(Stroke. 1997;28:2067-2077.)
© 1997 American Heart Association, Inc.
Articles |
Biophysical Mechanisms of Stroke
From the Endovascular Therapy Service, Department of Radiological Sciences, University of California at Los Angeles School of Medicine.
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Abstract |
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 Top Abstract IntroductionBiophysical Mechanisms of StrokeCarotid Artery BifurcationHemodynamics of a StenosisConclusionsReferences |
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Background Stroke is the third leading cause of death and the leading cause of long-term disability in the United States. Although a host of genetic, biochemical, physiological, anatomic, and histological factors have been implicated, to varying degrees, in the pathogenesis of stroke, biophysical factors are believed to play a significant role in the development, diagnosis, and therapy of stroke. The purpose of this review article is to identify, describe, and illustrate these causes and biophysical and hemodynamic mechanisms predisposing a person to stroke, which often form the basis for novel methods of diagnosis and therapy.
Summary of Review This mini-review begins by describing the physical principles that govern the flow of blood through normal and stenosed carotid artery bifurcations. In addition to the tortuosity, curvature, and tensile forces of the carotid artery bifurcation, the effects of biophysical phenomena from flowing blood such as viscous forces, pressure forces, velocity, kinetic energy, momentum, impulse, shear stress, and vibrational displacements exerted by the flowing blood on the vessel wall are conducive to abnormal flow behavior and patterns, degrading the vessel wall and creating the potential for stroke.
Conclusions Recent advances in the treatment of stroke are based on increasing knowledge of its underlying biophysical mechanisms, as well as on better-publicized advances in imaging instrumentation and procedures for the management and treatment of patients.
Key Words: biophysics • hemodynamics • stroke
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Introduction |
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TopAbstractIntroductionBiophysical Mechanisms of StrokeCarotid Artery BifurcationHemodynamics of a StenosisConclusionsReferences |
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Optimal function of the human brain is critically dependent on the flow of blood circulating through the brain. Under normal conditions, the brain is supplied oxygenated blood through the common carotid and the vertebral arteries. The common carotid arteries branch into the internal and the external carotid arteries in the neck. The external carotids supply the face, scalp, and most of the neck tissues. The internal carotids, which ascend on a deeper plane, divide into arteries that supply the anterior portion of the brain. The vertebral arteries, which travel close to the spine, supply the posterior portion of the brain. The anterior and posterior blood supplies connect in the circle of Willis, an arterial interchange at the base of the brain.
Blood flow then permeates through the brain tissue via intricate networks of capillary vessels, where it delivers oxygen and nutrients to the brain and removes cellular metabolic waste products before returning to the heart through the venous system. If arterial blood flow is obstructed or impaired at any point within this vascular route, portions of the brain relying on the occluded vessel for oxygenated blood become deprived of oxygen, initiating a cascade of mechanisms that result in brain tissue ischemia and eventually infarction.1 The extent of ischemia is dependent primarily on the proximity of the obstruction to the brain and the amount of collateral flow supplied to that region. The severe restriction or complete cessation of blood flow to the brain as the result of any cerebrovascular disease or neurological insult (brain injury) is commonly referred to as stroke.
Stroke is the third leading cause of death and the leading cause of long-term disability in the United States. Stroke afflicts approximately 500 000 people each year and places a substantial burden on the national healthcare system, costing an estimated 23.2 billion dollars.2 In addition, the mortality rate from stroke is 150 000 deaths per year. Given the magnitude and corresponding implications of stroke on the quality of life and public healthcare system, an intense effort is under way to determine underlying causes and mechanisms—particularly in this decade, the Decade of the Brain (1990 to 2000)—to better understand the causes of stroke and to devise more effective means of prevention and treatment.3
Although a host of genetic, biochemical, physiological, anatomic, and histological factors have been implicated, to varying degrees, in the pathogenesis of stroke, biophysical factors are believed to play a significant role in the development, diagnosis, and therapy of stroke. The purpose of this review article is to identify, describe, and illustrate these causes and biophysical and hemodynamic mechanisms predisposing a person to stroke, which often form the basis for novel methods of diagnosis and therapy.
Two types of stroke that have been identified and recognized clinically correspond to their characteristic mechanisms of flow obstruction or neuronal damage: ischemic stroke4 and hemorrhagic stroke.5 Ischemic strokes, which account for 80% of strokes, are caused by the obstruction or clogging of the major arteries in the cerebral circulation. Hemorrhagic strokes, which account for the remaining 20% of all strokes, occur as a result of rupture of vascular lesions within the cerebrovasculature, typically due to an aneurysm or a weakened blood vessel within an arteriovenous malformation. For each of these lesions, rupture causes hemorrhage, volumetric filling of blood into the surrounding space, and resultant compression of the surrounding tissues and vessels.
