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BLOOD FLOW IN ARTERIES
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ABSTRACT
Blood flow in arteries is dominated by unsteady flow phenomena. The cardiovascular
system is an internal flow loop with multiple branches in which a complex
liquid circulates. A nondimensional frequency parameter, the Womersley number,
governs the relationship between the unsteady and viscous forces. Normal
arterial flow is laminar with secondary flows generated at curves and branches.
The arteries are living organs that can adapt to and change with the varying hemodynamic
conditions. In certain circumstances, unusual hemodynamic conditions
create an abnormal biological response. Velocity profile skewing can create pockets
in which the direction of the wall shear stress oscillates. Atherosclerotic disease
tends to be localized in these sites and results in a narrowing of the artery
lumen—a stenosis. The stenosis can cause turbulence and reduce flow by means
of viscous head losses and flow choking. Very high shear stresses near the throat
of the stenosis can activate platelets and thereby induce thrombosis, which can
totally block blood flow to the heart or brain. Detection and quantification of
stenosis serve as the basis for surgical intervention. In the future, the study of
arterial blood flow will lead to the prediction of individual hemodynamic flows
in any patient, the development of diagnostic tools to quantify disease, and the
design of devices that mimic or alter blood flow. This field is rich with challenging
problems in fluid mechanics involving three-dimensional, pulsatile flows at
the edge of turbulence.
INTRODUCTION
The cardiovascular system primarily functions in nutrient and waste transport
throughout the body. The heart pumps blood through a sophisticated network
of branching tubes. The blood vessels distribute blood to different organs and
supply themselves with nutrition. The arteries, far from inert tubes, adapt to
varying flowand pressure conditions by enlarging or shrinking to meet changing
hemodynamic demands.
Blood flowunder normal physiologic conditions is an important field of study,
as is blood flow under disease conditions. The majority of deaths in developed
countries result from cardiovascular diseases, most of which are associated with
some form of abnormal blood flow in arteries. Many investigators have made
seminal contributions to this field, but their work cannot all be cited because of
page limitations. Instead, this review focuses on selected areas of cardiology
and vascular surgery by first discussing basic normal flows in arteries and the
biological responses to these flows. Then the review examines flows through
stenoses and the importance of their fluid mechanics to clinical medicine.
PHYSIOLOGIC ENVIRONMENT
Blood is a complex mixture of cells, proteins, lipoproteins, and ions by which
nutrients andwastes are transported. Red blood cells typically comprise approximately
40% of blood by volume. Because red blood cells are small semisolid
particles, they increase the viscosity of blood and affect the behavior of the
fluid. Blood is approximately four times more viscous than water. Moreover,
blood does not exhibit a constant viscosity at all flow rates and is especially
non-Newtonian in the microcirculatory system. The non-Newtonian behavior
is most evident at very low shear rates when the red blood cells clump together
into larger particles. Blood also exhibits non-Newtonian behavior in small
branches and capillaries, where the cells squeeze through and a cell-free skimming
layer reduces the effective viscosity through the tube. However, in most
arteries, blood behaves in a Newtonian fashion, and the viscosity can be taken
as a constant, 4 centipoise. Non-Newtonian viscosity is extensively studied
in the field of biorheology and has been reviewed by others (e.g. Chien 1970,
Rodkiewicz et al 1990).
GENERAL FLUID MECHANICAL CONSIDERATIONS
The existence of unsteady or pulsatile flow virtually throughout the cardiovascular
system forces the inclusion of a local acceleration term in most analyses.
The typical Reynolds number range of blood flow in the body varies from 1 in
small arterioles to approximately 4000 in the largest artery, the aorta. Thus the
flow spans a range in which viscous forces are dominant on one end and inertial
forces are more important on the other.
One-Dimensional Models
Pressure and flow have characteristic pulsatile shapes that vary in different
parts of the arterial system, as illustrated in Figure 1. The relationship between
the pressure waveform and total blood flow can be explained through a global
analysis of the fluid mechanics. The entire cardiovascular system may be
simplified using a lumped parameter or one-dimensional (1D) model of flow.
