a one-dimensional model of blood flow in arteries with friction, convection and unsteady taylor diffusion based on the womersley velocity profile

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A One-Dimensional Model of Blood Flow in Arteries with Friction, Convection and

Unsteady Taylor Diffusion Based on the Womersley Velocity Profile

Karim Azer

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy Department of Mathematics

New York University January 2006

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UMI Number: 3205629

Azer, Karim

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Acknowledgements

There are so many people I would like to thank, that have along the way, encouraged, supported and helped me in my doctorate studies I would like to thank my advisor, who has been such a tremendous help to me and without whom this work would not be possible I am very thankful to have Charlie as an advisor He has helped, supported and encouraged me through my qualifier exams, and the work that is presented in this thesis He is indeed an excellent advisor, outstanding teacher, and exceptional scientist and mathematician

I would like to thank the Applied Computer Science and Mathematics De- partment at Merck & Co., Inc for their support throughout my PhD studies, and especially Jeffrey Saltzman who has been a very big help, support, and an excellent mentor for me throughout my doctoral studies I would also like to thank Robert Nachbar and the rest of the staff of the Applied Computer

Science and Mathematics department at Merck for encouraging me and always

joking with me about life as a graduate student A special thanks to Vladimir Svetnik and Christopher Tong for their due encouragement and support during the latter years of my studies here at Courant

Amongst others, I would also like to thank Mette Olufsen for helpful discus-

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always going out of her way to assist and advise me on administrative issues at Courant

My wife Kelly has been so gracious and loving to me through this long and arduous journey Indeed I told her on several occasions that getting my PhD is not my accomplishment alone, by our accomplishment as a family She has been the every-supporter of me along the way Her tremendous sacrifices, profound love and persistent commitment all throughout have been and continue to be

unconditional

Being a parent myself, I can only begin to comprehend the depth of love and commitment that my parents, Samir and Nadia Azer, poured into me throughout the years Both my parents are very special to me, and I thank them for their enduring sacrifices and their persistent commitment to providing me with an excellent education all the way through college

My brother Joe is an immensely special person to me His sincere and caring

character are his trademarks I want to thank him for his encouragement, his

love and support throughout the years, and especially during the time I have been studying for my PhD

I have lived during a significant part of my college years with my grandparents Michele and Clementine Fares They have been a tremendous help, encouragement and support to me during those times They are both very dear and special to me I remember today and always will remember the many times I sat with my late grandfather for hours discussing and reflecting on deep and profound thoughts What stood out most about him was his love for history and beautiful poetry They have both had a significant impact on my life, and

I would like to thank them for their consistent and enduring love

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family I would also like to thank my grandparents Raphael and Madeleine Azer, for their sustaining love, and their commitment to encourage me and help

me along the way

I would also like to thank the rest of my family They have each had their turn in encouraging me, teasing me, and making life enjoyable and fun I would like to specifically thank Hani Fares and family, my cousins George and Joe and the Naggiar family

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Abstract

In this thesis, we present a one-dimensional model for blood flow in arteries, without assuming an a priori shape for the velocity profile across an artery We combine the one-dimensional equations for conservation of mass and momentum with the Womersley model for the velocity profile in an iterative way The pressure gradient of the one-dimensional model drives the Womersley equations, and the velocity profiles calculated then feed back into both the friction and

nonlinear parts of the one-dimensional model Besides enabling us to evaluate

the friction correctly and also use the velocity profile to correct the nonlinear terms, the velocity profiles play a central role in the calculation of the effective diffusion coefficient, and convection coefficient, in the theory of Taylor diffusion We present flow simulations using both structured tree and pure resistance models for the small arteries, and compare the resulting flow and pressure waves under various friction models Moreover, we present several simulations where some arterial tree characteristics were altered to model two disease conditions, namely hypertension and atherosclerosis

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derive a time-dependent diffusion equation for the mean concentration profile along the axial-direction in a pipe In the special case of time-independent flow, our result reduces to that of Taylor

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Contents

Dedication iv

Acknowledgements Vv

Abstract viii

List of Figures xiii

List of Tables xviii

1 Introduction 1

1.1 One-dimensional blood flow modeling with velocity profile 1 1.2 Taylor diffusion in time-dependent flow 4 1.3 Time-dependent Taylor diffusion driven by Velocity Profiles from

