the

morphology. It is therefore different from the Strahler system where the coding starts

from the leaves upstream taking into account the diameters of the parent-daughter pairs

(Yen et al 1984). The advantage of the B-tree, however, is that it provides a simple

description of the generations and facilitates computations based on generations.

4.1.1 Parent Daughter Relations

Zero Order Relation

In the zeroth order relation a parent does not have any offspring, thus it is a leaf. The

algorithm for calculation of leaf impedance takes into account, for this special case a

reflection coefficient. Since it is assumed that for practical purposes the vascular bed

86

extending from this leaf cannot be described for its individual elements physical

properties and morphologies, it can be lumped as a modified Windkessel (Figure 4.2).

If the parent characteristic impedance is Zp0 then the impedance mismatch between the

two elements, proximal element and the modified Windkessel can be expressed as

Γww po

w po

Z ZZ Z

=−+

(4.1)

where Zw is the Windkessel impedance and can be described as

Z Z Rj R Cw d

d

d d= +

+0 1 ω (4.2)

Rd and Cd are defined as the Poiseuille resistance and capacitance per unit length

respectively and given by equations 2.77 and 2.78.

Distal ElementProximal Element

Zdo

Cd

Zpo

Rd

Figure 4.2 The modified windkessel representing the termination at the leaf level.

Although there are different approaches to the modelling of the vessel wall

(Langewouters et al 1984, Elad et al 1988), it is often considered that the distal bed

pressure, Pd, is related to bed volume, Vd, in a non-linear fashion with an exponential

law (Kawasaki et al 1987)

87

P AedBVd= (4.3)

Assuming fixed oxygen requirements of the distal bed, thus constant blood flow,

pressure, Pd, is related to peripheral resistance, Rd, as

P Q Rd d d= (4.4)

Where A, B and flow (Qd) are constants. Then from (4.3) and later by (4.4)

dVdP BP

d

d d=

1 (4.5)

dVdP BQ R

d

d d d=

1 (4.6)

and since compliance is also defined as Cd=dVd/dPd then

R CBQd d

d= =

1 Constant (4.7)

Thus the time constant , τ=RdCd, of the modified Windkessel will be fixed under :

(i) The existence of an exponential pressure-volume relationship of the vessel wall

and

(ii) The fixed oxygen requirements of the arterial bed downstream (ie constant blood

flow).

By combining equation 4.1 with equation 4.2 and assumingZ Zp d0 0= for reasons of

continuity, Γw could be expressed as a function of ω (Figure 4.3).

88

CdRdτ=

R (R +2Z )d d do

Figure 4.3. The frequency dependent behaviour of reflection coefficient when the terminal element is a Windkessel representing the distal arterial bed. The reflection coefficient in this arrangement is dependent on frequency of which is determined by the windkessel time constant τ.

In this new expression, Γw approaches ( )dodd ZRR 2/ + as ω approaches zero while as ω

approaches infinity Γw approaches zero. From equation 4.1 and equation 4.2 it is seen

that the frequency dependent behaviour of Γw is determined by the time constant RdCd

as given by equation 4.7. Unique parameters, Γw and τ, set this way, used in this

termination mode to describe the lumped vascular properties downstream.

First Order Relation

In the first order representation one of the offspring of the parent does not exist. The

terminal impedance then becomes the remaining daughter's input impedance.

Second Order Relation

Both of the daughters exist. In this case the terminal impedance becomes the parallel

addition of the offspring impedances.

89

4.1.2 Computational Algorithm

In an assembly of a binary tree, the computations are possible for impedances, for

description of pressure-flow relations, and for transmission ratios for pressure-pressure

relations. This can be achieved by utilising the three possible configurations of parent-

daughter relations as described in 4.1.1. The computational algorithm for calculation of

the transmission properties and impedances of the entire B-tree becomes a

straightforward implementation of the following recursive procedure in Pascal language

(Algorithm 4.1)

Algorithm 4.1 Pseudo code for calculation of the arterial tree as viewed from source.

The coding for this configuration is M, MR, ML, MLR, MLL where ML is the dummy

element of the length ε, where 0<ε<<1 cm, ie ε is a finite length close to zero. For

example, using this model a long tube can be expressed as elements of the tree of

MRRRRRR indicating no diatomic branching (a single tube). In this respect, the code

for a conventional T-Tube representation is M, MR and ML. Using this nomenclature

of expressing the nodes, a branching structure of three elements from a single root can

be specified by inserting a dummy branch into the sequence. An example of such a

configuration is shown in Figure 4.4.

Procedure Calculate Tube Downstream(Node) Begin If Node is not Leaf then Begin Calculate Tube (Left Subtree of the Node) Calculate Tube (Right Subtree of the Node) End Calculate Terminal Impedance (Sisters) Use 4.1.1, 4.1.2 and 4.1.3 Calculate Characteristic Impedance Use equation 2.125 Calculate Propagation Coefficient Use equation 2.129 Calculate Reflection Coefficient Use equation 2.93 Calculate Input Impedance Use equation 2.127 Calculate Transmission Ratio Use equation 2.133 END

90

DummyElement

M

M

MR

ML

MLRMLL

Figure 4.4 A parent with three daughters, a possible representation of renal arterial branching of the abdominal aorta (left). The equivalent B-Tree representation of the branching node after inserting a dummy element of length ε, where 0<ε<<1 cm (right).

This kind of representation does not take into account the bifurcation angles and inlet

lengths, since they are not relevant parameters in this type of simulation. Apart from

these concerns, arithmetic operations and algorithmic calculations are continuous.

Algorithm 4.2 Pseudo code for calculation of arterial tree as seen from an arbitrary node.

The above computational algorithm calculates the input impedance downstream. If,

however, the calculation of upstream impedances and transmission ratios are required,

the calculation order has to be modified, which is given in Algorithm 4.2. This

Procedure Calculate Tube Upstream (Node) Begin If Node is not Leaf Begin Calculate Tube Downstream (Sister of the Node) Use Algorithm 4.1 Calculate Tube Upstream (Parent) End Calculate Terminal Impedance (Sister and Parent) Use 4.1.1, 4.1.2 and 4.1.3 Calculate Characteristic Impedance Use equation 2.125 Calculate Propagation Coefficient Use equation 2.129 Calculate Reflection Coefficient Use equation 2.93 Calculate Input Impedance Use equation 2.127 Calculate Transmission Ratio Use equation 2.133 End

91

obviously necessitates the insertion of a reflection coefficient for the main root,

ascending aorta in this case, indicating the source impedance to the B-tree.

4.2 ANATOMICAL DATA

The anatomical data for the human arterial system is obtained from the literature

(Avolio 1976a, Haimovici 1981). The elastance data, however, is found to yield a

relatively slow pulse wave velocity for the various segments. The data for elastances

has been altered to obtain the reported values in humans as well as changes induced by

ageing for various segments (McDonald 1968; Avolio et al 1983b). Using interpolated

data for twenty years of age, the pulse wave velocity in the trunk was found to be 6.2

m/sec, in the upper limbs 8.4 and in the lower limbs 10.9 m/sec. For the age of 60

years, the respective values obtained were 10.2 m/sec, 10.6 m/sec and 14.9 m/sec.

Modifications have also been made to relate the wall viscosity to the smooth muscle

content of the vessel wall. This has been achieved by relating the phase of the wall

viscosity, θ, to the wall thickness to diameter ratio, h/2R, by a function of the form

θ θ= 0 2hR

(4.8)

where the parameter θ0 is assumed to be 10 degrees. Since the vessel wall becomes

thicker towards the periphery, this formula assumes an increase of the viscous losses per

unit diameter length.

The overall data of the constructed human arterial segments are given in Table 4.1 with

their B-tree codes. A representative arterial tree constructed from this model is given in

Figure 4.5.

92

30 Cm

Figure 4.5. The model of the human arterial system employed in the thesis (prepared by Miss Lina L. Lee from data given in Table 4.1). Arterial segments are drawn to scale both for length and for diameter. Total number of elements are 142 of which 14 are dummy elements of length ε (see text for more information).

93

B-Tree Code Name L

(cm)

D

(cm)

h

(cm)

E

dyne.cm-2

(x106)

Yng Old

θ

(deg)

MLLR SUBCLAVIAN ARTERY(L) 6.80 0.80 0.07 7.28 10.19 7.00 MLLRL (D) SUBCLAVIAN 3 (L) 0.10 0.80 0.07 8.17 11.44 7.00 MLLRR (D) SUBCLAVIAN 1 (L) 0.10 0.80 0.07 8.17 11.44 7.00 MLLRLL INTERNAL MAMMARY (L) 15.00 0.20 0.03 10.40 14.56 12.00 MLLRLR VERTEBRAL ARTERY (L) 14.80 0.19 0.05 10.40 14.56 21.05 MLLRRL COSTO-CERVICAL (L) 5.00 0.40 0.05 10.40 14.56 10.00 MLLRRR (D) SUBCLAVIAN 2 (L) 0.10 0.80 0.07 8.17 11.44 7.00 MLLRRLR SUPRASCAPULAR (L) 10.00 0.20 0.02 10.40 14.56 8.00 MLLRRRL THYROCERVICAL (L) 5.00 0.20 0.03 10.40 14.56 12.00 MLLRRRR AXILLARY ARTERY (L) 6.10 0.72 0.06 8.58 12.01 6.67 MLLRRRRL THOROCO-ACROMIAL(L) 3.00 0.30 0.04 20.80 29.12 10.67 MLLRRRRR AXILLARY ARTERY (L) 5.60 0.62 0.06 8.58 12.01 7.74 MLLRRRRRL SUBSCAPULAR (L) 8.00 0.30 0.04 20.80 29.12 10.67 MLLRRRRRR BRACHIAL ARTERY (L) 6.30 0.56 0.05 9.74 13.64 7.14 MLLRRRRRLL CIRCUMPLEX SCAPULAR (L) 5.00 0.20 0.03 20.80 29.12 12.00 MLLRRRRRRL PROFUNDA BRACHII (L) 15.00 0.30 0.04 10.40 14.56 10.67 MLLRRRRRRR BRACHIAL ARTERY (L) 6.30 0.52 0.05 9.55 13.37 7.69 MLLRRRRRRRL SUP. ULNAR COLATERAL(L) 5.00 0.14 0.02 20.80 29.12 11.43 MLLRRRRRRRR BRACHIAL ARTERY (L) 6.30 0.50 0.05 9.56 13.38 8.00 MLLRRRRRRRRL INF. ULNAR COLATERAL(L) 5.00 0.12 0.02 20.80 29.12 13.33 MLLRRRRRRRRR BRACHIAL ARTERY (L) 4.60 0.48 0.05 9.73 13.62 8.33 MLLRRRRRRRRRL ULNAR ARTERY (L) 6.70 0.42 0.05 10.40 14.56 9.52 MLLRRRRRRRRRR RADIAL ARTERY (L) 11.70 0.32 0.04 10.40 14.56 10.00 MLLRRRRRRRRRLL ULNAR ARTERY (L) 8.50 0.38 0.05 10.40 14.56 10.53 MLLRRRRRRRRRLR INTEROSSEA ARTERY (L) 7.90 0.18 0.03 20.80 29.12 13.33 MLLRRRRRRRRRRR RADIAL ARTERY (L) 11.70 0.32 0.04 11.55 16.17 10.00 MLLRRRRRRRRRLLL ULNAR ARTERY (L) 8.50 0.38 0.05 11.14 15.60 10.53 MRR SUBCLAVIAN ARTERY(R) 6.80 0.80 0.07 7.28 10.19 7.00 MRRL (D) SUBCLAVIAN 3 (R) 0.10 0.80 0.07 8.17 11.44 7.00 MRRR (D) SUBCLAVIAN 1 (R) 0.10 0.80 0.07 8.17 11.44 7.00 MRRLL INTERNAL MAMMARY (R) 15.00 0.20 0.03 10.40 14.56 12.00 MRRLR VERTEBRAL ARTERY (R) 14.80 0.19 0.05 10.40 14.56 21.05 MRRRL COSTO-CERVICAL (R) 5.00 0.40 0.05 10.40 14.56 10.00 MRRRR (D) SUBCLAVIAN 2 (R) 0.10 0.80 0.07 8.17 11.44 7.00 MRRRLR SUPRASCAPULAR (R) 10.00 0.20 0.02 10.40 14.56 8.00 MRRRRL THYROCERVICAL (R) 5.00 0.20 0.03 10.40 14.56 12.00 MRRRRR AXILLARY ARTERY (R) 6.10 0.72 0.06 8.58 12.01 6.67 MRRRRRL THOROCO-ACROMIAL(R) 3.00 0.30 0.04 20.80 29.12 10.67 MRRRRRR AXILLARY ARTERY (R) 5.60 0.62 0.06 8.58 12.01 7.74 MRRRRRRL SUBSCAPULAR (R) 8.00 0.30 0.04 20.80 29.12 10.67 MRRRRRRR BRACHIAL ARTERY (R) 6.30 0.56 0.05 9.74 13.64 7.14 MRRRRRRLL CIRCUMPLEX SCAPULAR (R) 5.00 0.20 0.03 20.80 29.12 12.00 MRRRRRRRL PROFUNDA BRACHII (R) 15.00 0.30 0.04 10.40 14.56 10.67 MRRRRRRRR BRACHIAL ARTERY (R) 6.30 0.52 0.05 9.55 13.37 7.69 MRRRRRRRRL SUP. ULNAR COLATERAL(R) 5.00 0.14 0.02 20.80 29.12 11.43 MRRRRRRRRR BRACHIAL ARTERY (R) 6.30 0.50 0.05 9.56 13.38 8.00 MRRRRRRRRRL INF. ULNAR COLATERAL(R) 5.00 0.12 0.02 20.80 29.12 13.33 MRRRRRRRRRR BRACHIAL ARTERY (R) 4.60 0.48 0.05 9.73 13.62 8.33 MRRRRRRRRRRL ULNAR ARTERY (R) 6.70 0.42 0.05 10.40 14.56 9.52 MRRRRRRRRRRR RADIAL ARTERY (R) 11.70 0.32 0.04 10.40 14.56 10.00 MRRRRRRRRRRLL ULNAR ARTERY (R) 8.50 0.38 0.05 10.40 14.56 10.53

94

B-Tree Code Name L

(cm)

D

(cm)

h

(cm)

E

dyne.cm-2

(x106)

Yng Old

θ

(deg)

MRRRRRRRRRRLR INTEROSSEA ARTERY (R) 7.90 0.18 0.03 20.80 29.12 13.33 MRRRRRRRRRRRR RADIAL ARTERY (R) 11.70 0.32 0.04 11.55 16.17 10.00 MRRRRRRRRRRLLL ULNAR ARTERY (R) 8.50 0.38 0.05 11.14 15.60 10.53 MRL COMMON CAROTID (R) 8.90 0.74 0.06 7.28 10.19 6.49 MRLL COMMON CAROTID (R) 8.90 0.74 0.06 7.28 10.19 6.49 MRLLL CAROTID EXTERNAL (R) 5.90 0.36 0.05 10.40 14.56 11.11 MRLLR CAROTID INTERNAL (R) 11.80 0.30 0.04 10.40 14.56 10.67 MRLLLL SUPERIOR THYROID (R) 4.00 0.14 0.02 10.40 14.56 11.43 MRLLLR EXTERNAL CAROTID (R) 5.90 0.26 0.04 10.40 14.56 12.31 MRLLRL MIDDLE CEREBRAL (R) 3.00 0.12 0.02 20.80 29.12 13.33 MRLLRR CEREBRAL ARTERY (R) 5.90 0.16 0.03 20.80 29.12 15.00 MRLLLRL FACIAL ARTERY (R) 4.00 0.20 0.03 20.80 29.12 12.00 MRLLLRR EXTERNAL ARTERY (R) 5.90 0.16 0.03 20.80 29.12 15.00 MRLLRRR OPTHALMIC ARTERY (R) 3.00 0.14 0.02 20.80 29.12 11.43 MRLLLRLL LINGUAL ARTERY (R) 3.00 0.20 0.03 10.40 14.56 12.00 MRLLLRRL SUPERFICIAL TEMPORAL(R) 4.00 0.12 0.02 20.80 29.12 13.33 MRLLLRRR MAXILLIARY ARTERY (R) 5.00 0.14 0.02 20.80 29.12 11.43 MLR COMMON CAROTID (L) 8.90 0.74 0.06 7.28 10.19 6.49 MLRR COMMON CAROTID (L) 8.90 0.74 0.06 7.28 10.19 6.49 MLRRR COMMON CAROTID (L) 3.10 0.74 0.06 7.28 10.19 6.49 MLRRRL CAROTID INTERNAL (L) 11.80 0.30 0.04 10.40 14.56 10.67 MLRRRR CAROTID EXTERNAL (L) 5.90 0.36 0.05 10.40 14.56 11.11 MLRRRLL CEREBRAL ARTERY (L) 5.90 0.16 0.03 20.80 29.12 15.00 MLRRRLR MIDDLE CEREBRAL (L) 3.00 0.12 0.02 20.80 29.12 13.33 MLRRRRL EXTERNAL CAROTID (L) 5.90 0.26 0.04 10.40 14.56 12.31 MLRRRRR SUPERIOR THYROID (L) 4.00 0.14 0.02 10.40 14.56 11.43 MLRRRLLL OPTHALMIC ARTERY (L) 3.00 0.14 0.02 20.80 29.12 11.43 MLRRRRLL EXTERNAL ARTERY (L) 5.90 0.16 0.03 20.80 29.12 15.00 MLRRRRLR FACIAL ARTERY (L) 4.00 0.20 0.03 20.80 29.12 12.00 MLRRRRLLL MAXILLIARY ARTERY (L) 5.00 0.14 0.02 20.80 29.12 11.43 MLRRRRLLR SUPERFICIAL TEMPORAL(L) 4.00 0.12 0.02 20.80 29.12 13.33 MLRRRRLRR LINGUAL ARTERY (L) 3.00 0.20 0.03 10.40 14.56 12.00 M A.AORTA 4.00 2.90 0.16 6.58 13.29 4.41 ML AORTIC ARCH 2.00 2.24 0.13 6.98 14.43 4.64 MR BRACHIOCEPHALIC 3.40 1.24 0.09 7.28 11.65 5.81 MLL AORTIC ARCH 1.90 2.14 0.13 7.41 15.85 4.86 MLLL DESCENDING AORTA 7.20 2.00 0.12 7.41 15.85 4.80 MLLLL THORACIC AORTA 5.20 1.90 0.12 7.54 16.13 5.05 MLLLLL THORACIC AORTA 5.20 1.90 0.12 7.54 16.79 5.05 MLLLLLL COELIAC ARTERY 1.00 0.78 0.06 7.54 16.13 6.15 MLLLLLR ABDOMINAL AORTA 5.30 1.74 0.11 7.54 17.47 5.06 MLLLLLLL GASTRIC ARTERY 7.10 0.36 0.05 7.54 16.13 11.11 MLLLLLLR (D) COELIAC 1 0.10 0.78 0.06 7.54 16.13 6.15 MLLLLLRL (D) ABDOMINAL 1 0.10 1.74 0.11 7.54 17.47 5.06 MLLLLLRR SUPERIOR MESENTRIC 5.90 0.86 0.07 7.54 16.13 6.51 MLLLLLLRL SPLENIC ARTERY 6.30 0.56 0.05 7.54 16.13 7.14 MLLLLLLRR (D) COMMON HEPATIC 0.10 0.44 0.06 7.54 16.13 10.91 MLLLLLRLL RENAL ARTERY (L) 3.20 0.52 0.05 7.54 16.13 7.69 MLLLLLRLR (D) ABDOMINAL 2 0.10 1.74 0.11 7.54 17.47 5.06 MLLLLLLRRL HEPATIC ARTERY 6.60 0.44 0.05 7.54 16.13 9.09 MLLLLLLRRR GASTRIC ARTERY 3.20 0.52 0.05 7.54 16.13 7.69

95

B-Tree Code Name L

(cm)

D

(cm)

h

(cm)

E

dyne.cm-2

(x106)

Yng Old

θ

(deg)

MLLLLLRLRL ABDOMINAL AORTA 5.30 1.14 0.08 8.32 17.80 5.61 MLLLLLRLRR RENAL ARTERY (R) 3.20 0.52 0.05 7.54 16.13 7.69 MLLLLLRLRLL ABDOMINAL AORTA 5.30 1.14 0.08 8.58 18.35 5.61 MLLLLLRLRLR INFERIOR MESENTRIC 5.00 0.32 0.04 8.58 18.35 10.00 MLLLLLRLRLLL COMMON ILIAC (L) 5.80 1.04 0.08 9.24 17.52 6.15 MLLLLLRLRLLLL EXTERNAL ILIAC (L) 8.30 0.58 0.05 8.51 15.67 6.90 MLLLLLRLRLLLR INTERNAL ILIAC (L) 5.00 0.40 0.04 13.28 20.93 8.00 MLLLLLRLRLLLLL EXTERNAL ILIAC (L) 6.10 0.54 0.05 8.10 14.59 7.41 MLLLLLRLRLLLLLL FEMORAL ARTERY (L) 12.70 0.48 0.05 7.55 13.02 8.33 MLLLLLRLRLLLLLR PROFUNDIS ARTERY(L) 12.60 0.46 0.05 13.28 12.60 8.70 MLLLLLRLRLLLLLLL FEMORAL ARTERY (L) 12.70 0.46 0.05 7.40 12.75 8.70 MLLLLLRLRLLLLLLLL POPLITEAL ARTERY (L) 9.40 0.40 0.05 6.54 11.07 10.00 MLLLLLRLRLLLLLLLLL POPLITEAL ARTERY (L) 9.40 0.40 0.05 6.97 10.98 10.00 MLLLLLRLRLLLLLLLLLL ANTERIOR TIBIAL (L) 2.50 0.26 0.04 5.63 8.87 12.31 MLLLLLRLRLLLLLLLLLR POSTERIOR TIBIAL (L) 16.10 0.36 0.05 6.24 9.83 11.11 MLLLLLRLRLLLLLLLLLLL ANTERIOR TIBIAL (L) 15.00 0.20 0.04 4.67 6.87 16.00 MLLLLLRLRLLLLLLLLLLR PERONEAL ARTERY (L) 15.90 0.16 0.02 7.70 12.14 10.00 MLLLLLRLRLLLLLLLLLRR POSTERIOR TIBIAL (L) 16.10 0.36 0.05 6.68 10.53 11.11 MLLLLLRLRLLLLLLLLLLLL ANTERIOR TIBIAL (L) 14.00 0.20 0.04 5.03 7.93 16.00 MLLLLLRLRLLLLLLLLLLRR PERONEAL ARTERY (L) 14.90 0.16 0.02 9.39 14.80 10.00 MLLLLLRLRLLLLLLLLLLLLL (D) ANTERIOR TIBIAL (L) 0.1 0.20 0.04 5.03 7.93 16.00 MLLLLLRLRLLLLLLLLLLRRR (D) PERONEAL ARTERY (L) 0.1 0.16 0.02 9.39 14.80 10.00 MLLLLLRLRLLR COMMON ILIAC (R) 5.80 1.04 0.08 9.24 14.56 6.15 MLLLLLRLRLLRL EXTERNAL ILIAC (R) 8.30 0.58 0.05 8.51 13.41 6.90 MLLLLLRLRLLRR INTERNAL ILIAC (R) 5.00 0.40 0.04 13.28 20.93 8.00 MLLLLLRLRLLRLL EXTERNAL ILIAC (R) 6.10 0.54 0.05 8.10 12.77 7.41 MLLLLLRLRLLRLLL FEMORAL ARTERY (R) 12.70 0.48 0.05 7.55 11.90 8.33 MLLLLLRLRLLRLLR PROFUNDIS ARTERY(R) 12.60 0.46 0.05 13.28 20.93 8.70 MLLLLLRLRLLRLLLL FEMORAL ARTERY (R) 12.70 0.46 0.05 7.40 11.66 8.70 MLLLLLRLRLLRLLLLL POPLITEAL ARTERY (R) 9.40 0.40 0.05 6.54 10.31 10.00 MLLLLLRLRLLRLLLLLL POPLITEAL ARTERY (R) 9.40 0.40 0.05 6.97 10.98 10.00 MLLLLLRLRLLRLLLLLLL ANTERIOR TIBIAL (R) 2.50 0.26 0.04 5.63 8.87 12.31 MLLLLLRLRLLRLLLLLLR POSTERIOR TIBIAL (R) 16.10 0.36 0.05 6.24 9.83 11.11 MLLLLLRLRLLRLLLLLLLL ANTERIOR TIBIAL (R) 15.00 0.20 0.04 4.67 7.36 16.00 MLLLLLRLRLLRLLLLLLLR PERONEAL ARTERY (R) 15.90 0.16 0.02 7.70 12.14 10.00 MLLLLLRLRLLRLLLLLLRR POSTERIOR TIBIAL (R) 16.10 0.36 0.05 6.68 10.53 11.11 MLLLLLRLRLLRLLLLLLLLL ANTERIOR TIBIAL (R) 14.00 0.20 0.04 5.03 7.93 16.00 MLLLLLRLRLLRLLLLLLLRR PERONEAL ARTERY (R) 14.90 0.16 0.02 9.39 14.80 10.00 MLLLLLRLRLLRLLLLLLLLLL (D) ANTERIOR TIBIAL (R) 0.1 0.20 0.04 5.03 7.93 16.00 MLLLLLRLRLLRLLLLLLLRRR (D) PERONEAL ARTERY (R) 0.1 0.16 0.02 9.39 14.80 10.00

TABLE 4.1 Physical data for human arterial system together with B-Tree codes for major arteries. The dummy elements are marked by (D) in the B-Tree code. The name field codes (L) and (R) indicate left and right extremities while L and D are the length and diameter of the elements. Wall properties, h (wall thickness), E (wall elastance) and θ(phase of wall viscosity) are also given for young (yng) and old.

96

A BINARY TREE MODEL OF THE HUMAN ARTERIAL SYSTEM The alteration of wave shapes in the human arterial system due to ageing, vasoactive

drugs and physical manoeuvres are well investigated (Nichols and O'Rourke 1990). The

consensus on these phenomena is that the presence of wave reflection can help explain

these observations (O'Rourke et al 1992). However, this concept was challenged by

investigators who question the location of the sites of reflection or the presence of re-

reflection (Papagiorgiou and Jones 1988; Ting et al 1990). There is also scepticism of

the explanations provided by this theory on differential action of drugs (Kelly et al

1990b; Simkus and Fitchett 1990).

These concerns cannot be answered readily. Firstly there are the experimental

difficulties in introducing a single reflected wave and in following its retrograde

CHAPTER 5

97

transmission along the arterial system. Secondly, even if this could be achieved, the

numerous branches of the arterial system present interferences to the propagated wave

that would hamper the interpretation of the results. Yet, modelling of the arterial system

can assist in addressing these issues. By selective alteration of the physical properties of

different branches reflected waves can be induced and spatial and temporal relationship

could be obtained. This information allows the specific origin of wave reflection from

in vivo recordings to be determined. It is even possible to devise experiments that

specifically target these origins to verify model predictions.

In this chapter, the multi-branching B-tree model (Chapter 4) is extended to simulate the

branching structure of the human arterial tree. By using this multi-branching network,

the transmission and reflection of pressure waves are investigated. To address the role

of wave reflection on the arterial pulse, the B-tree model is subdivided into anatomically

separable compartments (upper limb, trunk and lower limb). Influence of reflected

waves originating from these compartments is then studied. By mapping the pressure

waves along a path that extends through these compartments, the distribution of

reflected waves is visualised and interpreted.

Since the pressure pulse in the upper limb is significantly important in clinical settings,

the investigations are extended to the upper limb. In this simulation the relation

between the upper limb pulse and the central aortic pulse is studied in the frequency

domain by transfer function analysis. The properties of the upper limb arterial system

(elastance, reflection coefficient, time constant and wall viscosity) are altered and their

influence on the transfer function and the wave shapes are also investigated.

98

5.1 PRESSURE WAVE PROPAGATION IN THE HUMAN ARTERIAL MODEL

In the simulations that follow, the wave propagation path used for investigation extends

from right arm (radial- brachial- axillary-subclavian arteries) to the trunk (ascending-

descending- thoracic- abdominal aorta) and then to the right leg (iliac-femoral-popliteal-

anterior tibial arteries), Figure 5.1. All side branches that are supplied by these arteries

are considered to be the beds specific to these compartments.

30 Cm

Figure 5.1 The path in which the propagation of waves are simulated. It extends from right arm (radial- brachial- axillary- subclavian arteries) to trunk (ascending- descending- thoracic- abdominal aorta) and then to the right leg (iliac-femoral-popliteal-anterior tibial arteries).

5.1.1 Pressure Contour Maps

To facilitate the interpretation of the model findings, data maps of pressure waves were

generated. These maps are formed by stacking the pressure waves, which are also

functions of time, in their spatial distance from the ascending aorta or from the input

(Figure 5.2).

99

Figure 5.2 3-D Pressure wave propagation map showing the spatial and temporal distribution of pressures along the path. The generated pressure wave in ascending aorta travels to the upper limb, to the trunk and the lower limbs. The left most wave is that which is recorded in the radial artery while the right most that in the anterior tibial artery. The pressure wave peaks due to wave reflection during its travel along the path.