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Biophysical Mechanisms of Stroke |
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TopAbstractIntroductionBiophysical Mechanisms of StrokeCarotid Artery BifurcationHemodynamics of a StenosisConclusionsReferences |
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Although the consequences of both ischemic and hemorrhagic stroke are similar in that a vessel obstruction and the resultant reduced blood flow to the brain may lead to neurological deficits and possibly death, the biophysical and hemodynamic mechanisms behind the obstruction of blood flow are different. Biophysical mechanisms for the development of obstructions that ultimately lead to stroke can arise by six distinct processes: atherosclerosis, embolus, thrombus, reduced systemic pressure, hemorrhage, and vasospasm.
Atherosclerosis
Atherosclerosis, commonly referred to as
"hardening of the arteries," is a pathological process in which
calcified lipid or fatty deposits from the flowing blood accumulate
circumferentially along the innermost intimal layer of the vessel wall
(Fig 1
). Atherosclerotic plaques are
found almost exclusively at the outer wall (hip) of one or both
daughter vessels at major bifurcations, including the
carotid.3 Atherosclerosis and the
development of arterial plaques are the product of a
host of independent biochemical processes including the oxidation of
low-density lipoproteins, formation of fatty streaks, and the
proliferation of smooth muscle cells.6 As the plaques
form, the walls become thick, fibrotic, and calcified, and the lumen
narrows, reducing the flow of blood to the tissues the artery
supplies.
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Atherosclerotic deposits promote the development of blood clots or the process of thrombosis due in part to flow obstruction and to high shear stresses exerted on the vessel wall by the blood. High wall shear stress might mechanically damage the inner wall of the artery, initiating a lesion. On the other hand, low wall shear stress might encourage the deposition of particles on the artery wall, promoting the accumulation of plaque. Turbulence has also been implicated in atherosclerotic disease both because it can increase the kinetic energy deposited in the vessel walls and because it can lead to areas of stasis, or stagnant blood flow, that promote clotting. In addition, the presence of atherosclerotic lesions introduces an irregular vessel surface that, as a result of turbulent blood flow, can cause the dislodgment of plaques of varying size into the bloodstream until the plaque lodges into a vessel of smaller size, preventing further passage of blood flow. Atherosclerotic thrombosis accounts for 33% of all stroke cases.
Embolus
An embolus represents gaseous or particulate (eg,
atheromata) matter that acts as traveling "clots." A
common example of emboli is a platelet aggregate dislodged from an
atherosclerotic lesion. The dislodged platelet aggregate is
transported by the bloodstream through the cerebrovasculature until it
reaches vessels too small for further propagation. The clot has nowhere
to go and remains there, clogging the vessel and preventing blood flow
from entering the distal vasculature. Although our discussion at the
present is focused primarily on the carotid arteries and associated
cerebrovasculature, emboli can originate from distant sources such as
the heart, lungs, and peripheral circulation, which could
eventually travel within the cerebral blood vessels, obstructing flow
and causing stroke. Other sources of emboli include atrial fibrillation
and valvular disease. The severity of stroke depends on the
size of the embolus and the location of the obstruction. The bigger the
embolus and the larger the vessel obstruction, the larger the territory
of brain at risk. Approximately 31% of all stroke cases are attributed
to emboli.
Thrombus
Thrombosis is an internal physiological
mechanism responsible for the clotting of blood. A thrombus is a blood
clot, an aggregation of platelets and fibrin formed in response
either to an atherosclerotic lesion or to vessel injury. In response to
vessel or tissue injury, the blood coagulation system is
activated, which initiates the following cascade of processes
transforming prothrombin and resulting in a fibrin
clot: Prothrombin
ThrombinFibrinogen FibrinFibrin
Clot
Although a host of mechanisms and causes are responsible for vessel injury, vessel injury can occur as a result of forces (shear stresses)7 coupled with the excess energy created by the turbulent flow8 9 10 exerted against the inner (intimal) lining of the vessel wall, particularly an atherosclerotic vessel wall. Approximately 33% of all stroke cases are attributed to thrombi.
Reduced Systemic Pressure
The previously described mechanisms of blood flow obstruction
leading to stroke occur along localized regions of the cerebral
arteries. It is assumed in this instance that the heart is functioning
normally under proper systemic pressure. Cardiovascular
diseases such as atrial fibrillation and myocardial infarction weaken
the cardiac wall and introduce abnormalities in the
physiological function of the heartbeat, which
ultimately result in reduced systemic pressure and conditions of
ischemia.
Hemorrhage
Blood vessels are typically structurally adept to withstand the
dynamic quantities required to maintain circulatory function. For
reasons that are not entirely understood, the vessel wall can become
fatigued and abnormally weak and possibly rupture. With vessel rupture,
hemorrhage occurs with blood seeping into the surrounding brain
tissue. As the blood accumulates within the brain, the displaced volume
causes the blood, now thrombosed, to ultimately compress the
surrounding vessels. The compression of vessels translates into a
reduced vessel diameter and a corresponding reduction in flow to
surrounding tissue, thereby enlarging the insult. Among the vascular
lesions that can lead to hemorrhagic strokes are aneurysms and
arteriovenous malformations (AVMs).