One-Dimensional Blood Flow Model 7

2 Background 9

2.1 Physiological Background 0.00000 ees 9 2.2 Mathematical Modeling Background 20

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3.1 One-Dimensional Large Arteries Model: Assumptions and Equa-

3.2 Womersley Theory 0 0 eee ee ee 27

The Integrated Model: 1D Model with Womersley Velocity

Profile 30

4.1 Matching the One-Dimensional Model with the Womersley Model 31 4.2 The One-Dimensional Model and the Womersley Model as One

Iterative Model for Solving the Equations of Blood Flow in Arteries 34

Boundary Conditions 37

5.1 Characteristics and Riemann invariants 37 5.2 Infow and biurcaton condiions 42

5.3 Structured Tree Outfow Condiion 44

Numerical Methods for the Integrated Model 51

6.1 Numerical Scheme 0.0000 0p eee eee 52 62 Convergence Results 0.0 0000 eee eee 54

Results Comparison of Model with Experiment, Comparison of Different Versions of the Model and Disease Modeling 61 7.1 Comparison of Simulation with and without Womersley Profiles 67 7.2 Comparison of Various Friction Models .2 72 7.3 Comparison of Structured trees and Pure Resistance Boundary

Conditions © cv kg vQ 75

7.4 Physiological Validations of the Simulated Flow 78 7.5 Effects of Altered Arterial Compliance on Pressure and Flow

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7.6 Effects of Altered Small Vasculature Resistance on Pressure and Flow Waveforms 00 eee es 91

7.7 Effects of Arterial Narrowing and Stiffening: Late Stage Atheroscle-

TOSIS 2 aaa Ẽ 94

7.8 Conclusions about Modeling the Arterial Pulse 101

8 Taylor Diffusion in Time-Dependent Flow [2] 102

8.1 Derivation of the Generalized Taylor Diffusion Equation 102

8.2 Application 0 ni 108

8.3 Conclusion 2 aqaa q aẶD 110

9 Taylor Diffusion Integrated with Blood Flow Model 114 9.1 Combining Taylor Diffusion with Blood Flow Model 115 9.2 Numerical Methods for Convection-Diffusion Equation 118 9.3 Convergence Results 2 0 ee ee 124 9.4 Taylor Diffusion Combined with Blood Flow Simulation in a Vessel130

10 Conclusion and Future Work 144

10.1 Conclusion ee 144

05281nn 20) 2 es 147

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List of Figures 2.1 2.2 2.3 2.4 2.5

General Structure of Arteries and Veins Reprinted from Text/Atlas

of Histology, Leeson, Leeson and Paparo, Page No 315, Copy-

right (1988), with permission from Elsevier 15 Comparison of Artery (right) and Vein (left) Histology Reprinted

from Text/Atlas of Histology, Leeson, Leeson and Paparo, Page No 321, Copyright (1988), with permission from Elsevier 16

Tunica Media of an Elastic Artery (Aorta) The Lumen (L) and Tunica Adventitia (TA) are shown with a thick Tunica Media Section in between them Reprinted from Text/Atlas of Histol- ogy, Leeson, Leeson and Paparo, Page No 316, Copyright (1988),

with permission from Elsevier 0.0.0.0 ee uee 17 Pressure Distribution in the Circulation Reprinted from Text- book of Medical Physiology, Vol 10, Guyton and Hall, Page No 145, Copyright (2000), with permission from Elsevier 18 Systemic and Pulmonary Circulations Reprinted from Textbook of Medical Physiology, Vol 10, Guyton and Hall, Page No 145,

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4.1 4.2 5.1 5.2 5.3 6.1 6.2 6.3 6.4 6.5 6.6 7.1 7.2

Matching of the One-Dimensional Flow with Womersley Flow when Womersley Model is driven by Dynamic Pressure Gradient Matching of the One-Dimensional Flow with Womersley Flow when Womersley Model is driven by Static Pressure Gradient Plus and minus Characteristics at the point of intersection in a bifUIrcation c c c Q Q Q vn ng kg kg ke va Separation of fow and vortex Íormation at the entrance region

of an expansion in area, such as occurs at a typical bifurcation

in the arterlal trĐ ee ee

Impedance boundary condition at the outflow from the left com- mon carotid artery, 60 ee

