George J Hademenos. American Scientist. Volume 85, Issue 3. May/Jun 1997.
Unlike an electron, a single red blood cell cannot go through two openings at once. Indeed, it is generally true that the physicist seeking to understand the circulation of blood can ignore most of the perplexities of 20th-century physics. Given the scale of the circulatory system and the speed of blood flow, neither quantum mechanics nor relativity applies. Instead the flow of blood through the heart and the vascular tree can be adequately described by the familiar mechanics of Newton and Galileo.
If the circulation of the blood obeys the laws of classical mechanics, however, this does not mean it is simple. An early experimental model of the vascular tree was a system of glass tubes filled with water. But unlike water, blood is not an “ideal” fluid. Instead it is a suspension of cells that, under certain circumstances, can behave in non-Newtonian ways. Moreover, flow through the vascular tree is pulsatile rather than steady; blood vessels taper and are elastic rather than rigid; and flow in any part of the densely interconnected system is influenced by flow in neighboring regions. When these factors and the fantastic geometrical complexity of the labyrinthine vascular system are taken into account, the equations of blood flow, while remaining classical in inspiration, quickly become too complex to be solved explicitly.
Confronted with a system of overwhelming complexity, scientists typically resort to the intelligent simplification, that is, to a model. The first mathematical model of the human circulation was the windkessel, or compression chamber, model developed by Otto Frank in 1899 to explain how pulsatile flow from the heart is converted into steadier flow in the peripheral circulation. Sophisticated analytical models, the descendants of the windkessel model, still provide insight into the functioning of the circulatory system. But they are increasingly supplemented by numerical models, which exploit the power of the computer to arrive at accurate approximations of values that satisfy systems of equations that would otherwise be unsolvable. The computer can also be used to produce stunning images that allow otherwise cryptic measurements or calculations to be grasped intuitively.
A physicist by training, I was hired by the UCLA School of Medicine to study blood flow. As I worked with radiologists at the medical school, however, I became interested in modeling blood flow under pathophysiological rather than abstract conditions. Over the past four years, I have developed biophysical models of three circulatory abnormalities that can lead to stroke: cerebral aneurysms, local outpouchings of the walls of arteries that supply the brain; cerebral arteriovenous malformations, which are abnormal capillary beds that include enlarged and weakened vessels; and stenoses, or partial blockages, of the arteries feeding the circulation of the brain.
To introduce the concepts of modern hemodynamics, this article describes the physical principles that govern the flow of blood through normal and stenosed carotid-artery bifurcations. Biophysical models contribute in many ways to our understanding of the circulation. They allow existing knowledge about a circulatory problem to be assembled and integrated, the sensitivity of the vascular system to different parameters of flow to be determined, parameters that are difficult or impossible to measure to be calculated, hypotheses about normal or abnormal circulation to be inexpensively tested, and gaps in knowledge to be pinpointed.
Although hemodynamics has been integrated into the clinical practice of medicine only slowly, this situation is changing. Already some biophysical models have sufficient fidelity to accurately predict a patient’s risk of hemorrhage or ischemia. And many of the new therapies that have been proposed for stroke are based on what could be called a physical rather than a clinical understanding of stroke.
Stroke is the third leading cause of death and the leading cause of long-term disability in the United States. Each year half a million people die from a stroke. The treatment of stroke places a substantial burden on the national economy, costing an estimated $23.2 billion a year. Because of the toll this condition takes on its victims and their families and on the nation as a whole, an intense effort is under way in this Decade of the Brain to better understand the causes of stroke and to devise more effective means of preventing and treating it.
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 and throat 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, divide into two cervical and five cranial branches each. The cervical branches supply the neck, and the cranial branches supply the posterior portion of the brain. The anterior and posterior blood supply connect in the circle of Willis, an arterial interchange at the base of the brain.
“Stroke” is a portmanteau term covering any disease or neurological insult that results in the marked restriction or cessation of flow through this supply system. There are two fundamentally different kinds of stroke: hemorrhagic and ischemic. Hemorrhagic strokes, which account for 20 percent of all strokes, take place when vascular lesions rupture, releasing blood into the surrounding brain tissue. The remaining 80 percent of strokes are ischemic in character, caused by the obstruction or clogging of the major arteries in the cerebral circulation. (Ischemia refers to decreased blood supply to a tissue, a potentially reversible condition; uncorrected, it leads to infarction, or tissue death due to anoxia.)