Brain Aneurysms
A brain aneurysm, shown in Fig 2, is a form of cerebrovascular disease
that manifests itself as a pouching or ballooning of the vessel wall.
The vascular dilatation develops at a diseased site along the
arterial wall into a distended sac of stressed and thinned
arterial tissue. The fully developed cerebral
aneurysm typically ranges in size from a few millimeters to
15 mm but can attain sizes greater than 2.5 cm.11 If
left untreated, the aneurysm may continue to expand until it
ruptures, causing hemorrhage, severe neurological complications
and deficits, and possibly death. In the United States, approximately
28 000 aneurysms rupture each year; approximately 50% of
these patients die or become permanently disabled as a result of the
initial hemorrhage, and another 25% to 35% die of a future
hemorrhage.
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Blood flow in most aneurysms is regular and predictable primarily according to the geometric relationship between the aneurysm and its parent artery.12 As blood flows within the parent artery with an aneurysm, divergence of blood flow, as occurs at the inlet of the aneurysm, leads to dynamic disturbances with a Bernoulli effect, producing increased lateral pressure and retrograde vortices that are easily converted to turbulence.13 Blood flow proceeds from the parent vessel into the aneurysm at the distal or downstream extent of the aneurysm neck, circulates around the periphery along the aneurysm wall from the neck to the top of the fundus (downstream to upstream), returning in a type of "isotropic shower" along the aneurysm wall toward the neck region, and exits the proximal or closest extent of the aneurysm neck into the parent vessel.14
As flow persists, areas of stagnation or vortices develop within a central zone of the aneurysm. These rotating vortices, formed at the entrance to the aneurysm at each systole and then circulated around the aneurysm, are caused by the slipstreams or regions of recirculating flow rolling upon themselves when they enter the aneurysm at its downstream wall during systole.15 The stagnant vortex zone occurs in the center and at the fundus or upper portion of the aneurysm and becomes more pronounced in larger aneurysms. It is this stagnant zone that is believed to promote the formation of thrombi or blood clots, particularly in giant aneurysms.
Brain Arteriovenous Malformations
The normal human circulation originates from the heart and
consists of a branching arrangement of arteries of continually
decreasing size until they feed into a capillary bed before exiting the
bed through small veins that increase in size before returning to the
heart. The capillary bed serves an important purpose in that its
vascular resistance slows the flow of blood considerably to allow
perfusion of oxygen and nutrients to surrounding tissue and removal of
cellular waste. In one form of cerebrovascular disease, AVMs, the
vessels constituting the capillary bed of the brain become malformed
during embryonic development and prohibit the opportunity for blood to
properly perfuse into the surrounding tissue.
AVMs, shown in Fig 3, are congenital
vascular lesions that occur as a result of capillary maldevelopment
between the arterial and venous systems.16
Approximately 0.14% of the United States population has an
intracranial AVM17 that poses a significant risk and
represents a major life threat, particularly to persons under
the age of 50 years. The vessels constituting the AVM are weak and
enlarged and serve as direct shunts for blood flow between the
high-pressure arterial system and the low-pressure venous
system, corresponding to a large pressure gradient and small vascular
resistance. The abnormal low-resistance, high-flow shunting of blood
within the brain AVM without an intervening capillary bed causes the
fragile dilated vessels in the nidus to become structurally abnormal
and fatigued, to further enlarge, and possibly to
rupture.18
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The abnormal microvessels of an AVM serve as passive conduits for blood flow from the arterial circulation directly to the venous circulation, bypassing their normal physiological function of brain tissue perfusion. The hemodynamic consequences of an AVM occur as a result of two interdependent circulatory mechanisms involved in the shunting of blood between artery and vein.19
In the normal cerebral circulation, blood flows under a high
cerebrovascular resistance and high cerebral perfusion pressure.
However, the presence of a brain AVM in the normal circulation
introduces a second abnormal circuit of cerebral blood flow where the
blood flow is continuously shunted under a high perfusion pressure
through the AVM, possessing a low cerebrovascular resistance and low
venous pressure. The clinical consequence of the abnormal shunt is a
significant increase in blood returning to the heart (
4 to 5 times
the original amount, depending on the diameter and size of the shunt),
resulting in a dangerous overload of the heart and possible cardiac
failure. Volumetric blood flow through an AVM ranges from 200 mL/min to
800 mL/min20 and increases according to nidus size.
The abnormal shunting of blood flow by brain AVMs rapidly removes or "steals" blood from the normal cerebral circulation and substantially reduces the volume of blood reaching the surrounding normal brain tissue. This phenomenon, known as cerebrovascular steal, depends on the size of the AVM and is the most plausible explanation for the development of progressive neurological deficits.21 Cerebrovascular steal could translate into additional neurological complications developed as a result of cerebral ischemia or stroke in neuronal territories adjacent to an AVM.
Vasospasm
When bleeding occurs in the subarachnoid space, the
arteries in the subarachnoid space can become spastic with a
muscular contraction, known as cerebral vasospasm.22 The
contraction from vasospasm can produce a focal constriction of
sufficient severity to cause total occlusion. The length of time that
the vessel is contracted during vasospasm varies from hours to days.