Input flow at the heart used in the convergence study

Ll and L2 convergence in the main branch

L1 and L2 convergence in daughter branch 1

L1 and L2 convergence in daughter branch 2

Pointwise convergence along the three-vessel tree used in the con- Vergence stUdy Qua Stencil of the two-step Richtmeyer version of the Lax-Wendrof method — Q Q Q Q Q Q Q Q vn gu v.v kg V va Input flow at the heart used in the simulation Data was mea- sured in the ascending aorta, approximately 2 cm into the vessel This data was provided by Mette Olufsen and recorded by E.M Pedersen and Y Kim at Skejby University Hospital in Denmark Illustration of the arterial tree simulated

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7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16

The systemic circulatory system Reprinted from http://users ren.com/jkimball.ma.ultranet /BiologyPages/C/Circulation.html,

with permission from Dr John W.Kijimbal Comparison of Flow with and without Womersley Profiles

Comparison of Pressure with and without Womersley Profiles

Velocity profiles at various locations in the arterial tree simulated using Womersley model combined with 1D model Comparison of flow waveforms simulated using different friction

modl§ 0Q LH ng gà gà vn

Comparison of pressure waveforms simulated using different fric-

tion models 2 ee

Comparison of flow waveforms simulated using two different models for the small arteries 2 - c c Q cv

Comparison of pressure waveforms simulated using two different models for the small arteries 2 0 ee ee Steepening of the pressure waveform as distance form the heart INcreaseS 6

Slight decrease in mean pressure as distance from the heart in-

CIEASES 6 Q Q Q Q Q Q Q LH Q Q r nu vn g v g v v v v kg va

Three-Dimensional View of the Pressure as a Function of Space and Tỉme uc Q Q Q c n cv Q vn g g v và và va Effect of Decreased Compliance on Pressure Waveform Effect of Decreases Compliance on Flow Waveform Effect Of Changing the Compliance of the Large Arteries on the Flow Waveforms in the Large Arteries

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7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25 7.26 8.1 8.2 8.3 9.1 9.2 9.3

Effect Of Changing the Compliance of the Small Vasculature on the Flow Waveforms in the Large Arteries 88 Effect Of Changing the Compliance of the Large Arteries on the

Pressure Waveform 2 va 89

Effect Of Changing the Compliance of the Small Vasculature on the Pressure Waveforms in the Large Arteries 90

Effect of Doubling Peripheral Resistance on Pressure Waveform 92 Effect of Doubling the Peripheral Resistance on Flow Waveform 93

Comparison of Flow in the Normal Arterial Tree with an Arterial

Tree where the Carotid Artery was Occluded 96 Flow in an Arterial Tree with an Occluded Carotid Artery 97

Comparison of Pressure in the Normal Arterial Tree with an Ar- terial Tree where the Carotid Artery was Occluded 98 Comparison of Flow in the Normal Arterial Tree with an Arterial

Tree where the Abdominal and Thoracic Aortas were Occluded 99

Comparison of Pressure in the Normal Arterial Tree with an Arte-

rial Tree where the Abdominal and Thoracic Aortas were Occluded100

Comparison of convection-diffusion solution with Taylor solution 111

Comparison of convection-diffusion solution with Taylor solution

in the slug flow limil ee en 112

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9.4 Pressure at a given x-location in the Brachial Artery 9.5 Velocity profiles at a given x-location in the brachial artery 9.6 Propagation of concentration distribution through the brachial artery; D = 0.1 =Ẻ Ko ủ cá cà ng cà kg kg KV NV ki k KỀA 9.7 Propagation of concentration distribution through the brachial

artery; D= joe ae

9.8 Propagation of concentration distribution through the brachial

artery, D=10m 0Q uc va vo

9.9 Effective diffusion coefficient; D = 0.1 8ˆ ¬

9.10 Effective diffusion coefficient; D = ya ee 9.11 Effective diffusion coefficient; D = 10 Ce 9.12 Effective diffusion coefficient as a function of space at various times; D = 19s" Ce ki và