Among the vascular lesions that can lead to hemorrhagic strokes are aneurysms and arteriovenous malformations. An aneurysm is a pouch, or balloon, in a vessel wall, typically only millimeters long when fully developed. Arteriovenous malformations, or AVMs, are congenitally malformed capillary beds. A normal capillary bed consists of blood vessels about eight-thousandths of a millimeter in diameter. The vascular resistance of these tiny channels slows the flow of blood sufficiently to allow oxygen and nutrients to diffuse into surrounding tissue. Instead of impeding the flow of blood, the enlarged vessels in an AVM shunt blood between the high-pressure arterial and the low-pressure venous systems. Aneurysms and AVMs can form anywhere in the human vasculature, but they pose the greatest risk when they occur within the brain.
The obstructions that cause ischemic strokes can arise by at least five different processes: atherosclerosis, embolus, thrombus, hemorrhage and vasospasm. Atherosclerosis is a pathological process characterized by yellowish plaques of cholesterol, lipids and cellular debris in the inner layers of the walls of arteries. 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.
By encouraging the aggregation of platelets, atherosclerotic plaques can contribute to the formation of either emboli or thrombi. An embolus is any traveling obstruction-commonly a platelet aggregate dislodged from a plaque but also potentially a bubble of gas-that is transported through the vasculature until it lodges in and blocks a vessel. A thrombus is a blood clot, an aggregation of platelets and fibrin formed in response either to an atherosclerotic lesion or to vessel injury.
Ischemia can also be a secondary consequence of spontaneous or traumatic hemorrhage. A hemorrhage into the fixed volume of the skull can compress blood vessels, enlarging the initial insult by reduring blood flow to the surrounding tissue. Bleeding into the space between the two membranes on the surface of the brain sometimes causes the sudden, transient spasm, called vasospasm, of the blood vessels there. This condition can persist for days, much longer than the six hours during which cerebral ischemia is thought to be reversible.
The Carotid-Artery Bifurcation
Understanding the physics of flow through a stenosed bifurcation is a matter of considerable human importance; fully 33 percent of all cases of stroke are caused by atherosclerotic lesions and thrombosis, and another 31 percent are attributed to emboli. Because most atherosclerotic lesions are found near bifurcations, the diversion of flow there is thought to play an important role in their formation.
Many aspects of fluid flow at a carotid-artery bifurcation can be captured by modeling it as a simple Y shaped structure, but others require models with more realistic geometry. The angle between the internal and external carotid arteries can vary from 30 to 120 degrees, and the daughter arteries may not be straight but instead may curve away from the parent artery. Above the bifurcation, the internal carotid swells into a pocket known as the carotid sinus, or bulb, which has its own distinctive hemodynamics.
The apex of the bifurcation and the sinus are common sites of circulatory disease. Aneurysms can occur at the apex. Atherosclerotic plaques are found almost exclusively at the outer wall (hip) of one or both daughter vessels at major bifurcations, including the carotid.
But the mechanism or mechanisms by which these lesions are created are still the subject of debate. The static mechanical properties of the arterial bifurcation may be one contributing factor. Just as stress tends to be concentrated in certain areas of bridges or buildings, stress tends to be concentrated in some parts of the bifurcation, and high stress concentrations seem to be associated with the formation of atherosclerotic lesions. The main culprit in the formation of atherosclerotic plaques, however, is probably hemodynamics, particularly the shear stress imposed by the moving blood on the vessel wall. 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 low blood flow, that promote clotting. It is the role of a biophysical model to adjudicate between these and other, sometimes contradictory hypotheses of atherogenesis.
Laminar Flow through a Bifurcation
Both mechanical and hemodynamic energy are concentrated at the apex of a bifurcation by its geometry. Neglecting for a moment the blood it carries, an artery can be visualized as an elastic cylindrical tube, or even more simply, as a rubber band. An arterial bifurcation, then, is analogous to three rubber bands linked together in the form of a Y.