However, regardless of the duration of vessel constriction during
vasospasm, reduction of blood flow induces cerebral ischemia,
thought to be reversible within the first 6 hours and irreversible
thereafter. It has been shown that vasospasm is maximal between 5 and
10 days after subarachnoid hemorrhage and can occur up
to 2 weeks after subarachnoid hemorrhage. The resultant
damage to brain tissue can be minimized with the administration of
pharmacological agents such as the vasodilator papaverine.
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Carotid Artery Bifurcation |
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TopAbstractIntroductionBiophysical Mechanisms of StrokeCarotid Artery BifurcationHemodynamics of a StenosisConclusionsReferences |
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The common carotid arteries, which bifurcate ultimately into a vast and complex arrangement of craniofacial-cerebral blood vessels, represent an important vascular location for atherogenesis. One reason for this is the geometry presented by the carotid artery bifurcation. Both mechanical energy and hemodynamic energy are concentrated at the apex of a bifurcation according to its geometry. An arterial bifurcation experiences highly variable regions of wall shear stress characteristic of flow separation. Regions of elevated shear stress are believed to cause injury to the endothelial cell layer of the vessel wall and predispose the vessel to disease. The body responds to the injury with platelet aggregation and clotting mechanisms, creating the potential for stroke.
In addition, the geometry of the bifurcation helps to determine the size of the vortices that form there. According to Fisher and Fieman,23 the vortices tend to be larger if the combined size of the daughter vessels is much larger than the parent vessel, if the bifurcation angle is large, or if the daughter vessels are curved. Several geometric/hemodynamic features of the carotid artery bifurcation that make it more vulnerable to potential processes causing stroke are (1) tensile forces exerted on the bifurcation apex, (2) hemodynamic forces exerted on the bifurcation apex, and (3) hemodynamic forces at vessel curvature.
Tensile Forces of an Arterial Bifurcation
The consequences of arterial bifurcation geometry can
be seen by resolving the tensile, or stretching, forces exerted by the
three arteries on the apex into their geometric components. The
resultant force exerted on the apex of a bifurcation can be determined
easily by resolving the tensile or stretching forces produced by the
parent artery, Fp, and the daughter arteries,
Fd1 and Fd2. The vector diagram of the forces
acting on the apex is given in Fig 4. At
static equilibrium, it is assumed that no other external forces are
acting on the apex, and thus the following is true:
 | (1) |
 | (2) |
 | (3) |
 | (4) |
 | (5) |
, and 5. It quickly becomes apparent that the larger the
bifurcation angle, the more the forces exerted by the daughter arteries
will offset one another and the less they will compensate for the force
exerted on the apex by the parent artery. The bifurcation angle of the
carotid artery bifurcation ranges from 30° to 120°.24
Regardless of the quantitative values of the forces, the point is that
there is a measurable force acting on the apex, particularly since the
bifurcation angle is never 0° or 180°. To investigate the role of
mechanical stress in atherosclerotic disease, Salzar et
al25 constructed computational models of the carotid
artery bifurcation subjected to a normal incremental pressure of
40 mm Hg (to represent the pulse) and loaded with
tractions equivalent to the in vivo longitudinal stress on the
arteries. They found that the stress at the apex was 9 to 14 times
greater than that along straight segments of an artery.
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d1 and
Hemodynamic Forces of an Arterial
Bifurcation
At a normal human arterial bifurcation, blood flow
proceeds from the parent artery, is separated at the bifurcation, and
is redirected into two daughter arteries of different radii stemming at
different angles from the apex. The apex of bifurcations is the site of
maximum hemodynamic stress in a vascular network
because of the impact, deflection, and separation of the blood flow
streamlines and vortex formation at the lateral angles.26
Fully developed flow within the parent artery is
represented by a parabolic approximation of the velocity
profile as:
 | (6) |
by the following:
 | (7) |
, it can be seen that the wall shear stress
increases as the radial position of the bolus profile moves from the
center of the vessel to the vessel wall. Thus, at the center of the
vessel, wall shear stress is zero and is maximum at the wall. As the
flow is divided at a bifurcation, the parabolic velocity profile is
dramatically skewed toward the apex of the bifurcation, as shown in Fig 5.3 In other words, the
layer with the highest velocity moves away from the center of the
vessel and toward the apex. For this reason, the flowing blood exerts
greater shear stress on the inner walls of daughter arteries than on
the walls of the parent vessel.
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The pressure exerted on the apex is much greater than the pressure exerted on all other sites of the vessel wall. In addition, experimental studies have shown shape changes at the bifurcation apex in response to changes in the transmural pressure.27 This pressure difference translates into a higher pressure gradient in the daughter arteries, resulting in a corresponding increase in the volumetric flow rate and a decrease in the blood flow velocity of the daughter arteries.