The maps obtained this way is called the pressure propagation maps in three

dimensions, P(P, z, t). When grided appropriately and interpolated, the surface

obtained yields the pressure value at a given location at a given time. Using this

method, one can obtain a visual expression of wave propagation and wave reflection.

Slicing the 3-D propagation maps with the pressure axis at different pressures, Pi, and

projecting these slices onto z-t plane, results in isobaric contour maps, similar to

pressure maps in weather forecasting or to topographical mapping of heights, Figure 5.3.

These maps are useful in estimation of wave propagation properties, characteristic

impedance and wave reflections. Pulse wave velocities, for example, can be estimated

along the constant Pi lines, by calculating dz/dt on these contour maps. Since the

reflected waves will travel in negative space but positive time, their presence will be

indicated by negative pulse wave velocities. Wave fronts in these maps will be

indicated by closely spaced Pi with respect to time. The strong impedance mismatches

will appear as closely spaced Pi lines along the constant z axis.

100

ISOBARIC MAP

RADIAL

A.AORTA

ILIACBIFURCATION

ANTERIORTIBIAL

TIME (milliseconds)

Figure 5.3 Isobaric map of Figure 5.2 showing the spatial and temporal distribution of pressure waves along the path. The pressure waves at the same level (3 mmHg) are joined by continuous lines. Closely spaced lines indicate a steep rise or fall of the pressure values (wavefronts). The generated pressure wave at the level of the aortic root travels along the path (arrows) both to upper limbs (down arrow) and to trunk and lower limbs (up arrow). The travel speed of the wave is determined by the pulse wave velocities of branches given by the Moens-Korteweg equation. The calculated values at different anatomical locations using this equation indicated by the heavy dark line. Deviations of the direction of wavefront from this line indicate the presence of reflected waves.

Figure 5.3, corresponding to Figure 5.2, can be deciphered precisely by inspection. The

left most contours are the lowest pressures in the system. As one moves right, with

increasing time, the contours become closely spaced at first, denoting the wave front,

then spaced approximately evenly apart. In the latter part of the map, the contours

gradually denote decreased pressure. In this figure, every contour is at steps of 3

mmHg. These are stacked together to form a hill on the 3-D map. At the anterior tibial

level, a summit occurs at around 260 milliseconds that can also be seen in the 3-D map.

The wave velocities here can be calculated by taking partial derivatives of the contours

with respect to time and space. This procedure produces similar values of wave

101

velocities as indicated by the similarity between the solid line in Figure 5.3 and the wave

front which is determined from Figure 5.4. Any wave front contour in this map which

matches with the wave velocity indicates a forward-travelling contour. When it

decreases in speed, it denotes a reflected wave is present in the waveform as seen in the

hump in the right of Figure 5.2.

5.1.2 Introduction of Reflected Waves

Wave reflection is a consequence of the impedance mismatch between the source and

the load. The alterations of the values of load impedances can be used to generate

reflected waves given that the source impedance remains constant. The alteration of

terminal impedances at the leaf levels of the B-tree can be used to generate reflected

waves originating from the terminations. If these waves are made to propagate

upstream, without altering the properties of the upstream branches, they will appear in

more proximal branches after being summed with waves from different origins. The

changes in the terminal impedances at the leaf levels can be thought of as changes in the

vascular lumen area (wall thickness to diameter ratio) or wall elastance (changes in wall

material properties) or both. Instead of specifying the type of parameter change, one can

define a reflection coefficient to lump all these factors into one. Previous investigations

have suggested that the reflections often originate from the arteriolar level and have

values of 0.85 under normal conditions (O'Rourke and Taylor 1966). Vasodilatation

decreases it to a lower value. However, since the model employed in this model could

not be extended to the arteriolar level branching, a lumped approach has been devised

which takes into account the numerous branches downstream. Presence of such a

configuration ensures a smooth decline in reflection coefficient modulus at higher

frequencies due to cancellation of reflected waves (Taylor 1966b). To simulate this

behaviour of the reflection coefficient, the leaves are assumed to be terminated with

modified Windkessels (see chapter 4).

102

5.1.3 Tracking Of Reflected Waves

Injection of a retrograde wave at each termination separately and following its upstream

travel is probably the most instructive method of all. Yet, for practical purposes,

extension of this to every branch in the model is tedious. However, one can lump

reflected waves originating from the terminal branches downstream and follow the

travel of this composite wave upstream. To obtain this composite reflected wave one

can make use of the superposition principle (the system is linear). It is known that when

the reflection coefficient is zero at the terminations, no reflections will be generated

from these leaves. However, when it is different from zero reflected waves are

generated. One can then take the difference of these two situations to obtain the net

effect of introduced positive reflection coefficient, hence the composite reflected wave.

Extending this logic to the entire arterial tree, a track of reflected waves can be obtained

by allowing certain branches to have zero and non-zero reflection coefficients at the

terminations and subtracting them. When displayed in a map form, the spatial and

temporal distribution will then be the track of reflected waves. The distribution of

reflected waves along a segment in this study is obtained by subtracting a surface where

terminations are altered from a control surface. The remaining surface can be expressed

as for a single tube

( )acfPP Γ−Γ=∆ (5.1)

Where Pf, forward wave, Γc and Γa are the control and altered reflection coefficients

respectively. If, for special case Γa =0 then

∆ ΓP Pf c= (5.2)

Thus one can estimate the intensity of reflected waves by inducing total vasodilatation

and subtracting it from the control conditions. Since this procedure requires a reflection

103

coefficient of absolute zero to be introduced at the terminations, it is only suitable in

modelling studies where this condition is fulfilled.

5.1.4 Wave Velocities and Reflection Coefficients

The wave velocities for these compartments were adjusted to simulate pulse wave

propagation in the arterial system of young and old human subjects. The values are

taken from published data (Ho 1982; Avolio et al 1983b, 1985) for age 20 and for 60

years for the mean values of the compartments (see chapter 4). The values from the

model for the spatial distribution are given in figure 5.4 for the young (Open Circles)

and for the old (Closed Circles). The terminal reflection coefficients of the

compartments were selected to be 0.0 for vasodilatation 0.85 for normal state and 0.95

for vasoconstriction. The time constant of the modified Windkessels (see chapter 4)

representing the leaves was chosen to be 250 milliseconds in each case (Liu et al 1989).

Figure 5.4 The pulse wave velocities of the model elements that are calculated using Moens-Korteweg equation. The simulation values for the young (Closed Circles) and for the old (Open Circles) show a relative stiffening of the peripheral branches. The mean values of upper limb, trunk and lower limb segments are: 10.9 m/sec, 8.0 m/sec and 9.0 m/sec respectively for the young simulation. Corresponding values for the old simulation are 11.7 m/sec, 12.9 m/sec and 11.3 m/sec.

104

5.1.5 Input to the Model

In generation of pressure waveforms a digitised flow wave from Nichols et al (1987)

was used. This wave is considered to have an ejection duration of 330 milliseconds at

heart rate of 72 beats/minute (Figure 5.5). The peak flow was scaled to yield a pulse

pressure of 40 mmHg for early systole (up to the first shoulder; Murgo et al 1980b).

Due to finite number of branches, the model could not reproduce the actual values of

peripheral resistances (see chapter 4). Therefore, a mean blood pressure of 100 mmHg

was assumed to exist at every branch of the arterial tree.

Figure 5.5. The flow waveform used as input at the ascending aorta in the model to generate pressure waveforms throughout the branching system. The peak value of the flow wave is normalised to yield a pulse pressure of 40 mmHg between end diastole and the first systolic shoulder. Flow waveform is digitised from Nichols et al (1987).

5.1.6 Input Impedance

The calculated input impedance of the arterial tree at the main root, ascending aorta,

simulating conditions of ageing (young, left; old, right), control (Γc =0.85),

vasodilatation (Γc =0.00) and vasoconstriction (Γc =0.95) are displayed in figure 5.6.

105

When compared to previous publications of the ascending aortic impedance (Murgo et

al 1980b; Nichols and O'Rourke 1990) there was a close agreement between experiment

and simulated impedance pattern. As suggested previously (O'Rourke and Avolio 1980)

ageing shifts the spectrum to the right (early wave reflection) with concomitant increase

in low frequency modulus (increased reflections). Increase in terminal reflection

coefficients (vasoconstriction) increases both the modulus and the phase delay at the

low frequency components. The opposite is observed with the decrease in terminal

reflection coefficients (vasodilatation).

ASCENDING AORTIC IMPEDANCE

YOUNG OLD

Figure 5.6 Ascending aortic input impedance of the arterial tree displaying simulations for the young (Left) and for the old (Right) model. Calculated values for control (Γc = 0.85, Circles), Vasodilatation (Γc = 0.0, Squares) and vasoconstriction (Γc = 0.95, Triangles) is also shown for each case.

106

5.1.7 Pressure Wave Transmission

The spatial and temporal distribution of the pressure waves is shown in figure 5.7. The

pressure pulse amplitude increases as waves travel peripherally due to increased wave

reflection. This increase in wave reflection is due to increases in wave velocity and

getting closer to the terminations. The presence of a second systolic hump that spans

the entire length of the path in the older subject's 3-D map can be made more visible

when plotted in isobaric map format.

YOUNG OLD

Figure 5.7 The 3-D pressure wave propagation map (Top) together with Isobaric contour map (Bottom) in the arterial tree for the case of control (Γc =0.85). Each map

is similar to that in Figures 5.2 and 5.3 and calculated for the young (Left) and for the old (Right).

107

5.1.8 Reflections From Compartments

Figure 5.8 displays the propagation of waves reflected from upper limb in both 3-D map

and isobaric map format. These maps are generated by assigning zero for terminal

reflection coefficients of the upper limb and subtracting the maps obtained from the

ones in Figure 5.7.

UPPER LIMBS

YOUNG OLD

Figure 5.8 The 3-D pressure wave propagation map (Top) together with Isobaric contour map (Bottom) in the arterial tree for the case of upper limb reflections only. Each map is similar to that in Figures 5.2 and 5.3 and calculated for the young (Left) and for the old (Right).

Compared to Figure 5.7, the terminations are the major source of reflections. Waves

travelling in the retrograde direction cross the ascending aorta to spread into the rest of

108

the circulation after being heavily attenuated. The impedance of the trunk and leg

compartments as seen from the upper limb are lower than the upper limb impedance,

thus the branch becomes negatively matched. This reverses of sign of reflection

coefficient at the junction of ascending aortic-brachiocephalic branch. Aging increases

the amount of reflections by apparent shortening of the upper limb vessels, thus

decreasing absolute attenuation in retrograde direction.

LOWER LIMBS

YOUNG OLD

Figure 5.9 The 3-D pressure wave propagation map (Top) together with Isobaric contour map (Bottom) in the arterial tree for the case of lower limb reflections only. Each map is similar to that in Figures 5.2 and 5.3 and calculated for the young (Left) and for the old (Right).

Reflections in the lower limbs have the same characteristics as those described for the

upper limb and shown in Figure 5.9. The phenomenon-of negative mismatch of the

109

limb-trunk junction also occurs here as seen from the figure. The waves appear not to

be able to propagate beyond the junction due to negative wave reflection. Aging

increase these reflected waves, due to increased wave velocity.

TRUNK

YOUNG OLD

Figure 5.10 The 3-D pressure wave propagation map (Top) together with Isobaric contour map (Bottom) in the arterial tree for the case of trunk reflections only. Each map is similar to that in Figures 5.2 and 5.3 and calculated for the young (Left) and for the old (Right). Note the increased amplitude seen in the upper limb and in ascending aorta when reflections are originating from the trunk compared to those when reflected waves are originated from periphery as seen in Figure 5.8 and 5.9.

The reflections from the trunk and their propagation are shown in Figure 5.10. The

reflections have a spatial origin around the abdominal aorta (30 cm from the ascending

aorta) and immediately after the wave front. They travel into both compartments (upper

limbs and lower limbs) without much attenuation. During this travel they also appear in

110

the ascending aortic waveform, boosting late systolic pressure. The same increase in

systolic pressure does not happen in the upper and lower limbs, due to blunting effect of

local reflections that are more prominent than these secondary waves. Aging does not

affect the origin of these waves but increase their intensity.

5.2. PRESSURE WAVE PROPAGATION IN THE UPPER LIMB

The modulus of the pressure transfer function between the ascending aorta and radial

artery displays a characteristic peaking (figure 5.11, top left). It has a DC (zero

frequency) value of unity rising to a peak value of 2.55 at 3.5 Hz. It then decreases

steadily with increasing frequency to attain a value of 0.65 at higher harmonics. The

negative phase of the transfer function (figure 5.11, bottom left) indicates the delay

between the frequency components of waves in the ascending aorta and the radial artery.

It starts from a DC value of zero to have an inflection point at around the peak of the

modulus. After subsequent oscillations it reaches an asymptotic value at higher

frequencies indicating a constant group delay.

Similar to second order system, this behaviour was described in relation to models of

the upper limb (Warner 1957; Gardner 1982). The upper limb often is likened to a

catheter system having a resonant frequency of 4 Hz and a damping coefficient of 0.3

(Gardner 1982). In these models however, wave propagation is not taken into account

although its presence is evident from the phase pattern of the transfer function. In a

typical second order system, modulus has a single peak with attenuation of 40 dB/octave

at higher frequencies. In the model constructed above, it is clear that there is more than

one peak and a logarithmic attenuation. The transfer function displays secondary

oscillations on the descending limb of the modulus and it never reaches zero value, even

111

asymptotically, at higher values. It rather approaches a value other than zero, around

0.5. On the other hand, phase of a second order system has an asymptotic value of -π at

higher frequencies which is not valid for this case. The phase never attains a value of -

π/2 at the peak modulus (break) frequency (Oppenheim et al 1983).

HAA RA−

Figure 5.11 The simulation results obtained in the arterial model of a "Young" subject showing the pressure waveform contours (right) and the transfer function (HAA RA− , left)

between ascending aorta () and the radial artery (•−•). The reflection coefficient, Γ, is 0.85 and time constant, τ, is 250 milliseconds at the terminations of the upper limb. The modulus of HAA RA− (top left ) peaks at 3.5 Hz exceeding unity to a value of 2.55

indicating amplification of the pressure wave. Note the oscillations in the phase values which settle to a constant value dictated by wave speed (bottom, left). The phase delay presents itself as time difference between the ascending aortic and radial pressure waves

The pressure waveform in the radial artery differs markedly from that in the central

aorta. Its peak is accentuated and the systolic pressure exceeds that in the central aortic

one due to reflected waves (Figure 5.11, Right). The foot of the wave is delayed due to

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finite wave velocity. The radial diastolic pressure at the foot of the wave is very close to

that in the aorta.

5.2.1 Effect of Change in Wall Elastance (E)

The effect of increase wave velocity (to simulate the age-related increase in arterial

stiffness) obtained by increase in elastance is to move the peak of transfer function

modulus to the right (Figure 5.12, top left).

HAA RA−

Figure 5.12 The effects of stiffening of the upper limb arteries on the transfer function (HAA RA− , left) and on the radial pressure waveform contours (right). The average pulse

wave velocities in the upper limb is varied from 10.9 to 15.3 m/sec in increments of 10%. The reflection coefficient, Γ, is 0.85 and time constant, τ, is 250 milliseconds at the terminations of the upper limb. Increase in elastance shifts the transfer function to the right with concomitant decrease in the phase delay (arrows). The general pressure pulse shape, however, is little effected.

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However, because of viscous attenuation due to wall and blood viscosity, the peaking

becomes less prominent. The phase becomes less negative with increasing frequency.

The pressure wave in the upper limb becomes steeper and the foot becomes less delayed

(figure 5.12, top right). The systolic pressure in the radial artery is not effected by these

changes.

5.2.2 Effect of Change in Reflection Coefficient (Γ)

When reflection coefficient decreases from 0.95 (vasoconstriction) to 0.85 (normal

vasoconstriction) then to 0.5 (vasodilatation) and to 0.0 (total vasodilatation), the

peaking of the transfer function modulus becomes less prominent (figure 5.13, top left).

Although pulse wave velocities were similar in each case, the frequency at which the

peak occurs gradually shifts left which can also be interpreted as decrease in pulse wave

velocity.(Figure 5.13, bottom left).

However, the asymptotic value of phase delay does not support this interpretation. The

zero reflection coefficient, that is a perfect impedance match at the terminations,

abolishes the peaking of transfer function modulus. The oscillations present in the

phase of the transfer function decreases with decreases in reflection coefficient.

Corresponding changes in the wave shape indicate that most of the energy in the

reflected waves is contained in the systolic peak of the pressure pulse (Figure 5.13, top

right). With the decrease in reflection coefficient, the pressure pulse becomes less and

less amplified approaching to the same value of ascending aorta at zero reflection

coefficient. Pulse pressure decreases and so does the systolic pressure. However, the

diastolic pressure does not change as much indicating that reflections are the major

cause of systolic peaking. Although it can be argued that presence of augmented

systolic pressure without simultaneous decrease in diastolic pressure should result in

increase in energy in the pulse, close inspection of the descending limb of the pressure

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wave reveals equal but opposite fluctuation in the pressure wave, thus fulfilling the

criterion that wave reflection does not augment the net energy content of the pulse.

HAA RA−

Figure 5.13 The changes in transfer function between ascending aorta and the radial artery (Left) and radial pressure waveform contour (Right) caused by changes in reflection coefficient. Decrease in reflection coefficient, Γ, from 0.95 (), 0.85 (•••), 0.50 (−−−) and to 0.0 (•−•) is accompanied by reductions in the pressure pulse amplification evident from the initial peak and the systolic pressure. In each case asymptotic values of phase delay were similar yet the peak amplification frequency decreased. The superimposed oscillations on the phase also decrease, indicating reductions in reflections.

5.2.3 Effect of Changes in Time Constant (τ)

The Windkessel time constant, τ, for the terminal load is varied from 0.6 seconds to

zero in steps of 0.2 seconds (Figure 5.14). This was done to simulate the decreased

compliance of small arteries at the terminations without arteriolar vasoconstriction. As

expected, when the time constant is zero (no capacitance) the load becomes purely

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resistive and thus introduces the highest amount in peaking (figure 5.14, top left). With

increase in the time constant however, the peak in the modulus becomes less and the

peak point shifts to the left. The phase becomes less oscillatory, without altering its

asymptotic behaviour. The pressure waves at the peripheral site, corresponding to this

simulation, display attainment of higher systolic pressures with decreases in time

constant (figure 5.14, top right). These findings also support the view that decreases in

compliance of the small arteries for a given vasoactive state actually increases the

reflected waves.

HAA RA−

Figure 5.14. The changes in transfer function between ascending aorta and the radial artery (Left) and radial pressure waveform contour (Right) due to changes in Windkessel time constant. Increase in time constant, τ, from 0.0 second () to 0.2 second (•••) and to 0.4 second (---) to 0.6 second (−•−) is accompanied by reductions in the amplification. In each case asymptotic values of phase delay were similar yet the peak amplification frequency decreases. The superimposed oscillation on the phase decreases indicating reductions in reflections.

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5.2.4 Effect of Change in Wall Viscosity (Θo)

To simulate the changes in viscous losses in the arterial wall, wall viscosity Θo is varied

from 0o to 15o in steps of 5o (Figure 5.15). The resultant transfer function indicates

that viscosity effects are important in producing oscillations in the modulus as first

suggested by Taylor (1966b). Although there are little changes in the low frequency

components of the transfer function moduli and therefore on the radial pressure

waveform, higher frequencies are heavily attenuated. This indicates that under

physiological conditions, ie normal heart rates, the only important part of the transfer

function in determining the wave shape is to the left of the peak amplification.

HAA RA−

Figure 5.15. The changes in transfer function between ascending aorta and the radial artery (Left) and radial pressure waveform contour (Right) caused by changes in wall viscosity. Increase in time constant, Θ0, from 0.0 degrees () to 5 degrees (•••) and to 10 degrees (---) to 15 degrees (−•−) is accompanied by reductions in the amplification at higher frequencies. In each case value of phase delay were similar yet the peak amplification decreases without much alteration in the actual pulse pressures.

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5.3 DISCUSSION

The human arterial system has been modelled in the past by various investigators

employing different techniques (see chapter 2). This study, however, is different from

those attempts, since it directs its attention to the propagation of the pressure waves

rather than pressure-flow relations. Although the mathematical and clinical implications

are the same, the former is readily applicable in clinical settings.

In this study, a technique of mapping the pressure wave propagation is also introduced.

Similar to the weather maps, this visualisation technique enhances the visual appeal of

otherwise complex data and does not suffer from lack of spatial information as present

in conventional graphs. Careful analysis of these maps provides information about the

wave velocities, reflection sites and the intensity of reflections. Coupled with

superposition principle, these maps can be useful in detection of reflected waves from

different compartments. Previously, the same information was obtainable by complex

analysis of the pressure-flow data (Nichols and O'Rourke 1990).

The findings of this study indicate that retrograde transmission of pressure waves from

the limbs is heavily attenuated. This supports the view that although reflected waves

become prominent when there is elastic non-uniformity, the retrograde transmission of

generated, reflected waves become heavily attenuated (Li et al 1984). Thus the arterial

system favours the forward transmission, tending to decouple the heart from the

periphery (Taylor 1964). The reflections from the limbs reach the ascending aorta often

heavily attenuated. However reflections originating the ones from the splanchnic

circulation are 30% or more of the forward travelling pulse. Thus, splanchnic

circulation being not only close to the ascending aorta, but also being favoured for

retrograde transmission plays an important role in determining ascending aortic

impedance. This assertion is supported by the blood flow data to those compartments.

It is well known that, under resting conditions, some 40% of the blood is used to perfuse

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the organs around the abdominal aorta (McDonald 1974). One would expect, therefore,

a higher influence of the vasoactive state of this compartment on the overall

haemodynamics.

The limited effect of lower limb vasculature on human arterial pulse compared to

influence of the trunk seems surprising. A close inspection of the arterial model (Figure

4.4) reveals that there are not many branches in the lower limb circulation. This is

especially true for the proximal arteries of the lower limb circulation that perfuses the

muscles around the thigh. However, the method employed in the delineation of the

reflected waves does take this fact into account. The alteration of reflection coefficients

from 0.85 (control) to 0.00 (complete vasodilatation) eliminates any load presented to

the common iliac, thus simulating the total vasodilatation of the distal branches

irrespective of their numbers. Still, there was not enough changes in the ascending

aortic impedance spectrum (see below) to suggest reasonable effect of this manoeuvre.

It is possible that when values less than zero were used expected impedance changes

might be generated. However, there is evidence that the reflection coefficient at the

aorta iliac bifurcation never becomes less than zero even with maximal vasodilatation

(Li 1984).

This concept was further supported by changes in the aortic impedance spectrum caused

by alteration of reflection coefficient of the compartments. The influence of reflections

from splanchnic circulation is far greater than from both limbs (figure 5.16, Squares).

The effect of reflections from both limbs is minor compared to the splanchnic

circulation. This result of the simulation warrants an investigation of the splanchnic

circulation. Chapter 10 of this thesis describes a study in which this hypothesis is

verified. Vasodilatation of this circulation caused by ingestion of glucose, explained the

reduction in reflected waves in the central waveform, similar to simulation results.

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ASCENDING AORTIC IMPEDANCE

YOUNG OLD

Figure 5.16 Ascending aortic input impedance of the arterial tree displaying simulations for the young (Left) and for the old (Right) model. Calculated values for control (Closed Circles), Vasodilatation of the upper limb (Open Circles), lower limbs (Triangles) and trunk (Squares) is also shown for each case.

These findings are further confirmed by superimposing the reflected waves originating

from each compartment (Figure 5.17). This technique enabled to isolate the path of

reflected waves which would otherwise be blunted by multiple reflections from different

compartments. Again, the ascending aortic and the upper limb pressure pulse is heavily

affected by the reflected waves from the trunk compartment.

When the pressure waves in the ascending aorta and in the radial artery is decomposed

into its components by separating the effects of reflections from different compartments,

it supports the view that the major reflecting site is in the trunk area (Figure 5.18, Top).

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Time (milliSeconds)

0 200 400 600 800Radial

Ant.Tibial

Aortic Bifurcation

A.Aorta

Brachial

Renal Branches

Femoral

Figure 5.17 Isobaric map showing the spatial and temporal distribution of travelling pressure waves along the path given in Figure 5.1. Arrows indicate the direction of travel. After the initial wave (Heavy Dark Lines) which travels both to the upper limbs and the lower body it is reflected in four major reflection sites, radial termination, splanchnic branches, aortic termination, femoral bifurcation and the tibial termination, (Dark lines). These reflected waves are re-reflected during retrograde travel (Light Lines). Although the reflected waves from both limbs appear to be trapped by reciprocal reflections, reflected waves originating from the trunk travel in all directions. The influence of these waves on ascending aortic wave contour is more pronounced than those from the peripheral limbs.

Reflected waves originating from the trunk area travels into the ascending aorta and into

the upper limb compartment without much alteration in its amplitude. In these

simulations they formed the secondary oscillations present in both composite waves.

Although peak systolic pressure in the aortic pulse is augmented with these reflected

waves, the radial systolic pressure has not. In the radial waveform, reflections from the

upper limb compartment boosts the initial systolic upstroke blunting the reflected waves

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ASC. AORTA RADIAL

TRUNK

LOWER LIMB

UPPER LIMB

COMPOSITE

Figure 5.18 Decomposition of pressure pulses in the ascending aorta (Left) and in the radial artery (Right) into reflected waves from different origins. (Top) The reflected waves originating from the trunk area augment the peak systolic pressure in the ascending aorta but appear in the radial artery pressure waveform as secondary oscillations. (Middle) Upper limb reflections are prominent and constitute the first peak in the radial pulse but have little influence on central aortic pressure pulse . Note the ripples in the reflected waves from the trunk and lower limb compartments indicating the presence of re-reflections. (Bottom) Reflections originating from the lower limb compartment have little effect on the wave shapes in both upper limb and in ascending aorta. (Dash dot lines indicate the instant of diastolic and peak systolic pressures in the composite wave while dotted lines indicate the level of mean pressure).

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from the trunk (Figure 5.18, Middle). As expected, the effect of lower limb

compartment is minor and negligible in both pulses (Figure 5.18, Bottom). Peak

systolic pressure in ascending aorta and in radial artery is defined by reflections from

two different origins. While local reflections originating from the upper limb

terminations defines peak systolic pressure in the radial artery, reflections originating

from the trunk area are responsible for the same phenomenon in the ascending aorta. It

is therefore possible to alter the peak systolic pressures recorded from these two sites

independent from each other by changing the characteristics of these two origins.

Reflections from the trunk undergoes secondary reflections in the upper limb arteries as

evident from the oscillations in the composite waveforms. Similar superimposed

oscillations are also observed in the lower limb reflected waves.

A close inspection of the intensity and the timing of reflected waves originating from

the upper limb also reveals that the early systolic peak in the upper limb will often be

different from the one at central aorta (Figure 5.18). However, the diastolic and the

mean pressures in these locations are approximately equal. These phenomena, caused

by the timing of reflected waves can be exploited in calibrating waveforms in both

arteries. Since pressure waves obtained by arterial tonometry could not be calibrated

with the current techniques, use of pressure waves obtained by other means, such as

invasive radial artery waveforms, enable these waveforms to be calibrated. This

technique involves equating the diastolic and mean pressures in both waveforms,

invasive and tonometric, to obtain a linear scale which would then be used to calibrate

the tonometric waveforms. In Section III this technique is employed to calibrate carotid

and radial tonometric waveforms.

In the latter part of the simulation it was shown that the pressure wave is amplified

during its travel along the upper limb. Although it has been known for a long time

(Warner 1957), this study is first to investigate the relevant determinants of this

amplification. As opposed to earlier attempts, in this study it was shown that the upper

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limb is different from second order system analogue, due to wave travel. The second

order system approximation can be justified if the wave velocity is high enough (Figure

5.12). The increase in wave velocity shifts the peak of the transfer function to the right

and decreases the amplification. Thus, second order approximation becomes feasible to

apply.

Amongst all the physical parameters investigated, the profound effect is observed when

the terminal reflection coefficient is varied. It generated the most radical changes in the

transfer function and thus the peripheral radial wave shapes. Yet the influence was not

linear and was little for changes from 0.5 to zero but substantial for changes from 0.95

to 0.80. The time constant, however, is linear in its effect to the transfer function thus to

the radial pressure wave. Since there is a direct relation between the time constant and

the reflection coefficient (chapter 4) for a given terminal compliance, this suggests a

stable transfer function under these conditions.

From a practical point of view, the data obtained for changes in wall viscosity indicate

that although there can be substantial changes in the high frequency components of the

transfer function, the peripheral pressure waveform is quite insensitive to these changes.

This is because the aortic pressure pulse is already band limited (Nichols and O'Rourke

1990).

In summary, the B-Tree model employed in this study predicted changes similar to that

observed experimentally in the human arterial system. Using this model and the

pressure maps, propagation of reflected waves from different compartments was

investigated. Amongst the compartments studied (upper limbs, lower limbs and the

trunk) reflected waves originating from the trunk influence the ascending aortic

impedance more than other compartments. Furthermore, the reflected waves from the

trunk travel into every other compartment and form the secondary oscillations in the

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pressure waveforms. Similar phenomena were not observed in other compartments

indicating the possible role played by trunk vasculature on the overall haemodynamics.