The consequences of this geometry can be seen by resolving the tensile, or stretching, forces the three arteries exert on the apex into their geometrical components. 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
To investigate the role of mechanical stress in atherosclerotic disease, Robert S. Salzar of the University of Virginia and his colleagues at the Heineman Medical Research Laboratories in Charlotte, North Carolina, constructed numerical models of the carotid-artery bifurcation and then loaded them with a normal incremental pressure of 40 millimeters of mercury (to represent the pulse) and 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.
The kinetic energy of the flowing blood also tends to be deposited at the apex of the bifurcation. Why this happens can be understood by considering the fundamental equation of hemodynamics, Poiseuille’s law, and its implications in the case of a bifurcation. Blood flows through a vessel because there is a pressure difference along the vessel. The relation between pressure and flow was established by J. L. M. Poiseuille, a French physician who conducted extensive experiments in which he used compressed air to force water through capillary-sized glass tubes. (He was unable to use blood because there was no known means to prevent its coagulation.) Poiseuille found that the pressure gradient needed to produce a given flow is a function of the dimensions of the tube and the viscosity of the fluid.
Although Poiseuille arrived at his equation empirically, it can be derived by assuming that each particle of blood moves at a constant velocity parallel to the vessel wall and that the force opposing this motion is proportional to the blood’s viscosity and to the velocity gradient perpendicular to the flow. Because blood near the wall of the vessel is slowed by viscous drag, the flow within the vessel quickly assumes the form of a series of concentric layers, or concentric tubes. The velocity of any one layer is constant, but the closer the layer is to the center of the vessel, the higher its velocity. When the flow within a vessel is fully developed, its velocity profile is parabolic: The velocity of the blood traveling along the axis of the vessel represents the center of the parabola, and the velocity of the blood traveling next to the walls is at the tail of the parabola. The shear stress each layer exerts on the walls of the vessel is inversely related to its velocity and directly related to its distance from the wall. Thus the blood flowing through the center of the vessel exerts no shear stress on the wall, and the blood flowing next to the wall exerts the maximum shear stress.
As the flow is divided at a bifurcation, the parabolic velocity profile is dramatically skewed toward the apex of the bifurcation. 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 outer walls of curved vessels are subjected to abnormally high wall shear stress for similar reasons. Just as bobsled riders are forced to the outside on a curve, in a curved vessel the velocity profile of the flowing blood is skewed to the outside wall. Thus the blood exerts greater shear stress on the outer than on the inner wall of the curve.
There is another consideration as well. According to Poiseuille’s equation, flow is sensitively dependent on vessel size. Indeed the flow volume rate depends on the fourth power of the vessel radius; doubling the radius of a vessel increases flow through it sixteenfold. The total cross-sectional area of daughter arteries is typically greater than that of the parent artery It follows that as the blood enters the daughter arteries, its velocity, and therefore its kinetic energy, decreases. 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.
The pulsatile nature of blood flow only exacerbates the vulnerability of the apex to mechanical and hemodynamic stress. The force exerted by the blood on the vessel walls is not constant but instead varies over one cardiac beat cycle. What matters is not the average force but rather the impulse, or the product of the force and the time interval over which the force is exerted. The smaller the area the blood strikes and the shorter the time over which it acts, the larger the impulsive force. The pointlike apex, then, is subjected to larger impulsive forces than other areas of the bifurcation.
One final feature of Poiseuille’s law with interesting consequences for biophysical modeling is its resemblance to Ohm’s law. According to Poiseuille’s law, the physical properties of a vessel determine how large a pressure gradient is needed to produce a given flow through it. The ratio of the pressure gradient to the flow rate, called the vascular resistance of a vessel, is analogous to the electrical resistance of a circuit component, which is defined by Ohm’s law as the ratio of the voltage drop across the component to the current through it. This electrical analogy turns out to be very useful because it allows the techniques of network analysis to be applied to the modeling of the densely interconnected vascular system. I recently used network analysis to model an arteriovenous malformation and to investigate whether impaired drainage of the malformation increased the risk of spontaneous rupture within it (see Figure 8).
