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.
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.
Biophysical Mechanisms of Stroke
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, 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.
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.
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.
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⇒Thrombin⇒Fibrinogen⇒ Fibrin⇒Fibrin 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.
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).
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.
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
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.
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.
Carotid Artery Bifurcation
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: Resolving the three force vectors into their geometric components: If the magnitude of one of the forces is known, the two other forces can be determined easily by simultaneously solving Equations 3, 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.
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: where R is the radius of the vessel and r is the radial position across the cross-sectional radius of the artery. Wall shear stress can be derived from Equation 6 by the following: From Equation 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.
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: Thus, as blood flow from the parent artery enters the daughter arteries, the larger total cross-sectional area of the daughter arteries promotes a decrease in blood flow velocity and a corresponding decrease in kinetic energy as a result of the frictional stress imposed by the blood flow on the artery wall. The kinetic energy, which scales as the square of the velocity, must be dissipated somewhere, and the skewed velocity profile suggests it will be dissipated largely by frictional stress imposed by the blood on the inner walls of the daughter vessels near the bifurcation.3 The dissipation of kinetic energy at the apex of the bifurcation results in structural fatigue of the artery wall and is an important factor in the origin of aneurysms.
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: where J is the impulse. The magnitude of the variable force is the ratio of the change in momentum to the change in time. The magnitude of the variable force, F, is: where Δp is the change in momentum and Δt 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: where vm is the mean blood flow velocity, d is the diameter of the vessel, ρ is the density of blood, and η 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, τ, in terms of the volumetric flow rate, Q, is given as: From the above relation, it can be seen that the larger the degree of tortuosity, the smaller the radius of curvature and the larger the wall shear stress. The tails of the velocity profile (the points of the profile in contact with the vessel wall) exhibit the maximum wall shear stress. As blood flows through a curved portion of a blood vessel, the velocity profile becomes skewed away from the radius of vessel curvature, as shown in Fig 7A⇓. Thus, the blood exerts greater shear stress on the outer wall than on the inner wall of the curved vessel.
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: where m is the mass of the body in uniform circular motion, a is the centripetal acceleration, v is the velocity of the body, and R is the radius of the circular path. As R decreases (eg, the curve becomes sharper), the centripetal force increases accordingly. The centripetal force is counteracted by a centrifugal force that is always directed outward from and perpendicular to the axis of rotation. When driving around a curve, one leans into the curve to counteract the centrifugal force. It is the centrifugal force acting against the blood particles that causes a shift in the center of mass and hence the velocity profile. Once the blood flow passes the curve, the influence of the centrifugal force is removed and the velocity profile returns to its axisymmetric form.
Hemodynamics of a Stenosis
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: where Q is the flow rate, ΔP is the pressure gradient, r is the inner radius of the vessel, L is the length of the vessel, and η 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, ΔP, according to: where b is a constant relating the change of ΔP with respect to Q (slope of the experimental flow curve), and a is the value of ΔP 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.
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⇓.
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: According to Bernoulli’s principle, the flow velocity, v, is inversely proportional to the intravascular pressure, P, which can be determined by Poiseuille’s law. The premise behind applications of Bernoulli’s principle to a stenosed vessel is that, since conservation of mass holds, Bernoulli’s principle can be applied to two points representing flow through different segments of a vessel. Therefore, Bernoulli’s principle can be expressed for the two points according to the following: one point (1) representing flow through a normal unobstructed region of the vessel and another point (2) representing flow through an obstructed region at the maximum point of stenosis.
Bernoulli’s principle expressed for these two points can be equated, relating hemodynamic parameters characteristic of the two distinct regions of flow: From a biophysical standpoint, we now have an expression for the loss of pressure due to a partial obstruction of the vessel. The ability to make such calculations is important because of the probability that clinical measurements and observations at or around a stenosis are susceptible to errors due to the small distances imposed by stenoses and limitations in the imaging modality and corresponding technique.
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 : where η 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 ρ 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.
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=ΔV/ΔP. 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 where E is the Young’s modulus of elasticity, h is wall thickness, R is mean vessel radius, and ρd 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: The solution of the above equation, the step-by-step derivation that can be found in any elementary engineering textbook,45 yields the following expression for the resonant frequency, ωr: The consequences of the resonant frequency to the problem at hand can be seen by increasing the damping coefficient, B. Increasing B lowers the resonant frequency of the atherosclerotic wall, making resonance more likely.
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.
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.
- Copyright © 1997 by American Heart Association
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