When extended to the upper limb, this model predicted that there is a more complex

relation between central aortic and radial pressure waveforms than a simple second

order system. The transfer function which defines the relationship between central

aortic and peripheral radial pressure waveform is insensitive to alteration of arterial

parameters at lower frequencies. At higher frequencies, however, the transfer function

might vary considerably but due to limited bandwidth of the central aortic pressure

waveform the peripheral radial pressure waveform changes little.

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MODEL VALIDATION AND PARAMETER ESTIMATION IN A SINGLE ELASTIC TUBE In previous chapters the model used to simulate the arterial tree was described in terms

of its organisation and implementation. Results were also presented for wave

propagation and effects of altering arterial properties. However, no model can be

complete without testing its accuracy in its assumptions and predictions. Similarly, the

implementation of model algorithm and analysis tools have to be tested. The integrity

of the hardware and developed software requires testing so that errors could be spotted

and corrected before proceeding with experimentation.

This chapter describes an experimental investigation using a single elastic tube

terminated with a real positive load aimed at validating the model described previously.

CHAPTER 6

126

This enables estimation of model parameters using an optimisation technique which

allows for global convergence of system parameters to experimental observations.

6.1 SINGLE ELASTIC TUBE EXPERIMENT

The experiment is similar to the classical tube experiments performed by early

investigators of arterial haemodynamics (Taylor 1957b, 1959b; Hardung 1964) , figure

6.1.

A/D and D/A Board

PressureTransducer

PressureTransducer Clamp

Amplifier

Pressure Generator

Data

PC

Elastic Tube

Figure 6.1 The schematic drawing showing arrangement of experimental hardware. A PC coupled with A/D-D/A board controls the experiment. It generates pressure waveforms through a servo controlled pressure generator in the tube. The simultaneous recordings of propagated pressure along the tube are made by identical pressure transducers which are amplified and digitised using the same PC. The digitised data are stored in computer hard disk for subsequent analysis.

This setup was chosen not only because of its ease of construction but also because there

is ample theoretical and experimental evidence for comparison (Milnor 1989; Nichols

and O'Rourke 1990). Briefly, a silicon elastic tube (Silastic, Dow-Corning), two metres

127

in length, was clamped at one end, while the other end was excited by a pressure source.

The tube had an internal diameter of 0.335 cm and a wall thickness of 0.065 cm. The

tube was relatively thicker and shorter than that described in previous reports (Taylor

1957a, 1959; Hardung 1964). Specific waveforms were generated by means of a servo

controlled loudspeaker (Biotek waveform generator Model 601A). Since the volume

displacement that could be attained by the drive mechanism was small (2 ml) within

linear range, it was important to ensure a stiffer tube at this length to restrict volume

displacement. The tube was filled with isotonic saline (0.9% wt/vol) which was boiled

and cooled to eliminate air bubbles in the fluid. The pressure generator with the tube

filled with this fluid was able to generate pressure frequencies up to 20 Hz, in line with

input fed to the waveform generator. The pressure generator was excited by waves

having fundamental frequencies of 0.65, 1.25, 2.0 and 2.5 Hz dumped out from an IBM-

AT computer in analogue form using an Analog to Digital/Digital to Analog board (DT

2801-A, Data Translation, Mass). The same computer and board were used to acquire

signals from pressure transducers that were conditioned by a suitable amplifier (Model

HS9, Medtel Instruments).

C5C4C3C2C1

R5R4R3R2R1

PressureWave

Generator

~

Figure 6.2 The arrangement of sites of pressure recording (R1=0 cm, R2=30 cm, R3=60 cm, R4=90 cm, R5=200 cm) and the sites for clamp application ( C1=100 cm, C2=120 cm, C3=140 cm, C4=160 cm, C5=200 cm).

Pressure waves were recorded using to disposable strain gauge manometers (Admac

100, Spectromedics). The transducers which are directly attached to 22 gauge

hypodermic needles are inserted at recording sites R1-R5 ,Figure 6.2. Since the

128

dynamic equivalence of the manometers was more important than true frequency

characteristics (the waves are all described in relation to pressure waves at site R1), the

test on this factor showed that they are identical up to frequencies of 40 Hz. The

pressures are recorded at sites R1 for input, and R2, R3, R4 and R5. Site R2 is

assumed to be fixed when expressing the influence of increased reflection due to early

wave reflection. To simulate different distances to the wave reflection site, ie early

wave reflection, the tube is clamped at different distances from the input (C1=100 cm,

C2=120 cm, C3=140 cm, C4=160 cm, C5=200 cm).

6.1.1 Estimation of Tube Elastance

The manufacturers specification for the Silastic tube used in this experiment did not

include values of wall elastance therefore it was necessary to determine tube elastance

by experimentation. The calculation of the elastance of the tube was compared with the

model predictions from the optimisation process. Although the calculation could be

achieved by using a single method, three separate methods previously reported (Patel

1972) are employed in the calculation.

Calculation of Elastance from Stress-Strain Relation by Longitudinal

Stretch

Upon completion of the experiment, a segment of the tube 100 cm in length was

subjected to stretch by hanging weights to one end while suspended from the other. The

resultant displacements were recorded. Assuming a Hookean substance for the wall

material, the elastance was derived by fitting a straight line to stress-strain relation

(Attinger 1964):

σ = =

FA

M gA.

(6.1)

129

where M is the mass in kg, g is the gravitational constant 9.81 m/sec2 and σ is the

tensile stress. The strain, ε , is described as

−×= 1100

100 mlε (6.2)

where lm is the length of the tube under mass M. The elastance then becomes

ESS =σε

(6.3)

Calculation of Elastance from Pressure-Volume Relations

A bolus of saline 0.5 ml is injected into the tube. The resultant change in pressure, ∆P ,

is then used to estimate elastance using the formula given by Bergel (Bergel, 1960).

( )2223

oi

oi

i

PV RRRR

RR

PE−∆

∆= (6.4)

where Ri and Ro are the inner and outer radius respectively. Both radii were calculated

after considering a circular cross-sectional area where expansion of tube in z direction is

ignored.

Calculation of Elastance from Wave Velocity:

Wave velocity can be used in conjunction with Moens-Korteweg equation to estimate

elastance

E c Rh

i o= 02 2 ρ (6.5)

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The value of wave velocity, c0, can either be obtained by foot-to-foot estimates

(Nichols and O'Rourke 1990) or by averaging phase velocities for frequencies above the

first minimum. An alternative method can be devised using the quarter wavelength

formula

c f l0 4= min (6.6)

Since phase velocity minimum, fmin, also denotes the minimum of the impedance

spectrum (see chapter 2.2.6 ), first minimum of the phase velocity spectrum or first

maximum of the transfer function modulus can be used to estimate fmin. Wave

velocities are estimated by foot-to-foot and quarter wavelength method and

subsequently to estimate of elastances of ECff and ECf min respectively.

6.2 ESTIMATION OF MODEL PARAMETERS

The arterial model described in Chapter 4 is used in conjunction with an optimisation

process to estimate various physical parameters of the tube. For this purpose, a uniform

transmission line model of the experimental set up was constructed using the B-tree

model described in chapter 4. The model parameters for the experimental case of

clamping at 200 cm (C5) are given in Table 6.1. The measurable parameters of the

model - diameters, wall thicknesses, and lengths- are entered as known parameters. The

elastance of the tube and reflection coefficient at the termination are assumed to be

unknown and thus estimated by the model.

Using pressure waves generated by the model and the ones actually measured along the

tube, an error function is defined to describe the deviation of model from the elastic

tube. For this purpose, recorded pressure waveform at site R1 is used as input to model

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at node M. The sum square difference between the pressure waves recorded at sites R2-

R5 and the ones calculated by the model for the corresponding sites is defined as the

forcing quantity. The process is iterated with different values of E and RF until values

less than 2 percent are achieved (Figure 6.3). At this value of E and RF the model is

assumed to represent the elastic tube. The entire iteration procedure is implemented as a

batch process. This enabled to predict model parameters from experiments described in

Section IV.

B-Tree Code

NAME L (cm)

D (cm)

h (cm)

Elastic Modulus

(dyne/cm2) x106

θ (Degrees)

Γ (Units)

M R1 0.01 0.34 0.07 E 0.0 MR R2 30.0 0.34 0.07 E 0.0 MRR R3 30.0 0.34 0.07 E 0.0 MRRR R4 30.0 0.34 0.07 E 0.0 MRRRR R5 110.0 0.34 0.07 E 0.0 RF

TABLE 6.1 Table of elements included in the model representing the entire tube and its measurement sites where the tube is clamped at C5=200 cm. Elastance , E, and reflection coefficient at C5, RF, are estimated by the optimisation process.

E, RF

InputPressure

Wave

RegressionCoefficient

Calculated

Measured

New Model Parameters

Linear Regression

B-Tree Model

Elastic Tube

Pressure

Pressure

Figure 6.3. Block diagram for estimation of elastic tube parameters using an iterative technique and an appropriate model.

132

6.3 RESULTS

6.3.1 Tube Elastance

The stress-strain relation obtained from longitudinal stretch was not perfectly linear,

concave towards strain axis, indicating a higher compliance for larger stresses (figure

6.4). The slope of the linear relation, ESS , was 48.25 x106 dyne/cm2. This value

compares well with the value obtained from pressure-volume relation. Injection of 0.5

ml fluid changed the pressure by 130 mmHg, yielding an EPV value of 44.65 x106

dyne/cm2. The foot-to-foot delay between sites R1 and R5 was 67 milliseconds, giving

a pulse wave velocity of 2985 cm/sec and by equation 6.5 the elastance ECff becomes

45.94 x106 dyne/cm2.

Stress

Figure 6.4. The stress-strain relationship of the elastic tube used in the experiment. The values are obtained by applying tension to the tube and by measuring the elongation.

The value obtained from equation 8.6 for recordings between sites R1 and R5 yields an

ECf min value of 42.04 x106 dyne/cm2 for a minimum of 4.2 Hz (Table 6.2). The close

133

similarity between different methods of calculation of the tube elastances allows one to

use any of the methods for estimation of wall elastance.

METHOD Elastance (x106 dyne/cm2)

Stretch (ESS ) 48.25 Pressure Volume (EPV ) 44.65 Foot-to-foot (ECff ) 45.44 Quarter Wavelength (ECf min ) 42.04 Mean ± SEM 45.09±1.28

Table 6.2. Data showing the comparison of different methods on the estimation of the elastance of the tube.

6.3.2 Convergence of the Model

The error introduced by the model converged to a minimum for different fundamental

frequencies (figure 6.4).

For curves rich in low frequency power, that is at lower fundamental frequencies, the

concavity of the error surface was less than that at higher fundamental frequencies.

However, the errors obtained in the latter were much higher than in the former. These

indicate a high sensitivity to the model parameters E and RF for high frequency

components. Not surprisingly, the predicted value for E by the model varied between

39.61 x106 dyne/cm2 and 47.27 x106 dyne/cm2, with an average of 42.18 x106

dyne/cm2. Convergence always occurred for errors less than 2 %.

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A B

C D

Figure 6.4. The sum square error surfaces between by the model predicted (Table 6.1) and measured pressure waves at site R5 during which the fundamental frequency of the input wave is varied (A=0.65 Hz, B= 1.25 Hz, C= 2.0 Hz, D= 2.5 Hz). Note the shallow minima at lower fundamental frequencies. The site of tube clamp is at C5 (200 cm) and the pressure recording is made in R2. Convergence always occurs with an error of less than 2 percent.

6.3.3 Predicted Pressure Waves

The pressure waves generated by the model for the optimum values are shown in figure

6.5.

135

A B

C D

Figure 6.5 Pressure waves predicted by the model (solid line) and measured (dotted line) at different fundamental frequencies (A=0.65 Hz, B=1.25 Hz, C=2.0 Hz, D=2.5 Hz) for the optimised model. The individual parameter optimisation process for these waveforms is given in Figure 6.4. The site of clamp (C5) is 200 cm away from the origin.

The upstroke and the contour of the pressure waves are all well matched. The waves

from the model (solid line) are not identical to experimental waveforms at peak systolic

pressures (dotted line). The difference in peak systolic pressure is dependent on the

fundamental rate, showing accentuation at or around 3 Hz (see below). However, this

difference never exceeds 15% of the calculated pulse pressure.

136

6.3.4 Transfer Function

Modulus

Early wave reflection caused by shortened termination increases the amplitude of the

modulus at the maximal point with accompanying rightward shift of the peak (top

figures in figure 6.6 through 6.8).

A B

Figure 6.6 Transfer function moduli (top) and the apparent phase velocities (Bottom) of the model (solid line) and of the experiment (circles) between sites R1 (0 cm) and R2 (30 cm) when termination is at 100 cm (A) and 120 cm (B) from the origin. The dashed line in the phase velocity plot (2955 cm/sec) represents the true wave velocity calculated from the mean elastance value of Table 6.2. Note the difference in scale of moduli in (A).

137

This change in modulus of transfer function between sites R1 and R2 shows the same

trend for both the experiment (circles) and the model (solid line).

A B

Figure 6.7. Transfer function moduli (top) and the phase velocities (Bottom) of the model (solid line) and of the experiment (circles) between sites R1 (0 cm) and R2 (30 cm) when termination is at 140 cm (A) and 160 cm (B) from the origin. The solid line in the phase velocity plot (2955 cm/sec) represents the true wave velocity calculated from the mean elastance value of Table 6.2.

Wave reflection also increases due to early wave reflection caused by shortened

distance. At higher frequencies, it increases the differences between recorded waves

while at lower frequencies the amplitude of augmentation decreases. This rightward

shift of the maxima caused by shortening of the tube beyond R2, can also be interpreted

as the stiffening of the segment between R1 and R2. The higher amplitude at this

138

maxima can also be interpreted as an increase in the reflection coefficient at the

termination R2.

Figure 6.8 Transfer function moduli (Left) and the phase velocities (Right) of the model (solid line) and of the experiment (circles) between sites R1 (0 cm) and R2 (30 cm) when termination is at 200 cm from the origin. The solid line in the phase velocity plot (2955 cm/sec) represents the true wave velocity calculated from the mean elastance value of Table 6.2.

These alternative explanations cannot be accepted readily, unless they are confirmed by

information gained from the phase plots. The model, on the other hand, reproduced the

essential features of the experimental findings. There was some discrepancy, however.

The model tended to underestimate the amount of the pressure wave amplification in

every frequency, sometimes as much as 15%. This was apparent from the synthesised

pressure waves, figure 6.5, where the model could not fit the exact amplitude by up to

15%. This finding is surprising, since there were no losses in the wall, wall viscosity

was zero and the estimated reflection coefficient was close to unity (actually 0.999).

This difference is probably due to the tethering of the tube (Taylor 1959). The

mathematical model uses strong tethering which increases viscous losses due to

increased coupling between wall and fluid, yet the tube was tethered slightly in this

experiment. Since there is substantial tethering in vivo situations, the model should

apply well in those circumstances.

139

Phase Velocity

Phase velocities, between sites R2 and the termination, predicted by the model (Bottom

figures in figures 6.6 through 6.8) agree well with experimental data (circles). They

both start from a high value for low frequencies, then fall to a minimum, followed by

maxima and minima around true wave velocity that is calculated using mean elastance

value in table 6.2. The deviation from this true wave velocity is caused by reflected

waves originating from the closed end termination. This is manifested by the frequency

of the second minimum being three times that of the first minimum (3.8 vs 11.8 Hz for

clamp at C5). When the clamp is moved closer to the measuring site (from C5 to C1),

the first minimum shifted rightward. There was simultaneous increase in the values of

phase velocities at lower frequencies. This agrees with model predictions that the

reflections will increase (higher values) with decreased distance (rightward shift) to the

reflecting site.

In the extreme case, C5, the distance to the reflecting site becomes more difficult to

calculate using the quarter wavelength formula. The very short distance makes the

minimum very dispersed hence making its detection more difficult. Due to introduction

of significant errors in calculations, the phase differences are small, at higher

frequencies the model predictions are different from the experimental findings. The

distances to the reflecting site as calculated from the phase velocity plots, change in the

same direction for model data and experimental data, indicating the implementation of

the B-tree model does not introduce errors.

6.3.5 Reflection Coefficient

The model predicted a terminal reflection coefficient of 0.95 or more for each

optimisation. This value is similar to the experimental setup, given that the reflection

140

coefficient is determined by using phase velocities. This value varied from 0.8 to 0.95

for proximal reflecting sites obtained by clamping the tube proximally. The values

obtained from the model for reflection coefficients for the same situation yielded the

same values, 0.8 to 0.95 as in experiments.

6.4 DISCUSSION

The elastic tube model has been used successfully by previous investigators (Taylor

1959, Hardung 1964) to verify theoretical predictions. They measured pressure and

flow along the elastic tube and calculated various model parameters. In this study, the

experiments are performed with omission of flow measurements. By programming the

experiments in a digital computer, attempts have been made to mimic the real

conditions. However, it is difficult to predict the behaviour of an algorithm

implemented in a computer. Such an implementation may be tested by simulating the

real events with real data. This approach was adapted in these experiments. A

performance criterion was defined and the simulation was optimised until this criterion

was achieved.

The real data was obtained from the dimensional measurements of the elastic tube

material and from the pressure measurements. Since an extra physical quantity,

elastance, is needed as well to define the material, different techniques were utilised to

estimate this quantity. It was found that these techniques estimated similar values with

close agreement. The model estimate closely approximated the actual values of wall

elastance under a wide variety of conditions. The same performance was obtained for

wave reflection coefficient. Although the initial values of these two parameters were

very much different from the actual values, the convergence was definite, free from

local maxima and minima. This suggests that continuous estimation of parameters may

141

be made with minimal computer resources. Since this approach uses only pressure

measurements it avoids the complexities brought by measurement of blood flow

velocity.

Proper implementation of the model in a system should necessitate measurement of the

tube's physical properties. In tube models, distances can be measured quite accurately as

well as contours of pressure waves. The wall thickness, diameters and wall viscosity

parameters could be fixed for certain segments of the arterial tree, freeing computer

resources from estimation of these parameters as well. Normalisation process by setting

a fixed diameter, wall viscosity and wall thickness, does not make much difference in

phase velocity estimates. The variation of diameter will effect the α number, which is

high under normal circumstances. Therefore α related parameters are asymptotic at this

range. The wall viscosity, however, is important in setting the transfer function moduli

values (Taylor 1966b, Avolio 1976a) as demonstrated in this experiment. Even small

viscous forces due to tethering can change the synthesised pulse pressures. In the model

presented here it was less than 15 percent.

It was also shown that the implemented model followed the changes in transfer function

modulus and impedance, as determined from the phase velocity estimates, reliably. This

was achieved under wide variety of conditions which included changes in fundamental

frequency and distance to the reflecting sites. As expected, when distance to the

reflecting site increased, the first minimum of the phase velocity spectrum and the

transfer function modulus shifted leftwards. Since the wave velocity was unchanged,

the distance from the reflecting site could be obtained using quarter wavelength formula.

However, when the true wave velocity is unknown, the use of either of these plots alone

may lead to misinterpretation of the phenomenon; that is, estimations made from the

modulus alone do not differentiate between increase in intensity of reflection coefficient

from distance to the reflecting site or change in wave velocity. The actual reflection

coefficient or the actual wave velocity can be overestimated.

142

Obviously, the parameter estimation could still be achieved if one has to construct

pressure waveforms at site R1 for corresponding inputs of R2-R5. However, in this

calculation, the problem of inverse filtering of a low pass filter occurs (Kenner 1978).

The high frequency components of the pressure wave measured downstream are

attenuated and are close to noise level due to viscous losses. The tube, therefore,

behaves like a low-pass filter at these frequencies and the information lost during this

process would not be regained by inverse filtering. To overcome this problem, a higher

fundamental frequency with enough power would have to be issued. In this study the

highest fundamental frequency used in this study was 2.5 Hz and the components of the

pressure wave at the 6th (15 Hz) and above harmonics were well below system noise.

There is also possibility that parameters estimated using this process may not be unique.

To avoid being attracted to some local minimum in the solution, the estimator first scans

the entire parameter space for detection of local minima. It then adjusts the iteration

steps accordingly. Since this global search approach uses extensive amount of computer

time, once the boundaries of local minima are known, the model assumes this

information as a priori and uses it to limit the parameter space. Iteration maps of the

calculation process are generated for the entire E and RF space, to test the hypothesis

that these calculated values are unique. Once it has been verified that there is a unique

solution by this method, no further maps are generated since it was consuming extensive

amount of computer time.

In summary, the model could reliably predict parameters such as reflection coefficient,

wall elastance and spatial distribution of pressure waves. It is also clear from these

findings that one can use the model to support experimental data to extend the

experimental findings. This can be achieved without significantly increasing the errors

introduced other than by measurement.

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SECTION III

EXPERIMENTAL STUDIES

In this Section, the features of the arterial pulse, measured in a central or peripheral

artery, are investigated systematically in human subjects. To explain features of the

pulse, methods are described to estimate the intensity of reflected waves using pressure

wave recordings. In general, however, the emphasis is placed on the formation of the

upper limb pressure pulse as upper limb arteries are readily accessible and pressure

pulses can be obtained non-invasively. Experiments were performed using wave

propagation parameters as a substitute for pressure-flow relations in real conditions.

In the first chapter, the upper limb system is described in subjects where the input to the

system could be varied. In a group of 18 subjects with implanted programmable cardiac

pacemakers, the input pulse shape to the upper limb arterial system was altered by

144

changing heart rate. Resultant changes in the output pulse were measured. Using

spectral techniques a unique expression was derived between the central pressure

waveform and the peripheral/radial pressure waveform.

In the second chapter, the effect of reflections from the terminations on upper limb

system were investigated. By applying gradual compression to the vessels of the palm

vasculature and by reducing the mean arterial pressure in the upper limb, the pressure

waveform and the intensity of reflected waves were varied and studied. The transmural

pressure in this vasculature was found to have a systematic effect on the system.

However, under normal circumstances, the system was similar to that observed under

total occlusion of this vasculature.

In third chapter, the effects of vasoactive drugs on this system were investigated.

Although there are documented reduction in secondary pressure oscillations caused by

these drugs , these were found not to be related to changes in upper limb properties.

These drugs affected the proximal vessel properties without altering the reflections

occurring at the termination of the upper limb.

The fourth chapter concentrates on the relationship between secondary oscillations on

the pressure wave and the arterial circulation as a whole. Administration of oral glucose

to a group of subjects caused reductions in secondary oscillations. However, there were

no changes in the upper and lower limb arterial system suggesting a possible role of

vasoactive state of the splanchnic circulation on these oscillations. This suggested a

bifurcation of the aorta into two circulations, one to the splanchnic and the other to the

lower limb. Representation of the arterial tree with a modified T-tube model using this

concept generated realistic pressure and flow waveforms.

145

DETERMINATION OF WAVE PROPAGATION CHARACTERISTICS IN THE HUMAN UPPER L IMB The clinical implications of pressure wave propagation of the upper limb are often

overlooked. Pressure wave in the upper limb is not only different in its amplitude but

also different in its shape to that in the central aorta (Kroeker and Wood 1955;

Remington and Wood 1956, Kelly et al 1989). Due to wave propagation and wave

reflection phenomena the travelling pressure wave is delayed and is amplified. Yet, the

pressure pulse in the upper limb is used to infer central pressure pulse and the left

ventricular afterload ( Pauca et al 1989).

Previously, the comparisons of pressure waves recorded in the upper limb to that in the

central artery are made by comparing the peak and mean pressures obtained using

Chapter 7

146

sphygmomanometric techniques (Borow et al 1982). The invasive data is obtained

during vasodilation (Kelly et al 1990, Simkus and Fitchett 1990), exercise (Rowell et al

1968), normal conditions (Kroeker and Wood 1955; Remington and Wood 1956) and

shock (O'Rourke 1970). However, these studies were limited by the range of heart rate

available to the investigators. Although it was possible to obtain a spectrum of rates in

many patients, this was not useful in investigating the variation brought by different

heart rates. When it was possible to change the heart rate by exercise (Rowell et al

1968), the alteration of global control mechanisms and the vasoactive state brought by

this manoeuvre made it impossible to analyse the data.

In this Chapter, results are presented demonstrating the clinical implications of wave

travel and wave reflection in eighteen subjects with implanted programmable

pacemakers. By pacing the hearts at different rates and taking simultaneous

measurements of pressure waves in the carotid and radial arteries together with cardiac

output, the changes in the central and peripheral pressure wave shapes were

documented. A transfer function was determined for each subject by relating the central

and peripheral radial pulses in the frequency domain. It was found that the

amplification of the pulse from central to peripheral sites depends on heart rate as first

suggested by O'Rourke (1970) and that the upper limb transfer function is essentially

constant in the frequency domain. Results were similar to those predicted by the upper

limb model presented in Chapter 5.

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7.1 METHODS

7.1.1 Subjects

Subjects for this study were drawn from a group attending a hospital pacemaker clinic as

outpatients. All were equipped with VVIR pacemakers (Telectronics, Optima-MP) and

were undergoing routine pacemaker functionality test. The population consisted of 12

males and 6 females whose age ranged from 19-82 years with a mean age of 57.6 years.

All patients were in good health and signed a consent form to participate in the study.

7.1.2 Measurements

The heart rate and cardiac output of the subjects were measured with a non-invasive

cardiac output meter (Bomed, NCCOM3-R7). This device operates on the principle of

impedance cardiography by injecting currents between two electrodes at 20 to 200 kHz

and detecting the voltage drop across the thorax with a different pair. With proper

calibration, this device has shown to give similar results to existing cardiac output

methods, ie. thermodilution and dye clearance (Herok 1992). Two tonometers, one at

the carotid artery, the other on the radial, continuously recorded the pressure wave at

these sites while a photoplethysmographic blood pressure measurement (FINAPRES)

was performed on the middle digit of the right hand. When the patient was relaxed and

all the recordings were stable, the heart rate was varied by 10 beats/minute from 60

beats/minute to 120 beats/minutes, while the established rate was maintained for 10

seconds to allow for a steady state to be achieved. After completion of the study, the

pacemaker was programmed to run at a rate of 70 beats/minute where another set of data

was collected.

The carotid and radial pressure data at different heart rates digitised on-line via an IBM-

compatible PC equipped with an A/D board (DT 2801-A, Data Translation Mass).

Acquired data (128 Hz, 12 bits per channel) was temporarily stored on the hard disk to

148

be later transferred to a WORM (Write Once Read Many) drive (Maxtor 800S) for

permanent storage. Using in-house software developed for this purpose and using ECG

triggering data segments of 15 second length were averaged.

Calibration of Waves

It is normally impossible to predict the calibration of non-invasive tonometric tracings

due to lack of knowledge of exact applanation pressure and orientation. However,

calibration is possible if one determines the pressure values by an independent technique

and assumes that (i) the mean arterial pressure is equal throughout the arterial tree and

(ii) the diastolic pressure values can be considered to be approximately equal in every

branch. These two assumptions are readily satisfied in this study. Firstly, the

measurements are made in the supine position and secondly at diastolic pressure level

the effects of reflected waves are shown to be minimal (see Chapter 5). In this study,

the calibration of tonometer tracings is performed by equating the mean and diastolic

pressures obtained from the FINAPRES device and the non-invasive recordings.

7.1.3 Data Analysis

Pressure Wave Amplification

To quantify the pressure wave amplification between the carotid pulse pressure, PPc,

and radial pulse pressure, PPr, in the time domain the following relation is defined.

100×=c

ramp PP

PPPP (7.1)

Pressure Wave Shapes

The augmentation index, AI, is used in this study to document changes in the radial

pressure waveform with increase in heart rate. The augmentation index is often used to

express the intensity of reflected waves (Murgo et al 1980b). This index is defined as

149

the ratio of peak systolic pressure increase due to wave reflection to the peak pressure at

around flow and given as:

dPP

PPAI−−

×=1

12100(%) (7.2)

Where P1, P2 and Pd are pressure at late systole, at early systole and at diastole. When

applied to the radial artery waveform, however, this index attains significantly lower

values from the centrally derived indices (Kelly et al 1989a). Because of the

mismatches, reflected waves originating at the palm level increase the first peak in the

radial artery without necessarily affecting the second peak (see Figure 5.17). Thus, the

augmentation index measured in the radial artery actually decreases due to an increase in

locally reflected waves. Furthermore this index attains negative values for the

peripheral waveforms. Therefore, this index is modified throughout the thesis by adding

an offset of 100 to the form defined by Murgo et al (Murgo et al 1980).

Transfer Functions

For an absolute determination of the relationship between the carotid and radial

waveforms transfer function of the upper limb RACAH − is determined. This allows a

unique way of expressing both the pressure pulse amplification and the wave shapes

under different pacing rates. The transfer functions are determined by relating the

harmonic components calculated using Fourier analysis (see Chapter 2).

Parameters Obtained from the Transfer Function

By applying equations derived for the single tube analog of the upper limb transfer

function, the resonant frequency and reflection coefficient can be estimated.

Resonant frequency

150

The resonant frequency of the upper limb is derived from the quarter wavelength

formula

l

cfn 4

0= (7.3)

where c0 is the pulse wave velocity obtained from foot-to-foot measurements and l is

the distance between measurement sites.