The Role of Turbulence
So far this discussion has assumed that the blood flows smoothly through the bifurcation, but this assumption does not always hold. There are two distinct regimes of fluid flow: laminar and turbulent. In laminar flow, each parcel of fluid follows a course nearly parallel to the adjacent ones. In turbulent flow, portions of the fluid move radially as well as axially, forming eddies and vortices. Turbulent flow is potentially more damaging to the circulatory system than 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.
The critical factors in the transition from laminar to turbulent flow were identified by the British engineer and physicist Osborne Reynolds in 1883. In the case of flow through a rigid cylindrical tube, they are the dimensions of the tube, the mean velocity of flow and the kinematic viscosity and density of the fluid. These variables can be used to express the ratio of the inertial and the viscous forces acting on a parcel of the fluid. This ratio, called the Reynolds number, is an index of the tendency of the flow to become turbulent. 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. For flow through cylindrical pipes, the critical Reynolds number is 2,000. For flow around a sphere, however, the critical Reynolds number is 1. The critical Reynolds number for a normal artery of the circulatory system is typically about 2,300, but in a bifurcation it is approximately 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, particularly if the internal carotid curves away from the common carotid, 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, transitional stages between laminar and turbulent flow. This condition has the distinct potential for the development of atherosclerosis and thrombosis.
The geometry of the bifurcation helps to determine the size of the vortices that form there. According to Mark Fisher and Sherry Fieman of the University of Southern California School of Medicine, the vortices tend to be larger if the daughter vessels are together much larger than the parent vessel, if the bifurcation angle is large or if the daughter vessels are curved.
Studying an experimental model of the bifurcation (isolated carotid arteries rendered transparent by chemical treatment), Mineo Motomiya and Takeshi Karino of McGill University Medical Clinic showed that the critical Reynolds number for the formation of the recirculation zone in the carotid sinus of the bifurcation is about 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. Turbulence plays an even greater role in flow through a stenosis, as we shall see.
The Stenosed Artery
So far I have discussed the hemodynamics of the carotid-artery bifurcation in its normal state. But just as a scattering of boulders can turn a stream into a rapids, so can a partial blockage dramatically alter the flow through a vessel. Indeed, depending on the degree of blockage, quite different physical laws may be needed to describe flow through a stenosed vessel.
In particular, when flow is restricted by a stenosis, the validity of Poiseuille’s law of fluid flow is compromised. According to Poiseuille’s law, flow volume scales as the fourth power of the vessel radius and is linearly related to the pressure gradient. In 1965 David Byar of the Colorado General Hospital in Denver and his colleagues performed a series of fluid-flow experiments that showed that in a stenosed vessel, the relation between flow volume and vessel radius or differential pressure is more complex. For instance, flow volume is relatively unaffected until the stenosis becomes severe, but then flow begins to drop precipitously.
As I have remarked, Poiseuille envisioned the blood as driven through the vascular tree by pressure gradients. There can also be a kinetic-energy difference between two points in the vascular tree, however, and this difference becomes important in the case of stenosed flow. In particular, 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.
In a sense, a stenosed vessel is the physiological equivalent of a Venturi tube, a short tube with a constricted throat that is used to determine fluid pressures and velocities by measurement of the pressure differential between the throat and the open tube. Indeed, Venturi-tube calculations provide a useful check on clinical measurements of pressure loss and flow velocities at a stenosis, which are susceptible to errors because of the scale over which the hemodynamic changes occur and the limitations of current imaging techniques.
Although the precise geometry of a blockage plays an important role in its hemodynamics, for analytical purposes the geometry is often represented by the clinical index of carotid-artery disease: percent stenosis. One to 39 percent stenosis is considered mild; 40 to 59 percent is considered moderate; 60 to 79 percent severe; and 80 to 99 percent critical.
Why is 80 percent stenosis potentially a critical stenosis? As an artery is progressively blocked by a developing plaque, the velocity of the blood flowing through it increases and, following Bernoulli’s law, the pressure gradient across it decreases. These trends continue until a critical stenosis is reached. At the critical stenosis, the increased turbulence around the blockage causes a sharp decrease in flow rate. The critical stenosis is specific to vessel geometry and hemodynamics, but in the major vessels of the human vasculature, it is typically about 80 to 85 percent.