The transfer of hemodynamic energy, momentum, and force
produced by increases in the pressure gradient and flow velocity acts
to physically degrade the artery wall in the surrounding apical
region.28 This can be visualized through a qualitative
analysis of the physical variables involved. The large
blood flow velocity in the parent artery translates into a large
kinetic energy acting directly on the bifurcation apex. The kinetic
energy of an incompressible fluid over a unit volume V is proportional
to the flow velocity squared, given mathematically by:
 | (8) |
The impact of the blood flow on the apex can be better understood by
introducing the physical concept of impulse. Impulse is, in effect, the
magnitude of an applied variable force that strikes a certain
surface area or point over a defined time interval, t2-t1,
representing one cardiac beat cycle and defined as:
 | (9) |
 | (10) |
p is the change in momentum and t is the change in
time. Thus, from the equation above, it can be reasoned that the
impulse becomes large in magnitude if the blood, flowing with a large
velocity, strikes a small area (apex) over a small time interval. This
equation and overall concept can be used to illustrate the forces
generated not solely by the magnitude of blood pressure but by sudden
surges in blood pressure due to physical exertion and daily activities.
A sudden increase or surge in blood pressure would increase the
magnitude of the variable force (Equation 10) exerted on the artery
wall, which, in turn, increases the magnitude of the impulse. Another factor of physical importance is the vibrational energy transferred to the arterial wall by the blood flow.29 Continual vibrations, a result primarily of the forced vibrations or the oscillations that occur in response to the periodic pressure pulse propagated at the onset of a heartbeat, tend to weaken the structural integrity of the bifurcation apex and magnify the existing state of destructive fatigue.
In addition to the kinetic energy imparted to the bifurcation apex, there is accompanying turbulence. In turbulent flow, portions of the fluid move radially as well as axially, forming eddies and vortices. The velocity profile in turbulent flow is much flatter over the central portion of the vessel and decreases much more sharply at the vessel wall. Turbulent flow is potentially more damaging to the circulatory system than smooth, laminar flow because the blood, instead of being directed parallel to the vessel walls, is directed toward them. Moreover, turbulent flow can create areas of stasis where clots can form.
Reynolds number, Re, is a dimensionless hemodynamic
parameter used to predict transitions from laminar flow to
turbulent flow and is given mathematically as:
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is the density of blood, and 
is the
viscosity of blood. At low Reynolds numbers, viscous forces dominate
and flow is laminar. At high Reynolds numbers, inertial forces dominate
and flow becomes turbulent. (When the flow is pulsatile rather than
steady, its tendency to become turbulent is described by another
nondimensional number, called the Womersley parameter.)
The Reynolds number that marks the limit of laminar flow, called the
critical Reynolds number, is a function of boundary
geometry.3 For flow through cylindrical pipes, the
critical Reynolds number is 2000. For flow around a sphere, however,
the critical Reynolds number is 1. In the human circulation, maximum
Reynold's numbers over one cycle range from 6000 to
<10-3 in transport from the heart to the
microcirculation.24 The critical Reynolds number for a
normal artery of the circulatory system is typically 2300, but in a
bifurcation it is 600 and can be as low as 400, greatly increasing
the risk of turbulence.
The Reynolds number plays an especially important role in determining
the pattern of flow in the sinus bulb, one of the sites where
atherosclerotic plaques are commonly found. Because the velocity
profile of the blood is skewed toward the inner wall of the
bifurcation, an area of low-velocity flow develops within the sinus.
This region, when acted on by the transverse pressure gradient that
arises because the flow is changing direction, tends to develop a zone
of recirculation or a vortex, represent transitional stages between
laminar and turbulent flow. Studying an experimental model of the
bifurcation, Motomiya and Karino30 showed that the
critical Reynolds number for the formation of the recirculation zone in
the carotid sinus of the bifurcation is 170, well below the
physiological value of 600. This finding indicates
that there is probably a standing recirculation zone in the sinus under
normal physiological conditions. This condition has
the distinct potential for the development of
atherosclerosis and thrombosis. Turbulence plays an
even greater role in flow through a stenosis, as will be
seen.
The arterial bifurcation affects the hemodynamic properties of the flowing blood by lowering the Reynolds number in some cases by half depending on the bifurcation angle or the angle between the two daughter arteries.31 The reduction of Reynolds number increases the likelihood of turbulence.
Over a period of years, oscillatory hemodynamic forces, continually striking at the apex of the bifurcation, exert large shear stresses against the endothelial surface and the underlying elastin network, which may cause focal or localized degeneration of the internal elastic lamina and may lead to aneurysm formation.28 The effect of the hemodynamic forces may be compounded by the presence of abnormal physiological conditions such as chronic high blood pressure (arterial hypertension).
Hemodynamic Forces at Vessel Curvature
A number of points along the human carotid artery exhibit
curvature, particularly before and after the bifurcation apex (Fig 6). The importance of vessel curvature in
the carotid artery can be demonstrated by considering the
hemodynamic parameter wall shear stress.
The wall shear stress, 
, in terms of the volumetric flow rate, Q, is
given as:
 |
. Thus, the
blood exerts greater shear stress on the outer wall than on the inner
wall of the curved vessel.