Reflection coefficient

An estimate of the complex reflection coefficient is derived from the phase velocities

analogous to equation 2.148 and described as

∞

∞+ +

−=Γ

cccc

i

i0 (7.4)

where ci is the pulse wave velocity at frequency 'i' and c∞ is the foot-to-foot estimate of

true wave velocity obtained from the pressure waves. This technique assumes c∞ to be

proportional to Z0,characteristic impedance, and ci to be proportional to the terminal

impedance, ZT, value at frequency 'i'. Similarly, the pulse wave velocity at very long

wavelengths, c0, or ones close to zero frequency, c0+ , is considered to be related to the

peripheral resistance values. These velocity values were then used to estimate a

compatible reflection coefficient value with those previously reported (O'Rourke and

Taylor 1966). Estimation of the value of c0 for this purpose normally requires

utilisation of random excitation techniques to increase the resolution of the phase

velocity spectrum. Since application of this technique in humans was impossible, c0+ ,

the phase velocity at very low frequencies, is substituted for c0 and estimated by linear

extrapolation of phase velocities of second and first harmonic to zero. This linear

extrapolation technique has already been described in conjunction with estimation of

151

frequency independent values of elastance from frequency dependent terms in smooth

muscle preparations (Gow 1972).

Reduced Model Of the Upper limb Arterial System

Since it was possible to obtain reasonably high fidelity recordings from these patients,

an attempt was made to obtain parameters describing the arterial properties of the upper

limb. For this purpose, a parameter estimation technique similar to that performed in

the elastic tube experiment (Chapter 6) was employed. However, in these sets of

simulations a reduced model of the upper limb involving few branches was constructed

in order to decrease computation time (Table 7.1).

B-Tree Code

NAME L (cm)

D cm

h (cm)

E (x106 ) (Dyne/cm2)

Θ (Degree)

Γ (units)

τ (Sec)

M Asc. Aorta 0.10 0.94 0.08 Em θm Γm τm MR Innominate 9.10 0.94 0.08 Em θm Γm τm MRR Brachial 22.0 0.60 0.18 Em θm Γm τm MRRR Radial 41.0 0.34 0.16 Em θm Γm τm MRRRR Termination 0.10 0.34 0.16 Em θm Γm τm

Table 7.1 Reduced model of the upper limb used in estimation of transmission characteristics. Em , θm , Γm and τm are the model parameters to be estimated from the best fit between pressure waves generated by the model and those measured using linear regression analysis.

Input to this model was considered to be the carotid pressure waveform. Similar to the

elastic single tube experiment, the model was iterated with different elastance, Em , wall

viscosity, θm , reflection coefficient, Γm , and time constant, τm , values to yield a best

fit between model derived and actually measured radial waveform. The model values

obtained this way are considered to be the values representative for the arterial segments

in the upper limb vasculature. Since same elastance values are used for each element of

the model, the wall thickness to diameter ratio is adjusted to account for progressive

increase in pulse wave velocities (Table 4.1). This resulted in seemingly unrealistic

152

values for wall elastance values. However, corrections could be made if the actual wall

thickness to diameter ratios were known by multiplying wall elastic modulus with the

appropriate ratio.

Model limits are then compared with these values obtained by analysing modulus and

phase velocities of transfer functions.

7.1.4 Statistics

Differences from the baseline heart rate are analysed using paired t-test. A p value of

less than 0.05 is accepted statistically significant. Data are presented as mean ± standard

error of mean unless otherwise stated.

7.2 RESULTS

Figure 7.1 shows an example of pressure waveforms recorded in the carotid and radial

arteries for a single patient during the pacing protocol. Under control conditions the

pulse pressure in the radial artery was higher than that found in the carotid artery.

Pacing the heart from 60 to 120 beats/min with increments of 10 beats/min decreased

the pulse pressure in the carotid artery. The pulse pressure in the radial artery either

increased or remained unchanged. This made it difficult to asses the pulse pressure in

the carotid artery from the radial artery. When heart rate was reduced to 70 Beats/min,

the difference in pulse pressures was restored to baseline values. This finding indicates

that heart rate is the only determinant of this amplification. Similar changes are also

observed in all patients (Table 7.2).

153

HEART RATE

Figure 7.1 Pressure recordings obtained from one patient showing carotid (solid line) and radial (dotted line) pressure pulses. Pulses are averaged and calibrated by FINAPRES device. The heart rate increased from 60 to 120 with increments of 10 beats/min, then decreased back to 70 beats/min. Note the decrease in pulse pressure in the carotid while not much increase is observed in the radial.

When data from all subjects are pooled and expressed in terms of AMPPP , it revealed a

non-linear relationship between the heart rate and the pulse pressure difference (Figure

7.2). There was an increase in AMPPP with heart rate (from 114±7% (mean ± SD) at 70

beats/min to 123±15% (mean ± SD) at 110 beats/min).

Pacing of the heart also caused significant changes in the pressure waveforms (Figure

7.3). There was a significant drop in the augmentation index, AI, with increase in

pacing rate (from 79±10% at 70 beats/min to 50±7% at 100 beats/min, p<0.001). At

much higher rates (110 and 120 beats/min) the algorithm used to detect the second peak

(see Chapter 12) failed in all instances due to absence of this peak. This finding alone

154

Carotid Pulse Pressure (mmHg)

Radial Pulse Pressure (mmHg)

P No Age

60 B/min

70 B/min

80 B/min

90 B/min

100 B/min

110 B/min

120 B/min

70 B/min

60 B/min

70 B/min

80 B/min

90 B/min

100 B/min

110 B/min

120 B/min

70 B/min

1 19 34 35 38 43 38 43 * * 43 42 42 44 44 41 * * 2 37 43 39 39 39 36 37 35 39 45 41 46 47 44 45 45 54 3 30 52 44 40 38 38 34 32 45 55 51 51 55 49 51 49 64 4 35 46 45 40 40 39 44 * 35 54 54 54 56 56 59 * 53 5 64 81 75 71 69 90 * * 93 82 80 76 79 100 * * 98 6 59 53 49 48 56 51 49 51 49 56 56 60 65 63 62 60 64 7 65 * 50 45 46 43 36 39 60 * 61 56 56 56 54 53 73 8 59 52 57 57 54 52 47 41 63 56 65 64 66 64 61 59 74 9 45 * 53 49 48 43 43 42 59 * 65 62 64 60 59 54 64 10 61 59 50 41 42 44 47 46 52 59 53 47 46 53 54 53 56 11 71 54 62 59 56 55 63 64 61 62 67 65 64 60 67 72 65 12 82 30 32 31 31 30 30 26 37 34 37 34 34 34 36 29 46 13 74 47 40 37 35 42 40 36 51 53 47 45 42 51 49 46 58 14 71 63 60 49 54 46 48 48 81 70 68 53 58 55 60 62 76 15 77 66 58 56 54 53 54 52 68 68 62 65 62 64 64 62 73 16 82 41 43 46 38 53 52 46 46 41 44 47 39 49 50 49 47 17 23 24 30 28 28 25 25 24 33 32 36 37 38 38 32 32 43 18 78 88 86 75 67 65 62 54 80 98 94 93 86 85 82 72 95 Mean 57.33 51.50 50.44 47.17 46.56 46.83 43.25 42.67 54.50 55.57 56.83 55.39 55.61 56.94 52.75 53.67 63.00 SEM 4.73 4.13 3.32 2.88 2.65 3.30 2.42 2.71 4.05 4.15 3.49 3.29 3.24 3.60 2.86 3.08 3.67

Table 7.2 Patient details and pressure pulses obtained at baseline and during pacing. Missing values (*) are due to lack of baseline data or the impossibility of pacing at a higher rate.

155

PPAMP

Figure 7.2 The pulse pressure amplification between carotid and radial sites during pacing. The pressure wave is normally amplified during its travel in the upper limb. There is also small but significant increase in amplification with higher heart rates. (Bars represent ± 1 SEM, * p<0.05, † p<0.001 compared with 70 beats/min).

RADIAL AI

Figure 7.3 The augmentation index in the radial artery pressure waveform At rates higher than 100 beats/min, the second shoulder was impossible to be detected. (Bars represent ± 1 SEM, † p<0.001 compared with 70 beats/min).

156

HCA RA−

60708090

10011070x

Figure 7.4 (Left) The transfer function between carotid pulses and the radial pulses of Figure 7.1 during pacing the heart form 60 beats/min to 110 beats/min and back to 70 beats/min. (Right) The transfer function between carotid and radial artery sites for the entire study group, compare this to one given in Figure 5.11. Bars represent 95% confidence interval.

reveals that major change in the pressure wave is in its first peak brought by increased

wave reflection from the palm vasculature.

The left panel in Figure 7.4 displays the transfer function modulus and phase for the

recordings given in Figure 7.1. Although there were significant differences in the

amplitude and the shape of individual pulses, radial pulses were related to carotid pulses

in a unique way in the frequency domain. As expected, there is a consistent

amplification of harmonics around 4 Hz. The graph on the right shows all patient data

that are pooled to obtain a generalised transfer function for the study population. The

close correspondence of this transfer function with the model predictions (Figure 5.11)

157

indicates the applicability of arterial model presented in Chapter 5 into upper limb

arterial system.

Patient No

fn (Hz)

Γ0+ (Units)

Em (X106) (dyne/cm2)

θm (Degrees)

Γm (Units)

τm (sec)

1 3.83 0.02 1.69 20.50 0.31 0.07 2 3.23 0.24 1.25 22.88 0.40 0.08 3 4.38 0.37 2.64 13.25 0.59 0.09 4 4.27 0.67 2.84 15.86 0.59 0.06 5 4.50 0.08 2.37 16.00 0.11 0.04 6 3.45 0.24 1.19 11.88 0.35 0.06 7 3.84 0.40 2.35 13.43 0.54 0.02 8 3.69 0.42 1.84 12.75 0.43 0.06 9 3.96 0.46 2.35 10.43 0.57 0.07 10 4.57 0.15 2.63 7.50 0.22 0.05 11 4.51 0.17 2.60 19.63 0.15 0.06 12 3.79 0.23 1.80 13.88 0.24 0.08 13 4.32 0.40 2.44 11.63 0.34 0.08 14 3.85 0.24 1.96 12.00 0.27 0.07 15 4.17 0.13 2.13 15.38 0.21 0.05 16 3.43 -0.06 1.28 25.13 0.04 0.04 17 3.71 0.39 1.90 14.50 0.64 0.07 18 3.85 0.27 1.75 17.25 0.39 0.07 Mean±SEM

3.96±0.09

0.27±0.04

2.06±0.12

15.22±1.03

0.36±0.04

0.06±0.00

Table 7.3 Measured and model generated upper limb model parameters. Resonant frequency fn and Γ0+ reflection coefficient are derived from quarter wavelength formula while Em , θm , Γm and τm are derived from model iterations.

Table 7.3 displays the values estimated from the transfer functions using quarter

wavelength formulas and with the model iterations. The resonant frequency is close to 4

Hz and the reflection coefficient estimate is 0.27 by the analysis of transfer functions.

Similar values are obtained from the model iterations for reflection coefficient.

However, the elastance values are relatively small with respect to documented values.

158

The small value of τm indicates that the termination of upper limb behaves more like a

closed end tube.

7.3 DISCUSSION

These results (Table 7.2) confirm the model predictions which suggests the presence of

pressure wave amplification along the upper limb. In this study, the pressure pulse is

not only amplified but its contour also altered while it travels along the upper limb

vasculature. As heart rate increased by pacing these changes became more dramatic.

These observations indicated the presence of frequency dependent amplification of the

pressure pulse in the upper limb and warranted the utilisation of spectral techniques for

studying this phenomenon.

The transfer function which relates the input to the output, ie central aortic to upper limb

pressure pulse, usually obtained by relating the harmonic components of input and

output pulses. However, the technique is limited by the heart rate involved which would

only resolve the spectrum at fundamental frequencies. In this study the continuous

spectrum is obtained by scanning the transfer function spectrum with alteration in heart

rate. Scanning of arterial impedance spectrum by pacing the heart is employed

previously in determination of ascending aorta (Taylor 1966c) and pulmonary artery

(Bergel and Milnor 1965). Use of this technique in pressure transfer function is new

and enables to compare the model predictions with the experimental findings.

The application of this technique to the pressure pulses revealed the uniqueness of the

transfer function in the upper limb, (figure 7.4, Left). Although, there were differences

in both the amplitude and contour (Figures 7.1, 7.2 and 7.3), the harmonic components

of the pulses are related in a unique manner. Similar results were obtained in each

159

patient by pacing the heart. The transfer functions were found to be similar to the one

predicted by the model (Figure 7.4, Right). As suggested by the model, the peak of the

modulus occurs at around 4 Hz, reaching to a value of around 2.2 at this frequency.

Analysis of transfer functions using single tube analogy indicated a resonant frequency

of 3.97 Hz and a terminal reflection coefficient of 0.27. The iteration of the model to

predict the properties of vessel wall and the terminal element confirmed these results.

Although the predicted wall elastance value is nearly one order of magnitude less than

that reported (2.06 vs 10 to 20 x106 dyne/cm2, Table 7.3 vs Table 4.1), the wall

viscosities are comparable (15.22 vs 10 to 20 degrees, Table 7.3 vs Table 4.1). The

modulus of reflection coefficient (0.36 vs 0.27, Γm vs Γ0+) and time constant (0.06 vs

0.00, θm vs real termination) also agrees. The discrepancy between the model predicted

and reported wall elastances are probably due to inaccurate estimation of wall thickness

to diameter ratio.

Based on the transfer function in the upper limb, a higher dependence of amplification

to the heart rate is expected. This behaviour can be explained by invoking the effects of

pacing on both ventricular filling and ejection. The pressure wave amplification does

not only depend on the transfer function, but also the components of the flow wave

constituting the fundamental harmonic. If the amount of energy contained in the

fundamental harmonic is decreased, the estimate of amplification by this technique will

be blunted. The VVIR method of pacing does not allow the atrium to contract in

synchrony with the left ventricle, thus not enhance left ventricular end-diastolic

dimensions. This decreases the contractile force developed by the left ventricle since it

is proportional to the end diastolic volume (Frank-Starling mechanism). As a result, the

stroke volume and the energy contained in the flow pulse decreases. Normalisation of

pulse pressure measured in the radial artery to the stroke volume, however, takes this

factor into account, Figure 7.5.

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PP/SV

Figure 7.5 The radial pulse pressure for a given stroke volume indicating the dissociation of pulse pressure to that of stroke volume. (Bars represent ± 1 SEM, † p<0.001 compared with 70 beats/min)

The presence of amplification between the central and upper limb vessels and presence

of similar transfer function across different individuals suggests that radial artery

waveforms are not reliable for patient monitoring. When the heart rate can be as high as

90 beats/min under routine invasive monitoring situations, the peripheral pulse is an

unreliable guide to stroke volume and cardiac work. This assertion becomes more valid

when the heart rate is high and this amplification of the components of the peripheral

pressure pulse becomes as much as 200 per cent of the central pulse (Figure 7.5).

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QUANTIF ICATION OF THE INTENSITY OF REFLECTED WAVES IN THE UPPER L IMB The upper limb pressure pulse is used in every aspect of patient management and for

diagnosis and treatment of hypertension. Yet in nearly all of these circumstances, the

effect of wave reflection in the upper limb arteries as a determinant of arterial blood

pressure has never been addressed properly. There is a lack of documentation of the

origin of reflections in the upper limb and quantification of relative intensities of

reflected energy. This chapter attempts to document the effect of reflected waves in the

upper limb pressure waves by analysing the pressure waves recorded in the central

(Carotid) and peripheral (Radial) arteries.

It was predicted in Chapter 5 that the systolic part of the upper limb pressure wave is

contaminated with reflected waves originating from upper limb terminations. Being

CHAPTER 8

162

different from the ones in the central aortic waveform they are responsible for peak

systolic pressure in the upper limb.

In the following experiment, these predictions were investigated by altering the intensity

of reflected waves originating from the palm of the hand by altering the impedance

match at this level. Reflection coefficient at this termination were altered by

modifications of terminal impedance by graded compression of the palm vasculature

and/or the characteristic impedance by raising the arm. This in turn resulted in

alteration in the intensity of reflected waves. To estimate the degree of wave reflection,

the same upper limb model employed in Chapter 7 was also used. By predicting

pressure wave propagation parameters and by utilisation of non-invasive techniques to

measure blood pressure waveforms, this model enabled the intensity of reflected waves

to be calculated without the need of employing complex flow measurements.

8.1 METHODS

8.1.1 Subjects

Thirty-three subjects who had undergone routine coronary artery bypass surgery were

recruited from the Cardiothoracic Department, St. Vincent's Hospital (Table 8.1). None

had peripheral vascular disease or arterial stenosis. They were screened for

hypertension (mean arterial pressure <120 mmHg) and hypotension (mean arterial

pressure >60 mmHg). The study was approved by the research ethics committee of the

hospital and all patients gave consent prior to the study.

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Mean±SEM Range Age (year) 58.2±1.6 42-74 Sex 25 Males and 8 Females Weight (kg) 80.1±2.4 52-102 Height (cm) 171±1.7 148-1.86 MAP (mmHg) 84.1±2.1 63-111 Heart Rate (beats / min) 93.1±2.05 75-122 Type of Operation 26 CAG, 2 MVR, 4 AVR, 1 ASD Time after Operation (Hours) 6:08±0:26 1:45-11:00 Temperature (°C) 37.2±0:26 32.8-38.8

Table 8.1 Summary data for patients enrolled in the study ( MAP = Mean arterial pressure, MVR = Mitral valve replacement, AVR = Aortic valve replacement, ASD = Atrial septal defect, CAG = Coronary artery graft)

8.1.2 Procedure and Data Acquisition

Changes in the terminal load were introduced through graded occlusion of the hand

vasculature by means of a sphygmomanometer cuff. By increasing the pressure in the

cuff, the physical properties of the underlying arterial bed are modified due to

transmission of this external pressure to the arterial wall and hence decreasing the wall

tension. This decrease in wall tension introduces changes in the lumped characteristics

of the distal arterial bed. The changes in the proximal characteristic impedance were

introduced by raising the arm analogous to the cuff occlusion manoeuvre, the reduction

in mean arterial pressure caused by raising the arm decreased the transmural pressure

and thus increased the compliance of the proximal vessels and decreased the

characteristic impedance, since both are inversely related.

The procedure involved wrapping an adult-size sphygmomanometer cuff around the

hand of the arm which had an indwelling radial artery catheter. While the central

pressure waveform was being registered, the cuff was rapidly inflated to 200 mmHg. It

was then deflated slowly (2 mmHg/sec) to zero pressure. This cycle of inflation and

164

deflation was performed three times to allow for data averaging. The same procedure

was repeated while the arm was raised from its baseline level to reduce the registered

mean arterial pressure by about 15 mmHg.

As a routine procedure, ECG (5-lead) and invasive arterial pressure were used to

monitor the heart rate and blood pressure via a bedside monitor (Spacelabs-PC). The

amplifier of this monitor has an output of 1 volt/100 mmHg. The pressure monitoring

catheter system was composed of 220 cm polyethylene tube attached to a disposable

radial arterial cannula (18-20 gauge) and disposable strain gauge transducer. Although

the natural frequency of this system was more than 60 Hz, the attachment of catheters

and connectors greatly reduced this. The investigation of the actual and achievable

frequency response of a typical system was performed by sinusoidal excitation of the

recording assembly with a function generator (Hewlett Packard 3311A) connected to a

pressure generator (Biotek model 501A). The pressure waveforms obtained from this

bench top experiment were subjected to frequency analysis to estimate the amplification

and phase delay (Figure 8.1). The transfer function obtained this way was later used to

correct the manometer system by inverse filtering.

Figure 8.1 The frequency response of the catheter system including the amplifiers and the digital filters.

165

Before any measurements were taken, the catheter was checked and flushed with sterile

normal saline (0.9% wt/vol) to minimise trapped air bubbles and blood clot. An

estimation of frequency response of the catheter system was then performed by the

modified pop test. This test involves tapping the end of the catheter and estimating the

period of the oscillations. Whenever the estimated oscillation was below 20 Hz the

system was considered to be unsuitable. Subsequent flushing and eliminating the

bubbles consistently improved the characteristics of the system. The manometer was

then moved from the rack mounting the entire transducer assembly to the cannulation

site. This process allowed the registration of blood pressure at the radial arterial level

rather than usual heart level. Since the entire measurement took 15 minutes to

complete, routine blood pressure monitoring was performed in the contralateral arm

using an oscillometric technique.

Applanation tonometry was used to record the blood pressure waveform in the carotid

artery (Central). Since it was impossible to predict the calibration of these non-invasive

carotid tracings (due to lack of knowledge of exact applanation pressure and

orientation), they were calibrated by equating the mean and diastolic pressures of the

invasive radial and carotid tracing. This technique assumes that the mean arterial

pressure is equal throughout the arterial tree and at diastolic pressure level the effects of

reflected waves are minimal and thus pressure values can be considered to be

approximately similar in every branch (see Chapter 5). Inevitably this technique led to

errors when the arm was raised, but appropriate change of mean pressure due to this

manoeuvre was taken into account when calibrating these recordings. A separate

tonometer which was calibrated previously was attached to the sphygmomanometer cuff

connectors in order to monitor instantaneous cuff pressure during the procedure

All pressure and ECG signals were digitised on-line with an IBM compatible computer

using an A/D board (DT 2801-A, Data Translation, Mass). Acquired data (128 Hz, 12

bits per channel) was temporarily stored on the hard disk and later transferred to a

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WORM (Write Once Read Many) drive (Maxtor 800S) for permanent storage. Using

software developed for this purpose data segments of 8 second length during the

manoeuvre were averaged using ECG triggering. The data were then subjected to

transfer function analysis (see Chapter 2) and expressed with respect to transmural

pressure which is defined as

crt PPP −= (8.1)

where Pt, Pr and Pc are the mean transmural, radial and cuff pressures during the pulse

period.

8.1.3 Estimation of Intensity of Reflected Waves

Although the effects of reflected pressure waves on the formation of pressure waves are

well documented (Nichols and O'Rourke 1990; O'Rourke et al 1992) quantification of

wave reflection still presents major challenges. The amount of wave reflection may be

quantified by measuring pressure and flow at a site invasively, and by applying the

equations given in Chapter 2. Historically, these equations have been used either in the

ascending aorta to estimate the amount of global reflections as seen from the heart

(O'Rourke and Taylor 1967a) or in large conduit arteries to quantify the reflections from

major beds (O'Rourke and Taylor 1966; Li et al 1984). However, investigations

directed towards the applicability of these equations in small conduit arteries have been

scarce, not only because of the perceived lack of importance of reflected waves but also,

because of the difficulties associated with flow measurement techniques in small calibre

peripheral arteries.

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(i) Estimation from Phase Velocities

An estimate of complex reflection coefficient is derived from the phase velocities

similar to one described in chapter 7 is obtained. This equation is analogous to equation

2.148 and described as

∞

∞+ +

−=Γ

cccc

i

i0 (8.2)

Where ci is the pulse wave velocity at frequency 'i' and c∞ is the foot-to-foot estimate

of true wave velocity obtained from the pressure waves. This technique assumes c∞ to

be proportional to Z0, characteristic impedance, and ci to be proportional to the

terminal impedance, ZT, value at frequency 'i'. The pulse wave velocity at very long

wavelengths, c0 , or ones close to zero frequency, c0+ , is considered to be related to the

peripheral resistance values. The c0+ , the phase velocity at very low frequencies is

estimated by linear extrapolation of phase velocities of second and first harmonic to

zero. This technique yields values similar to that obtained with parameter estimation

method (see Chapter 7).

(ii) Estimation from Pulse Pressures

The pulse pressure amplification, AMPPP , between central, PPc is the central pulse

pressure, and peripheral sites, PPr , radial pulse pressure, is also calculated as an index

of reflections

100×=c

rAMP PP

PPPP (8.3)

(iii) Estimation from the Model

Another set of reflection coefficient estimates were obtained by the parameter

estimation technique similar to the one performed in Chapter 7. Briefly, a parameter

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estimation technique similar to the one performed in elastic tube experiment (Chapter 5)

is employed in a reduced model of the upper limb involving few branches (Table 8.2).

Input to this model is considered to be the carotid pressure waveform. Similar to elastic

single tube experiment, model output is compared against the in vivo measured radial

waveform while model parameters, Em , wall viscosity, θm , reflection coefficient, Γm ,

and time constant, τm , are continuously being altered. The model values representing

the best fit between these two pressures are considered to be the values representative

for the upper limb circulation.

B-Tree Code

NAME L (cm)

D cm

h (cm)

E (x106 ) (Dyne/cm2)

Θ (Degree)

Γ (units)

τ (Sec)

M Asc. Aorta 0.10 0.94 0.08 Em θm Γm τm MR Innominate 9.10 0.94 0.08 Em θm Γm τm MRR Brachial 22.0 0.60 0.18 Em θm Γm τm MRRR Radial 41.0 0.34 0.16 Em θm Γm τm MRRRR Termination 0.10 0.34 0.16 Em θm Γm τm

Table 8.2 Reduced model of the upper limb used in the estimation of transmission characteristics. The model parameters Em , θm , Γm and τm are estimated by the best fit between recorded and model generated pressure waves.

8.1.4 Statistics

Differences of indices from the zero transmural pressure control state are analysed

using Student's paired t-test. A p value of less than 0.05 is accepted statistically

significant. Data are presented as mean ± standard error of mean unless otherwise

stated.

8.2 RESULTS

A typical run obtained during the study on one patient is shown in Figure 8.2.

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Figure 8.2 A representative tracing obtained in one subject showing the radial (Solid), Carotid (Dotted) pressure waveforms and transmural pressure calculated using equation 8.1. Each pulse in the tracing is the average of 8 second recordings of three runs appended with respect to transmural pressure. (Negative transmural pressures indicates cuff pressures are higher than the intra-arterial mean pressure).

CAROTID RADIAL

Figure 8.3 Representative tracings obtained in one subject showing the carotid and radial pressure waveforms against transmural pressure (TP) in the hand. Arterial pressure data (AP) are stacked and interpolated to obtain continuous values for different transmural pressures (TP).

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CONTROL

TP (mmHg)

PP (mmHg)

PPAMP (%)

c∞ (m/sec)

c1 55. (m/sec)

c0+ (m/sec)

Γ0+ (Units)

-100.00 52.89±1.63 126.10±2.77 780.23±13.80 926.13±53.58 1138.51±98.72 0.14±0.03 -80.00 51.36±1.64 125.52±2.37 797.92±14.77 904.98±34.05 1117.81±62.65 0.13±0.02 -60.00 51.98±1.77 120.87±2.68 782.42±12.71 835.32±28.42 998.33±49.79 0.10±0.03 -40.00 47.76±1.47 120.00±2.63 798.05±13.37 813.32±28.42 955.73±46.02 0.03±0.03 -20.00 48.12±1.46 115.00±2.16 782.57±12.15 773.31±22.85 891.75±36.40 0.00±0.02 0.00 45.14±1.61 112.20±1.88 808.61±14.18 737.71±20.49 820.50±29.96 0.00±0.02 20.00 42.53±1.53 109.37±1.92 796.90±11.88 704.80±18.30 772.93±30.30 -0.03±0.02 40.00 45.51±1.70 114.09±2.03 801.49±15.43 763.78±23.39 862.60±30.96 0.03±0.02 60.00 44.62±1.26 118.76±2.09 784.84±12.18 776.31±22.40 894.06±33.49 0.05±0.02

ARM RAISED

TP (mmHg)

PP (mmHg)

PPAMP (%)

c∞ (m/sec)

c1 55. (m/sec)

c0+ (m/sec)

Γ0+ (Units)

-100.00 49.89±2.14 119.90±3.21 765.30±15.88 775.99±36.71 916.55±58.10 0.07±0.03 -80.00 49.14±1.99 121.96±2.99 766.27±13.22 798.49±27.22 952.85±53.75 0.07±0.03 -60.00 48.38±1.79 117.37±2.35 769.34±11.76 775.48±28.48 924.91±47.24 0.07±0.02 -40.00 46.32±1.44 115.30±2.56 797.95±13.18 767.03±25.91 897.41±44.30 0.04±0.03 -20.00 43.82±2.02 109.74±2.33 777.21±12.18 699.01±28.12 794.02±42.11 -0.01±0.02 0.00 39.97±1.52 108.61±2.39 774.23±16.99 671.74±20.11 729.36±29.99 -0.06±0.03 20.00 39.99±1.66 107.68±2.36 772.86±13.54 662.09±21.15 702.95±32.42 -0.05±0.02 40.00 42.42±2.28 107.93±2.36 771.45±14.06 654.00±17.37 703.13±22.19 0.01±0.02 60.00 43.79±1.57 113.78±1.65 789.52±11.81 718.12±15.25 823.79±24.25 0.06±0.02

Table 8.3 The effect of transmural pressure on wave reflection indices under control conditions (Top) after raising the arm (Bottom). (Tp = transmural Pressure; PP = Pulse pressure; PPAMP = Pulse pressure amplification between carotid and radial sites; c∞ , c1 55. and c0+ are the foot to foot, at 1.55 Hz and at near zero hertz components of the phase velocities respectively; Γ0+ is the reflection coefficient estimates from

Equation 8.2. (Data are given as mean ± standard error of the estimate).

During the cuff deflation period, ie increase in transmural pressure, the amplitude of

radial pulse pressure initially decreases. It later increases when the cuff pressure falls

below the intra-arterial level. Since no simultaneous pressure swing is observed in the

carotid waveform, this phenomenon is presumed to be local to the upper limb. This is

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further confirmed after averaging 8 second recordings of carotid (Figure 8.3, Left) and

radial pressure waveforms (Figure 8.3, Right) and expressing them with respect to

transmural pressure in the hand.

