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Annu. Rev. Biomed. Eng. 1999. 01:299–329Copyright 1999 by Annual Reviews. All rights reserved

1523–9829/99/0820–0299$08.00 299

Fluid Mechanics of Vascular

Systems, Diseases, and Thrombosis

David M. Wootton1

and David N. KuG.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology,

Atlanta, Georgia 30332–0405; e-mail: [email protected],

[email protected]

Key Words platelets, shear, arteriosclerosis, stenosis, intimal hyperplasia

Abstract The cardiovascular system is an internal flow loop with multiplebranches circulating a complex liquid. The hallmarks of blood flow in arteries arepulsatility and branches, which cause wall stresses to be cyclical and nonuniform.Normal arterial flow is laminar, with secondary flows generated at curves andbranches. Arteries can adapt to and modify hemodynamic conditions, and unusualhemodynamic conditions may cause an abnormal biological response. Velocity profileskewing can create pockets in which the wall shear stress is low and oscillates indirection. Atherosclerosis tends to localize to these sites and creates a narrowing of the artery lumen—a stenosis. Plaque rupture or endothelial injury can stimulate throm-bosis, which can block blood flow to heart or brain tissues, causing a heart attack orstroke. The small lumen and elevated shear rate in a stenosis create conditions thataccelerate platelet accumulation and occlusion. The relationship between thrombosisand fluid mechanics is complex, especially in the post-stenotic flow field. New con-vection models have been developed to predict clinical occlusion from platelet throm-bosis in diseased arteries. Future hemodynamic studies should address the complexmechanics of flow-induced, large-scale wall motion and convection of semisolid par-ticles and cells in flowing blood.

CONTENTS

Introduction ..................................................................................... 300Physiologic Environment.................................................................... 300

Flows in Specific Arteries................................................................... 302

The Carotid Arteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

The Aorta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Flow at the Left Coronary Artery Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Flows in the Heart and Great Vessels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Biological Responses to Hemodynamics ............................................... 305

Hemodynamics of Stenoses ................................................................ 308

1Department of Biomedical Engineering, Johns Hopkins University School of Medicine,

Baltimore, Maryland 21205



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Diagnosis of Disease. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Shear-Dependent Thrombosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Arterial Thrombosis .......................................................................... 310

Cellular and Molecular Mechanisms of Thrombosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Hemodynamics and Thrombosis ...... ...... ..... ...... ...... ...... ...... ...... ..... ...... 312

Hemodynamics in Advanced Atherosclerosis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Hemodynamics and Thrombus Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Shear and Platelet Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313Shear-Linked Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Modeling Clinical Thrombosis ..... ...... ..... ...... ...... ...... ...... ...... ...... ..... ... 317

Model Based on Ex Vivo Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

A Model of Occlusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Conclusions ..................................................................................... 322

INTRODUCTION

Nutrient and waste transport throughout the body is the primary function of the

cardiovascular system. The heart serves to pump blood through a sophisticatednetwork of branching tubes. The flow is not steady but pulsatile. The blood vessels

distribute blood to different organs while maintaining vessel integrity. The arteries

are not inert tubes but adapt to varying flow and pressure conditions by growing

or shrinking to meet changing hemodynamic demands.

It is important to study blood flows during disease as well as under normal

physiologic conditions. The majority of deaths in developed countries are from

cardiovascular diseases. Most cardiovascular diseases are associated with some

form of abnormal blood flow in arteries. This review focuses on some selected

areas of importance to cardiology.

PHYSIOLOGIC ENVIRONMENT

The fluid blood is a complex mixture of semisolid and liquid material. Blood is

composed of cells, proteins, lipoproteins, and ions by which nutrients and wastes

are transported. Red blood cells (RBCs) typically comprise 40% of blood by

volume. In most arteries, blood behaves in a Newtonian fashion, and the viscosity

can be taken as a constant 4 centipoise (cP) for a normal hematocrit. The non-

Newtonian viscosity is extensively studied in the field of biorheology and has

been reviewed by others (e.g. 21, 89).

Blood flow and pressure are unsteady. The cyclic nature of the heart pump

creates pulsatile conditions in all arteries. The aorta serves as a compliance cham-

ber that provides a reservoir of high pressure during diastole as well as systole.

Flow is zero or even reversed during diastole in some arteries such as the external



FLUID MECHANICS AND THROMBOSIS 301

carotid, brachial, and femoral arteries. These arteries have a high distal resistance

during rest, and flow is on/off with each cycle. Flow during diastole can also be

high if the downstream resistance is low, as in the internal carotid or the renal

arteries.

Pulsatile flows dominate many of the problems in the cardiovascular system.

The existence of unsteady flow forces the inclusion of a local acceleration term

in most analyses. In contrast to unsteadiness, several features of biological flows

may often be neglected as being of secondary importance for particular situations.

These include vessel wall elasticity, non-Newtonian viscosity, slurry particles in

the fluid, body forces, and temperature. Although each of these factors is present

in physiology, the analysis is greatly simplified if they can be justifiably neglected,

which is the case in most arterial flows.

Biologists are often concerned with the local hemodynamic conditions in a

particular artery or branch. The important fluid mechanic parameter is often a de-

tailed local description of the fluid-wall shear stress in a blood vessel for a given

pulsatile flow situation. The three-dimensional nature of many of these unsteady

flows has provided an important challenge to computational methods, because

the computational time required is enormous.

The arterial system is tortuous and must branch many times to reach an end

organ. The cross-sectional area along the axis may enlarge at branch points,sinuses, and aneurysms. However, if the area diverges, the flow must decelerate,

and an adverse pressure gradient can exist. In this situation, flow separation is

possible and typically occurs along the walls of the sinus.

As blood flows across the endothelium, a shear stress is generated to retard

the flow. The wall shear stress is proportional to the shear rate c (velocity gradient)

at the wall, and the fluid dynamic viscosity l: s lc. Shear stress for laminar

steady flow in a straight tube is

1 3s 32lqp D ,

where q is volume flow rate, and D is tube diameter. This approximation is a

reasonable estimate of the mean wall shear stress in arteries. For situations in

which the lumen is not circular or the blood flow is highly skewed, as it is at

branch points, shear stress must be determined by detailed measurements of veloc-ity near the wall. Shear stress is not easily measured for pulsatile flows. The

velocity and velocity gradient must be measured very close to a wall, which is

technically difficult. The gradient will depend highly on the shape of the velocity

profile and the accurate measurement of distance from the wall. For blood flow,

the viscosity very near a wall is not precisely known because the red cell con-

centration is reduced. Thus, arterial wall shear stress measurements are estimates

and may have errors of 20%–50%.