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This phenomenon occurs in common everyday experiences such as driving a
car or bicycling around a curve or sledding in a luge (Fig 7B). Either
example typically involves circular motion of an object or person that
is maintained by a centripetal or "center-seeking" force given
by:
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Hemodynamics of a Stenosis |
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TopAbstractIntroductionBiophysical Mechanisms of StrokeCarotid Artery BifurcationHemodynamics of a StenosisConclusionsReferences |
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The previous section discussed anatomic and hemodynamic factors presented by the carotid artery bifurcation in its normal state that could predispose an individual to atherosclerosis. The implementation of vascular disease such as atherosclerosis further complicates and compounds the potential dangers implicated in stroke. The carotid artery and hemodynamic effects were considered on a global basis; however, we would now like to focus our discussion on the local hemodynamic effects presented by an occlusion or stenosis.
The luminal reduction presented by a vascular stenosis introduces significant changes in the vessel geometry as well as the hemodynamics before, during, and after the formation of stenosis. The geometry of a stenosis presents an irregularity in the vessel contour, the consequences of which manifest themselves into alterations of normal hemodynamics.
Hemodynamics through an arterial stenosis is a problem of paramount importance in discussions of cerebrovascular disease and has been studied extensively.32 33 34 35 36 The basis for the hemodynamics at a vascular stenosis is Poiseuille's law, which in effect states a linear relationship between volumetric flow rate and pressure gradient under normal circumstances, and Bernoulli's principle, which states an inverse relationship between pressure and flow velocity.
Let us first consider Poiseuille's law of fluid flow. Blood flow in a
vessel can be approximated sufficiently according to Poiseuille's
formula:
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P is the pressure gradient, r is the
inner radius of the vessel, L is the length of the vessel, and is
the blood viscosity. According to Poiseuille's law, flow volume scales
as the fourth power of the vessel radius and is linearly related to the
pressure gradient. As a consequence, doubling the radius of a blood
vessel increases flow rate 16-fold; halving the radius decreases flow
rate 16-fold. The linear relationship between flow rate and the pressure gradient holds true only as a first-order effect and is valid only to a "point" where the hemodynamic relationship becomes nonlinear or drastically changes in form. This point corresponds to the transition from laminar flow to turbulent flow. The validity of Poiseuille's law of fluid flow is compromised when the fluid becomes non-Newtonian or turbulent or when the vessel radius is restricted to the stenotic region as opposed to the entire vessel.37 More specifically, the reasons behind the inapplicability of Poiseuille's law include (1) non-Newtonian viscosity of blood, (2) turbulence, (3) pulsatile driving pressures, (4) kinetic energy transformations, and (5) distensibility of vessels.38
One is then left to consider quantitative values of flow rate as the
pressure gradient increases continually within the stenosis
past the point of nonlinearity. Byar et al37 performed a
series of fluid flow experiments investigating the influence of all
hemodynamic parameters contained within
Poiseuille's law and found that fluid flow, Q, as it pertains to a
stenosed vessel, is related to the pressure gradient, P, according
to:
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P with respect to
Q (slope of the experimental flow curve), and a is the value of P at
zero Q (graphic intercept of the experimental flow curve).
Interestingly enough, although they were carefully considered in
experimentation, there is no external dependence on fluid viscosity,
vessel radius, and vessel length. The influence of these factors,
however, is incorporated into the mathematical function and associated
constants a and b. As the stenotic region of a vessel becomes critical (sufficient to cause a significant reduction of flow), the flow rate decreases, the pressure gradient across the stenosis decreases, and the flow velocity increases, as illustrated by Bernoulli's principle. Whenever there is a change in the velocity of blood, such as would occur in a tube that widens or narrows abruptly, some of the blood's kinetic energy is converted into pressure, or the pressure is converted into kinetic energy. The conversions are described by Bernoulli's law, named after the Swiss physicist and mathematician Daniel Bernoulli. Bernoulli's principle expresses how energy is conserved in a fluid through a trade-off between kinetic energy and pressure: More rapid flow is associated with lower pressure and slower flow with higher pressure.
In the normal cardiovascular system, blood vessels
narrow or widen only gradually, and the pressure gradients far outweigh
the small interconversions of kinetic energy and pressure. In disease
states such as stenosis, however, the Bernoulli effect becomes
quite marked. In a stenosed vessel, the more rapid flow of blood
through a narrower lumen decreases the pressure gradient across the
constriction (Fig 8). When pressure drops
in any segment of the arterial system, it is due to both
resistance from the stenosis and the conversion of potential
into kinetic energy. The pressure drop due to energy lost in overcoming
resistance is irreversible, since the energy is dissipated as heat;
however, the pressure drop due to change or transformation of potential
to kinetic energy as a vessel narrows is reversed when the vessel
widens again.