The radial systolic pressure decreases initially to reach a saddle point at around zero

transmural pressure then increases subsequently with an increase in transmural pressure

(Figure 8.3, Right) without any significant change in the carotid waveform (Figure 8.3.

Left). This characteristic change in radial pressure waveform was reproducible in all

patients (Table 8.3).

When individual pulse pressure data are pooled against transmural pressure in the hand,

the minimum pulse pressure near zero transmural pressure becomes evident in both

control (Figure 8.4, circles) and arm raised (Figure 8.4, squares) conditions.

PULSE PRESSURE

Figure 8.4 The effect of transmural pressure in the hand on radial pulse pressure under control (circles and solid line) and arm raised conditions (squares and dotted line). The right most points represent the baseline conditions. Data are pooled in bins of 20 mmHg and averaged, bars represent ± 1 standard error,* p<0.05, ** p<0.0001).

172

There is a lesser degree of change in the radial pulse pressure during arm raise than

during control. This is due to different baseline mean arterial pressures (83.11±2.31

mmHg versus 63.73±1.58 mmHg, p<0.0001) and pulse pressures (50.79±1.96 mmHg

versus 44.36±2.61 mmHg, p<0.05). At zero transmural pressure both pulse pressures are

reduced (control 45±1.61 mmHg, raised 39.97± 1.52 mmHg).

PPAMP

Figure 8.5 Changes induced by cuff occlusion on the pulse pressure amplification in the upper limb under control (circles, solid line) and arm raised conditions (circles, solid line). The right most points represent the baseline conditions. Data are pooled in bins of 20 mmHg and averaged, bars in the graph represent ± standard error, * p<0.05,.

Since changes in the actual values of the radial pulse pressure might be contaminated by

the input values (that is alteration in central pulse pressure), they are expressed in terms

of pressure amplification using Formula 8.3 (Figure 8.5). Under control conditions

(Circles) the AMPPP has a value of 125.41±1.12% which decreases to 112±1.88%

(p<0.001) at zero transmural pressure and increases back to 126.10±2.77% at -100

mmHg (p = NS). Raising the arm reduces the AMPPP to 116.32 ± 1.3% with respect to

control (p<0.05) while change in transmural pressure affects the AMPPP the same way

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as control (108.61±2.39% at 0 mmHg, p<0.0001 and 119.9 ± 3.21% at -100 mmHg

p<0.05) (Squares).

PHASE VELOCITY

CONTROL ARM RAISED

Figure 8.6 Changes induced by cuff occlusion on the phase velocity estimates in the upper limb under control (left) and arm raised conditions (right). The foot-to-foot velocity estimates, c∞ (circles) are constant across different transmural pressures both in the distal bed (abscissas) and the proximal conduit arteries (left and right). The first (c1.55, squares) and the extrapolated zero harmonic (c0+, diamonds) show the maximum variation to both pressure changes. The right most data points in both graphs represent the baseline conditions. Data are pooled in bins of 20 mmHg and averaged, bars represent ± standard error of mean.

The phase velocities at the first harmonic (1.55 Hz) and those extrapolated for the zero

harmonic for both control and arm raised display similar patterns (Figure 8.6). The

foot-to-foot wave velocity estimate, c∞, remains the same both in response to cuff

occlusion and when the mean pressure in the arm is lower (Figure 8.6 ). The first and

extrapolated zero harmonics are greater with respect to the condition where the arm is

raised (Figure 8.6 Right) indicating a higher mismatch between the proximal and distal

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beds (see below). The distal bed impedance which is proportional to the phase velocity

(Chapter 2) at baseline, estimated from the extrapolated zero harmonic, c0+, is 1.3 times

to the proximal beds characteristic impedance, c∞, under control conditions. Raising

the arm reduces this to 1.1 times that of c∞.

The reflection coefficient estimates calculated at extrapolated zero harmonic phase

velocities using equation 8.2 show changes parallel to that of the pressure pulse

amplification data (Figure 8.7). The reflection coefficient estimate, Γ0+, follows

changes in the PPAMP during control (circles) and arm raised (squares) conditions. It is

lower when the arm is raised than during control and it has the exactly same saddle

point at around zero transmural pressure. This near zero values of the reflection

coefficient estimate at around zero transmural pressure indicates a better match than that

for baseline.

Reflection Coefficient Estimate (Γ0+)

Figure 8.7 Changes induced by cuff occlusion on the reflection coefficient estimate in the upper limb under control (Circles) and arm raised conditions (Squares). The changes due to arm raising and the hand occlusion are in parallel with the ones derived from the pulse pressure amplification (see Figure 8.5). The right most points represent the baseline conditions. Data are pooled in bins of 20 mmHg and averaged, Bars in the graph represent ± standard error.

175

These parallel changes are further confirmed by the linear regression between the pulse

pressure amplification to that of the reflection coefficient estimate (Figure 8.8). There is

a direct relation between the pulse pressure and the reflection coefficient estimate

derived from the phase velocity spectra. Although both are derived in different domains

( AMPPP in time domain and Γ0+ in frequency domain) the regression equations between

the two were y=83.78x+113, R2=0.72 (p<0.0001) for under control (Figure 8.8, Left)

and y=84.37x+112.7, R2=71 (p<0.0001) for arm raised conditions (Figure 8.8, Right).

ALL PATIENT DATA

CONTROL ARM RAISED

Figure 8.8 The regression between the reflection coefficient estimate, Γ0+ , and the pulse pressure amplification, AMPPP , under control (Left, y=83.78x+113, r2=0.72,

p<0.0001) and arm raised conditions (Right, y=84.37x+112.7, r2=71) in the upper limb.

On the average one hundred iterations were required by the model to obtain a regression

coefficient of R2=0.98±0.02 for each pair of pressure pulse (Carotid and Radial). The

summary data for model estimated elastances, wall viscosities, reflection coefficients

and time constants are given in Table 8.4 for each transmural pressure.

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CONTROL

TP (mmHg)

Em (x106 ) (Dyne/cm2)

θm (Degrees)

Γm (Units)

τm (Seconds)

-100.00 1.21±0.06 23.69±1.88 0.53±0.05 0.13±0.01 -80.00 1.25±0.05 23.88±2.24 0.43±0.08 0.15±0.02 -60.00 1.17±0.07 28.03±2.90 0.51±0.06 0.18±0.03 -40.00 1.19±0.07 26.86±1.86 0.43±0.06 0.15±0.02 -20.00 1.31±0.11 24.65±2.58 0.40±0.05 0.17±0.02 0.00 1.14±0.07 31.94±1.95 0.36±0.04 0.23±0.02 20.00 1.20±0.08 28.61±2.18 0.34±0.05 0.20±0.02 40.00 1.12±0.06 28.90±2.54 0.31±0.04 0.20±0.01 60.00 1.13±0.08 24.31±1.64 0.40±0.05 0.18±0.03

ARM RAISED

TP (mmHg)

Em (x106 ) (Dyne/cm2)

θm (Degrees)

Γm (Units)

τm (Seconds)

-100.00 1.16±0.06 24.93±2.17 0.57±0.05 0.15±0.02 -80.00 1.16±0.06 26.15±2.07 0.51±0.05 0.17±0.02 -60.00 1.16±0.08 27.35±2.34 0.42±0.07 0.17±0.02 -40.00 1.12±0.07 27.60±1.79 0.45±0.05 0.17±0.02 -20.00 1.17±0.09 28.01±2.21 0.47±0.05 0.18±0.02 0.00 1.11±0.07 29.48±1.84 0.37±0.05 0.18±0.02 20.00 1.12±0.08 29.59±2.03 0.36±0.03 0.21±0.02 40.00 0.99±0.06 33.35±2.02 0.40±0.03 0.22±0.02 60.00 1.15±0.08 25.30±1.65 0.37±0.05 0.16±0.02

Table 8.4 Model parameters, Em , θm , Γm and τm estimated from the pressure waves recorded in the radial artery and in the carotid artery (Mean±SEM). Correlation coefficients, R2, for each fit is in excess of 0.98.

Although the estimated reflection coefficients Γm were similar, the elastance Em values

were lower and the wall viscosity θm values were higher than those calculated in

Chapter 7. Similarly, the time constants τm were higher than those in the previous

investigation (Chapter 7). Raising the arm had little effect on each of value

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investigated. However, in each of the transmural pressures the pulse pressure

amplification is linearly related to model parameters:

for control

mmmampPP θτ ×−×−Γ×+= 01.002.120.038.1 , R2=0.66, p<0.0001

for arm raised

mmmampPP θτ ×−×−Γ×+= 01.083.022.039.1 , R2=0.75, p<0.0001

8.3 DISCUSSION

Often systolic pressure and/or rate of rise in pressure in the upper limb is taken as an

index of myocardial contractility, and pulse pressure as an index of stroke volume (

Pauca et al 1989). However, this study indicates that these indices are definitely

dependent on the distal bed properties specifically the palm vasculature. Previous

studies found that systolic pressure in the radial artery is dependent on heart rate

(O'Rourke 1970). The compression applied to the radial artery distal to the cannulation

also effects the pressure pulse amplification (Pauca et al 1989). This second effect is

not surprising in the light of earlier studies (O'Rourke and Taylor 1967a; Van den Boss

et al 1977; Newman et al 1979). The occlusion of any artery creates a positive

reflection site which is close to unity which in turn augments the reflected waves.

This present study is significantly different from earlier studies. Since the procedure

involved partial occlusion of the arteries under the cuff, it was possible to create a whole

range of terminal impedance mismatches. Intuitively, one would expect this mismatch

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to be proportional to cuff pressure (higher cuff pressures introducing higher

mismatches). However, the outcome is more complex. The pulse pressure reduces

initially with increase in compression (Figure 8.3). It then increases beyond a minimal

point with further increase in the compression of palm vasculature.

These seemingly paradoxical findings can be explained, however, if one uses transmural

pressure rather than the cuff pressure to express the relation (Figure 8.9). It follows that

at some intermediate cuff pressure the transmural pressure approximates to zero. At this

small positive transmural pressure the wall stress approximates to zero and the vessel

wall becomes more compliant (point B in Figure 8.9) due to non-linear elastance of the

wall (Gow and Taylor 1968). If a modified Windkessel for the palm vasculature is

assumed first order approximation, increased compliance decreases the impedance of

this bed. Given that the proximal properties of the radial artery remains unchanged as

there is no change in pulse wave velocity, the decrease in terminal impedance leads to a

decrease in impedance mismatch. By equation 2.93 this decreases the reflection

coefficient at this point. This will in turn reduces the intensity of reflected waves which

often constitute a major part of the systolic upstroke.

With further increase in external pressure however, the wall stress becomes compressive

and both the intraluminal area of the vessels in the bed and its compliance decreases

(point C in Figure 8.9). As a result, the impedance of the bed will increase both due to

decrease in parent-daughter diameter ratio (Taylor 1966b) and the increased Windkessel

impedance. This in turn increases reflected waves and augments pulse pressure (Figure

8.5).

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Figure 8.9 The non-linear transmural pressure-volume curve of the arterial bed under the hand cuff. The point A represents the normal distending pressure of the bed where the compliance is lower and thus the reflection coefficient is higher. Increase in cuff pressure decreases the transmural pressure reaching to zero at point B where the bed becomes extremely compliant and the reflection coefficient is at a minimum. Further increases in cuff pressure decrease the volume further, asymptotically approaching a constant value. At those pressures (point C) the bed is again less compliant and thus higher reflections can be expected

These conclusions are confirmed by the estimated Windkessel time constant, τm, which

relates the compliant element to the resistive element (Chapter 5). There was a

significant increase in time constant as determined by the model accompanied by a

decrease in reflection coefficient and increase in wall viscosity at zero transmural

pressure levels (Table 8.4). While deriving this parameter, it was assumed that the

elastic properties of the wall are exponentially related to strain and the effective

resistance of the lumen is constant. Therefore, when all these changes are taken into

account it becomes apparent that the actual change in compliance is underestimated by

this technique.

Interestingly, the pulse pressure attained by maximum compression or vascular

occlusion, is close to the values under control conditions. This indicates that the

impedance of palmar bed is close to the values attained by maximum vasoconstriction.

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Since the pressure volume curve is S-shaped (Figure 8.9) and under normal

physiological conditions the arteries are positively stressed, the compliance values are

close to those obtained by vascular occlusion (point A versus C in Figure 8.9). Thus, by

altering the transmural pressure in the palm one can scan the pressure volume (stress-

strain) relation of the vessels involved where the effect will be visible from the changes

in pulse pressure or systolic rate of rise.

Under normal circumstances however, pulse pressure, systolic pressure or rate of rise

may not be a quantifiable parameter for estimation of reflection coefficient. This is

because factors other than the vessel wall properties affect these parameters, such as

stroke volume and systemic input impedance. One way to overcome this difficulty is by

expressing the relation in terms of the pulse pressure amplification (Equation 8.3). As

expected, this technique yields similar results (Figure 8.4 vs Figure 8.5).

The phase velocity plot spectrum is similar to the input impedance spectrum and takes

into account the presence of reflected waves (Chapter 2 and Figures 8.3-4). Since this

technique does not require calibrated pressure waveforms it forms an ideal case for non-

invasive systems that are capable of recording pulse waveforms. This study shows the

applicability of this technique to estimate the intensity of wave reflection using

tonometers (Figure 8.6 and Figure 8.7). Estimation of the zero frequency intercept was

pursued in this study to account for changes in heart rate in different individual subjects.

Obviously one could derive reflection coefficient estimates at each frequency using

equation 8.2 but this was necessary for comparisons to be made amongst different

individual subjects with different heart rates. Pulse pressure amplification and the

reflection coefficient estimates obtained this way are closely related (Figure 8.8), an

indication that they are compatible. An inspection of the regression equation reveals

that at the unity value of reflection coefficient estimate, the pulse pressure amplification

reaches 196% which is approximately the theoretical limit of 200%.

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The values obtained for reflection coefficient estimates by this technique (Table 8.3 and

Figure 8.7) are far less than the ones reported in the literature (O'Rourke 1967). This

difference can be explained in terms of the vasoactive state of the patient group that was

studied and the assumptions made for the equation employed. Nearly all the patients

were receiving vasoactive drugs, nitroglycerine (5 mg/hour) and sodium nitroprusside (5

mg/hour) as a routine after the operation, through which the reflections are expected to

be lower (O'Rourke et al 1992). Although they were allowed to warm up (37.1°C)

before performing the experiments (they were investigated at an average of 6:08 hours

after surgery), they were still hypotensive (MAP = 83 mmHg) and vasodilated, pulse

wave velocity estimates for the arm in this patient population (Table 8.3) were much

lower than the ones reported in the literature (Avolio et al 1983b, 1985) (780 cm/sec vs

1060 cm/sec).

Methodological errors in estimation might also be introduced due to the shape of the

phase velocity spectra and the sampling interval. It was assumed that the phase

velocities are proportional to the frequency thus a linear extrapolation technique was

used for the whole range of frequencies. In fact a linear relationship is more relevant for

these frequencies. However, when exponential extrapolation rather than linear

extrapolation was attempted for this purpose, the reflection coefficient estimates

differed as little as 20%. Another problem might arise due to the presence of minima in

the phase velocity spectrum. The extrapolation technique would produce errors if the

first and second harmonics were separated with a minimum value. This would

introduce an under estimation of the zero intercept. Since the patients recruited for this

study had often a higher heart rate, 93±1 beats/min (Table 8.1), the second harmonic

was at 3.1 Hz. The position of the minimum estimated using quarter length formula

(Equation 2.138) yields a value of 3.3 Hz which is not significantly higher than the

second harmonic. Nevertheless, even under these restrictions equation 8.2 still yields

values of reflection coefficient that change in the appropriate direction with pulse

pressure amplification.

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The purpose of raising the arm was to decrease the proximal characteristic impedance

and thus increase reflected energy. This manoeuvre caused a minor decrease in pulse

wave velocity in the proximal vessels, which did not reach statistical significance (60

cm/sec, p = NS). Moreover it is a recognised method to decrease the mean pressure in

the distal bed (Warnes et al 1983). This obviously altered the entire relation between

the proximal and distal vessels to an unknown point. Yet, when the cuff inflation was

introduced and results were expressed in terms of transmural pressure, compensation

was expected for the effect of arm raising on properties of distal vessels. However,

there was a slight but consistent decrease in apparent reflection coefficient which

warrants further study (Figure 8.8).

The general arterial model applied to the upper limb circulation was used to simulate

experimental conditions. The model predicted values for reflection coefficients that

were comparable with those estimated from phase velocity plots but were often higher.

Furthermore it was also possible to simulate changes in the compliance of the distal

beds by the time constant parameter of the model. The correspondence between the

model derived and the actual radial waveform was also satisfactory. The model

parameters explained 98 per cent of the features of the radial pressure waveforms. The

remaining 2 per cent could be introduced by experimental noise or the arbitrarily chosen

diameters and lengths or by the inherent assumptions made in the mathematical

abstraction. Nevertheless, the iteration technique helped to estimate parameters of

physiological significance within a reasonable iteration time. This technique can

therefore be extended to real situations.

In general, the pressure waves travelling in the upper limb are reflected back due to a

major reflecting site residing in the palm. This bed often exhibits a higher load for the

radial artery under extreme conditions, vasodilation and maximum compliance, thus

contributing to the formation of the upstroke of the pressure wave. This property of this

specific circulation, that is mostly a positive reflection coefficient led previous

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investigators to believe that the upper limb behaves like a second order system (Warner

1957; Gardner 1982; Chadwick et al 1986; Schwid et al 1987). For this kind of

approximation to be valid, the terminal load has to be higher than the proximal load. As

shown in the present study, this condition is satisfied even under the extreme

vasodilatory state of these patients. Yet, when there is a change in the transmural

pressure, for example by raising or lowering the arm, the terminal load changes its

characteristic, ie. the lumped parameters of the second order system. The damping

coefficient and resonant frequency of the second order system will reflect these changes.

However, it will not convey the information necessary to interpret the actual

phenomenon.

The transmission line analogue, however, explains these changes correctly. It will point

out the changes in the terminal load as deduced from the reflection coefficient. When

properly estimated from modelling studies, this terminal load characteristic opens new

ways of assessing passive changes from the active ones. Since the transmission line

analogue does not require the absolute amplitude to determine phase velocities, only

high fidelity recording systems are required. Tonometry or other high-fidelity non-

invasive techniques are ideal for this purpose.

This study also implies that one can predict the mean arterial pressure in the radial artery

by a simple manoeuvre as analogous to the oscillometric pressure measurement

technique (Posey et al 1969, Ramsey 1976). After plotting the pulse pressure in relation

to the cuff pressure, mean arterial pressure should correspond to the value of cuff

pressure when the pulse pressure becomes a minimum. When it is not possible to obtain

absolute pulse pressure, this technique can use the relative upstroke of the pressure

waveform. This index should also decrease when cuff pressure is at or close to the

mean arterial pressure.

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It is also possible to plot stress-strain relation of the arterial bed under the cuff as given

in Figure 8.6, by plotting the reflection coefficient against transmural pressure or cuff

pressure. This will then display the terminal impedance mismatch that can be converted

to the elastance of the bed by simple assumptions. Using this technique, one can then

easily document the effects of drugs and aging on this specific vascular bed.

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EFFECTS OF VASODILATORS ON UPPER L IMB WAVE PROPAGATION PROPERTIES Vasoactive agents alter the systemic input impedance and thus the left ventricular

afterload in a favourable direction (Nichols and O'Rourke 1990). They affect different

parts of the vasculature both at the level of conduit or cushion arteries, the arterioles or

the veins. These drugs are designed to induce changes on the vessel wall by directly or

indirectly acting on the smooth muscle. By causing changes in vessel wall properties -

diameter, thickness and elastance - these drugs may alter the wave propagation

properties of the vessel. This can become extremely important if they interfere with the

pressure wave transmission in the upper limb where blood pressure is routinely

monitored. In current clinical practice, the implications of wave transmission in the

upper limb are generally overlooked not only under normal conditions (see Chapter 7)

but also under the effect of these drugs. Recent reports on the discrepancies between the

CHAPTER 9

186

central and upper limb pressure measurements before and after the administration of two

very potent vasodilators, nitroglycerine and sodium nitroprusside (Kelly et al 1990b,

Simkus and Fitchett 1990) do not deal with this phenomenon. Although these

investigators attempted to explain these observations by differences in the wave shape in

the arteries, neither a comprehensive explanation of the phenomenon nor a description

of the effects of these drugs on the upper limb wave propagation properties is provided.

In the following experiments, an attempt will be made to quantify the pressure wave

transmission in the upper limb under control conditions and after using nitroglycerine

(NTG) and sodium nitroprusside (SNP). Using techniques developed in previous

chapters parameters of and upper limb model will be calculated. Comparisons will then

be made between the experimental findings and the model predictions made earlier in

Chapter 5.

9.1 METHODS

9.1.1 Nitroglycerine (NTG)

Since it was possible to obtain the original experimental data that appeared in the

original publication (Kelly et al 1990b) thanks to the courtesy of the authors, no attempt

was made to perform the same experiments under identical conditions. The data

comprised the patients' age, sex, and morphometric measurements, digitised recordings

of the invasive micromanometric pressure (the ascending aorta and brachial artery) and

the non-invasive tonometric (Radial ) pressure waves before and after NTG. Calibration

signals were also available for all signals.

187

Analysis of the upper limb transmission properties was performed on averaged

waveforms by relating the harmonic components of the ascending aortic - brachial

BAAAH − and ascending aortic - radial RAAAH − pairs. The transfer functions were

expressed in relation to frequency in modulus and phase format. Because of the

differences in heart rates, the transfer function values were not obtained at similar

frequencies. Thus, to obtain representative spectra for the control and for the NTG

administration, the spectra were clustered into bins of fundamental harmonics derived

from mean heart rates. Averaging the contents of these bins and performing linear

interpolations between these mean values yielded representative spectra for each case.

Comparisons using Student's t-test were also performed on these bins to estimate the

degree of variation due to the effect of NTG.

A model of the upper limb arterial vasculature for each patient was also constructed to

estimate the physical parameters responsible for the measured pulses. This was done by

employing the same approach as was described in Chapter 7. Briefly, the reduced

model of the upper limb arterial system was constructed for each patient (Table 7.1).

Input pressure waves were supplied to this model was that recorded in the central artery

(ascending aorta). The model was then iterated with different elastance, Em , wall

viscosity, θm , reflection coefficient, Γm , and time constant, τm , values to yield a best

fit between the waveform derived from the model and measured distal pressure

waveform (brachial or radial). Values for wall elastance and terminal reflection

coefficients corresponding to the minimum of the sum square difference between

calculated and measured distal waveform are regarded as the model parameters.

An estimate of the complex reflection coefficient is derived from the phase velocities

similar to that described in Chapter 7. The resonant frequency of the upper limb is also

estimated using the technique given in Chapter 7. The effect of NTG on these

parameters calculated from the transfer function and from the model were analysed.

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9.1.2 Sodium Nitroprusside (SNP)

The data for this agent were obtained from the original publication of the study (Simkus

and Fitchett 1990). The aortic and radial tracings in the publication were digitised using

a digitising tablet (Summasketch, Summagraphic). Since this process involves a noise

level higher than expected due to digitisation by hand, it is taken not to include the noisy

part of the waveform. This was done by excluding the frequency components that fall

below a noise threshold which was estimated by averaging components 9-16 hz.

Because of the absence of the morphometric data for the subjects reported in this

publication, a constant length, 62 cm, was used for ascending aorta to radial artery. This

figure was subsequently used in the modelling part of the study. Apart from these

changes made, the analysis of the waveforms and modelling of the upper limb

proceeded as in the case of NTG.

9.1.3 Statistics

Differences from the control state are analysed using paired t-test. A p value of less than

0.05 is accepted statistically significant. Data is presented as mean ± standard error of

mean unless otherwise stated.

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9.2 RESULTS

9.2.1 Nitroglycerine

Table 9.1 summarises the parameters that were calculated and derived from the model

for each patient. Since neither the estimated values of time constant nor the wall

viscosity are changed, these values are not reported.

Control

NTG

Patient. No

fn (hz)

Γ0+ (Units)

Em (x106 dyne/cm2)

Γm (Units)

fn (hz)

Γ0+ (Units)

Em (x106 dyne/cm2)

Γm (Units)

1 2.58 0.22 1.12 0.56 1.50 0.24 * * 2 1.79 0.35 1.01 0.78 3.76 0.77 0.85 0.67 3 2.48 0.52 1.35 0.67 1.98 0.43 0.78 0.78 4 1.95 0.38 0.95 0.67 2.85 0.35 1.15 0.84 5 2.12 0.63 1.24 0.84 2.85 0.80 0.82 0.73 6 2.42 0.41 * * 2.58 0.58 0.78 0.84 7 2.53 0.40 1.95 0.62 2.31 0.47 1.11 0.84 8 2.36 0.45 1.12 0.78 3.10 0.91 1.33 0.78 9 2.14 0.71 1.31 0.73 2.33 0.56 0.75 0.73 10 2.50 0.54 1.02 0.73 2.15 0.39 0.86 0.84 11 3.40 0.66 1.10 0.73 2.47 0.45 1.21 0.78 12 2.63 0.49 1.49 0.73 2.59 0.01 0.78 0.62 13 2.18 -0.05 2.49 0.73 2.48 0.36 1.31 0.67 Mean ± SEM

2.39 ± 0.11

0.44 ± 0.06

1.35 ± 0.13 0.71 ± 0.22

2.36 ± 0.15

0.49 ± 0.07

0.98 ± 0.07# 0.76 ± 0.22

Table 9.1 Calculated resonant frequencies, fn and reflection coefficient estimates, Γ0+, and model derived elastances, Em , and reflection coefficients, Γm , under control and under NTG conditions for ascending aorta to radial system. The missing (*) values denote cases where R2 is less then 0.94. (# p<0.05).

The model-derived parameters for ascending aorta to radial artery system demonstrated

a slight reduction in the estimated elastances (Figure 9.1 Left). This change is in line

with decrease in pulse wave velocity apparent from phase of the transfer function. The

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reflection coefficient estimate either by the model or by the analysis of transfer function

pointed towards a consistent but statistically insignificant increase (Figure 9.1 Left).

NTG

Em Γm

Figure 9.1 The model (ascending aorta to radial) estimated elastance (left) and the reflection coefficients (right) of the group of patients before (open bars) and after NTG (closed bars). Bars indicate 1 SEM).

The grouped, binned and average upper limb transfer function between ascending aorta

to brachial artery BAAAH − (Figure 9.2, Left) indicates a peaking similar to one predicted

by the model (Chapter 5). NTG (open boxes, dotted line) alters the modulus of this

transfer function by an overall reduction of the amplification of the transfer function.

Since he major reflection site in the upper limb (ie palm vasculature, Chapter 8) is distal

to the brachial arterial site the influence of the impedance of the proximal branches will

be more apparent in this transfer function (see below). There is also a slight reduction

in the frequency where the peak modulus occurs. As expected, the upper limb transfer

function RAAAH − (Figure 9.2, Right) was similar to the one predicted in Chapter 5. and

subsequently presented in Chapter 7. It is steeper with frequency for the phase and

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lower in frequency for the maximum amplitude with respect to BAAAH − . This is

predictable on the basis of greater distance for the radial artery (equation 2.138). NTG

does not considerably alter the modulus of this transfer function in contrast to the

BAAAH − . Similar to RAAAH − there is a slight steepening in the phase indicating a

decrease in pulse wave velocity.

HAA BA− HAA RA−

Figure 9.2 The transfer function HAA BA− (Left) and HAA RA− (Right) of the entire

study group under control conditions (circles and solid line) and after administration of NTG (boxes and dotted line). Bars represent ±1 SEM and dash dot line represents no amplification.

9.2.1 Sodium Nitroprusside

Table 9.2 summarises the model derived and calculated parameters for each patient

under control and after administration of SNP. There was no change in any parameter

derived from the model.

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When transfer function, RAAAH − , data is pooled and averaged for the control case

(Figure 9.3, Solid line), it is apparent that this population is not very much different

from that in Figure 9.1 (Right, Solid line). It has the same prominent peak at 4 Hz. The

effect of SNP on the propagation properties of the upper limb is similar to that of NTG

(Figure 9.3, dotted line).

Control

SNP

Patient No

fn (hz)

Γ0+ (Units)

Em (x106 dyne/cm2)

Γm (Units)

fn (hz)

Γ0+ (Units)

Em (x106 dyne/cm2)

Γm (Units)

1 3.63 0.73 2.50 0.73 2.49 0.66 1.55 0.78 2 2.49 0.18 1.18 0.56 2.39 0.05 0.92 0.56 3 5.38 0.08 * * 4.98 0.71 2.87 0.78 4 2.74 0.52 1.32 0.62 2.34 0.51 0.88 0.56 5 2.20 0.47 0.85 0.73 1.95 0.51 0.72 0.67 6 2.44 0.57 * * 1.84 0.53 0.79 0.84 7 3.50 0.28 2.51 0.56 3.52 0.45 2.54 0.56 8 3.01 0.38 1.32 0.56 3.09 0.31 1.25 0.62 9 2.85 0.50 1.54 0.67 2.32 0.46 1.02 0.62 10 3.20 0.59 2.10 0.73 2.35 0.49 1.21 0.73 Mean ± SEM

3.14 ± 0.29

0.43 ± 0.06

1.67 ± 0.22 0.65 ± 0.03

2.73 ± 0.30

0.47 ± 0.06

1.38 ± 0.24# 0.67 ± 0.03

Table 9.2 Calculated Resonant frequencies, fn and Reflection Coefficient Estimates, Γ0+, and model derived Elastances, Em , and Reflection coefficients, Γm , under Control and under SNP. The missing values (*) denotes cases where R2 is less then 0.94.