At the lumenal surface, shear stress can be sensed directly as a force on an

endothelial cell. In contrast, cells cannot sense flow rates directly. Determination

of the flow rate would require knowledge of blood velocities far away from cells

in the artery wall, as well as some way to integrate the velocities to give the



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volume flow rate. Thus, it is natural for endothelial cells to sense and respond to

shear stress.

Arteries will typically adapt to maintain a wall shear stress of 15 dyn/cm2

(41). This appears to be true for different arteries within an animal, between

animal species, as well as after large changes within a single artery. The blood-

wall shear stress modulates diameter adaptive responses, intimal thickening, and

platelet thrombosis. The wall shear stress is thus central to the vascular response

to hemodynamics.

The other major hemodynamic force on an artery is the transmural pressure

across the thickness of the wall. Arteries have a mean pressure of 100 mmHg,

whereas veins have pressures of 10 mmHg. The hoop stress can be estimated

by Laplace’s Law as

1r 0.5PDt ,

where t is wall thickness, D is vessel inner diameter, and P is transmural pressure,

for vessels with circular lumens that are not too thick (38). It is possible that the

primary determinant of smooth muscle cell response is the local strain of these

cells. The arterial wall may remodel in response to both static and cyclical loading

conditions by secretion and organization of collagen and elastin, respectively(88).

FLOWS IN SPECIFIC ARTERIES

There are four major arteries that are subject to the most clinical disease. These

include the carotid bifurcation, the abdominal aorta, the left coronary artery, and

the heart and proximal aorta.

The Carotid Arteries

The carotid arteries are located along the sides of the neck. These arteries supply

the brain and face with blood. Atherosclerosis, which develops right at the bifur-

cation, causes the majority of strokes in patients. The branch is unique in that

there is an anatomic sinus or expansion at the origin of the internal carotid. Themean Reynolds number is300, and the Womersley parameter is4. The daugh-

ter branches are 25 off-axis of the parent artery, on average.

Measurements of velocity have been made in machined plastic models of this

bifurcation by laser Doppler anemometry (65). Secondary flows are produced

downstream of the bifurcation (Figure 1). Velocity profiles obtained by laser

Doppler anemometry and computational fluid dynamics quantify the extent of

reverse velocities at the outer wall of the internal carotid sinus (Figure 2). A

region of transient flow separation is created along the posterior wall of the carotid

sinus, which is prominent during the downstroke of systole. The artery wall in

the sinus region would experience oscillations in near-wall velocity and a low

mean wall shear stress. Atherosclerotic plaque is highly localized to a small area




FIGURE 1 Hydrogen bubble visualiza-

tion of flow through a model carotid bifur-

cation illustrating the laminar flow at the

flow divider and separation of flow at the

posterior wall of the internal carotid sinus.

The separation region of transient reverse

velocities is also the site of secondary vor-

tex patterns. (Reprinted with permission of the American Heart Association, Inc.)

within this sinus region and correlates with low wall shear stress with coefficients

greater than 0.9, p 0.001. Comparison of the unsteady, three-dimensional in

vitro results against in vivo measurements with Doppler ultrasound confirms that

the assumptions of the modeling are valid (66). Several groups have recently used

computational fluid dynamics to study the effects of wall elasticity and non-

Newtonian viscosity (4, 86). These effects are small in comparison with the ana-

tomic and flow variations between patients (79, 83).

The Aorta

The aorta is the large vessel from the heart that traverses the middle of the abdo-

men and bifurcates into two arteries supplying the legs with blood. The renal

arteries have a low resistance so that two-thirds of the entering flow leaves the

abdominal aorta through these branches at the diaphragm. Curiously, atheroscle-

rotic disease extends along the posterior wall of the relatively straight abdominal

aorta downstream of the renal arteries in all people. Little disease is ever present

in the upstream thoracic aorta.

In vitro measurements in a glass-blown aorta model show that outflow con-

ditions combine with curvature to create an oscillation in velocity direction at the

posterior wall of the aorta, with a corresponding low average wall shear stress

(77). The area of low wall shear stress correlates very well with the location of



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FIGURE 2 a. Axial velocity

profiles in the sinus region of a

three-dimensional model of the

carotid bifurcation, using laser

Doppler anemometry ( LDA) and

computational fluid dynamics

(CFD). b. Flow in the carotid

sinus is unsteady with a transientreverse flow at the outer wall

shown in this three-dimensional

plot of velocity vs diameter posi-

tion and time. (Reprinted from 65

with permission from Elsevier

Science, Ltd.)

atherosclerotic plaque measured in autopsy specimens, p 0.001 (35, 77). As

verification, measurements of in vivo flow in humans exhibit the same skewing

and time-varying velocity profiles as are produced in the model (76).

Flow at the Left Coronary Artery Bifurcation

Flow at the left coronary artery bifurcation is complicated by several important

features (10). First, the left main coronary artery is quite short, leading to an

entrance type flow at a small Womersley parameter of 3. Second, the flow wave-

form in the left coronary artery is reversed in comparison with that in most arter-




ies, having more flow during diastole. Flow can be reversed during systole. The

high pressure in the myocardium during systolic contraction causes the blood

flow to reverse direction in the coronary arteries. Third, the bifurcation does not

lie in a single plane but curves around the heart while branching. The curvatures

likely set up secondary flows during part of the cardiac cycle. The actual fluid

dynamics have been characterized with large-scale experimental models (103)

and spectral-element computational modeling (50, 51). Comparison of the flow

field with maps of atherosclerotic disease locations yields a strong correlation

between oscillations in shear stress and probability of plaque ( r 0.85, p

0.001) (51). Surprisingly, variations in branch angle do not alter the overall flow

field regimes in a dramatic way (50). However, changes in the coronary flow

waveform affect the magnitudes of oscillation significantly (50).

Flows in the Heart and Great Vessels

Flows in the heart and great vessels are dominated by inertial forces over viscous

forces. Reynolds numbers at peak systole are 4000. The flow in the aorta and

pulmonary trunk is similar to an entrance-type flow, which is not developed.

Consequently, the core of the flow can be considered as an inviscid region away

from a developing boundary layer at the wall. The pressure and velocity patterns

in a complex chamber of the heart can be modeled in three dimensions, evenincluding a moving boundary condition that develops tension (80, 113).