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In addition, there is a buildup of excess pressure proximal to the
stenosis. The trends of these hemodynamic
parameters continue until a critical stenosis is
reached. The critical stenosis is defined as the percent
stenosis at which intravascular flow approaches zero and the
pressure approaches its maximum value. The critical stenosis is
unique to vessel geometry and hemodynamics but has been
shown to occur generally at approximately 80% to 85% obstruction of
the major vessels in the human vasculature.39 40 At the
point of critical stenosis, a sharp decrease of flow rate is
observed as a result of the increased turbulence proximal to the
stenosis, as shown in Fig 9.
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As the stenosis progresses to occlusion, the pressure drop across the stenosis reaches 100% of the maximum and the flow rate is zero. In addition, the prestenotic pressure is equal in magnitude to that at the origin of the parent vessel. As an illustrative example of Newton's Third Law, the stenosis exerts an equal yet opposite force against the hemodynamic forces generated by the systemic blood pressure driving blood through the obstructed vessel. At 100% stenosis, the pressure gradient is not exactly zero due to the additional energy losses. However, no relations, experiments, or adequate explanations exist that elucidate and accurately characterize these energy losses, leading one to approximate the pressure gradient.
Bernoulli's principle describes the total energy of a flowing fluid
per unit volume through a rigid vessel:
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Bernoulli's principle expressed for these two points can be equated,
relating hemodynamic parameters
characteristic of the two distinct regions of flow:
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If we return to our discussion on Bernoulli's principle, the inclusion
of viscous forces into the fluid flow problem requires modifications to
the original equation describing Bernoulli's principle. Incorporating
terms representing the contribution of stenosis
length to pressure drop across stenosis and the contribution of
distal luminal expansion of the pressure drop, the equation of a
pressure drop across the stenosis is36 :
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is blood viscosity, L is vessel length, R is vessel
radius, A1 is cross-sectional area of normal
arterial lumen, A2 is cross-sectional area of
stenosed arterial lumen, v1 is velocity of
blood in unstenosed artery, and is the density of blood. Since ischemic strokes are the most common and most likely the result of atherosclerotic lesions developed along the arterial wall, we will now consider the influence of an atherosclerotic lesion on the vessel wall and corresponding interactions between the flowing blood and vessel wall. As mentioned earlier, an atherosclerotic lesion is an irregularly distributed mass of calcified fatty deposits that narrows the arterial lumen and stiffens the affected portion of the vessel wall, creating a region of rigid tissue countered on either end by vascular wall that has retained its elastic behavior. This, in effect, places a mechanical load on the vessel wall, causing significant changes in the biophysical and biomechanical proper-ties of the vessel, ultimately resulting in reduced distensibility.
Reduced Volumetric Blood Flow Rate
The first and probably most obvious of these biophysical changes
caused by flow through the stenotic lesion is the reduction of
blood flow. The consequences of reduced blood flow are a possible
decrease in blood flow volume circulating through the brain and the
occurrence of stasis and thrombosis. As the blood flow capacity of the
affected artery is reduced, other arteries compensate by dilating,
thereby increasing blood flow and maintaining adequate levels of brain
tissue perfusion.
However, problems originate at the site of the occlusion. Blood is a fluid that must be in continual motion to function properly. As blood flow is reduced through the stenosis, recirculation zones form distal to the stenosis, causing the flow to become stagnant. The stagnation of blood in these zones can trigger clotting mechanisms that lead to thrombosis, one of several mechanisms by which less-than-critical stenosis at the carotid artery bifurcation can cause stroke elsewhere in the cerebral vasculature.3 The mechanism by which blood clots as a result of reduced motion is termed stasis, and the resultant clot is the thrombus. It should be noted that the thrombus does not adhere strongly to the vessel wall and itself can be dislodged into the blood stream as an embolus and result in stroke.
Increased Blood Flow Velocity
The effects of increased blood flow velocity occur according to
three different mechanisms.
First, increased blood flow velocity induces high kinetic energy at the stenosis, exerting a significant hemodynamic force against the normal portion or poststenotic region of the vessel wall. The increased blood flow velocity through the stenosed region of the abnormal vessel exhibits unique characteristics and is termed jet flow. Jet flow, commonly used to describe flow exiting a hypodermic needle or catheter under fairly large pressures, represents the turbulent nature of flow following a constricted area of the vessel. Prolonged impingement of the blood flow at this magnitude of force could induce structural fatigue and corresponding changes in the vessel wall, resulting in distension of the vessel and ultimately leading to the development of an aneurysmal dilatation. The distension distal to the stenosis, also known as poststenotic dilatation, is believed to be due in part to the conversion from high kinetic energy to high potential energy, as given by Bernoulli's principle.