The corresponding elastances and reflection coefficients when plotted as bar graphs are

given in Figure 9.4, indicating that there is no significant change in these parameters

which can be ascribed to SNP.

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HAA RA−

.

Figure 9.3 The transfer function RAAAH − of the entire study group under control

conditions (open circles, solid line) and after administration of SNP (closed circles, dotted line). Bars represent ± 1 SEM and dash dot line represents no amplification.

Em Γm

Figure 9.4 The ascending aortic-radial model estimated elastance (left) and the reflection coefficients (right) of the group of patients before (open bars) and after (closed bars) SNP (Bars indicate ± 1 SEM)

194

9.3 DISCUSSION

The data presented here are important in its implications on two fronts. Firstly, in

different patient populations the transfer function between the central and upper limb

measurement sites is relatively similar. This similarity is consistent with the model

predictions (Chapter 5) and with the other studies (Chapters 7 and 8). This is most

relevant to clinical situations where the consistency is quite important. Since blood

pressure is mostly measured in the upper limb, these findings imply that the errors in

measurement due to different populations will be minor.

Secondly, the relative stability of the upper limb propagation properties under two very

potent drugs, NTG and SNP, allow one to consider the possibility of using these transfer

functions as constant terms under these conditions.

However, it is rather surprising to find such a consistency in transfer functions during

vasodilation. It may be due to alteration of proximal and distal vessel properties in the

same ratio as evidenced from little change in reflection coefficient. If the changes in

wall thickness to diameter ratio caused by these drugs are accompanied by equal and

opposite changes in the wall elastance by smooth muscle relaxation, then, no change in

the reflection coefficient and the proximal elastances is observed. Actually, Cox (1978)

observed this with smooth muscle activation where changes in wall elastance were

offset by changes in wall thickness diameter ratio. Although it seems paradoxical, it has

been suggested that this might be partly due to arrangement of collagen and elastin in

the vessel wall (Nichols and O'Rourke 1990). Activation of smooth muscle reduces the

diameter but relieves tension in the series collagenous element, thus unloading it. When

compared to the original diameter, this effect can be interpreted as a reduction in

elastance (Gow 1972). In the model employed however, no allowance was made neither

for this non-linear behaviour nor for the alterations in the wall geometry. Thus, any

change caused by the model as changes in elastance might be due to a combination of all

195

these parameters. For example, model used in analysing the effects of SNP on wall

elastance might be obscured by changes in wall thickness to diameter ratio.

The simulation results displayed a reduction in elastance by NTG but not with SNP

although the trend was similar. The reflection coefficient estimates derived from the

model did not decrease. In fact there was a tendency to increase indicating no change in

the terminal vessel properties. These surprising findings might be interpreted as changes

on the passive rather than the active mechanical properties of the vessel wall. Indeed, in

both cases the mean blood pressure was reduced slightly but significantly (from 98±3

mmHg to 90±3 by NTG, from 80±3 to 69±2 mmHg by SNP). Yet, based on the

reflection coefficient estimates, both through model and through transfer functions,

these findings suggest that these vasoactive drugs do not alter the mechanical properties

of the upper limb. Therefore, it is possible that any observed changes in the wave shape

are due to changes in the input to the upper limb system. It is further possible that these

drugs alter the central circulation more than the upper limb one. Based on these

findings, the effect of the drugs should be investigated in the central circulation,

probably trunk area, rather than the upper limb.

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EFFECT OF THE SPLANCHNIC CIRCULATION ON THE FORMATION OF THE ARTERIAL PULSE The arterial model presented in Chapter 5 suggests that reflections from the lower part

of the aortic trunk play an important role in the formation of the central pressure

waveform. It also suggests that vascular beds in the lower trunk, ie mainly the

splanchnic bed, have considerable influence on systemic input impedance as opposed to

beds of the extremities.

Experimental evidence on vasodilators, NTG and SNP, on upper limb arterial properties

(Chapter 9) suggested that these drugs have little effect on modification of properties of

the brachial-radial arterial system. However, these drugs alter the pressure waveforms

dramatically (Yaginuma et al 1986; Kelly et al 1990b, Simkus and Fitchett 1990)

CHAPTER 10

197

without necessarily altering the systolic pressure measured in the upper limb. These

findings and the model simulations suggest that the circulation in the trunk area,

especially splanchnic, has a special affinity to vasoactive drugs. By the action of these

drugs, substantial changes in the arterial waveforms can be induced which can be

detected in the upper limb.

It is known that the blood flow to the digestive organs increases after ingestion of

glucose (Chou 1983, Jern 1991). Thus, ingestion of glucose can be the convenient

stimulant which might reproduce the action of vasoactive drugs on the arterial

waveforms. In this study, this hypothesis was investigated by administering glucose

orally to human subjects to cause vasodilation in the splanchnic circulation (Chou 1983,

Takagi et al 1988, Kato et al 1989, Jern 1991). These caused reductions in the reflected

waves, as manifested by reductions in the augmentation index, while no changes in the

arterial properties of upper and lower limbs were observed. Subsequent implementation

of a reduced model of the systemic circulation suggested that vasodilation of the

splanchnic circulation, induced either by vasodilators or by glucose, reduces the

reflected waves from this circulation, thereby altering the oscillatory load as seen from

the left ventricle.

10.1 METHODS

10.1.1 Subjects

Ten normal healthy subjects participated in this study. None had a history of myocardial

infarction, stroke or diabetes. The characteristics of the study group are given in Table

10.1. The nature, purpose and risks of the study were explained to the subjects and a

198

signed consent form was obtained from each. The study was approved by the Research

Ethics Committee of St Vincent's Hospital.

Mean±SEM Range Age 43.6 ± 5.15 28 - 77 Sex 5 Male 5 Female Weight (kg) 62 ± 3.52 40 - 78 Height (cm) 165 ± 2.27 155 - 181

Table 10.1 The clinical characteristic of the patients recruited for the study.

10.1.2 Protocol

Experiments were performed in the morning in the post-absorptive state after an

overnight fast (10-12 hours). Each subject was studied on two separate occasions within

a period of two weeks. On each occasion they ingested either glucose (75g glucose + 30

ml of cordial + 300 ml of water) or placebo (30 ml cordial + 300 ml water). The

subjects were studied in a quiet, dimly lit and well-ventilated room whilst in the supine

position. They were asked to relax and avoid unnecessary communication with the

examiners whilst they were allowed to establish a semi-recumbent position during

ingestion of the solution. To eliminate observer and subject bias, the solutions were

randomised in a double-blind fashion. At the end of the analysis of the recordings, the

code assigned to the solutions was broken to allow further analysis.

10.1.3 Measurements and Data Analysis

Electrocardiogram (3-leads) and finger blood pressure (FINAPRES, Ohmeda) were

recorded continuously throughout the study. After instrumentation, the examiners (3 in

all) were allowed to practise non-invasive recordings at carotid, both radial and femoral,

199

and dorsalis pedis sites, to gain familiarity with the arterial site. This practice period

normally lasted for 15 minutes before the test. The subjects were asked to maintain this

relaxed position for at least 30 seconds to allow for recordings to be done without any

interference. After a training period of both patient and examiner, a baseline period of

30 minutes was spent while the tonometric recordings were being taken simultaneously

every five minute. At the end of this baseline period, the solution was administered and

further recordings were taken at each five-minute interval which lasted for sixty

minutes. Each tonometric recording throughout the study was concluded with blood

pressures measured in the arm with a semi-automated oscillometric device (Dynapulse,

Pulsemetric Inc, CA). At the end of the study, the distances from the sub-sternal notch

to the recording sites (carotid, radial, femoral and dorsalis pedis) were recorded to allow

for pulse wave velocity calculations.

The data were digitised on-line at a rate of 128 samples/sec, with a 12-bit A/D converter

(Data Translation, DT 2801-A) using an IBM-AT compatible computer. The data were

then transferred to a Write-Once-Read-Many (WORM, Maxtor 800S) drive for

permanent storage.

Data were analysed off-line after averaging segments of 20 second length of each

recording to obtain representative waveforms for each recording session. Blood

pressure waveforms were calibrated by setting mean and diastolic pressures of each non-

invasive waveform to that of the FINAPRES recordings (see Chapter 5). Pulse wave

velocities of the arm (carotid-radial), trunk (carotid-femoral) and leg (femoral-dorsalis

pedis) were calculated by foot-to-foot estimate method. The augmentation indices in the

carotid and radial artery were calculated to obtain an index of wave reflection. Transfer

functions between the carotid-radial, carotid-femoral and femoral-dorsalis pedis sites

were calculated to estimate the segmental propagation properties. All data were then

grouped into bins of 15 minutes which were later averaged and analysed.

200

10.1.4 Statistics

Statistical analysis was performed between derived indices by 2-way analysis of

variance (ANOVA) for repeated measures with subjects as the random factor. Data

averaged in bins of 15 minutes and expressed as mean ± SEM, unless otherwise noted.

10.2 RESULTS

Although, there was no significant difference between the calculated baseline values one

agent was consistently found to reduce the augmentation index in the carotid artery

(Figure 10.1).

PLACEBO GLUCOSE

Baseline Ingestion

1 Sec

Baseline Ingestion

1 Sec

Figure 10.1 A representative carotid tracing obtained from one patient illustrating the experimental protocol and the changes in pressure wave shapes. After a run in period of 30 minutes either placebo (Left Panel) or Glucose (Right Panel) is ingested at time zero (Arrow). Recordings are made at each 5 minute and subsequently averaged over 15 minute periods. Note the decrease in augmentation of the late systolic peak in carotid waveform after glucose ingestion.

201

This effect of the agent started 15 minutes after the ingestion and lasted longer than 60

minutes. When the code was broken, this was found to be glucose. Under the action of

glucose, the second peaks in both radial and carotid arteries were reduced in amplitude,

which was not reproduced by placebo ingestion. The placebo, however, increased the

mean blood pressure without affecting any other features of the pulses. The cause of

this increase in mean blood pressure was not investigated.

When pressure data from all arteries were binned and pooled, it was possible to observe

the effect of glucose on different arterial pulses (Figure 10.2, Table 10.2). The

secondary fluctuations in the central and upper limb arterial pulses decreased in

amplitude more than the lower limb pressure waveforms. Yet, the foot to foot delays in

all segments remained identical to the baseline indicating no decrease in pulse wave

velocities.

CONTROL GLUCOSE

Figure 10.2 Averaged pressure tracings obtained at baseline (Left) and after glucose ingestion in a subject (Right). There was a significant reduction in the central pressure augmentation, accompanied with reductions in pulse pressure in all arteries except the radial.

202

Accompanying these, there was a slight but significant increase in the heart rate (62.0 ±

2.6 beat/min) to (66.9 ± 3.0 beat/min) (p<0.01) at 30 minutes after ingestion of glucose,

but not with placebo (58.9 ± 2.4 vs 58.4 ± 2.3, p = NS). Blood pressure (systolic,

diastolic, mean) did not change with glucose but did so with placebo (mean BP 90.9 ± 3

mmHg to 98.9 ± 3.1 mmHg) (p<0.01) (Figure 10.3, Left).

Control Glucose p value Systolic BP (mmHg) 135.31±5.76 142.04±6.17 NS Diastolic BP (mmHg) 71.52±3.22 70.53±3.14 NS Mean BP (mmHg) 91.58± 3.06 91.46± 3.32 NS Heart Rate (beats/min) 62.0± 2.6 66.9 ± 3.0 <0.01 Carotid AI (%) 112.3 ± 6.3 102.3 ± 7.4 <0.01 Radial AI (%) 65.2 ± 5.9 59.4 ± 6.9 <0.05 Arm-Pwv (cm/sec) 1039.73± 45.9 1050± 43.41 NS Trunk-Pwv (cm/sec) 811 ± 79.15 741.45± 44.28 NS Leg-Pwv (cm/sec) 1016.78± 42.16 1004.14± 37.51 NS Γ0+ -Arm 0.35± 0.05 0.37± 0.03 NS Γ0+ -Trunk 0.41± 0.04 0.35± 0.06 <0.05 Γ0+ -Leg 0.57± 0.05 0.58± 0.05 NS Γs - model 0.67 ± 0.03 0.42±0.05 <0.001 Γl - model 0.53±0.08 0.68±0.05 NS

Table 10.1 Table of calculated parameters (mean ± SEM) during control and 30 minutes after ingestion of glucose. Γ0+ represents the reflection coefficient estimated from the phase velocity, while Γs and Γl are derived from iteration of a reduced model and represent the reflection coefficients at the splanchnic and lower limb circulations respectively (see text for more explanation).

Although segmental pulse wave velocities are not altered with glucose ingestion, both

carotid and radial augmentation decreased (Figure 10.4, Right). Carotid augmentation

decreased from 112.3 ± 6.3% to 102.3 ± 7.4%, p<0.01; radial decreased from 65.2 ± 5.9

to 59.4 ± 6.9, p<0.05. There was a slight increase in carotid augmentation index after

placebo ingestion but it did not reach statistical significance (114.2 ± 6.2% to 117.6 ±

6.8%, p = NS).

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Mean BP Heart Rate

* ** *

* **

Figure 10.3 Changes in mean blood pressure (Left ) and heart rate (Right) introduced by placebo (closed symbols, solid line) and by glucose ingestion (open symbols, dotted line). Arrow represents the time of ingestion. Bar represents Mean ± SEM, * p<0.01.

Carotid AI Radial AI

# * *

##

Figure 10.4 Changes in carotid augmentation index (Left) and radial augmentation index (Right) introduced by placebo (closed symbols, solid line) and by glucose ingestion (open symbols, dotted line). Arrow represents the time of ingestion. Bar represents Mean ± SEM, * p<0.01, # p<0.05.

204

The transfer function in the segments investigated did not differ for the arm and the leg

as expected from the modelling studies, yet there was a significant change in the trunk

segment (Figure 10.5). The peak modulus of the transfer function decreases and the

phase velocity showed lesser amplification at lower frequencies indicating decrease in

reflected waves. Since the foot-to-foot pulse wave velocities remained unchanged, the

characteristic impedance of the aorta was assumed to remain the same (Chapter 9,

Nichols and O'Rourke 1990).

Amplification Phase Velocity

Figure 10.5 Changes in carotid to femoral transfer function modulus (Left) and phase velocity (Right) under control (closed symbols, solid line) and by glucose ingestion (open symbols, dotted line). Bars represents Mean ± SEM

Using phase velocity data and equation 8.4, reflection coefficients are estimated for each

compartment. There was no change in the reflection coefficient estimates obtained this

way in the upper limb and the lower limb, yet the reflections for the trunk segment

decreased significantly (0.41±0.04 to 0.35±0.06, p<0.05) (Figure 10.6, Right).

205

Γ0+

Figure 10.6 Changes introduced by ingestion of glucose on reflection coefficient estimate in the trunk segment (left) and the leg segment (right). This estimate is determined from the phase velocity data using equation III.6.4. (Bars represent Mean ± SEM, see text for more explanation).

10.3 DISCUSSION

This study demonstrates that glucose ingestion significantly lowers augmentation index

without altering the propagation properties of the upper and lower limb, as evidenced

from the transfer functions. This is accompanied by a slight but significant increase in

the heart rate, with no change in arterial pressures (systolic, diastolic and mean)

measured in the upper limb. Placebo, on the other hand, induced a slight but not

significant, increase in the augmentation index and a slight increase in the mean arterial

pressure. The changes in wave shapes and segmental properties bear resemblance to

those documented with vasodilators (Kelly et al 1990b; Simkus and Fitchett 1990).

They also support model findings in Chapter 5, where vasodilation of the trunk arteries

caused major changes in second peaks in pressure waves. The experimental findings of

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Chapter 9 are also reproduced, that is that upper limb propagation properties are not

affected by vasodilator drugs. These seemingly contradictory findings on arterial

haemodynamics can be explained by the action of ingestion of food.

It was shown previously that blood flow to the digestive organs (ie coeliac, superior

mesenteric, Moneta et al 1988; renal beds, Avashti et al 1987) increases after a meal in

humans. Because of decreases in resistances of coeliac, superior mesenteric (Takagi et

al 1988) and gastric (Kato et al 1989) beds, central peripheral resistance decreases

(Chou 1983, Jern 1991). Ingestion also affects heart rate, cardiac output, blood pressure

and left ventricular performance indices (Fagan et al 1986; Cornyn et al 1986; Kaspar et

al 1987). These changes in haemodynamics are dependent on the composition of the

meal (Moneta et al 1988; Jansen et al 1990). The ingestion of glucose causes the

maximum changes (Jern 1991; Jansen et al 1989). The changes documented in this

study due to glucose ingestion, are yet another confirmation of these early findings.

However, this present study appears to be the first to couple these changes with pulsatile

haemodynamic measurements performed non-invasively.

These changes related to ingestion of food with vasodilators are not incidental however.

Murrell, who first suggested the use of nitroglycerine as a remedy for angina pectoris,

quoted in 1879 "I took my pulse, and found it was much fuller than natural, and

considerably over 100. The pulsation was tremendous, and I could feel the beating to

the very tips of my fingers. The pen I was holding was violently jerked with every beat

of the heart" (Murrell 1879). This indicated that nitroglycerine actually increases the

heart rate with accompanying sensation of pulses in the finger tips. This effect was

corroborated in recent studies that showed that NTG affects pre-arteriolar arteries

(O'Rourke et al 1992). Surprisingly, a similar feeling can be observed after eating. In

1908, Oliver noted "Observation has shown that the ingestion of food invariably raises

the arterial pressure in the distal area (last phalanx of finger or thumb). This rise

begins as a rule, within five or ten minutes, after the inception of food and it attains its

207

maximum development of from 15 to 20 mmHg in an hour, then it declines and in the

course of from one and a half hours to two and a half hours it gradually settles down to

its minimum level" (Oliver 1908). If one replaces the word "food" by nitroglycerine, the

above quote actually describes the action of nitroglycerine on the arterial pulse as first

described by Murrell. These historical qualitative accounts are probably the first

descriptions provided for common mechanism for nitroglycerine and food on the arterial

pulse.

The changes brought about by a meal (decrease in splanchnic bed resistance, cardio-

acceleration and decrease in reflected waves) could be attributed to the action of insulin

(Jern 1991). Insulin was shown to induce vasodilation in animals (Liang et al 1982) and

in humans (Scott et al 1988) and isolated arteries in vitro studies (Alexander 1977).

However, direct injection of insulin into the brachial artery was shown not to induce

increases in the forearm blood flow in humans (Natali et al 1990) suggesting that the

action of insulin is elsewhere. It was also reported that abnormal insulin mechanism

may affect the arterial wave shape especially the latter part of the pulse (Lax and

Feinberg 1959). This study extends these early findings by describing the action of

insulin on the arterial waveform by augmentation index. The pressure waveforms in the

central arteries and this index have been shown to be related to the impedance patterns

(Murgo et al 1980b, Takazawa 1987). Higher indices are related to reflected waves that

arise from oscillations of the input impedance spectrum. It was shown that this index

decreases with vasodilators (O'Rourke et al 1992) and with certain manoeuvres (Murgo

et al 1980a), yet increases with aging (Kelly et al 1989a). This study suggests that

ingestion of glucose modifies the vasoactive state of the splanchnic circulation.

The effect of the splanchnic circulation on aortic impedance and thus pressure

waveforms are often overlooked. McDonald in his monograph (1974) pointed out this

fact by stating that "the consideration of these (organ) flows illustrates more

dramatically the functional role of the branches than do the dimensions of the branches.

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The direct line of the anatomical aorta unconsciously leads us to isolate it conceptually.

From the functional point of view a better perspective would be maintained if the aorta

was thought of as terminating at the diaphragm in a group of large branches supplying

the important organs of the abdomen". This conceptualisation of the lower body has

been ignored in the past by investigators who searched for reflection sites in the human

arterial system. Attempts were often directed towards the terminal aorta and sometimes

to the femoral artery (Nichols and O'Rourke 1990). However, evidence that the aorta

"terminates" at the level of diaphragm is mounting. It was first suggested by Alexander

(1952) who proposed the presence of a standing wave in the aorta with its node at

diaphragm level. Ryan et al (1956) attributed this to influence of vessels supplied by

the abdominal aorta to the peripheral resistance. Later, Latham et al (1985) proposed

that the major reflecting site to be at around the diaphragm level, while Ting et al (1990)

found a reduction in pulse wave velocity around this region. The modelling studies

presented in Chapter 5. also support the view that this region behaves like a functional

origin for reflected waves. It was shown that alteration in reflected waves by decreases

in reflection coefficients of vascular beds around the trunk area generates the most

profound changes in the arterial waveform and in the systemic input impedance. The

same changes could not be observed by altering the terminal loads at the limbs. This is

not accidental. The vessels perfusing the splanchnic circulation and the capacitance

vessels are shown to vasodilate more than other beds in response to SNP infusion (Risoe

et al 1990).

There is also evidence from other sources, pointing out the importance of this region in

formation of the arterial pulse. Chinese folklore, which tries to relate organ flows to the

arterial pulse (Fung 1984, Dai et al 1985, Xue et al 1989), has been shown to have some

basis (Young et al 1988, Wang et al 1991) especially under conditions of ingestion,

stress and temperature. It is possible that all these observations are related to the

alterations of reflected waves originating from this bed.

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10.3.1 Reduced Model of the Arterial System

The relative importance of the splanchnic bed on the formation of the pulse and

ascending aortic impedance warrants special consideration of this bed in interpreting

and modelling of the pulse and impedance patterns. Usage of traditional reduced

models, T-tube and two-tubes in series, does not take into account the importance of this

bed. Often the aorta has been treated in these models to extend from the heart to the

bifurcation as opposed to its functional role. Both of these models considers the lower

limb as an extension of the trunk. In reality, the aorta can be represented as a single tube

extending from the root to the diaphragm where it splits into two, one being the

splanchnic circulation, the other being the lower limb circulation. The evidence for this

is from the observation of the transfer functions obtained from the upper limb, trunk and

lower limb segments in this study. Although there was no change in the upper limb and

lower limb transfer functions, there was considerable change in the trunk reflections

(Figure 10.6). If the traditional view was true, there is no reason why there should be

changes in the transfer function in the trunk, since the terminal load, ie. lower limbs,

remains unchanged. This can only be possible if the lower limb circulation is parallel to

another circulation, the splanchnic, in which the properties are altered. Using this

interpretation of the arterial system, a reduced model was constructed where the blood

flow to upper limbs and head with respect to that in the lower limbs and splanchnic bed

is ignored (Figure 10.7).

In this model, two parallel branches simulating the splanchnic bed and the lower limbs

and which are terminated with modified Windkessels (see Chapter 4) form the

termination to the trunk element (thoracic aorta). The parameters of this configuration

can readily be estimated from morphological measurements and anatomical relations.

The length of the trunk LT, can be estimated as the distance from sub sternal notch to

the Xiphisternum. The elastance ET, can be estimated from pulse wave velocity of the

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Reduced Model of Arterial System

Γl

Γs

Trunk

Legs

Splanchnic

Figure 10.7 Reduced model of the arterial system based on the experimental findings of this study. Neglecting the flow to the head and upper limbs the trunk segment bifurcates into two at the diaphragm level one to the splanchnic circulation the other is to the legs. These elements are terminated with modified windkessel represented by complex reflection coefficients.

trunk (sub sternal notch - femoral recording site) and the aortic diameter, DT, from the

body surface area (BSA) as (Keats 1990)

BSADT 717.1= (10.1)

Wall thickness, hT, assumed to be 0.07 of the DT. The length of the splanchnic bed

tube, LS, can be estimated as the distance from Xiphisternum to Umbilicus, and of

lower limb tube, LL, from Xiphisternum to femoral artery. Elastance of lower limb, EL,

is estimated from pulse wave velocity in the leg (femoral-dorsalis pedis) by using

Moens-Korteweg equation, while that of the splanchnic bed, ES, was taken as

(ET+EL)/2. The diameters of the splanchnic element, DS, and of lower limbs, DL, are

obtained as from the distribution of cardiac output as (McDonald 1974)

TS DD 74.0= (10.2)

211

TL DD 61.0= (10.3)

while hS and hL are assumed to be 10 percent of their diameters.

By altering the terminal reflection coefficients, ΓS and ΓL (splanchnic and lower limb

respectively), the model is assumed to simulate leg exercise, baseline and ingestion.

Traditionally, model fits are checked against the synthesis of reasonable pressure

waveforms it generates. However, it has been suggested that the flow waveforms are

better templates for predicting the fit of any model to the arterial system (Beneken

1972). Since measurements made in this study do not allow a flow waveform to be

measured, the error function is defined from the synthesised flow wave instead of

comparing it to the measured wave. For this purpose, a carotid pressure waveform was

used as input to the model and an aortic flow wave was synthesised. Using this flow

wave, the error function is defined as the deviation of the diastolic portion of the flow

wave from zero. The relative flatness and zero value of ascending aortic flow waves

have been used previously for estimation of model parameters (Fry et al 1957).

When model is iterated with different values of ΓS and ΓL ,the error surface obtained

displayed always a single global minimum, (Figure 10.8). The values of ΓS and ΓL

corresponding to this minimum represent the model values where the diastolic portion

of the synthesised aortic wave is flat. Since, the iteration for the entire error surface was

time consuming, a minimum searching algorithm based on partial derivatives was

utilised to locate this point (solid circles on the error map in Figure 10.8) which reached

to the same global minimum with reduced number of iterations.

The corresponding pressures and flows for this map is given in Figure 10.9. Although it

is difficult to ascertain the true shape of the flow wave, the flat diastolic flow is in

agreement with reported recordings. The relatively small positive values can be

attributable to the usage of carotid pressure wave instead of the ascending aortic and to

212

the crude estimates of physical dimensions. These flows suggests that the flow wave

remains similar except a slight peaking and shortened ejection duration.

Iteration Map

Figure 10.8 An example of the entire error surface's contour map during iterating the model parameters ΓS , and ΓL . The numbers on contours indicate the percentage error for a given ΓS - ΓL combination. The concentric contours in the middle indicate the

global minimum for this error surface. The solid dots and arrows indicate the path followed by the recursion steps to reach to this global minimum.

The modulus of corresponding impedances (Figure 10.10) shows reduction in lower

frequencies, in line with phase velocity estimates presented in Figure 10.4. The

modulus of the first harmonic is decreased without significant alterations in the zero

crossing of the phase. When these data are compared to those documented in the

literature dealing with effects of nitroglycerine on pulsatile haemodynamics (Yaginuma

et al 1986, Fitchett et al 1988), the correspondence of alterations is clear. This suggests

that changes are due to similar mechanisms.

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Carotid Pressure Synthesised Flow

Figure 10.9 The pressure waveforms used to predict the model parameters by iteration of the reflection coefficients (Left). For each pressure pulse (control: solid line and glucose ingestion: dotted line) a flow waveform (Right) is generated by altering the model parameters. The flatness and zero value of the diastolic flow, form the criteria of the best fit.

Figure 10.10 The model impedance patterns derived by iteration obtained under control (solid line, filled symbols) and after glucose ingestion (Dotted line, open symbols). There is a reduction in both the phase angles and of the lower frequency components the modulus (compare this to Figure 5.16).

214

The estimated reflection coefficients by this method for pressure waves at baseline and

30 minutes after ingestion of glucose for each patient are shown in Figure 10.11. Again,

reduction in the reflection coefficient of the splanchnic circulation is the best predictor

of these changes.

REFLECTION COEFFICIENT

Figure 10.11 The model predicted changes introduced by ingestion of glucose on reflection coefficients of splanchnic element (left) and the leg element (right). (Bars represent 1 SEM, see text for more explanation).

In this study, it was demonstrated that ingestion of glucose alters the central wave shape

by reducing reflected waves. Since the reflected waves from both upper and lower

limbs are not altered in the mean time, it is concluded that vascular beds supplying the

splanchnic circulation vasodilate after glucose ingestion. Since these reductions in

reflected waves mimic the ones brought about by vasodilators, the apparent action site

of these vasodilators is in splanchnic circulation.

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SECTION IV

APPLICATIONS

The amount of information contained in the arterial pulse is often underestimated,

wrongly addressed or at times over-emphasised under clinical conditions (Wright and

Wood 1958; Apter 1963; Bourgeois et al 1974; Wei and Chow 1985; Gerber et al

1987). This has lead to misconceptions mainly due to the lack of information on factors

responsible for the formation of the pulse. The absence of readily accessible tools for

non-invasive measurement of the arterial pressure pulse and interpretation of the

pressure data has exacerbated this problem. However, tools and theories related to the

phenomenon of the arterial pulse have been available in various laboratories for nearly

three decades, but were scarcely applied in the clinical environment (Freis and Kyle

1968). This was because of the complexities involved in the mathematics and

instrumentation in application of these techniques, as well as lack of interest from

216

equipment manufacturers who often respond to specific demands. Because of these

factors, both the clinician and the patient had to resort to sphygmomanometry, and the

information gained from that procedure which has not changed appreciably since its first

introduction in 1905. The only exception to this was the introduction of 24 hour blood

pressure measurement devices due to recent developments in microprocessors. The

opposite trend was encountered in places where accurate invasive measurement of

pressure was possible. Since the sites of measurements and their effects were ignored,

attempts to correlate pressure waves obtained this way to the condition of the patients

was thwarted. Even if this was achieved the conclusions drawn were inconsistent with

laboratory investigations. The most notorious of these inconsistencies is the notion

derived from the Windkessel concept. High systolic pressure was considered good

because it denoted a vigorous heart while the high diastolic pressure was considered bad

since it denoted a high afterload.