Developed turbulence is believed to be the source of bruit, noise that can be heard through a stethoscope placed over the stenosed area. A recent study by a group headed by Y. Kurokawa of Ube Industries Central Hospital in Yamaguchi, Japan found that the bruit frequency was less than 850 hertz if an arterial stenosis was less than 70 percent and greater than 800 hertz if the stenosis was greater than 70 percent, a finding consistent with the idea that critical stenosis is preceded by the development of strong turbulence.
The first and probably most obvious of the changes caused by a stenotic lesion is a reduction in the volume of flow through the lesion. As the pressure gradient across the stenosis diminishes, the vessels on the far side of the blockage dilate to keep the flow at adequate levels, but this compensatory mechanism has limits. Once the vessels have stretched as far as they can, any decrease in the diameter of the stenotic vessel results in a reduction of mean flow. As flow through the stenosis decreases, recirculation zones form downstream of it. The stagnation of blood in these zones can trigger clotting mechanisms that lead to thrombosis, one of several mechanisms by which lessthan-critical stenosis at the carotidartery bifurcation can cause stroke elsewhere in the cerebral vasculature.
The increased velocity of flow through the lesion, in turn, can contribute to the disease process in at least three ways. First, if the constriction is great enough, blood may rush from the constricted region in the form of a jet. The length of the jet depends on the diameter of the stenosis and its length. A long stenosis produces a longer jet and a wider stenosis a shorter jet. The jet subjects the wall of the vessel to high stresses when it reconnects with it, potentially high enough to cause structural fatigue. Such a mechanism may explain why it is common to find a distended area of a vessel, and sometimes ultimately an aneurysm, just past a stenosis.
Second, the critical Reynolds number of the stenosed vessel is lower than that of a healthy vessel. Since the velocity of the blood is also higher, flow through the stenosed vessel has a greater tendency to become turbulent. Turbulent flow allows the blood’s kinetic energy to be transferred into the cracks and crevices of plaque, potentially dislodging chunks of plaque into the bloodstream. Third, the pattern of flow through the stenosed vessel leads to a region of high shear stress along the top of the lesion and regions of low shear stress along its sides. The shear stress, together with the kinetic energy dissipated by turbulent flow, can also dislodge particulate matter from the plaque.
But this is not the end of the story. A calcified atherosclerotic vessel is also less elastic than a healthy vessel, and this has several consequences. The elastic walls of healthy vessels help to propel blood through the vasculature; a pulse of blood distends the walls of the vessel, and as the walls recoil after the pulse passes, the elastic energy stored in them is reconverted to energy of flow, giving the blood an additional push. Indeed, it turns out that the velocity with which a pulse travels through an artery is a function of the elasticity of the vessel wall. The two variables are related by an equation known as the Moens-Korteweg equation, after the scientists who contributed to its derivation.
In an inelastic atherosclerotic vessel, the ability of the vessel to expand in order to accommodate the volume of ejected blood at the onset of the cardiac cycle diminishes, and consequently recoil can be damped to the point that blood flow through the diseased vessel is substantially reduced. In order to maintain proper levels and rates and blood flow, the systemic pressure must be raised: the heart must work harder.
Moreover, a stenotic lesion partially reflects the pressure wave. Pressure waves travel without reflection in a tube that is uniform in diameter or elasticity, but whenever there is a discontinuity in vascular resistance, such as would be caused by a narrower lumen or a less elastic wall, the wave is partially reflected. Any reflection constitutes a diversion of energy from the pressure wave and a potentially damaging disturbance of flow.
As the blockage becomes complete, partial reflection can create a waterhammer effect, the same effect that produces the banging noise heard in a water pipe when a tap is abruptly turned off. The sound is caused by 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 pipe. If hammering took place at a stenosis, the more inelastic the vessel walls, the more intense the hammering would be.
Finally, atherosclerosis also makes the vessel more vulnerable by altering its frequency response. Plaque loads the vessel wall, depressing its natural frequency of vibration, much as a weight on a violin string would lower its pitch. The frequency of a pressure wave in a blood vessel is normally much lower than the natural frequency of vibration of the vessel wall. The frequency of pulsatile blood flow is about 450 hertz, and the natural frequency of vibration of the normal vessel wall lies between 1 and 2 kilohertz. Because of the frequency difference, the wall responds only feebly to the driving force of the pulse. It is conceivable, however, that atherosclerotic changes could depress the natural frequency of the wall to one close to the frequency of the pulse. The wall would then respond strongly to the pulse, a condition called resonance. Indeed, the resulting vibrations of the vessel wall might be violent enough to cause its rupture.