The presence of atherosclerotic lesions produces a reduction in
the vessel diameter, which in turn promotes abnormal
hemodynamics. The abnormal hemodynamics
are believed to be responsible for the dilatation or pouching of the
vessel wall, which occurs just past the lesion. Several factors may
contribute to the development of poststenotic dilatations,
including (1) the conversion of the high kinetic energy of the swiftly
moving bloodstream into high potential energy or lateral pressure; (2)
shocks of impacts of alternating high and low pressure; (3) the
increase in lateral pressure caused by the lower velocity due to the
widening of the vessel, according to the Bernoulli equation; and (4)
high-frequency pressure fluctuations within a turbulent field. The
hemodynamics through a poststenotic dilatation
is shown in Fig 10.
|
Second, increased blood flow velocity coupled with the irregular
geometry causes a decrease in Reynold's number and a corresponding
tendency for the blood flow to become turbulent. In turbulent flow, the
kinetic energy produced by the flowing blood is transferred into the
cracks and crevices presented by the abnormal plaque
distribution with potentially enough accumulated energy over time to
dislodge a portion of the plaque into the bloodstream, where it now
becomes a particulate embolus (see Fig 8). In addition, the developed
turbulence is believed to be the source of bruits, or audible sounds
detected with a standard stethoscope placed over the stenosed
area.41 42 Although the bruit frequencies vary, a recent
study by Kurokawa et al43 revealed a frequency <850 Hz
for arterial stenoses <70% and >800 Hz for
stenoses>70%.
Third, increased blood flow velocity causes a high shear stress along the upper portion of the lesion and a region of low shear stress along the tails and bottom portion of the lesion. The shear stress acts in conjunction with the kinetic energy created by the turbulent flow to create a potentially dangerous situation.
Another possible hemodynamic phenomenon that can occur as the direct result of increased blood flow velocity due to a sudden change in the vessel diameter is the water hammer effect. In reality, it is the sudden conversion of the kinetic energy of the blocked flow to pressure and the reflection of the resulting pressure wave between the ends of the vessel.3 If it occurs, the intensity of the water hammer effect increases as the vessel wall becomes more inelastic. As the fluid impinges on the constricted area of the vessel, hemodynamic energy is expended in forcing the fluid through the constriction and distensions of the vessel wall, causing the rapid changes in pressure and resulting in the water hammer effect.
Compliance Mismatch
The change in elasticity induced by the atherosclerotic lesion is
termed compliance mismatch. Compliance, an indirect measurement of the
vessel wall elasticity, is the change in volume with respect to the
change in pressure or C=V/P. The implications of a compliance
mismatch can be visualized by considering the elastic function in
response to blood flow. As the pulsatile flow strikes the wall of a
normal vessel, the elastic wall reacts with recoil in response to the
hemodynamic forces, further propelling the blood along
the vasculature. In the atherosclerotic region of the vessel, the
recoil response is substantially reduced or eliminated, depending on
the extent and distribution of the lesion, which may not be sufficient
to force blood around the lesion. The heart also has to exert more
force and produce more work to maintain proper levels and rates of
blood flow.
As the incoming hemodynamic pressure wave
propagates through the blood vessel containing the atherosclerotic
lesion, part of the wave is transmitted through the patent portion of
the vessel, while the remaining part of the wave reflects off the
lesions and propagates in a direction opposite to that of the original
incoming pressure wave. The hemodynamic pressure wave
propagates along the vessel wall with a wave velocity, c, given
by44
 |
d is blood
density. This is known as the Moens-Korteweg equation. As the elastic
modulus of the atherosclerotic vessel decreases, so does the wave
velocity, resulting in a localized deposition of kinetic energy
proximal to the lesion.
The reflected pressure wave is critically damped within the boundaries
created by the normal and atherosclerotic lesion of the vessel wall.
The increase in damping reduces the natural frequency of the vessel
wall. The natural frequency of the normal vessel is 1 to 2 kHz, and
the frequency of the pulsatile blood flow is 450 Hz. Because of the
frequency difference, the wall responds only feebly to the driving
force of the pulse.3 However, the vessel wall frequency
could be reduced, depending on the distribution and extent of the
stenotic lesion, to equal that of the blood flow, making
resonance and subsequent rupture a physical possibility. If we set
aside issues of geometry for the moment, the vibrational displacement
of an elastic object subjected to a periodic driving force can be
expressed mathematically by the differential equation:
 |
r:
 |
|
Conclusions |
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|
TopAbstractIntroductionBiophysical Mechanisms of StrokeCarotid Artery BifurcationHemodynamics of a StenosisConclusionsReferences |
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The clinical origin and causes of stroke are attributed not to one single factor but a host of factors acting either independently, in combination, or in succession. These factors involve possibly genetic, biochemical, physiological, anatomic, and histological components. However, biophysical factors are believed to play a significant role in the development, diagnosis, and therapy of stroke. The subject matter presented in this mini-review addressed primarily the biophysical processes implicated in the origin and mechanisms of stroke. Recent advances in the treatment of stroke are based on increasing knowledge of its underlying biophysical mechanisms, as well as on better-publicized advances in imaging instrumentation and procedures for the management and treatment of patients.
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Footnotes |
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Reprint requests to George J. Hademenos, PhD, Department of Physics, University of Dallas, 1845 E Northgate Dr, Irving, TX 75062.
Received June 26, 1997; revision received July 10, 1997; accepted July 10, 1997.
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References |
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TopAbstractIntroductionBiophysical Mechanisms of StrokeCarotid Artery BifurcationHemodynamics of a StenosisConclusionsReferences |
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