The analysis of hemodynamics in this representative set of arteries enables

one to develop a general understanding of the fluid mechanics in the normal

cardiovascular system. It should be remembered that arteries are not fixed tubes.

They are biological organs, which remodel over time.

BIOLOGICAL RESPONSES TO HEMODYNAMICS

The artery reacts to the dynamic changes in mechanical stress. Several physiologic

responses are essential to the maintenance of normal functioning of the circulatory

system. The responses of arteries to the hemodynamic environment may createnormal adaptation or pathological disease.

Hemostasis is the arrest of bleeding. Trauma is a common occurrence, and the

body must be able to deal with this possibility. In this hemodynamic environment

of high shear stresses, hemostasis is maintained primarily by platelet adherence

and activation. Platelets pass quickly over the injury site, and adherence must

occur in milliseconds.

On a longer time scale, an artery can respond to minute-to-minute changes in

hemodynamics. The blood vessels must adapt to differing physiologic demands

and conditions from changes in blood pressure and flow. This response is typically

governed by the need to control systemic vascular resistance, venous pooling,

and intravascular blood volume.



306 WOOTTON KU

Arteries adapt to long-term increases or decreases in wall shear stress. The

response to increased wall shear stress is to vasodilate and then remodel to a

larger diameter with the same arterial structure. This situation is commonly seen

with the creation of an arteriovenous fistula for hemodialysis access. Decreased

flow rates will cause a thickening of the intimal layer to reestablish a normal wall

shear stress (41). Eventually, the artery may maintain a thickened intima or

remodel to a normal artery of smaller diameter.

On an even longer time scale of weeks to months, arteries will remodel their

intima and media layers. The medial thickness is influenced by the local amount

of hoop stress and nutrition. As described above, as the blood pressure increases,

the hoop stress will proportionally increase (22). Because the formation of a

lamellar unit requires the proliferation of smooth muscle cells and the creation of

a highly organized extracellular structure, the process may take several days.

Alterations in the pulsatile pressure lead to changes in organization of the elastin

and collagen structure within the media (41, 88).

The effects of flow, shear stress, and stretch on arteries in vivo have been

studied by several groups. Flow can be augmented through an artery by the crea-

tion of an arteriovenous fistula. Such increased flow causes a dilation of the artery

until the wall shear stress reaches the baseline level of the artery (60, 116). This

baseline appears to be

15–20 dyn/cm

2

for most arteries in a wide range of species (41). Conversely, restricted flow through an artery produces a smaller-

diameter vessel (68).

Several pathological states may arise from an excessive or uncontrolled

response to a hemodynamic stimulus. Long-term hypertension produces a gen-

eralized medial thickening of blood vessels. Studies of intimal hyperplasia in a

canine model clearly indicate that low shear stresses can accelerate intimal thick-

ening. Shear stress can also be varied in a single artery by using tapered vascular

grafts with differing diameters. In this case, intimal thickening follows from low

shear stresses even for a constant flow rate as depicted in Figure 3a (94).

Atherosclerotic disease forms over decades. Atherosclerosis is highly localized

to only a few places in the systemic vasculature. The primary locations of ath-

eroma are at the carotid artery sinus, the coronary arteries, the abdominal aorta,

and the superficial femoral arteries. In each of these arteries, there are localizedsites where the mean wall shear stress is very low and oscillates between positive

and negative directions during the cardiac cycle. Comparisons of the sites of

disease with the local hemodynamic conditions reveal a consistent curve where

low wall shear stress is strongly correlated with atherosclerotic intimal thickening

(Figure 3b) (51, 67, 77). Typically, most intimal thickening is found where the

average wall shear stress is 10 dyn/cm2 and follows the curve shape shown for

intimal hyperplasia and arterial adaptation. Thus the biological pattern of arterial

response to shear stress appears to be consistent and preprogrammed.

Currently, a field of cellular and tissue engineering is developing that attempts

to subject cultured cells and tissues to well-defined stresses in an in vitro envi-




FIGURE 3 a. Neointi-

mal hyperplasia thickening

vs wall shear stress in a

dog arterial graft. The

inverse relationship indi-

cates more thickening at

low shear stresses. b. Ath-

erosclerotic intimal thick-ening vs wall shear stress

in human carotid arteries.

The reciprocal relationship

holds for mean and maxi-

mum wall shear stresses

and correlates directly

with oscillatory shear

stress.

ronment. The creation of flow chambers that recreate physiologically realistic in

vivo stresses is an important area of research (53, 75).

The effects of hemodynamics on convective mass transfer should not beneglected. Most biologically active molecules are convected from one site to

another. These molecules may be nutrients, wastes, growth factors, or vasoactive

compounds. Systemic hormones reach an artery by convection and then may

diffuse through the wall, with the intima as a major barrier. However, convective

mass transport may be a limiting factor for small molecules such as nitrous oxide

and oxygen, which diffuse rapidly through the wall. Such convection would be

impaired in areas of flow separation or reversing wall shear at sites prone to

atherosclerosis (70, 71). Alternatively, biologically active molecules released by

endothelial cells may have an effect downstream if the molecules are trapped in

a boundary layer near the wall.



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FIGURE 4 X-ray contrast angiogram of a dis-

eased carotid bifurcation illustrating the focal

nature of a stenosis. The stenosis (arrow) will

reduce blood flow and pressure to the brain. (From

Strandness DE and van Breda A, 1994. Vascular

Diseases: Surgical and Interventional Therapy.

Reprinted with permission of Churchill Living-

stone Inc.)

HEMODYNAMICS OF STENOSES

When arteries become severely diseased, the arterial lumen becomes restricted

over a short distance of about 1 cm. This constriction is commonly referred to as

a stenosis. An example of an atherosclerotic carotid artery stenosis is depicted in

Figure 4.

In clinical medicine, percent stenosis is commonly defined as percent occlusion

by diameter, as follows:

% stenosis (D1D2)/D1 100%,

where D1 is upstream diameter and D2 is the minimum diameter in the stenosis.

As disease advances, the percent stenosis increases.

Stenotic flows have been well characterized by a number of investigators.

Some important summary features are that flow separation (Figure 5b) occurs in

the expansion region at Reynolds numbers of 10 for a 70% stenosis, a strong

shear layer develops between the central jet and the recirculation region, the

critical upstream Reynolds number for turbulence is 300 (114), turbulence

intensity levels reach up to 100% of the upstream velocity values, and the tur-

bulence is high for

1.5–6 diameters downstream (69).For stenoses 75%, flow is limited severely by two mechanisms. Intense tur-

bulence downstream of the stenosis creates large pressure losses. In addition, low

pressure at the stenosis throat, owing to a Bernoulli-type pressure drop, can cause

local collapse in severe stenoses.