Numerical Models of Stenosis
Most numerical models of blood flow are based not on the analytical equations introduced in this article but rather on the Navier-Stokes equations, fundamental equations of motion that describe the trajectory of the blood at every point in the modeled vessel. The equations assume that the pressure exerted to move a particle of fluid is exactly balanced by inertial forces associated with its acceleration plus the viscous forces that must be overcome in moving it. Under these circumstances, the laws of fluid motion can be expressed in three equations that state the forces related to the longitudinal, radial and angular pressure differences. In addition to the Navier-Stokes equations, a model typically includes means of ensuring the conservation of mass and energy (or “continuity”) within the system and boundary conditions, such as the assumption that the fluid layer next to the wall has a velocity of zero.
A few examples of recent work with numerical models must suffice to suggest their utility and their limitations. Remarking that in vivo studies of cardiovascular disease are difficult and expensive to conduct and are limited to the easily accessible arteries, N. Stergiopulos and colleagues at the Swiss Federal Institute of Technology recently constructed a computer model that used a simplified one-dimensional form of the Navier-Stokes equations to describe an arterial system consisting of 55 arterial segments.
The scientists found that arterial stenoses caused changes in modeled arterial pressure and flow patterns similar to those seen clinically. For example, the model faithfully duplicated the effect of stenoses of varying severity on two commonly used diagnostic tests for stenosis: the pulsatility index, defined as the ratio of peak to trough flow to the mean flow; and the systolic pressure ratio, the ratio of systolic pressure in an artery containing a stenosis to a reference systolic pressure assumed to be unaffected by the stenosis. (Flow rates can be measured noninvasively with an ultrasonic flowmeter.) The authors concluded that “computer models could be used to study various aspects of cardiovascular disease and, consequently, help explore possible diagnostic techniques.”
Another new numerical model, by Gheng Tu and his colleagues at the Universite Catholique de Louvain in Belgium, puts the stenosed vessel under the microscope. Tu remarks that although flow disturbances associated with moderate to severe degrees of stenosis can be detected in certain vessels through the use of Doppler ultrasound, there is no method for detecting minor arterial stenosis, in part because detailed knowledge of the flow pattern through a stenosed vessel is lacking. His model, a two-dimensional numerical model consisting of finite-element approximations of the Navier-Stokes equations, is designed to identify these flow patterns.
Having built the model, Tu and his colleagues conducted numerical experiments to determine the effect of stenoses on both steady and pulsatile flows. Studies of steady flow showed, for example, that vortices developed downstream of medium or severe stenosis, and that the flow reversed direction downstream of the stenosis near the wall. Remarkably, in some cases the wall shear stress at the stenosis was 55 times greater than that at a nonstenotic region.
Studies of pulsatile flow through a 70-percent stenosis showed that the flow pattern changes dramatically during one cardiac cycle. Even when the Womersley parameter was small (there was little risk of turbulence), a vortex developed downstream of the stenosis at the beginning of systole and grew as the flow accelerated. When the flow reversed direction near the wall at the beginning of diastole, another, smaller vortex developed upstream of the stenosis. Both vortices grew in size, detached from the stenosis and filled most of the artery before they were dissipated by the main flow at the end of the cardiac cycle. The scientists remark that the model’s results are realistic enough that it could be used “to predict quickly stenotic flow patterns on an individual basis,” but only in a straight vessel. Flow through a bifurcation is complex enough that a three-dimensional model is needed to capture it accurately
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. The fidelity of the numerical models of blood flow, which is limited by computational power, has increased dramatically in the past 20 years, a trend that can be expected to continue. As the Stergiopulos and Tu models suggest, numerical models are already able to provide insights into the diagnosis and treatment of stroke that would be difficult or impossible to attain by other means. In the future they may well be instrumental in improving the outcome of patients afflicted with cerebrovascular disease.