Previous chapters demonstrated the consistency of the transfer function in the upper

limb as opposed to lower limb and the trunk under a wide variety of conditions. Using

this feature of the upper limb arterial system, it is possible to synthesise the central

aortic pressure waveform from peripheral upper limb pressure recordings. Previous

chapters also shed light on the features of the pressure waveform and the inherent

information content. One may therefore apply these concepts to design of a system for

automated analysis of the pressure waveform. Coupled with synthesis technique, this

system might find use in clinical environments.

In this section, methods for the synthesis of the central pressure waveform from the limb

pressure measurements are described. These methods are then implemented in a

computer program by which certain features of the arterial pressures are extracted and

reported.

217

THE SYNTHESIS OF THE CENTRAL PRESSURE WAVEFORM FROM THE PERIPHERAL PULSE IN THE UPPER L IMB As demonstrated in the experiments described in Section III, there are substantial

differences between peripheral and central pressure waveforms due to wave propagation

and wave reflection. More importantly, these differences are dependent not only on the

arterial properties as dictated by age, anatomy and vasoactive state, but also on the heart

rate. The effect of dependence on the latter becomes apparent when one considers the

use of ad hoc correction and estimation techniques such as linear regression analysis and

statistical methods for estimation of central from peripheral recordings.

CHAPTER 11

218

It would not be of much concern whether the deviation introduced by these techniques

was large, provided that the central pressure waveform was not carrying more

information than that in the peripheral waveform. The central pressure waveform has

been shown to be important in calculation of left ventricular vascular coupling

parameters (Yin 1987). It is also useful if left ventricular performance indices (Sagawa

1981) could be calculated using this waveform instead of left ventricular pressure. It is

therefore not surprising to find on going attempts to synthesise the central aortic

pressure pulse from peripheral radial/brachial recordings using available techniques.

These varied from synthesis of the pulse from upstroke of the waveform by ECG gated

sphygmomanometers (Rodbard et al 1957; Johnston and Kannar 1974; Marmor et al

1987) to application of standing wave concepts (Warner 1955) and to straightforward

inverse filtering (Lasance et al 1976). When none of these were available, correlations

between the blood pressure measured in the brachial artery to that of central were

investigated using different sphygmomanometric techniques (Borow et al 1982).

In this chapter, pressure waves from previously published central and peripheral sites

are analysed. The analysis is repeated after synthesis of the central pressure waveforms.

The synthesis of central pressure waveform involves the use of the transfer function

derived in chapter 8. It is shown that these transfer functions could be used both in

frequency domain where a single pulse could be synthesised off-line, and in the time

domain where a series of pulses could be obtained on-line. The frequency domain

technique is compared to these reported independently and shown to be in close

agreement both in estimated pressure shape and in peak systolic pressure. Extension of

this technique by convolution windows based on these transfer functions is used to

generate on line ascending aortic pressure waveform form peripheral recordings. It will

be shown that this approach could enhance non-invasive determination of cardiac

properties, when used with other techniques (echo cardiography, radionuclide scans).

219

11.1 METHODS

The resynthesis of central pressure from the peripheral pulse assumes that the system is

in steady state oscillation and linear. This allows the application of linear system

identification techniques. In such a system, a transfer function describes the input-

output relations explicitly similar to that presented in previous chapters for upper limb

wave propagation characteristics. Once the system is characterised by relating the input

and output relations in the frequency domain, it can then subsequently be used to find an

input from an output or vice versa.

11.1.1 Synthesis in the Frequency Domain

Since it was shown in Section III that the upper limb transfer function is relatively

constant under different conditions such as ageing, vasodilation and following physical

manoeuvres, it can be used to calculate an input from an output signal. To perform this

operation, the output (ie the peripheral pressure waves) must be transformed into the

frequency domain and subsequently divided by the transfer function (Equation 11.1),

since the output is obtained by the multiplication operation between the input and

system identifier, transfer function, (Oppenheim et al 1983). The resultant frequency

domain representation of the input can then transformed back to time domain to obtain

synthesised aortic pressure waveform.

×=

− )(1)()(

ωωω

PPAAHPPAA (11.1)

In this expression AA, PP and HAA-PP are the ascending aortic, peripheral pressures

and transfer function between the two. The symbol "× " denotes multiplication.

220

Data used for synthesis are obtained from published recordings where accurate aortic

and/or brachial or radial artery pressure waves were given. These were recorded

simultaneously or at short intervals apart. They were obtained from the studies

representing vasodilation (Simkus and Fitchett 1990), exercise (Rowell et al 1968),

normal conditions (Kroeker and Wood 1955; Remington and Wood 1956) and shock

(O'Rourke 1970) to simulate the wide variety of conditions that an individual might

encounter. The aortic and radial/brachial tracings in these reports were digitised by

hand using a digitising tablet (Summasketch Summagraphic). Since this process

involves a noise level higher than expected due to digitisation by hand, it is taken not to

include the noisy part of the waveform. This was done by excluding the frequency

components in the calculation that are far below a noise threshold that was estimated by

averaging components 9-16 Hz.

To synthesise the central waveforms, the digitised peripheral recordings were first

smoothed through a Hamming window (Hamming 1977) and later transformed into the

Fourier (frequency) domain. After eliminating the frequency components with

amplitude below the noise level, they were divided by the transfer functions as discussed

in Chapter 9 in the complex domain. Transforming the resulting complex values

yielded the central pressure waveform. There was considerable attenuation and scatter

in high frequency components of the transfer functions. When complex division was

applied, the attenuation produces an erroneous amplification of high frequency

components. Because of the low power content of high frequency components of the

arterial pulse, it was justifiable to correct for this artefact by assigning the value of unity

to the frequency components of the inverse transfer function whose values are greater

than unity.

The synthesised central pressure waveforms are then compared in the time domain with

the recorded waves. Since systolic pressure was part of the peripheral pressure

waveform most influenced by wave reflection, linear regression analysis was performed

221

on recorded central and peripheral systolic pressure together with the calculated aortic

systolic pressure. Furthermore, visual comparisons were made to ascertain the fit

between calculated wave shapes.

11.1.2 Synthesis in the Time Domain

The method of synthesis of central pressure waveform from peripheral recordings in

frequency domain requires the entire peripheral pulse to be recorded, transformed into

frequency domain, multiplied by the inverse of the transfer function and then

transformed back to the time domain. These steps are not only time consuming, but it is

also impossible to synthesise the aortic waveform on-line. For practical applications it

is important to obtain a continuous trace of central waveform. In physiological

manoeuvres where beat-to-beat changes can occur, such as tilt table testing, Valsalva

manoeuvre and aortic counterpulsation the former approach needs extensive computer

power. These limitations are overcomed by performing operation in the time domain

since multiplication in the frequency domain becomes convolution in the time domain

and vice versa.

In time domain the frequency domain representation of equation 11.1 is

( )thtPPtAA PPAA−∗= )()( (11.2)

where "∗" denotes convolution operation. The function ( )th PPAA− is the time domain

representation of

− )(1

ωPPAAH which can be obtained by inverse Fourier

transformation

− )(1

ωPPAAH.

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11.2 RESULTS

11.2.1 Frequency Domain

As already documented in these studies (Kroeker and Wood 1955; Remington and

Wood 1956, Rowell et al 1968, O'Rourke 1970, Simkus and Fitchett 1990), there were

substantial differences between the wave shapes of central and peripheral pulses. These

differences are pronounced at peak systole due to reflected waves effecting specifically

the peripheral systolic pressures

The systolic pressure differences between peripheral and central aortic pulses ranged

from a few mmHg to as high as 60 mmHg during exercise (data from Rowell et al

1968), Figure 11.1, Left.

Before Synthesis After Synthesis

Peripheral Systolic Pressure (mmHg) Peripheral Systolic Pressure (mmHg)

Figure 11.1 The relationship between the peripheral and central systolic pressure difference, ( )AAPA SPSP −∆ , and peripheral systolic pressure before (Left) and after

corrections are made (Right). Although initially a a linear relationship exist, the synthesised pressure becomes independent of peripheral pressure ( 0001.0 67.0 10.2032.0 2 <=−= pRxy ). Dotted lines represent 95% confidence

intervals.

223

Importantly, these differences were linearly dependent on the actual peripheral pressure,

suggesting the influence of reflected waves on peripheral pressure wave contour. At

first, it could be thought that this relation might be useful in determination of the central

systolic pressure from peripheral recordings by simple linear regression analysis.

However, the relatively small correlation between these two pressures (R2=0.67) makes

it almost impossible to employ linear techniques for this purpose. The difference

between the central and calculated systolic pressure became independent of the

peripheral systolic pressure when the resynthesis technique was applied (Figure 11.1,

Right). This indicates that synthesised central pressure waveform correlates better with

the actual pressure and the scatter should be explained by factors other than the wave

reflection.

This finding is also supported when the actual and synthesised pressure waveforms are

compared (Figure 11.2).

CONTROL NITROPRUSSIDE

Figure 11.2 The radial (thin solid line) measured aortic (thick solid line) and the synthesised aortic pressure (dotted line) waves, before (top) and after (bottom) administration of sodium nitroprusside. Although there were significant differences between the radial and central aortic pressure waves, the synthesised aortic wave approximates the actual measured wave. Data are obtained from Simkus and Fitchett 1990.

224

Although significant differences in both the shape and the peak systolic pressures were

present between peripheral radial and central aortic pulses before and after

administration of sodium nitroprusside, the synthesis method eliminated the major

discrepancies. During synthesis of the central pulse, however, the high frequency

components of the pulse are lost due to filtering and windowing effects. This indicates

that more work has to be done to reduce these effects. Yet, the point of incisura as well

as the shoulders and the upstroke were still identifiable as the sudden inflection points in

the pressure wave.

Figure 11.3 Graph showing directly recorded (closed circles) and synthesised (open circles) ascending aortic pressure compared with recorded brachial or radial pressure (open squares) in data published by (A) Simkus and Fitchett 1990 under control and (B) after nitroprusside infusion, (C) by Rowell et al 1968 under control and during graded degrees of exercise, (D) by O'Rourke 1970 during shock, (E) by Remington and Wood 1956 under control and (F) by Kroeker and Wood 1955 under control conditions.

When data were grouped within the sources from which they were obtained (Figure

11.3, Table 11.1) it was seen that the synthesis technique often decreased the systolic

overestimation. However, in four cases it predicted a lower central systolic pressure

225

compared to the actual measured pressure. Close inspection of these recordings revealed

that the mean arterial pressures in the peripheral sites were much lower than that in the

central aorta. This might be caused either by the digitisation technique or by the actual

pressure drop in the upper limb, since the zero hertz components of the modulus of the

transfer functions were always equal to unity.

Systolic Pressure (mmHg) Pulses Radial or

Brachial Asc.Aorta (Calc) Asc.Aorta

(Meas) Simkus and Fitchett ,1990 (Control, n=10)

116±3* 103±3 101±3

Simkus and Fitchett , 1990 (Nitroprusside, n=10)

105±3* 93±3 88±8.3

Rowell et al 1968 (Graded degree of exercise, n=4)

187±18= 143±11 141±12

O'Rourke et al 1970 (Shock, n=1) 85 75 73 Remington end Wood, 1956 (Control, n=1)

139 133 132

Kroecker and Wood, 1955 (Control, n=1)

165 138 142

Total (n=27) 124±7* 106±5 104±5

Table 11.1 Comparison of measured peripheral, measured aortic and synthesised aortic systolic pressures from different investigators. *=p<0.0001, ==p<0.005 compared against cental aortic values).

The maximum reduction in synthesised systolic pressure occurred in the data reported

by Rowell et al (C in Figure 11.3) where the heart rate was more than 120 beats/min. It

was expected, however, that under conditions involving high hearts rates the errors

introduced by the variability of the transfer functions would be apparent. Yet, either due

to harmonic composition of the peripheral pressure wave or due to pre-processing of the

pressure waveforms, this problem does not introduce severe error in the synthesis.

226

11.2.2 Time Domain

The convolution window that has been used to synthesise aortic from brachial is given

in Figure 11.4. It has a zero mean indicating no change in mean value and a delay of

128 milliseconds indicated by the peak of the window. This delay (45º / Hz) has to be

taken into account for each sample acquired.

( )th AABA−

Figure 11.4 Convolution window used for synthesis of ascending aortic from brachial pressure wave.

An example for on-line calculation of aortic from radial pressure waves using

convolution window based on transfer function between radial and aortic is given in

Figure 11.5. The on-line method yields similar results to the off-line method.

Although the difference between the radial (Middle, solid line) and aortic pressures

(Bottom, Solid Line) were high, the window reduced this difference considerably

(Bottom, Dotted line). There was however a linear delay caused by application of

convolution to the data. Although real aortic wave occurs earlier than the peripheral

waves ( of the order of 60-100 milliseconds) the synthesised wave appears on the output

128 milliseconds later. Because of this processing delay it is impossible to obtain the

actual temporal relationship between the two waves. However, for display purposes a

227

value of pulse transmission may be empirically added to the raw signal so that two

waves can be aligned temporally. These findings are confirmed in a study performed in

12 patients. Results are similar to frequency domain synthesis and well compared to

beat-to-beat and patient-to-patient data (Table 11.2, figure 11.6).

ECG

Radial

Aortic

Synthesised

50

mmHg

Figure 11.5 On-line calculation of ascending aortic wave from radial pressure waves. Data is recorded using high fidelity Millar micromanometers. The radial pressure wave(middle panel)is shifted by 50 mmHg up for display purposes. The synthesised aortic pressure wave (bottom panel, dotted line) agrees well with actually recorded aortic pressure wave (bottom panel, solid line), although it introduces a delay of 128 milliseconds to the data.

228

Systolic (mmHg) Diastolic (mmHg) Pulses Radial Asc.Aorta

(Calc) Asc.Aorta

(Meas) Radial Asc.Aorta

(Calc) Asc.Aorta

(Meas) 9.0 133.2±9.0 122.0±8.5 127.2±3.7 67.8±8.0 71.5±8.0 68.2±1.6 6.0 125.7±4.6 120.5±4.3 126.4±5.1 70.1±3.9 74.5±2.7 74.5±5.9 7.0 122.5±4.6* 105.4±4.2 110.0±4.4 62.9±3.9 65.4±3.9 66.3±3.1 7.0 151.1±3.4* 131.8±2.9 133.3±3.5 78.0±1.7* 79.6±1.7* 73.0±1.2 8.0 137.5±1.9* 122.6±2.5= 118.8±1.7 76.2±3.0 79.4±2.8= 76.2±0.7 6.0 123.5±13.6 116.8±13.6 121.3±1.7 64.0±5.1 66.2±4.8 65.3±2.1 6.0 128.8±4.6* 115.0±3.7 109.7±2.1 59.1±2.6* 61.4±2.0= 65.4±1.1 8.0 164.4±5.9* 153.1±5.5 151.3±7.2 70.0±3.2 73.2±3.1 70.1±3.2 8.0 124.9±3.4* 103.1±4.2 102.3±2.4 59.3±4.4= 65.2±4.6 68.7±0.9 8.0 166.4±9.2 156.8±9.6 163.2±3.6 78.0±7.9 81.4±8.4 84.5±2.1 10.0 131.1±3.1= 113.1±2.8* 124.9±3.9 69.2±1.6 71.7±1.8= 68.7±3.6 10.0 123.2±4.4* 99.2±4.1 101.8±2.7 59.0±3.7 62.1±3.4 61.4±1.6

Total 136.0±15.6* 121.6±18.1 124.2±18.6 67.8±7.1 71.0±6.9 70.2±6.2

Table 11.2. Results obtained by time domain synthesis method on series of peripheral radial pressure pulses in 12 patients. The technique reduces both beat to beat (indicated by row values) and patient to patient (indicated by column values) differences in both systolic and diastolic pressures. Radial pressure waves are measured by tonometer and calibrated against brachial pressure. Due to technical difficulties data was not recorded simultaneously. (Data are given as Mean±SD, *=p<0.001, ==p<0.05 compared against cental aortic values).

11.3 DISCUSSION

It is shown in this study that a single transfer function can be used to estimate the central

waveform from the peripheral upper limb pulse. This transfer function which is

obtained from an earlier study (Chapter 9), has a local peak at around 4 Hz which is

predicted by the model (Chapter 5) and subsequently demonstrated by experimentation

(Chapters 7). Being higher than unity at frequencies of physiological significance, it

dictates the amplification of components of the pressure wave.

229

The usage of a single transfer function for this purpose could be questioned. This

transfer function might be expected to vary between individual subjects with ageing,

vasodilation, exercise and under certain physical manoeuvres. However, the simulated

effect of ageing, as produced by increase elastic modulus, or vasodilation, as produced

by decreases in global reflections, on this transfer function is minor (Chapter 5). Some

of these predictions of the model were later confirmed by experiments on human

subjects. It was shown that, although both have profound effect on global reflections

neither the vasodilator agents nor the ingestion of glucose has a major impact on this

transfer function (Chapter 9 and 10). Another major determinant of the transfer

function, the terminal reflection coefficient, was shown in the model to affect the

pressure wave amplification (Chapter 8). However, it was determined that this factor is

stable and varies little under physiological conditions. All these points justify the use of

single transfer functions for aortic to radial and aortic to brachial pressure wave

propagation systems for different subsets of patients.

Transfer functions per se do not dictate the specific contour of the peripheral waveform,

but state the relationship between the input and the output. It is therefore important to

quantify the input to the system used for reconstruction of a pressure waveform.

Clearly, the pressure waveform at the output will only be composed of harmonics of the

input pulse, amplified or attenuated and delayed (ie all as consequences of a linear

system). The relative consistency of the transfer function at lower harmonics makes

more sense, if it was to be used to synthesise the central waveform, when one notices

the power present in these harmonics of the pressure waveform. As already shown

(Nichols and O'Rourke 1990) the pressure pulse often contains much of its energy

(approximately 90 per cent) at frequencies below 4 Hz (Figure 11.7).

This frequency in turn is the one where the transfer function happens to be more reliable

and not showing much scatter (Section III). Thus, if one uses this transfer function, it is

probable that the most of the central pressure wave will be synthesised reliably since

230

both the power content of the pulses and the reliability of transfer function is high at

frequencies below 4 Hz.

Figure 11.7 Graph showing the transfer function between the carotid artery to radial (Solid line , Closed Circles) and the percentage total power in the carotid waveform (Dotted line, Open squares). Although there is amplification of the frequency components above 3 Hz, the power in these harmonics constitute less than 10% of the total power. Thus radial pressure waveform is not sensitive to alteration in the transfer function at higher frequencies.

The resynthesis is inevitably an operation of low pass filtering of the peripheral pulse

(the inverse operation of amplification is attenuation). This operation, unfortunately,

reduces the amount of intermediate frequency components in the peripheral pulse and

the synthesised pressures become rounder (Figure 11.2). Since the transfer function

approximates unity at high harmonics, the features with steep changes are retained. It is

therefore possible to extract these features from the synthesised pulse. In cases where

this is impossible, such as the incisura, values derived from the original pulse can be

used (Gallagher et al 1992). This becomes extremely useful for the purposes described

in the introduction of this chapter.

231

5 10 15 20

1

2

3

-5

0

-10

Frequency (Hz)

Phase (Radians)

|Amp|

Figure 11.8 The transfer function between ascending aorta and brachial artery used by Lasance et al (1976) to synthesise the central from the brachial pressure waveform for calculation of cardiac output. Note the similarity between this and the transfer function of the brachial artery under control conditions displayed in Figure 9.2.

These findings are also in line with these reported in the literature (Falsetti et al 1974;

Krovetz et al 1974) where the effect of harmonic correction by linear techniques was

analysed. The synthesis technique of undistorted pressure waveforms recorded by

catheter systems with poor frequency response is basically similar to that employed in

this study. Although the system that was analysed in this study was practically different

(catheters versus upper limb circulation), the analogy is striking. When similar inverse

filtering technique is used in catheters having resonant frequencies of 7-10 Hz, it was

possible to reduce the systolic overshoot from 27% to 6% of the actual (Krovetz et al

1974). Therefore this study confirms the assumption that the important parameters are

the behaviour of the transfer function and the harmonic content of the pulse. Another

study that was essentially similar to that presented in this chapter was made in 63

subjects to calculate a representative transfer function between the ascending aorta and

the brachial artery (Lasance et al 1976). This generalised transfer function (Figure 11.8)

was then applied to the original data to synthesise the central from the peripheral pulse

232

with the aim of calculating cardiac output. The calculated transfer function and the

correspondence of the synthesised and measured waves were similar to those obtained

independently and presented in this chapter and earlier in this thesis.

In conclusion, it is appears that to a first approximation a single transfer function can be

used in human adults to synthesise central pressure waveform from peripheral upper

limb recordings. The synthesised waveforms bear a close resemblance to measured

waves. However, due to the low pass filtering effect of the process, some features of the

central pulse may not be reproduced. The upper limb transfer function employed in this

technique is considered to be independent of age and vasodilators. In earlier chapters, it

was also shown to be relatively stable even under extreme conditions, such as occlusion

of the hand and reactive hyperaemia. However, performance of the technique might be

further increased if transfer functions are personalised using anatomical and

physiological quantities. Application of time domain representation of transfer function

(convolution windows) enables to calculate central from peripheral recordings on-line.

This procedure could be used where central aortic features are needed on a beat-by-beat

basis.

233

A SYSTEM FOR ON-LINE ANALYSIS OF BLOOD PRESSURE WAVEFORMS Results of investigation described in previous chapters have lead to development of a

system which integrates the current techniques of blood pressure waveform registration

and feature extraction. The system comprises a computer with appropriate input/output

devices, such as screen, keyboard and disk(s); an analogue to digital board to convert

signals from various pressures and triggering sources, and software to gather, analyse

and report the pulses using this hardware. Since it is of vital importance to run in near

real-time in the clinical environment, provisions are made to accelerate the entire

process while minimising costs.

The basis of the system is the acquisition of pulses from a peripheral site (radial, carotid,

femoral, brachial, axillary, subclavian and dorsalis pedis artery) or from ascending aorta

CHAPTER 12

234

by means of a pressure registration device (tonometry, volume clamping or invasive

catheters) and analysing the waves due to existing knowledge about the pulse. Pressure

wave propagation parameters such as reflection coefficients and pulse wave velocities

are calculated to quantify the effects of different drugs or manoeuvres on the arterial

vasculature. It also contains a database management system based on pressure

recordings which can be coupled with other diagnostic and therapeutic tools. To

achieve all these goals, the Diagnostic Applanation Tonometery (DAT) software

includes routines for signal acquisition, signal conditioning, feature extraction,

parameter calculation, database maintenance and report generation (Figure 12.1).

Feature Extraction Unit

ReportWriter

DatabaseEngine

Signal Conditioning Unit

Data Acquisition Unit

The DAT System

HELP

UNIT

Figure 12.1 The components of the Diagnostic Applanation Tonometery (DAT) system showing the hierarchy and interplay of separate units.

This chapter describes the implementation of the system and algorithms used for

analysis. The information on installation, operation and maintenance of the system is

described in detail by the user's manual provided in Appendix I.

235

12.1 IMPLEMENTATION

12.1.1. Data Acquisition

The peripheral pulse is acquired at a rate of 128 Hz using an A/D converter. The

sampling rate is fixed at this frequency since it not only facilitates operation of Fast

Fourier transforms (FFT) using techniques described by Blackman-Tuckey algorithms

(Blackman and Tuckey 1958) but also it is well above the bandwidth of pressure signals.

This choice of frequency is low enough to reduce overhead in data storage and analysis

while high enough from aliasing errors. By doing so, it is guaranteed that the same

harmonic will always lie in the 1 Hz window. The clock pulses necessary to generate

this sample interval is generated by the computers (IBM-PC) timer chip, (Intel 8255)

(although a clock on an A/D board would also suffice) generating an interrupt to be

serviced. The use of the system clock allows utilisation low cost A/D boards which do

not posses internal timer or interrupt generating capabilities. Since the timer interrupts

were used to synchronise the data acquisition steps, they were not available to the

operating system. Before data acquisition module takes control, the vector pointer at

address $0000:0020 Hex for timer interrupt ($08 Hex) handler of BIOS was changed to

the interrupt servicing routine of the DAT system (Figure 12.2). This technique

modified the priority scheme of the timer interrupts. Since same timer interrupts are

also used by DOS in time keeping activities, BIOS and DOS routines are activated at

regular intervals to update the system clock.

At each interrupt, the CPU reads data from the A/D ports representing the pressure (Ps)

and if present, the trigger (Ts), Figure 12.3. The channel numbers for these signals are

assumed to be sequential, the channel for Ps being the first, and given in the

configuration information. The Ps is then copied to a circular buffer of 10 second

length. Thus to fill the buffer once, 1280 samples are necessary. If the trigger signal is

236

PC TimerHardware

Layer

SoftwareLayer

BIOS

DOS

DAT AfterBefore

Figure 12.2 The modification of original interrupt vector table for timer interrupts. Arrows indicate the order of execution of code after each received interrupt. Under normal circumstances the priority is to the BIOS, DOS and then the user program. Alteration of the vector sequence also alters the existing priority.

not present, then Ps is also copied to a buffer of equal type and size for trigger

information, otherwise Ts will be copied into it. On-line calculation of aortic pressure

waveform is also performed during the acquisition step of the pulse. The Ps is directed

to a convolution buffer, where it is convolved with the window representing the inverse

transfer function (see below) to yield an ascending aortic pulse (Cs). The Cs is stored

into the same size circular buffer. If D/A channels are available, an identical copy of Cs

and Ps are dumped out from these ports. The Ps is then displayed on the screen in a

rotating drum sequence. If present the derivative of the trigger signal (Ts) is also

displayed. The same process is repeated continuously until terminated by the user.

For recording sites between the ascending aorta and either carotid, radial, femoral,

brachial, axillary, subclavian or dorsalis pedis a transfer function is determined either by

direct calculation from in vivo data or from the model. From this, a convolution

window is calculated for each transfer function (see chapter 11). To obtain a cut off

237

frequency at 16 Hz, the resultant window is then stored in a look-up table of 33 samples

long for each patient. Normally, this window function yields an estimate of the central

pressure value for each sample after (W+1)/2 convolutions, (W is the window length).

This operation introduces a linear delay of sampling period*(W+1)/2.

Cs

Ts

Ps

Triggersource

PeripheralPressure

Ps

Asc. Aorta

Interrupt

Display

Display

PC Timer

Ts Buffer

Ps Buffer

Cs BufferConvolutionWindow

A/D

D/A

Figure 12.3 Block diagram of the data acquisition section of the DAT system. Peripheral pressure waveform and trigger source is fed into to the respective buffers where they are kept and displayed. Peripheral pressure is convolved with the respective filter and stored as central pressure waveform. Due to presence of D/A converters the pressure signals (raw and convolved) are dumped out. Entire events are synchronised with PC timer interrupts. Dotted lines indicate optional paths.

12.1.2 Signal Conditioning

Upon the user's request data from buffers (Cs, Psand Tc) are transferred to the database

records, excluding the last 2 seconds to allow for data interruption which may occur

during the initiation of the request. The PC timer is then stopped and updated by the

system parameters while real-time clock registers are also updated. The contents of all

buffers are smoothed using 7-point moving average filters to limit the bandwidth of the

signals to 20 Hz (Oppenheim et al 1983). The data string in buffer Tc is then

238

differentiated and further smoothed using first forward differentials and 3-point moving

averages to find the triggering points and to eliminate the noise amplified during

differentiation. Maximal and minimal thresholds are defined as the 60% of the

maximum and minimum derivatives in the entire buffer content. This value for

threshold is determined empirically after trying for several pulses. The positions of the

onset of pulses are then determined by comparing them against the maximum threshold.

The segments within these maximal and minimal thresholds are marked. The

corresponding data from Ps and Cs are averaged to yield averaged recordings of

peripheral and synthesised waveforms. The averaged Ps is then calibrated using given

pressure values. The numerical values for Ps and Cs are then used to estimate overall

gain and offset of the system. The calibration values obtained this way are subsequently

used to calibrate the synthesised aortic pulse.

12.1.3 Feature extraction

Wiggers (1928) described 14 different features present in the arterial pulse contour.

Investigation described in previous chapters indicates that meaningful information can

be obtained a much smaller subset, This system extracts five time relative points on the

waveform from which parameters relating to the heart and arterial system are

determined, (Figure 12.4). They form the positions describing the foot, first shoulder,

second shoulder, incisura and the duration of the pulse. Since these features are

geometric and could be described as inflection points along a curve, they can be

identified using differentials of different orders (Figure 12.5). Of all the points on the

pressure waveform, particular importance is given to the systolic onset (foot of the

pulse), maximum of the first derivative (upstroke of the pulse) and systolic point

(maximum of the pressure pulse). All other points can then be expressed in relation to

these points, thus providing consistency in comparisons.