FIGURE 5 Steady flow

through a moderate steno-

sis (50% diameter reduc-

tion, 1.2 cm long, 4-mm

diameter, Re 160) (98).

a. Stenosis configuration.

b. Streamlines show sepa-

ration distal to the throat.c. Wall shear rate. Shear

rate increases sharply in

the converging stenosis,

reaching a peak just

upstream of the throat.

Shear rate is negative and

low in the post-stenotic

recirculation region.

Two important clinical consequences arise from the collapse of stenoses. One

is that the flow rate can be limited by choking, beyond that of purely turbulent

losses. This flow limitation or critical flow rate has long been observed by phys-

iologists and described as the coronary flow reserve that is limited even withdecreases in distal resistance. Estimates of coronary flow reserve should include

this choking flow limitation as well as other forms of viscous losses (46, 95). A

second consequence is that of the imposed loading conditions on an atheroscle-

rotic plaque. Stenotic flow collapse creates a compressive stress that may buckle

the structure. The oscillations in compressive loading may induce a fracture

fatigue in the surface of the atheroma, causing a rupture of the plaque cap.

Because plaque cap rupture is the precipitating event in most heart attacks and

strokes, the fluid-solid mechanical interactions present in high-grade stenosis may

contribute to the catastrophic material failure (74).

Diagnosis of Disease

There are a wide variety of clinical applications for hemodynamic studies of

stenoses. One area of investigation revolves around the diagnosis of severe ste-

nosis. The most accepted clinical predictors of impending heart attack, stroke,

and lower-limb ischemia are based on the presence of hemodynamically signifi-

cant stenoses. Currently, the best indicator for surgical treatment of arterioscle-

rosis is the degree of stenosis. Although X-ray angiography is currently the

standard, cost and morbidity are distinct disadvantages.

Doppler ultrasound can be used to measure the increased velocities in the

stenotic jet and back out a percent stenosis. This technique is widely used to

determine levels of stenosis in carotid artery disease, with an accuracy of 90%.

Doppler ultrasound can also be used to measure the flow waveform in the leg



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arteries. Normal arteries have a characteristic triphasic pattern, whereas diseased

arteries with a stenosis exhibit a blunted monophasic pattern.

Recently, magnetic resonance imaging (MRI) has been proposed as a less

expensive, less morbid alternative to X-ray angiography (115). In contrast to

Doppler techniques, which require an acoustic or optical window, MRI uses an

electromagnetic window that does not contact the flow. Thus, much more of the

body can be studied.

Shear-Dependent Thrombosis

Stenotic flows become critical to clinical medicine in the acute symptoms of

atherosclerosis. After the plaque cap ruptures, the revealed contents of the ath-

eroma stimulate a blood-clotting reaction called thrombosis. For the arterial sys-

tem, thrombosis is initiated by the adherence of platelets at the surface with rapid

accumulation of additional platelets. Although a number of confusing in vitro

experiments are described in the literature, studies with nonanticoagulated blood

through stenoses indicate that platelets stick at the throat of the stenosis. The

adherence and accumulation of these platelets are shear dependent—with more

accumulation at higher shear rates. The time scale of adhesion is on the order of

milliseconds. Likewise, the adhesion strength must be enormous because the shear

stresses on the platelet are large and increasing as the throat fills with clot. Thefollowing sections explore some of the relationships between thrombosis and

hemodynamics and how these relationships may be used to understand the risk

of clinical thrombosis.

ARTERIAL THROMBOSIS

Thrombosis is the formation of a blood clot, called a thrombus, inside a living

blood vessel. The mechanisms of thrombosis are identical to the mechanisms of

hemostasis, the clotting system that protects the body from excessive blood loss.

A thrombus is composed primarily of two blood cell types, platelets and RBCs.

The cells are bound together by molecules in the cell membrane of the platelets,called membrane glycoproteins (GPs), by a variety of plasma proteins, and by a

network of polymerized plasma protein called fibrin.

Arterial thrombosis is an extremely significant health problem because it is

linked to the onset of acute clinical symptoms in atherosclerosis. Thrombus super-

imposed on ruptured atherosclerotic plaque is commonly found in autopsy studies

of heart disease (24–27). Thrombosis is also associated with carotid artery plaque

rupture in stroke and transient ischemic attack (24, 85). Platelets and fibrin emboli

are frequently found in the myocardium (heart muscle) of victims of heart disease

(28, 37). Clinical studies confirm the link between thrombosis and atherosclero-

sis—antithrombotic drugs significantly reduce the risk of clinical ischemia (40,

81, 108).




FIGURE 6 Thrombosis in late-stage ath-

erosclerosis. a. Plaque rupture exposes sub-

endothelium to the blood, causing platelet

adhesion. b. Platelet aggregation forms a

platelet plug. c. Coagulation and platelet

aggregation may cause occlusion.

Cellular and Molecular Mechanisms of Thrombosis

Thrombosis is a complicated interaction of platelets and plasma proteins. At a

functional level acute thrombosis is described by three platelet functions (adhe-

sion, activation, and aggregation) and the coagulation cascade (Figure 6). These

mechanisms can occur simultaneously and have multiple interactions, with the

enzyme thrombin playing a central role.

Adhesion Thrombosis is triggered when a thrombogenic surface is exposed to

blood (Figure 6a). Thrombogenic surfaces include most artificial surfaces, the

subendothelial and medial layers of blood vessels, and subendothelial components

of atherosclerotic lesion such as fibrous plaque cap and atheromatous core (30).

Platelets adhere to proteins in the surface via platelet membrane GPs (62).

Subendothelial tissue and atheroma contain collagen, to which platelets bind via

glycoprotein GPIa/IIa (91), and von Willebrand factor (vWF), to which platelets

bind via two GPs. GPIb mediates a rapid but transient binding to vWF, whereas

GPIIb/IIIa mediates more permanent binding (96). GPIIb/IIIa can bind to many

other plasma and vessel wall proteins, including fibrinogen, fibrin, fibronectin,

thrombospondin, and vitronectin (62).