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List of Tables

7.1 This table contains the arterial tree dimensions used in the sim-

ulations The data in this table was provided by Mette Olufsen [48] The radius used for each vessel is the average of inlet radius

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Chapter 1

Introduction

1.1 One-dimensional blood flow modeling with velocity profile

The goal of this work is to use a one-dimensional model to study the flow of blood through the arterial system Our model allows us to solve for the flow and pressure waveforms at any point along the arterial tree, but also, as we shall see later, to solve for the velocity profile at any particular cross-section of an artery in the tree These profiles respond to changes in the characteristics of the arterial tree, such as arterial wall compliance, in a way that distinguishes normal and abnormal behaviors

Arteries are the means by which oxygenated blood is carried away from the heart and distributed to body tissue Physiologically, blood flow through tis-

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decelerate the flow in these low-resistance vessels [29, 50] The arterial tree is

a complex system of vessels, branching at bifurcation points The vessels are tapering, narrowing as one follows a vessel downstream away from the heart The vessel walls are compliant, maintaining flow through them at all times The compliant vessel walls store volume during systole and return that volume during diastole, thus maintaining a more nearly steady flow than would other- wise be the case The total cross-sectional area of the aorta is about 2.5 cm?, which then branches out to a total of approximately 20 cm? at the level of the

small arteries, and 40 cm? at the level of the arterioles (10, 26] Thus, at each

bifurcation, the total cross-section of the daughter branches is greater than the cross-section of the parent branch, even though the cross-section of each indi- vidual daughter branch is less than that of the parent Flow through the arterial system is pulsatile, because of the beating of the heart, and is met with resistance at the distal ends of the arterial tree This resistance is generated by the arterioles because of their relatively small cross-sectional areas

Studying blood flow in arteries and veins has many direct applications in medicine, including hypertension and atherosclerosis These conditions may both develop in patients with diabetes An early sign of diabetes is a change in the waveform of the arterial pulse: the disappearance of a secondary peak known as the dicrotic wave [20, 37] This was discovered by Henry Lax nearly

50 years ago Accurate simulation of the arterial pulse is therefore important

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dimensional models are used when there is the need for computing the wall shear stress When geometric efects are not the main interest, our one-dimensional model can be used, instead of a three-dimensional model to compute shear stress at the wall This is because we use Womersley theory in conjunction with the

one-dimensional model to obtain local, time-dependent velocity profiles The

main advantages of using a one-dimensional model are speed and ease of use, but also being able to simulate a more complete arterial network as opposed to a small section of the tree This enables one to study the changes in the wall shear stress in diseased arterial networks, as well as the wave propagation effects in such networks

Estimating the velocity profile has many advantages over simply assuming a certain profile throughout the arterial tree Knowing the velocity profile allows us to compute the flow throughout an arterial tree more accurately because of the more accurate representation of the velocity profile Moreover, other than being able to estimate the shear stress at the wall and correcting the nonlinear terms with the profile information, the velocity profile can also be used when modeling the transport of solute throughout the arterial tree using time-dependent Taylor diffusion [2] The velocity profile across arteries is useful in other applications, such as the mass transport of lipids to and from the arterial wall It is therefore of great importance to be able to estimate the velocity profile at any given location along an artery at any given time during the cardiac cycle

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This gives a more accurate and realistic estimation of the velocity profile and therefore also of the shear stress at the wall The one-dimensional theory allows us to solve for the pressure and flow along an artery, assuming a flat velocity profile Womersley theory gives the actual velocity profile, under the assumption that flow is independent of axial position along an artery Even though these two theories might seem to be mismatched, they prove to work well together In particular, the one-dimensional theory generates the pressure gradient that is needed as input to the Womersley theory, and the Womersley theory provides the velocity profile which is used in the computation of the friction and nonlinear terms of the one-dimensional theory

In this thesis, we present a numerical algorithm for solving one-dimensional blood flow equations and show how to calculate velocity profiles and also how to use the information from these profiles to correct the convection term in the momentum equation We show that the algorithm is second order accurate and present a simulation based on experimental data from humans We also present comparisons of flow and pressure estimated using both structured tree and simple resistance models and under various friction models The comparisons show that structured trees have superior properties when modeling flow in an arterial tree over a simple resistance method