239

Features of the Arterial Pulse

Second Shoulder

First Shoulder

Foot of the Pulse

Pulse Duration

T 1

T 2

T i

T f

T T

Aortic valve closure(Incisura)

Figure 12.4 The basic features of the arterial pulse. After the foot of the pulse indicating the onset of ejection determined from the trigger source, the pressure wave rises to an initial peak where it forms a shoulder. It then proceeds to a second shoulder which often constitutes the peak pressure in the elderly. The former point is related to timing of peak flow while the second shoulder to reflected waves. The end of ejection is associated with closure of the aortic valve which is often seen as a distinct incisura on the aortic pressure pulse.

The flow chart describing the feature extraction process is shown in Figure 12.6. The

point Tp is the systolic pressure point and is found by scanning the averaged pressure

pulse for its maximum point. Max dp/dt is the maximum point on the first derivative

curve and systolic onset can be determined by taking the first derivative of the pulse and

locating a zero crossing from negative-to-positive that precedes the Max dp/dt point.

The point Zc3rd+- is the third derivative's first zero crossing point from positive to

negative after the Max dp/dt point and it correspond to the peak flow in an "elderly"

(Type A) pulse Figure 12.7 (Murgo et al 1980b; Kelly et al 1989a).

240

Third Derivative

Pressure Waveform

TfTi

T Max dp/dt

Tp

T zc3rd-+

T zc3rd+-

First Derivative

T Max3rd

Figure 12.5 A typical waveform (solid line) together with the first and third derivatives (dotted and dash dot line) used in extracting features (see text for more explanation).

Tp>Tmax3rd+60msec

(Tzc3rd-+)<(Tp +60

msec) or (Tzcrd-+ )>

T2= Tzc3rd-+T2=0T2 =Tp

T1= Tmax3rd T1 =Tp

Type= B or CType=A

Find Tmax3rd, Tzc3rd+

Find 1st Derivative Max Point(Tmaxdpdt) and Wave

Foot (Tf)

Find Systolic Max (Tp)

START

(Tf+300 msec)

Figure 12.6 Flow chart for extraction of time domain features in the pulse.

241

The Zc3rd-+ point is the point where the second shoulder is expected to occur in a

"young" pulse (Type B or C) and could be determined as the first negative-to-positive

zero crossing point after the Tp on the third derivative curve. The point of incisura

representing the end of ejection can be determined once the second shoulder is found. It

is often at the Zc3rd+- point after the second shoulder. Once these times are found, all

the other important features can be determined from them using the flow chart.

Type A Type B or C

Figure 12.7 The types of pulses defined by Murgo et al (1980b) in describing the typical waveform in central aorta. Type A pulses have a distinct shoulder in late systole which augments the systolic pressure and seen mostly in elderly patients. Type B or C pulses are seen in young patients where no augmentation in the pulse can be observed. Although these descriptions are categorical there is a continuos spectrum of pulses can be observed due to different degrees of reflected waves present in the composite pulse. Augmentation index describes the amount of reflected wave in the pressure pulse.

12.1.4 Calculated Parameters

From averaged peripheral and synthesised central pulses the following values are

calculated:

Ejection Duration

242

ED Secs T Ti f( ) = − (12.1)

Heart Rate

HR Beats TT( min) = 60 (12.2)

Augmentation Index

AI P PP Pd

d(%) = ×−−

100 2

1 (12.3)

Pressure at First Shoulder

P mmHg P T1 1( ) = (12.4)

Pressure at Second Shoulder

P mmHg P T2 2( ) = (12.5)

Pressure at Diastole

P mmHg P Td f( ) = (12.6)

Pressure at Systole

P mmHg P Ts p( ) = (12.7)

Mean Diastolic Pressure

243

MDP mmHg

P

T T

ii T

T

T i

i

T

( )( )

=−

=∑

(12.8)

Mean Arterial Pressure

MAP mmHgP

T

ii

T

T

T

( ) = =∑

1 (12.9)

Mean Systolic Pressure

MSP mmHg

P

T T

ii T

T

i f

f

i

( )( )

=−

=∑

(12.10)

Tension Time Index

( )fi TTMSPHRBeatsmmHgTTI −××=)min.( (12.11)

Diastolic Time Index

( )iT TTMDPHRBeatsmmHgDTI −××=)min.( (12.12)

Subendocardial Viability Ratio

( )TTIDTISVI ×= 100% (12.13)

Augmented Pressure

AP mmHg P T P T( ) = −2 1 (12.14)

244

Reflection Transit Time

RT Sec T T( ) = −2 1 (12.15)

Maximum Rate of Rise

=

dtdPMaxSecmmHgdtdPMax )( (12.16)

Reference Age ( ) 81.33100642.0RA(Years) +−×= AI (12.17)

Reference Age is the age corresponding to the augmentation index as derived from

ageing studies, Kelly et al 1989a and where "P[]" indicates the element of the pressure

waveform array.

12.1.5 Database Engine

Since it is assumed that there will be further enquires regarding the recordings in the

future based on epidemiological and short term research, a database engine is installed

as part of the software. The data preserved are the raw and averaged data for each

peripheral and calculated central aortic waveforms as well as indices and values derived

from them. A separate database is linked to the recording's database by the name of the

patient and the time of visit (Table 12.1). This second database keeps the information

regarding the patient particulars such as age, sex and anatomical and diagnostic data

(Table 12.2). Type definitions for each record are given in tables 12.3 and 12.4.

To facilitate retrieval of archived information, the transaction time is kept to a minimum

by indexing both the "PATIENTS" and "RECORDS" databases on their key fields

(Table 12.1) using AVL-trees (Adelson-Velski and Landis 1962). AVL-trees are

245

derivations from the B-trees (Chapter 4) where it is ensured that a maximum of log2N

searches would be sufficient to access a particular record (N is the maximum number of

elements in the tree). B-trees lack this insurance due to the unbalancing of the tree when

sequential insertions are made. AVL trees however, take this into account by balancing

the B-tree at each insertion and deletion so that there will be log2N levels present in a B-

tree at all times. Using this algorithm, the software is able to locate a record from a list

of 65536 recordings in at most 16 attempts (Shammas 1988). Provisions are also made

to ensure integrity and distribution of the database across several platforms by obeying

strict rules imposed by the system in accessing the database using indexes.

FIELD TYPE Patient Name Small String Date of visit Small String Medication Array[1..2] OF Small String Notes Small String Operator Id Word Trigger Present Boolean Diagnosed Boolean Gain of System Real Offset of System Real Trigger Series Array[1..128] of Integer Recorded Pulses Array[Raw..Convolved] of Pulse

Buffer Heart Rate Real Augmented Pressure Real Reference Age Real Tension Time Index Real Diastolic Time Index Real Subendocardial Viability Ratio Real Mean Systolic Pressure Real Mean Diastolic Pressure Real End Systolic Pressure Real

Table 12.1 The template for record of database file "RECORDS". This file is indexed with Patient Name field and Date of visit and linked to PATIENTS database. For explanation of defined types see tables 12.3 and 12.4.

246

FIELD TYPE Patient Name Text String Patient ID String[8] Patient Age Integer Sex String[2] Address Small String Distances Array [Radial.. Aortic] of real Convolutions Convolution Types

Table 12.2 The template for record of database file "PATIENTS". This file is indexed with Patient Name field and related to RECORDS database. For explanation of defined types see tables 12.3 and 12.4.

FIELD TYPE Pulse Name Pulse Types Tonometer Input Series Pulse Stream Averaged Pulse Pulse Length of Averaged Pulse Integer Systolic Pressure Real Diastolic Pressure Real Mean Pressure Real Time to Foot Of Wave Real Ejection Duration Real Max dp/dt: Real Systolic Onset Real Time to Incisura Real Reserved Real Time to First Shoulder Real Time to Second Shoulder Real Reverse Shoulder Index Real

Table 12.3 The type defined for any pulse. It contains information about the original series of pulse, averaged pulse and the derived indices. This buffer is duplicated for both raw and convolved signals and corresponds to PS and CS buffers in Figure 12.3.

247

New Type Definition Small String String[40] Pulse Array[0..255] of Real Pulse stream Array[0..1023] of Integer Convolution Window Array [-16..16] of Real Pulse Types Radial, Carotid, Femoral, Brachial,

Axilla, Subclavian, Dorsalis Pedis, Aortic

Convolution Types Array[Radial..Aortic] of Convolution Window

Signal Types Trigger, Raw, Convolved

Table 12.4 The type definitions based on ordinal definitions supplied by Pascal language used in record definitions.

12.2 THE TRIAL OF THE SYSTEM IN A CLINICAL ENVIRONMENT

This system is currently being evaluated in several laboratories around the world. Early

implementation of this system in an outpatient clinic in Sydney has accumulated more

than 6000 recordings of nearly 800 patients over two years. Clinical utilisation of this

system and the database has been undertaken by Dr. David Gallagher as a component of

his thesis for Doctor of Medicine in collaboration with Prof. MF O'Rourke in this

laboratory. Experience has shown that the system is reliable. An experienced user

spends 20 seconds to record and analyse 10 seconds run of pulses using an IBM-XT. A

hardcopy of the report is generated in 90 seconds with conventional printer hardware.

The average access time to any recording is undetectable (less than 12 lookups).

Currently it is undergoing extensive usage to detect effects of vasodilators. It is hoped

that following these investigations, the system with the necessary hardware and software

will be implemented as a stand alone unit in the near future, thus enabling clinicians to

complement conventional sphygmomanometer recording using sophisticated pulse

waveform analysis.

248

SECTION V

SUMMARY AND CONCLUSIONS

The models employed and experimental studies performed and reported in this thesis

were directed towards a better understanding of the human arterial pulse as a carrier of

information relating to physical and wave propagation properties of arterial system. To

understand the causes of alterations in pulse contour, the phenomenon of pressure wave

propagation was investigated by characterising the arterial system in terms of pressure-

pressure relations. This approach was different from many previous studies in pulsatile

haemodynamics where the system was characterised in terms of pressure-flow relations.

The investigation concluded with approaches that not only describe the information

content of the pulse but also ways to quantify it.

249

Throughout the thesis, the arterial system is considered to be operating within the

physiological range of pressures. This approach enabled the use of linear techniques in

employing tubular models of the arterial system. A linear approach has been shown to

be adequate to quantify the specific features of the propagating arterial pulse under

normal physiological conditions. The extra information that might be gained by

adopting a non-linear approach did not justify the increased complexities of

computation. Non-uniformity of the arterial system was simulated by connecting

uniform tubes with different physical properties. Although the computational procedure

used in this model is similar to that used in earlier models, there are basic differences in

both the model construction and its implementation. For a better computational

efficiency, the present model uses a binary tree structure for the arterial tree. The

anatomical dimensions and physical parameters of the arteries were also updated using

recent data. There were also major differences in the terminations of the branches

which were loaded with modified Windkessels in contrast to previous models where

real elements were used explicitly. Using complex reflection coefficients derived from

this approach a wide variety of interventions were simulated. As part of the

experimental investigation the model was used to estimate significant model parameters

using optimisation techniques.

The advantage of using models in experimental settings was demonstrated by exploring

the propagation of reflected waves in the arterial system. The intensity of reflected

waves was modulated by altering the reflection coefficients at specific terminations.

Although limited to major arteries, this technique has been successfully applied in many

previous experimental investigations. In these previous studies however, individual

reflected waves were not isolated and their travel was not specifically tracked in the

arterial system. Using maps similar to those employed in weather forecasting, the

overall propagation of reflected waves from different origins was expressed as a

function of spatial dimension. The investigation concentrated on both the entire arterial

system and the arterial vasculature of the upper limb.

250

In the first part, the effects of wave travel and wave reflection were simulated. The

pressure pulse in different segments and the reflected waves from the upper limbs, trunk

and lower limbs were isolated. The study into the propagation of these waves

demonstrated the importance of vascular beds around the trunk on global arterial

circulation. The waves originating from the trunk segment formed the secondary

pressure oscillations in every arterial branch. In contrast, the effect of both limbs on

global function was found to be minor. They dictated the initial peak of the pressure

waveforms. These two findings indicate that pulse shape in different arteries may react

differently to different interventions.

In the second part, the formation of the upper limb pressure waveform is examined in

detail. Since these waveforms are widely utilised in clinical situations this detail

treatment was deemed necessary. Under normal conditions the pulses in this limb were

found to be considerably different from those in the central aorta. These differences

were also augmented under certain circumstances such as ageing, vasodilatation and

exercise. Although a second order system representation has often been suggested

previously, the upper limb system was found to be substantially different from a second

order system. It was found to be better modelled as an elastic tube terminated with a

complex load. Positive wave reflections from this termination induce a peaky transfer

function, similar to that of second order systems. However, the resultant phase and

modulus of this transfer function differ considerably from the resonant behaviour of the

second order system. The study then concentrated on the sensitivity of upper limb

system to altered arterial parameters. The upper limb system was found to be generally

insensitive to alterations in wall properties and the terminations within physiological

limits. The changes in peripheral waveforms and the transfer functions were also found

to be minor.

Since no model can be complete without testing its accuracy in its assumptions and

predictions, an experiment was arranged. The same experiment was also used in testing

251

the hardware and developed software. The software had to be tested to verify the coding

of the algorithms. Using iterative techniques the model accurately estimated pressure

waveforms and the physical properties.

The transfer function in the upper limb was characterised in subjects where the input

could be varied. Although in response to changes in heart rates both the input pulse and

the output pulse were altered, a unique expression of transfer function between the

central waveform and the peripheral radial waveforms was derived by resorting to

spectral techniques. Results of this study confirmed the assumptions made about the

linearity of the upper limb arterial system. Model values describing the upper limb

arterial system were derived by analysing the transfer function. Similar values were also

calculated by using parameter estimation techniques.

Reflections from the terminations of the upper limb were altered by compressing the

vessels of the palm vasculature. The systolic pressure and the intensity of reflected

waves increased in parallel with occlusion pressure. This finding confirmed that the

initial part of the upper limb waveform is contaminated by reflected waves originated

from the palm vasculature. Reduction in mean arterial pressure in the upper limb

caused by raising the arm decreased the intensity of reflections. However, when the

externally applied pressure to these vessels became equal to the intra-arterial pressure,

the intensity of reflected waves was decreased. This is consistent with known non-

linear pressure volume relations of the arterial wall. These findings indicate that under

normal conditions the intensity of reflections from this bed is comparable to that

induced by total occlusion of these arteries thus not subject to great variation.

Further investigations were made on patients receiving vasoactive drugs, in order to

assess their effects on wave shapes and propagation properties on the upper limb.

Nitroglycerine (NTG) and sodium nitroprusside (SNP) reduce secondary pressure

oscillations in all arteries. Results obtained in the upper limb indicated that these affect

252

mainly the proximal vessel properties. There was no evidence to suggest reductions in

the reflections occurring at the termination of the upper limb. There was little change in

the primary wave amplitude while the secondary wave amplitude was reduced indicating

the reduction in the intensity of reflected waves in the trunk. This indicates that these

drugs have a differential action by altering the properties of the vascular beds of the

trunk.

These experimental observations on the nature of the primary and secondary peak led

toinvestigations into the trunk segment. In a group of subjects, the oral administration

of glucose significantly altered both the intensity of second peak and the transfer

function of the trunk segment. However, the transfer functions of the upper and lower

limb segments changed little. This differential action of glucose is basically similar to

the action of vasodilators indicating involvement of common mechanisms. To explore

this possibility, a modified T-tube model was constructed to represent the arterial tree.

In this reduced model the lower T-tube was further bifurcated into two circulations, one

to the splanchnic and the other to the lower limb. This reduced model explained these

observations while generating realistic flow waveforms for each state.

The relative stability of the transfer function in the upper limb was exploited to generate

central aortic pressures from pressure recording in the radial artery. When used as a first

order approximation, inverse filtering using a generalised transfer function yielded

reasonable central pressure waveforms. This approach was found to be applicable under

conditions such as vasodilator therapy, exercise or certain physical manoeuvres.

The on-line resynthesis technique together with feature extraction, analysis and storage

of pulses recorded from different sites of the arterial system is an important practical

outcome of this investigation. A cheap, reliable and methodologically sound system

was designed and implemented using available hardware and software tools. This

system has been tested in an out-patient clinic for assisting in current pharmacological

253

management of cardiovascular disease. It was also extended to the critical care area,

where the extra information provided by features of the arterial pulse aid in assessing

patient's condition.

In summary the following conclusions are made:

1. Because of the experimental difficulties in isolating the individual components of

the reflected pressure waves, mathematical models of the human arterial system are

sine qua non in investigating the consequences of pressure wave propagation. It is

also possible to characterise a real system using mathematical models in conjunction

with parameter estimation methods.

2. A functional origin of reflected waves resides in the aortic trunk segment of the

human arterial system. The vasoactive state of the arterial beds of this segment is

the major determinant of the impedance patterns found in the central aorta in

contrast to the vasculature of upper and lower limbs.

3. Reflected waves originating from the arterial beds of the trunk travel into every

arterial branch and constitute the secondary oscillations seen in the central aortic and

upper limb pulses. In contrast, reflected waves from upper and lower limbs are

mainly confined to individual segments and form the initial peak of the pressure

waveforms in these arterial beds. These two distinct components of the arterial

pulse could explain the well documented differences in central and peripheral upper

limb pulses.

4. By relating the spectral components of input and output pressure waves, the wave

propagation properties of any vascular system can be explicitly described by transfer

functions. The information about the intensity and timing of reflected waves can be

obtained not only by analysis of the modulus but also from the phase of the transfer

254

functions. The latter method is superior to the former when the pressure waves are

uncalibrated but recorded with high fidelity instruments. Being such an instrument,

the tonometer is best suited for this method in measurement of altered transmission

properties.

5. In the upper limb arterial system the intensity and timing of reflected waves, ie

reflection coefficient at the palm vasculature, is the most important determinant of

the peak amplitude of the modulus of transfer function. Other factors modify the

characteristics of the transfer function by altering the components at high

frequencies. However, these have little effect on the peripheral wave contour since

the central pressure waveform is band limited.

6. The peaky transfer function in the upper limb has severe implications in clinical

monitoring. Although a linear relationship between the systolic pressure and the

stroke volume is often assumed, simple alterations in heart rate causes significant

deviations from this linear assumption.

7. Under normal circumstances the terminal reflection coefficient in the palm

vasculature approximates to that obtained by total occlusion of this bed. It reduces

to zero or even becomes negative when the transmural pressure experienced by this

bed approximates to zero. This indicates relative stability of the upper limb transfer

function when intra arterial pressure is within normal or above normal range.

8. Vasoacting agents such as nitroglycerine and sodium nitroprusside do not produce a

significant decrease in the reflection coefficient in the palm vasculature. In fact,

they produce a functional increase in reflection coefficient due to increased proximal

arterial compliance. They, however, decrease the reflection coefficient in the trunk

vasculature. This explains the paradoxical disparity of wave shapes after the

administration of these drugs.

255

9. Ingestion of glucose causes similar changes in the arterial pressure wave contours

observed after administration of vasodilators. It causes reductions in the secondary

peak without necessarily effecting the primary peak. These changes in the wave

shapes can be substantiated by analysis of transfer functions of the upper limb, trunk

and lower limb segments which indicates significant reductions in the properties of

the vasculature of the trunk segment.

10. The human arterial tree can be modelled as a modified T-Tube to explain the wide

variety of wave shapes and impedance patterns. As opposed to T-Tube models

constructed for other species, in this model of the human arterial tree the lower

element of the T further bifurcates into two circulations, one to the splanchnic and

the other to the lower limb. This model can be used to obtain realistic impedance

patterns and possible stroke volumes from measurements of arterial pressure waves

along the tree.

11. It is possible to generate central aortic pressures from the peripheral upper limb

pressure recordings. This method uses the relatively stable transfer function of the

upper limb. The technique can be applied to synthesise on-line central pressure

waveforms under conditions such as vasodilator therapy, exercise or certain

physiological manoeuvres. This enables the determination of central aortic

waveform indices from recordings of peripheral upper limb pressure wave contour.

256

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279

PUBLICATIONS

(M.Karamanoglu)

Full Journal Papers

1. Karamanoglu M, O'Rourke MF, Avolio AP and Kelly RP (1992). An analysis of the relationship between central aortic and peripheral upper limb pressure waves in man. European Heart J (in press)

2. Kelly RP, Karamanoglu M, Gibbs HH, Avolio AP and O'Rourke MF (1990). Non-invasive blood pressure registration as an indicator of ascending aortic pressure. J Vasc Med Biol 1:241-247

280

Letters

1. O'Rourke MF, Avolio AP, Karamanoglu M, Gallagher D and Schyvens C (1990). Brachioradial delay. Lancet, 336:1377-1379

Papers In Conference Proceedings

1. Karamanoglu M, Avolio AP, O'Rourke MF (1990). Simulation of the effects of lower body wave reflection on aortic and brachial pulse in man: Implications to therapy. Proc.of IEEE-EMBS 12:1154

Book Chapters

1. O'Rourke MF, Avolio AP, Kelly RP and Karamanoglu M (1992). Systolic pressure in central and peripheral arteries; Monitoring of central from the brachial or radial pulse. In "Vasodilatation in conduit arteries: Logical strategy in cardiovascular disease". O'Rourke MF, Safar M and Dzau V(eds). Edward Arnold, London.

Abstracts

1. Avolio AP, Kelly RP, Karamanoglu M, O'Rourke MF (1989). Measurement of brachial artery systolic pressure underestimates systemic effects of nitroglycerin: role of pulse wave transmission in the upper limb. J Vasc Med Biol 1:157

2. Karamanoglu M, Gravlee G, Kelly RP, Avolio AP, O'Rourke MF (1989). Pressure wave propagation in the upper limb- clinical implications. Aust NZ J Med 19:548

3. Karamanoglu M, Kelly RP, Gravlee G, Avolio AP, O'Rourke MF (1989). Clinical implications of pressure wave propagation in the upper limb. Circulation 80:Suppl 2:p542

4. Avolio AP, Karamanoglu M, O'Rourke MF (1990). Simulation of effects of vasodilators as detected from measurement of arterial pressure in the arm. Aust. NZ. J. Med Suppl I:328

5. O'Rourke MF, Avolio AP, Karamanoglu M, Kelly RP (1990). Derivation of ascending aortic pressure waveform from the brachial pressure pulse in man. Aus. NZ. J. Med 20:Suppl I:329

281

6. Karamanoglu M, Avolio AP, O'Rourke MF (1991). Real-Time estimation of aortic pressure wave contour from non-invasive measurements of the peripheral pulse in the upper limb. Aust. NZ. J. Med. 20:Suppl I:329

7. Karamanoglu M, Gallagher DE, Schyvens C, Avolio AP, O'Rourke MF (1991). A system of on-line analysis of pressure waveforms. Aust. NZ. J. Med 21:526

8. Gallagher DE, Karamanoglu M, Huang G-H, Avolio AP, O'Rourke MF (1991). Assessment of interobserver variation on peripheral pressure waveform analysis using a semi-automated system. Aust. NZ. J. Med 21:527

9. O'Rourke MF, Avolio AP, Karamanoglu M, Kelly RP (1991). Derivation of ascending aortic pressure waveform from the brachial pressure pulse in man. Medical & Biol Eng & Comp 29:Suppl I,157.

10. O'Rourke MF, Avolio AP, Karamanoglu M (1992). Use of transfer functions to determine central from peripheral pulse. J Am. Coll. Cardiol 19:227A.

11. O'Rourke MF, Karamanoglu M, Avolio AP (1992). A modelling analysis of aging and vasoactive drugs on the human arterial pulse. J. Am. Coll. Cardiol. 19:74A

12. Gallagher DE, O'Rourke MF, Avolio AP, Karamanoglu M, Kelly RP (1992). Determination of left ventricular ejection time from the radial pulse. J Am. Coll. Cardiol 19:74A.

13. Gallagher DE, Karamanoglu M, Avolio AP, O'Rourke MF (1992). Functional origin of wave reflection in the human arterial system. Aust. NZ. J. Med. (in press)

14. Kuchar DL, Platt HL, Karamanoglu M, Gallagher DE, Thorburn CW (1992). Effect of vagal blockade on baroreceptor responsiveness assessed using the Valsalva manoeuvre. Aust NZ. J. Med (in press).

15. Kuchar DL, Platt HL, Karamanoglu M, Thorburn CW (1992). Power spectrum analysis of the electrocardiogram does not predict the occurrence of syncope during tilt testing. Aust NZ. J. Med (in press)

282

16. O'Rourke MF, Karamanoglu M, Lee L, Cutfield G, Avolio AP, Gallagher DE, (1992). Derivation of central aortic from radial artery systolic pressure using a non-occlusive cuff to attenuate wave reflection in the hand. Aust NZ. J. Med (in press)

17. Gallagher DE, Karamanoglu M, Herok G, Avolio AP, Baird DD, O'Rourke MF (1992). Peripheral pressure measurements may be unreliable for calculation of cardiac properties with change in heart rate. Aust NZ. J. Med (in press).

18. Gallagher DE, O'Rourke MF, Avolio AP, Karamanoglu M, Kelly RP (1992). Left ventricular ejection time can be determined from radial pressure pulse. Aust NZ. J. Med (in press).

19. Gallagher DE, Karamanoglu M, Avolio AP, O'Rourke MF (1992). Accuracy of a finger clamp device (Finapres) for assessment of arterial pulse wave contour and of wave velocity. Aust NZ. J. Med (in press).

20. Gallagher DE, Karamanoglu M, Young JA, Kelly RP, Avolio AP and O'Rourke MF (1992). Apparent alteration in the upper limb pressure wave transmission with changing heart rate. Circulation 86:1830.

283

APPENDIX I

284

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TABLE OF CONTENTS

INTRODUCTION 1

GETTING STARTED 3Hardware Requirements 3Software Requirements 3Installing the Data Acquisition Boards 3Installing the Software 4Executing the Program 5

OVERVIEW 7What is DAT ? 7Theory of Operation 7Features of DAT 7

PATIENT INFORMATION INPUT FORM 9Introduction 9Information Fields and Their Usage 10Keys and Functions 11

INPUT OF THE PULSE 13Acquisition of the Pulse 13Calibration of the Pressure Pulses 13Sound Generation 13Triggering 14Selecting the Pulse 14Feature Extraction 14Keys and Functions 15

REPORT GENERATION 17Introduction 17Viewing of Patient Recordings 17Generating Reports of Recorded Pulses 17Printing a Hard Copy 19Explanation of Report 19Keys and Functions 21

ii

ADVANCED TOPICS 23Configuration File 23Command Line Interface 25

TECHNICAL CONCEPTS 29Overview 29Data Acquisition 30Signal Conditioning 31Feature Extraction 32Calculated Values 33Database Engine 34

HOW TO ? 37Import Existing Tape Data to DAT 37Conduct Different Studies Simultaneously 37Combine Different Databases 37Perform Statistical Analysis of the Data 38Print Unattended 39Use Reports in Word Processors 39Record Waveforms Without Actually Seeing Them 40Measure Pulse wave Velocity 40

WHAT TO DO IF ? 41Cannot Run DAT Program 41DATHELP.MSG is Missing 41Cannot Register the Pulse. The System Crashes 41Cannot See Any Data on the Screen 42The System Crashes 42Cannot Find a Recording Registered Previously 42

iii DIAGNOSTIC APPLANATION TONOMETRY INTRODUCTION •••• 1

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DIAGNOSTIC APPLANATION TONOMETRY INPUT OF THE PULSE •••• 14

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REPORT GENERATION

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DIAGNOSTIC APPLANATION TONOMETRY REPORT GENERATION •••• 18

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T

T

T

( ) = =∑

1

Pressure at First Shoulder

P mmHg P T1 1( ) =Pressure at Second Shoulder

P mmHg P T2 2( ) =Augmented Pressure

AP mmHg P T P T( ) = −2 1Mean Diastolic Pressure

MDP mmHg

P

T T

ii T

T

T i

i

T

( )( )

=−

=∑

Mean Systolic Pressure

MSP mmHg

P

T T

ii T

T

i f

f

i

( )( )

=−

=∑

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AIP PP Pd

d(%) = × −−

100 2

1Tension Time Index

TTI mmHg Beats HR MSP T Ti f( . min) = × × −1 6Diastolic Time Index

DTI mmHg Beats HR MDP T TT i( . min) = × × −1 6Subendocardial Viability Ratio

SVIDTITTI

%0 5 = ×100

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Maximum Rate of Rise

Max dP dt mmHg Sec MaxdPdt

( ) = çåäâ

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DIAGNOSTIC APPLANATION TONOMETRY HOW TO ? •••• 38

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DIAGNOSTIC APPLANATION TONOMETRY HOW TO ? •••• 40

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DIAGNOSTIC APPLANATION TONOMETRY WHAT TO DO IF ? •••• 41

WHAT TO DO IF ?

Cannot Run DAT Program

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The clock speed of the PC is higher than

12 MHz5HGXFH WKH FORFN VSHHG RI WKH 3&

CHAPTER 9

DIAGNOSTIC APPLANATION TONOMETRY WHAT TO DO IF ? •••• 42

Cannot See Any Data on the Screen

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Channel number does not match the

analogue input(GLW OLQH 4 RI '$71&)*

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Cannot Find a Recording Registered Previously

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The path to DAT.CFG file is not correct8VH 6(7 FRPPDQG WR VHW D SDWK WR '$71&)* ILOH

DIAGNOSTIC APPLANATION TONOMETRY WHAT TO DO IF ? •••• 43