Activation Activation refers to platelet functions triggered by chemical or physi-

cal agonists (stimuli). Chemical agonists include ADP, thrombin, thromboxane

A2 (TxA2), fibrillar collagen, platelet-activating factor, and serotonin (23). Shear

stress (in the presence of vWF, ADP, and Ca2) can activate platelets (52). Plate-

lets may also be activated by biomaterials via the complement system (39, 59).

Perhaps the most important activation function is a conformational change in

GPIIb/IIIa that allows it to bind to plasma proteins. GPIIb/IIIa activation has been

estimated to occur within 0.1 s (84) and is required for aggregation and permanent

adhesion to vWF (96).



312 WOOTTON KU

Activation causes shape change with pseudopod extension, which increases

the strength of adhesion and may decrease the resistance of platelets to aggre-

gation. Activation triggers upregulation and local clustering of GPIIb/IIIa, which

may also strengthen adhesion. Activated platelets contract, consolidating loose

cells and fibrin into compacted thrombus, and release granular contents (23).

Several activation functions are positive feedback mechanisms for activation

of other platelets (23). Activated platelets synthesize platelet agonist TxA2 and

release ADP from dense granules; inhibition of either the ADP (87) or TxA2 (64)

activation pathway significantly reduces thrombus growth. Activated platelets

also catalyze thrombin production (23).

Aggregation Aggregation is essential to formation of a platelet plug (Figure

6b). Aggregation can proceed via several mechanisms. At low to moderate shear

rates, activated platelets can bind to other activated platelets via fibrinogen or

fibrin and GPIIb/IIIa. At higher shear rates, platelets aggregate primarily via vWF

(3, 58, 73). It is not clear whether activation occurs before or after initial vWF

binding. Incoming platelets may be activated by passing through an agonist

‘‘cloud’’ of thrombin, TxA2, or ADP before interacting with an adherent throm-

bus (57, 84). Alternatively, vWF on the surface of aggregated and activated plate-

lets could support binding of unactivated platelets via GPIb, followed byactivation and permanent binding via GPIIb/IIIa.

Coagulation Exposure of a thrombogenic surface is also likely to trigger the

coagulation cascade, leading to fibrin coagulation (Figure 6c). In normal hemo-

stasis, injury exposes tissue factor, which rapidly leads to thrombin generation.

Tissue factor is found at high concentrations in the necrotic core of the atheroma

(110) and may be exposed by plaque rupture. Coagulation may also be triggered

by exposure of collagen or an artificial surface (23).

The ultimate reaction in the coagulation cascade converts prothrombin to

thrombin. Thrombin cleaves fibrinogen so that it can polymerize to form fibrin,

which traps red cells in the clot and supports platelet adhesion. Thrombin is also

one of the most potent platelet agonists, causing activation, release of granular

contents, and irreversible aggregation (23). This interaction between platelets andthrombin is important in thrombosis lasting longer than 10 or 15 min (48, 49,

61).

HEMODYNAMICS AND THROMBOSIS

Thrombosis is fundamentally linked to hemodynamics because blood transports

cells and proteins to the thrombus and applies stresses that may disrupt the throm-

bus. In this section, we review how blood flow conditions affect the rate and

localization of platelet accumulation, platelet activation, and fibrin coagulation.




TABLE 1 Mean blood flow parameters for human arteries commonly subject to occlusive

thrombosis in atherosclerosis

Vessel (reference)Diameter

(mm)Average flow

rate (ml/s)

MeanReynolds

number

Mean wallshear

ratea (s1)

Mean wallshear stressa

(dyne/cm2)

Femoral artery (44) 5.0 3.7 280 300 11

Common carotid (65) 5.9 5.1 330 250 8.9

Internal carotid (65) 6.1 5.0 220 220 8

Left main coronary (51) 4.0 2.9 240 460 16

Right coronary (50) 3.4 1.7 150 440 15

aMean wall shear rate and shear stress are estimated from the Poiseuille profile.

Hemodynamics in Advanced Atherosclerosis

Most thrombosis experiments with controlled flow report the wall shear rate,

c s / l. In major arteries subject to occlusive clinical thrombosis (Table 1), mean

shear rate normally ranges from 200 to 500 s1 and mean flow Reynolds numbers

range from 100 to 400. Where atherosclerosis creates a stenosis, shear rate

increases to a peak just upstream of the stenosis throat (Figure 5c). The wall shear

rate may be estimated by using scaling based on the Reynolds number and geom-etry (99). Peak shear rate increases with Reynolds number and stenosis severity,

to 10,000 s1 for moderate stenoses and 100,000 s1 for severe stenoses.

Distal to the stenosis, a recirculation region may develop, with unusually high

residence time and low shear rate.

Hemodynamics and Thrombus Composition

The composition of a thrombus depends on local flow conditions. In static and

low-shear recirculating flow, the bulk of a thrombus consists of RBCs trapped in

fibrin. But in unidirectional flow at shear rates of 100 s1 and higher, the bulk

of an acute thrombus consists of platelets (18, 92, 101). At arterial and stenotic

shear rates, mechanisms of platelet adhesion, activation, and aggregation domi-

nate, and the thrombus size can be estimated by counting the number of plateletsthat accumulate.

Shear and Platelet Accumulation

Platelet Accumulation Rate Increases with Shear Rate An increase in platelet

accumulation is directly related to the shear rate. This has been observed in vitro

for platelet deposition on subendothelium (106) and collagen-coated surfaces (3,

8, 93, 100). The effect of shear rate has also been demonstrated in human (11,

92), baboon (72), and porcine (8) ex vivo experiments, which are dominated by

the aggregation phase of thrombosis. The rate of platelet accumulation on fibrillar



314 WOOTTON KU

FIGURE 7 Average platelet accumulation rate in ex vivo baboon (72), pig (8), and

human (11, 92) experiments, as a function of peak wall shear rate. Platelet accumulation

rate on collagen I is averaged over 15 min, measured on tubes (●) and stenoses () (72),

and in U-channels () (8). Platelet accumulation rate on collagen III over 5 min (,)

(11, 92), estimated from thrombus volume by a linear correlation of data published for the

same system (93), platelets/thrombus volume 9 1010 platelets/ml.

collagen increases for shear rates from 100 to at least 10,000 s1 (Figure 7). At

low shear rates, accumulation is roughly proportional to shear rate. At higher

shear rates, there may be a divergence from this trend. One experiment shows a

decrease in deposition rate between 10,000 and 32,000 s1 (11), whereas another

experiment shows an increase in deposition rate between 4,300 and 20,000 s1

(72).