1.2 ‘Taylor diffusion in time-dependent flow

In 1953, Sir Geoffrey Taylor considered the problem of measuring the concen-

tration of a solute in a slowly moving flow in a rigid pipe [61] Taylor assumed a Poiseuillian velocity profile of the form uo(1 — ), where ug is the maximum

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coordinate He showed that in the frame of reference moving with velocity +, the concentration of the solute satisfied the diffusion equation, with a diffusion coefficient of oe where D is the coefficient of molecular diffusion for the solute

In this thesis, we generalize Taylor’s result to time dependent flows and derive

the corresponding diffusion equation for the mean concentration of the solute

in an axisymmetric setting Some related work includes the dispersion of solute

in turbulent flow in a pipe (62], dispersion of solute in a pipe where the distri-

bution of the solute is described in terms of its moments in the flow direction

[57], studies of the dispersion coefficient of a chromatographic system where the velocity field is time periodic [27, 30] and a revision of the Taylor limit where

the validity of the theory is extended to better resolution and earlier times [63]

Calculating the concentration profile of an agent at a given site of action is important in certain Magnetic Resonance Imaging (MRI) studies Consider a

drug that is fast-acting (compared to total circulation time) that is administered

in some vein As an example, take the target area to be some region in the brain Using our time-dependent Taylor diffusion equation and supposing one has a complete circulatory model, one can calculate the concentration profile at the

site of action in the brain Dynamic susceptibility contrast (DSC) MRI studies

for imaging the brain are examples of studies where knowing the concentration profile of the administered agent at the site of action is important In DSC MRI, a bolus injection of a paramagnetic contrast agent is administered, and its passage through the vasculature is monitored by serial measurement of the MR signal loss in the surrounding tissue The scan time is typically very short, on the order of a few seconds, and the time it takes for an administered agent

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used to introduce contrast in the MRI image Gd-DTPA does not penetrate the blood brain barrier, but stays in the vasculature The primary interests in a DSC MRI study is to extract the blood volume map and perfusion map for the region of interest Since the scan time and the time taken to reach the site of action

from the site of administration is short (a few seconds) compared to the time of one complete circulation (approximately one minute in humans), how the agent

is diffused in the vasculature is important when calculating the concentration

profile In calculating the cerebral blood flow (CBF), the concentration profile

of interest is the concentration of contrast agent entering the region of interest,

and is often referred to as the arterial input function (AIF) The AIF is typically

estimated from a major artery, such as the middle cerebral artery, and assumed to be the input to the tissue However, ignoring dispersion and delay effects

that take place from the site of measurement of the AIF to the tissue site can cause significant errors in the quantification of CBF [8, 7] Since DSC MRI

is commonly used in clinical studies of cerebral ischemia, these errors lead to inaccurate information on stroke Using the time-dependent Taylor diffusion equation we derive, and a circulation model, one can calculate the AIF at the site of action and therefore derive more accurate perfusion maps

Researchers at the Toshiba Stroke Research Center and Aerospace Engineer-

ing and Radiology used an insoluble agent to improve flow measurement and

quantification of arteriovenous malformation (AVM) [71] The improvement on

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of the soluble agent is driven both by convective transport and diffusion Suppose a substance is present at time t = 0 with initial concentration f(z) in a rigid straight pipe, where the velocity field is given by V(r,t;a) and z is the distance along the axis of the pipe The spread of the substance in the fluid is primarily due to the combined effect of molecular diffusion in the radial direction and convection parallel to the axis of motion Assume that we are in a regime where variations in the concentration caused by convective transport occur in a time frame that is much slower than the time it takes for appreciable radial concentration differences to dissipate We show that the axial concentration profile of the substance is dispersed according to a process that obeys the

diffusion equation, with a certain variable diffusion coefficient, when considered

in a particular frame of reference The general time-dependent diffusion equation we derive reduces to Taylor’s result in the time independent case We then apply the resulting generalized equations to calculate the mean concentration along a pipe where the velocity field is one that comes from axisymmetric flow

with a pressure gradient that is sinusoidal in time

1.3 Time-dependent Taylor diffusion driven by

Velocity Profiles from One-Dimensional Blood

Flow Model

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the wall The other is to correct the nonlinear terms in the momentum equation In the second part of this thesis, we show how Taylor’s result on dispersion of

a solute in a rigid pipe carrying steady flow can be generalized to one having unsteady flow [2] In the last part of this thesis, we show how to use the velocity