The most well-known model is that of the Windkessel (shown in Figure 2),
which has been used to explain the rapid rise and gradual decrease of the flow
and pressure waveforms (e.g. McDonald 1974).
Velocity Profiles
Vascular biologists are currently more concerned with the local hemodynamic
conditions in a given artery or branch than simply the flow waveform predicted
by IO models. A detailed local description of these pulsatile flows is needed.
The fluid-wall shear stress in a blood vessel for a given pulsatile flow situation
often needs to be determined. Fully developed pulsatile flow in a straight or
tapered tube can be expressed analytically (Womersley 1955). A physiologic
pressure or flow waveform can be expanded as a Fourier series, and the harmonic
components of velocity can be summed to yield the unsteady velocity
profiles. Although in the past this summation was carried out by hand with
tables, now it can be easily calculated using a simple computer program such
as Mathematica (He et al 1993). Figure 3 shows an example of velocity profiles
for a femoral artery of a dog.
Entrance Regions
Flow from the heart comes from a large pressure reservoir into successively
smaller tubes. The flow is not fully developed near some of the origins of
arteries. Flow in these regions is similar to an entrance flow with a potential
core and a developing boundary layer at the wall. The velocity profiles are
blunt near the center, and the centerline velocity accelerates as the boundary
layer retards velocity near the wall.
FLOWS IN SPECIFIC ARTERIES
Four parts of the arterial tree can serve as prototypic examples of hemodynamics:
the heart and proximal aorta, the abdominal aorta, the carotid bifurcation,
and the left coronary artery. These vessels exhibit flow characteristics seen in
most of the arterial tree and are important because they often become diseased.
Flows in the heart and great vessels are dominated by inertial forces rather
than viscous forces. Reynolds numbers at peak systole are on the order of 4000. The flow in the aorta and pulmonary trunk is similar to an entrancetype
flow that is not developed. Consequently, the core of the flow can be
considered an inviscid region that is surrounded by a developing boundary
layer at the wall. The pressure and velocity patterns in a complex chamber of
the heart can be modeled in three dimensions, including a moving boundary
condition that develops tension (Peskin & McQueen 1989, Yoganathan et al
1994). Alternatively, in vitro models of the heart and great vessel anatomy can
be studied in the laboratory. Flow can now be measured directly in the human
heart using techniques such as catheters, Doppler ultrasound, and magnetic
resonance velocimetry.
Hemostasis
Hemostasis is the arrest of bleeding. Trauma is a common occurrence, and
the body must be able to deal with it. Therefore, hemostasis must occur on
a very short time scale of milliseconds to minutes. When an artery is injured
through trauma, blood quickly squirts out through the hole. The high outflow
results in high shear stress, leaving collagen and tissue factor exposed. In this
hemodynamic environment, hemostasis is initiated primarily through platelet
activation and adherence. Platelet adhesion is modulated as much by shear rates
as by such biological factors as density of Glycoprotein Ib (GPIb) receptors,
von Willebrand factor (vWF) concentration, and exposed collagen (Hellums
1993, Markou et al 1993). For higher shear rates, platelet deposition can
increase 100-fold, as illustrated in Figure 8 (Badimon et al 1986, Markou
et al 1993, Fernandez-Ortiz et al 1994). Recent studies indicate that plateletto-
platelet aggregation may also be shear dependent. Conversely, when blood
is stagnant, it will be clotted by a cascade of coagulation proteins. Coagulation
is affected by the length of time blood is exposed to very low shear stress.
Under this condition, thrombosis, or clotting of blood, is strongly modulated
by the change in hemodynamic conditions that stimulates hemostasis (Hubbell
& McIntire 1986). More study on hemostasis is needed before the effects of
local hemodynamics can be fully understood.