Platelets Adhere Preferentially in High-Shear Regions Shear also appears to

affect where platelets are deposited. Platelet accumulation on collagen-containing

stenotic surfaces is highest at the stenosis throat, where shear rate is highest (8,

72), for peak shear rates ranging from 1,300 to 20,000 s1 (72). On smooth

artificial surfaces by contrast, platelet accumulation may be depressed in high

shear regions (15, 97).

Shear-Linked Mechanisms

The correlations between shear and platelet accumulation may be explained in

terms of several shear-linked mechanisms: platelet transport, platelet activation,

and embolization.

Transport Platelet transport is important in acute thrombosis because plateletaccumulation on highly thrombogenic surfaces in vivo may be transport limited

or transport modulated for shear rates up to at least 20,000 s1 (111). Transport




in blood is still not completely understood, partly because there is no fundamental

theory to predict dispersion in a concentrated suspension like blood. But exper-

imental studies have identified two mechanisms that influence the rate of platelet

interaction with a thrombogenic surface: (a) RBC motion, the dominant mecha-

nism, increases small-scale transport by several orders of magnitude (44); (b)

nonuniform platelet concentration may increase platelet transport by a factor of

1–10. Both of these mechanisms increase platelet transport as shear rate increases.

RBC motion RBCs exhibit randomlike transverse motion and rotation in shear

flow (43), which displaces plasma and platelets and increases lateral transport.

The rate of platelet transport has often been quantified by an effective diffusivity,

derived by fitting experimental data to a species transport model of platelet adhe-

sion (e.g. 106). Power law correlations in the form De D1 (c/c1)n, where c is

the shear rate and c 1 s1) give power n and coefficient D1 that are functions

of hematocrit and the stiffness and size of the RBCs. For platelets or chemical

solutes in anticoagulated human blood at 40% hematocrit, n ranges from 0.49 to

0.89, and D1 ranges from 109 to 3 108 cm2 /s (1, 5, 107, 109). From these

correlations, De ranges from 2 108 to 3 107 cm2s1 at shear rate 100

s1 and from 5 107 to 1 105 cm2s1 at shear rate 10,000 s1, 1 to

4 orders of magnitude above the thermal diffusivity for platelets in plasma (1.6 109 cm2s1).

Estimates of De vary by up to an order of magnitude between experiments.

One source of variation may be the variability of platelet adhesion rates with

differences in anticoagulation and platelet handling; the adhesion rate begins to

affect the deposition rate in vitro for shear rates 300 s1 (107). This difficulty

can be avoided by using a global model of enhanced diffusivity in sheared con-

centrated suspensions (117). Assuming that RBC rotation is relatively unimpor-

tant, the effective diffusivity De is the sum of the RBC dispersion and the thermal

diffusivity:

D D D (1)e R s

where D R is the RBC dispersion coefficient and Ds is the thermal diffusivity of

the solute or platelets in the stationary blood. The dispersion coefficient for RBCsis correlated to experiments by

(2) D R a2cc p(1 p)m

with c 0.15 0.03 and m 0.8 0.3, where a is the RBC major radius, c

is the shear rate, and up is the hematocrit. For platelets, De is essentially propor-

tional to c for c 10 s1. The model is consistent with transport rates for a

variety of solutes in whole blood and was later confirmed for macromolecule

transport (63).

Nonuniform Concentration RBCs are concentrated in the center of a blood

vessel, and appear to force increased platelet concentration toward the vessel wall.



316 WOOTTON KU

This effect has been studied most heavily in narrow vessels (e.g. 43, 104) but has

also been observed in 3-mm-diameter tubes at arterial and higher shear rates (2).

Platelet concentration at the wall increases with increasing hematocrit, shear rate,

and platelet concentration. For example, in a 3-mm tube with a 40% hematocrit

and a 0.25 billion/ml average platelet count, near-wall concentration is a factor

of 2 to 4 higher than the average platelet count as the shear rate increases from

240 s1 to 1200 s1 (2).

Both RBC motion and enhanced platelet concentration link high shear to

increased platelet deposition. As long as molecular mechanisms of adhesion and

aggregation are rapid enough to permit platelet incorporation into a thrombus,

increasing shear will drive more platelets into the thrombus, resulting in more

rapid thrombus growth.

Activation The role of shear stress activation in clinical thrombosis is not clear.

The threshold shear exposure required for platelet activation in vitro has been

measured for shear rate (in whole blood) ranging from 103 to 107 s1 (52) and

fit to a platelet stimulation function, PSF (16), such that PSF s t 0.452, where

s is shear stress in dynes per square centimeter and t is exposure time in seconds.

The threshold for shear-induced activation is PSF 1000 (16). High shear stress

activates platelets with short exposure, whereas lower shear stress activates plate-lets over longer durations.

A platelet flowing through a stenosis in vivo is exposed to high shear stress,

but the exposure time is at least one order of magnitude lower than the threshold

for shear-induced platelet activation (16). Shear stress exposure may not be

directly responsible for platelet activation in most cases of relatively severe ath-

erosclerosis, if activation is required for the initial interaction between a circu-

lating platelet and growing thrombus. Shear stress exposure time will exceed the

activation threshold only if a platelet adheres to a stenosis.

Even if shear stress is not the sole activating agonist in vivo, the history of

shear stress exposure may change the threshold of platelet activation by chemical

agonists (42, 45). Compared with flow that has a gradually changing shear rate,

stenotic flow with a rapid increase in shear stress may significantly increase plate-

let activation and platelet deposition (54, 111, 112).

Embolization Another feature of thrombosis is embolization, the removal of

parts of the thrombus owing to fluid mechanical stress. A theoretical model has

been developed for embolization in steady and pulsatile flows (14), based on

models of drag on a protrusion into steady (12) and pulsatile (13) flow. The stress

on the thrombus depends on the particle Reynolds number, Re p c H p2 / t, where

H p is the thrombus height and t is the kinematic viscosity. For small thrombi

( Re p 1), stress on an isolated thrombus is four- to fivefold the wall shear stress

of the approaching flow. For larger thrombi, stress becomes a function of throm-

bus height, and stress increases rapidly as the thrombus grows.




The missing part of the model is quantitative data on the stress required for

platelet removal from a surface. Mechanical properties of platelets are the subject

of ongoing study (47), so the critical stress for embolization may soon be within

reach, using a combination of modeling and experiments.