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Chapter 2 Background

2.1 Physiological Background

The history of the circulatory system, and the early history of medicine and science in general is tremendously interesting and filled with intriguing sto- ries about ancient myths and the struggle of man to grasp the complexity and intricacy of the general circulatory system In 1628, William Harvey wrote “Movement of the Heart and Blood in Animals”, where much of the general functions of the heart and the arteries, as we know them today, were first de-

scribed (28, 21] Through many examinations of the anatomy of animals, Harvey

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mathematical perspectives [6] Some of these are Hales, known to first measure arterial pulse and the Windkessel model for arterial resistance; Euler, who established the general equations of fluid motion for ideal fluids and applied them to blood flow in arteries; Cauchy, who developed the general formulation, which applies to all continua [59]; D Bernoulli, who formulated what is known today as the Bernoulli equations; Navier, who first found the Navier-Stokes equations as we know them today from model studies and Stokes who later presented a correct derivation for the equations; Poiseuille, who established Poiseuille’s law

and Young who studied and contributed to the theory of elasticity [54]

2.1.1 The Circulatory System

The circulatory system consist of two main components: the blood vascular system and the lymph vascular system The lymph system begins in the tissue, consists of capillaries and different sized lymph vessels which return lymph from the tissue space back into the blood stream via the large veins in the neck The blood circulatory system is composed of the systemic circulation, also known as the greater or peripheral circulation, and the pulmonary circulation The pulmonary circulation supplies blood to the lungs The systemic circulation supplies blood to the rest of the body, hence the name greater circulation The functional parts of the circulation are

1 Arteries 2 Arterioles 3 Capillaries 4 Venules

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The main functional distinction between arteries and veins is that arteries carry blood away from the heart while veins carry blood towards the heart Arteries form an efferent network of vessels Anatomically, arteries have thicker stronger walls, and are much less distensible than veins Arteries carry blood under much higher pressure than do veins, as shown in figure 2.4 Veins have thin walls and serve the purpose of a blood reservoir Figure 2.2 is a histological comparison of a typical artery on the right and vein on the left One can see that the media section of the artery, where there are smooth muscle and elastic tissue fibers, is thicker than the media section of the vein

Arterioles act as control conduits of blood from the arteries into the capillaries The muscular walls of the arterioles greatly change the amount of blood going into the capillaries, having the ability to greatly dilate or completely close down The flow is controlled by structures known as precapillary sphincters Arterioles cover a total of approximately 40 cm? throughout the circulatory

system [26]

Capillaries have the thinnest walls, composed of endothelium They are cat- egorized into three groups, based on the structure of the endothelial cell wall The three groups are continuous, fenestrated and sinusoidal Their functional

role is to provide nutrients, oxygen and hormones carried in the blood to the

interstitial fluid Their walls are permeable to allow for the transport of nutrients and small molecules Capillaries are so narrow that only a single red blood cell can go through at a time The diameter of a typical capillary is approx-

imately 5 to 10 ym [39, 5] The total cross-sectional area of the capillaries is approximately 2500 cm? [26] Figure 2.5 shows arteries, veins and the capillary

bed connecting them

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to form larger veins De-oxygenated blood is ultimately carried in to the largest veins, namely the vena cava, through which then it is carried in to the right atrium of the heart Veins range in diameter from 1 mm to 1.5 cm

Artery and vein walls are made up of three layers or coats: 1 Tunica Intima is the innermost coat,

2 Tunica media is the middle coat,

3 Tunica adventitia is the outermost coat,

as shown in figure 2.1 The intima consists of the endothelial lining, a suben- dothelial layer of connective tissue and an internal elastic membrane which is the interface to the tunica media The tunica media, as shown in figure 2.3, is mostly laminae of elastic tissue with scattered smooth muscle cells Elastic and collagen fibers, together with proteoglycans, are found in between the laminae Finally, the tunica adventitia is of approximately the same thickness as the tunica media It is composed mainly of loose connective tissue with longitu- dinally oriented collagen and elastin The tunica adventitia is connected to the