Differences in platelet embolization stress may explain the difference between

platelet accumulation patterns on collagen (72) or damaged artery (7) and accu-

mulation on Lexan (97). Platelets probably adhere more strongly to collagen in

the natural surfaces than to the smooth Lexan surface and can support larger

thrombi without embolization. Ultrasound measurements of embolization from

knitted Dacron or collagen surfaces in ex vivo experiments show low emboliza-

tion rates (111).

Recirculation and Residence Time The effect of hemodynamics on thrombosis

is well documented in uniform or unidirectional shear flow. But separated flow

occurs at bifurcations, and downstream of stenoses that occur in atherosclerosis.

In regions of complex flow, the relationships between flow and thrombosis are

not very clear.

Convection patterns and high residence times may modulate thrombosis in

separated flows. In vitro experiments show increased platelet accumulation near

the reattachment point in Lexan stenoses, presumably caused by increased con-vection (15). Platelets may recirculate in the separated region long enough to

become activated and form small aggregates. Residence time and convection

patterns have also been related to fibrin polymerization in shear flow (36, 82).

Based on steady-flow experiments, residence time on the order of at least 10

s is required for significant shear-induced aggregation (56) or fibrin polymer

accumulation (82). In physiologic pulsatile flow, 10-s residence is quite long; for

example, 99% of particles are washed out of the recirculation zone of a 75%

or 95% area reduction stenosis within 10 s (19). One potential location for phys-

iologic high residence time would be along the trailing edge of a sharp geometric

flow separator, which could be created by a tear or flap following plaque rupture,

or by a poorly designed prosthetic valve. A sudden expansion, which has a geo-

metric flow separator, creates an environment favoring a fibrin-rich red thrombus

(18). High residence time could also occur distal to a flow-limiting acute plateletplug, in which case occlusion becomes the primary cause of high residence time

and fibrin coagulation.

MODELING CLINICAL THROMBOSIS

Despite the well-documented role of thrombosis in clinical ischemia, thrombosis

risk is not used as a surgical indicator in atherosclerosis. Stenosis severity is used

to identify patients who are good candidates for surgical treatment because it

correlates with risk of ischemia (6, 20, 78). The relationships between shear rate,

stenotic flow, and thrombus growth rate are probably reflected in the clinical



318 WOOTTON KU

statistics. However, using only stenosis severity misses patients with moderate or

mild stenoses (50% diameter reduction), who have a significant risk of ischemic

attack and death (17, 25, 105).

Clinically it is important to know the likelihood that thrombosis will lead to

occlusion following plaque rupture or ulceration. A model of occlusion risk could

be combined with a model of plaque rupture risk to decide which patients are

good candidates for surgical treatment and which patients can be managed med-

ically. A clinical thrombosis model has not been developed yet, owing to the

complexity of thrombosis and the wide range of data produced by different throm-

bosis experiments. But there is enough experimental data available to begin build-

ing a model, based on a theoretical mechanistic thrombosis model. The model

can estimate the time required for a thrombus to occlude a vessel, based on

hemodynamics and geometry, and the occlusion time can be used as a measure

of risk of occlusion in the event of plaque rupture.

Model Based on Ex Vivo Experiments

Only a few experiments are representative of clinical occlusive thrombosis. In

vivo and ex vivo experiments, which use nonanticoagulated blood or mildly hep-

arinized blood, include all of the major mechanisms of thrombosis and can occur

over time periods long enough for occlusion of a major artery. In contrast, invitro experiments require anticoagulation and often involve extensive blood sam-

ple manipulation, and platelet deposition rates are much lower in in vitro than in

ex vivo experiments (9, 30, 92). For model development, ex vivo experiments

have an advantage over in vivo experiments; precise control of flow rate and

thrombogenic surface geometry allows the shear rate to be calculated.

Relevant thrombosis experiments need to match the hemodynamic environ-

ment of stenotic arteries subject to clinical occlusive thrombosis. The controlling

hemodynamic variable is the shear rate on the thrombogenic surface, influenced

by the shear rate history of platelets flowing over the thrombogenic surface in

stenotic flow. The Reynolds number does not appear to be significant for forward

flow, apart from its direct effect on the shear rate.

Several experiments approximate the hemodynamic and hematologic environ-

ment expected in atherosclerosis in vivo. Baboon (72) and porcine (7, 8, 30) exvivo experiments span physiologic shear rates and approach a physiologic Rey-

nolds number, using little or no anticoagulation; the baboon experiments also

have well-characterized stenotic flow (72). Human ex vivo experiments (11, 92)

match physiologic shear rates without anticoagulation, although the duration of

the experiments is limited. These experiments can be used to guide and test a

model of occlusive thrombosis. Well-characterized in vivo canine experiments

can also be used to test an occlusion model (101).

Model Development

Several experiments provide insight into occlusion when platelets may adhere to

the entire lumen surface, a relatively severe injury. In stenotic geometry, the




FIGURE 8 The characteristic time course of platelet accumulation on collagen-coated

tubes, in ex vivo baboon experiments () (72). The experiment can be divided into three

phases: ( I ) an accelerating phase, lasting for about 5 min, ( II ) an acute phase, lasting 50

min, and ( III ) a slow phase, which extends to the end of the experiment, characterized by

a lower accumulation rate than the acute phase. The total platelet accumulation can be

approximated by a 5-min delay, constant accumulation at the acute rate, and no accumu-

lation during the slow phase (solid line).

stenosis throat is the location of most rapid platelet accumulation and of occlusion

(72, 101). For 4-mm–inside-diameter stenoses at a 100-ml/min flow rate, occlu-

sion occurs for smaller lumen sizes (2.7 to 3 mm) and for higher shear rates

(600 s1) (72). Occlusion occurs consistently and rapidly for narrower lumens

and higher shear rates (33, 101).

A theoretical model can help scale experimental data to other flow conditions.

Occlusion can be estimated by predicting the size of the thrombus, which is

proportional to the number of accumulated platelets, because platelets comprise

the bulk of the thrombus. The time course of platelet accumulation in ex vivo

experiments (72) is dominated by an acute phase, which eventually decelerates

to a slow phase (Figure 8). In some experiments, a platelet plug occludes the

lumen, slowing flow and platelet accumulation, but in other experiments, the rate

of accumulation is limited by a drop in the aggregation rate. To first order, the

final size of a thrombus is proportional to the acute rate of platelet accumulation

and the duration of the acute phase of platelet deposition.

The first objective of the model is to estimate the acute rate of platelet accu-

mulation, as a function of hematologic and hemodynamic variables. Several good

theoretical thrombosis models have been developed to understand thrombosis

experiments. Some models treat platelets as discrete particles (31, 55, 84, 90,

102); this approach has the potential to be more accurate as molecular models of adhesion are developed, but can become complicated. Current particle models

idealize or ignore the particle-fluid interactions and thrombus shape or are explor-



320 WOOTTON KU

atory tools. Other models treat platelets as a continuous chemical species (29, 32,

34, 106, 112) reacting with a reactive surface. Species transport models are quan-

titative and relatively simple but have not been applied to clinical thrombosis and

occlusion.

A modified species transport model has been developed to compute platelet

accumulation rates based on hemodynamics, geometry, platelet count, and aggre-

gation rate (111, 112). Unactivated platelets in the blood are treated as a chemical

species, which is transported by convection and shear-enhanced diffusion (117).

Near-wall platelet concentration is enhanced by a factor of two above average

platelet concentration, consistent with experiments in similar-sized tubes (2).

Platelets at the surface are incorporated into the thrombus by a first-order reaction

step that includes both aggregation and activation. Flow and transport equations

can be solved analytically in tubular flow. For a stenosis, a commercial compu-

tational fluid dynamics package is used to compute the flow field and platelet

accumulation rate. This approach predicts the acute platelet accumulation rate on

collagen-coated tubes and in the upstream, converging, and throat sections of

collagen-coated stenoses (111, 112) of differing stenosis severity (Figure 9c). The

platelet accumulation rate is highest at the stenosis throat and increases with

increasing percent stenosis (Figure 9b). The model is less successful in recircu-

lating post-stenotic flow, but in experiments the maximum platelet deposition rateis located in the throat section (7, 72), where occlusion occurs (101), so the model

is applicable to predicting occlusion in stenotic flow.

A Model of Occlusion

A model of acute platelet accumulation rate can be used to estimate thrombus

size and occlusion risk if the duration of the acute phase can be predicted. Unfor-

tunately, the mechanisms that are responsible for reducing the accumulation rate

are not well studied. Embolization has been assumed an important limiting mech-

anism, but embolization loss is difficult to measure, and large emboli appear to

be infrequent in ex vivo experiments (111). A model of embolization has been

derived (14), but the embolization stress is unknown. In addition, systemic

changes may reduce the rate of platelet activation, or the concentration of plateletactivation agonists may decrease locally as the thrombus size increases.

Absent a clear mechanism to limit thrombus growth, the occlusion time can

be estimated from the acute accumulation rate and lumen diameter, assuming that

the acute phase does not end. This extrapolated occlusion time can be used as a

risk indicator; a short occlusion time indicates a higher risk of occlusion when

there is plaque rupture.

For a fully reactive surface, occlusion occurs when the thrombus height reaches

the lumen radius. The occlusion time is

D C lumen thT t , (3)occlusion d

2 jlumen




FIGURE 9 Model of platelet accumulation on 4-mm-inner-diameter collagen-coated

stenoses (111). a. Collagen-coated surface consists of straight segment from x 1.2

cm to 0.6 cm, a cosine-shaped stenosis from 0.6 cm to 0.6 cm, and a straight segment

from 0.6 cm to 1.2 cm. b. Acute platelet accumulation rate ( j*) vs axial location ( x) in

50%, 75%, and 90% area reduction stenoses. j* peaks just upstream of the throat. Peak

accumulation rate increases with increasing stenosis severity. c. Average platelet accu-

mulation in the stenosis throat section ( x 0.48 cm to x 0.48 cm), for 50% (),

75% (), and 90% () area reduction (72), compared with model (lines).



322 WOOTTON KU

where Dlumen is the diameter of the vessel lumen (throat diameter in a stenosis),

C th is the concentration of platelets in arterial thrombus [estimated to be 75 billion/

ml from ex vivo experiments (111)], and is the ratio of thrombus height to

thrombus cross-section area, which accounts for the roughness of the thrombus

(estimated to be 2 based on experimental occlusion times in stenoses). A 5-min

delay (t d ) is used to model the effect of the accelerating phase of thrombosis. The

acute rate of platelet accumulation jlumen is computed by using the species trans-

port model. Equation 3 estimates occlusion times of 16, 28, and 66 min for 90%,

75%, and 50% area reduction stenoses, respectively. In experiments, occlusion

times were 18–25 min for the 90% stenosis and 25–35 min for the 75% stenoses

(72), relatively close to the model. The 50% stenosis does not occlude, so an

occlusion time somewhere between 30 and 60 min indicates a low risk of occlu-

sive thrombosis.

The model predicts increased risk of occlusion (decreasing occlusion time)

with increasing shear rate, decreasing lumen diameter, and increasing platelet

count. Because shear rate increases and lumen diameter decreases with increasing

percent stenosis, the correlation is consistent with clinical studies linking risk of

ischemia and benefit of surgery with percent stenosis. Platelet count is a hema-

tologic parameter that should also have a strong influence on risk of occlusion,

based on this model.

Future Directions

The knowledge that shear affects platelets is already being applied to the design

of cardiovascular devices, to minimize shear stress and residence time in blood

pumps, cardiopulmonary bypass devices, and prosthetic valves.

Clinical application of an occlusive thrombosis model depends on a better

understanding of mechanisms that limit thrombus growth after the acute aggre-

gation phase that is typically observed. Embolization and systemic negative feed-

back may contribute to subocclusive thrombus under some flow conditions. A

second requirement for an occlusive thrombosis model is a risk model for plaque

rupture. Combined understanding of plaque rupture and thrombosis, along with

measurements of degree of stenosis, could increase the accuracy of screening

patients for surgical treatment of atherosclerosis.

CONCLUSIONS

The study of hemodynamics is a rich field that allows one to characterize the

biological responses to mechanical forces. Specific arteries exhibit flow charac-

teristics that are three-dimensional and developing. Diseased arteries can create

high levels of turbulence, head loss, and a choked flow condition in tubes that

can collapse. The pulsatile nature of the flow creates a dynamic environment with

many interesting fundamental fluid mechanics questions. The fundamental knowl-

edge can be used to predict and change blood flow to alter the course of disease.




Shear stress and shear rate have emerged as important parameters that modulate

both chronic and acute biological responses.

The relationships between thrombosis and fluid mechanics are complicated. A

species transport model can be used to estimate clinical thrombosis risk based on

the hemodynamic environment. Future studies will be driven by the need to

understand the complex effect of hemodynamics on cells and the design of new

devices to modulate this effect.

Visit the Annual Reviews home page at http://www.AnnualReviews.org.

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