surrounding tissue of the organ through which the vessel is passing In larger arteries, vasa vasorum or small arteries are found in the adventitia (39, 5) Their

purpose is to supply the adventitia and media with nutrients

It is important to note that the composition of the artery wall does change as

a function of distance from the heart [10] Typically, the media is dominated by

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collagen and elastin in the tunica media Collagen fibers are stiffer than elastin fibers When the vessel is subjected to low stress, most of the load is borne by the elastin, since the collagen fibers are relaxed At higher stresses, the collagen

fibers become straight and start taking over control of the overall stiffness of the vessel [10] The wall therefore becomes stiffer at higher strain since collagen

fibers are stiffer than elastin fibers However, arteries are not purely elastic, but are visco-elastic There is a reaction delay of the deformation of the artery wall to a given stress, as opposed to an instantaneous response in the purely elastic case In the visco-elastic arteries, there is an initial response, which is

then followed by a continued deformation to the initial stress It is thought that mainly the smooth muscle cells and also the collagen fibers in the tunica media

of the artery wall are responsible for the visco-elastic behavior of artery walls

[10]

Blood vessels are the conduits through which blood circulates the body Blood makes up about 7 % of the body weight, or about 5L in volume Plasma is the liquid part of blood; the solid part is made up of three cell types, namely erythrocytes, leucocytes and thrombocytes Erythrocytes, also known as red blood cells, are what give blood its distinctive red color There are typically 4

to 6 million red blood corpuscles per mm? of blood volume [5] and they com-

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500 thousand platelets per mm? of blood volume [10] The primary function of platelets is to coagulate to minimize hemorrhage through a clotting process and to patch local damaged areas in the endothelial lining of the vessel wall The progression of this localized patching of platelets results in what is known

as a thrombus Leukocytes are colorless cells, also known as white blood cells

There are approximately 4 to 11 thousand leukocytes per mm’ of blood volume

There are five kinds of leukocytes with approximately constant proportions [5]:

1 Neutrophils - 55 to 60 percent Eosinophils - 1 to 3 percent Basophils - 0 to 0.7 percent mm 6® Lymphocytes - 2ð to 33 percent 5 Monocytes - 3 to 7 percent

Leukocytes play a crucial role in the fight against infection by forming antibodies

and eliminating micro-organisms

The mechanical properties of blood are determined primarily by red blood

cells At low shear rates (less than 1 s~!), blood exhibits a non-Newtonian

behavior with apparent viscosity being shear-dependent At physiological shear

rates, however, apparent viscosity of blood asymptotes to a constant value [4]

Blood is typically considered as a suspension of flexible red blood cells in a Newtonian fluid, namely the plasma It is partly the red blood cells that give blood its non-Newtonian character at lower shear rates

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Subandothotial intemal lọrnal

contnocllve tlesue elastic membrane elastic membrane

€ndothenu

Tualca tim:

Tunica media 4

về

Tunica advsnttia4 li cae Hann Yaga

i vasorum

Figure 2.1: General Structure of Arteries and Veins Reprinted from Text/Atlas

permission from Elsevier

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Figure 2.2: Comparison of Artery (right) and Vein (left) Histology Reprinted

from Text /Atlas of Histology, Leeson, Leeson and Paparo, Page No 321, Copy- right (1988), with permission from Elsevier

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Figure 2.3: Tunica Media of an Elastic Artery (Aorta) The Lumen (L) and Tunica Adventitia (TA) are shown with a thick Tunica Media Section in between

them Reprinted from Text/Atlas of Histology, Leeson, Leeson and Paparo,

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Ề i : s oe | 2 hết 2 ¢ 1Ề!,l1313 L3: Beek pe ee: E 233 a eB ey -8 a Ề ¢ Ÿ 8 gS 7h 818 3 : : ỹ : 8, Systemic Pulmonary

Figure 2.4: Pressure Distribution in the Circulation Reprinted from Textbook of Medical Physiology, Vol 10, Guyton and Hall, Page No 145, Copyright

(2000), with permission from Elsevier

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Pulmonary circulation—9%

Aorta

Superior

vena cava Heart—7% — Tô

Arteries~=13%

Inferior Systemic

vena cave vessels

Arterioles and capilanasen7% Veins, venules, and vanous Sinuses—64%