Part V
89
Introduction
Blood is composed of blood cells suspended in plasma. The viscosity of blood varies with clinical conditions that influence blood cell aggregation and the hematocrit, and involves hemodynamic changes in vessels directly [1– 3] . Generally, in arteries with diameters larger than 3 mm, the viscosity of blood is essentially constant when the shear rate exceeds 100 s −1 [4] . By contrast, in capillaries with diameters smaller than 400 m m, the decrease in the vessel diameter reduces the viscosity of blood by redistributing blood cells at the center of the vessel [5] . Therefore, we should consider non-Newtonian blood flow behavior according to the flow conditions in blood vessels. This section briefly describes the basics of fluid mechanics that might be useful for understanding cerebral hemodynamics.
Definition of a Fluid
A fluid is defined as a substance that deforms continuously under the application of a shear (tangential) stress no matter how small the shear stress may be [6] . In most cases, fluids comprise the liquid and gas phases of the physical forms in which matter exists. The distinc- tion between fluid and solid phases is clear when we compare their deformation over time when a shear stress is applied (Fig. 1 ). When the shear force F is applied to a plate, the deformation of the block continues to increase with time, whereas that of a solid is propor- tional to the shear stress applied.
J.C. Kim
Institute of Medical and Biological Engineering , Medical Research Center, Seoul National University College of Medicine , 199-1 Dongsoong-dong, Jongno-gu , Seoul 110-744 , Republic of Korea E.B. Shim (!)
Department of Mechanical and Biomedical Engineering , Kangwon National University , Chuncheon , Kangwon 200-701 , Republic of Korea
e-mail: [email protected]
Hemodynamics
Jeong Chul Kim and Eun Bo Shim
B.-K. Cho and T. Tominaga (eds.), Moyamoya Disease Update, DOI 10.1007/978-4-431-99703-0_14, © Springer 2010
90 J.C. Kim and E.B. Shim
Viscosity
Fluids may be classified as Newtonian and non-Newtonian fluids according to the relationship between the applied shear stress and the shear rate (Fig. 2 ). Fluids in which the shear stress is directly proportional to the shear rate are Newtonian fluids. All fluids in which the shear stress is not directly proportional to the rate of deformation (the shear rate) are non-Newtonian fluids.
The most common Newtonian fluids are water and air under normal conditions. The relationship between shear stress ( t ) and the shear rate (d u /d y , x -velocity gradient in the y -direction) in a Newtonian fluid can be described as Newton’s law of viscosity.
yx du,
= k dy
t (1)
where the subscript yx denotes the vector normal to the plane in which the shear stress is applied ( xz -plane) and the direction of deformation (the x-direction). The constant of proportion- ality, m , is the dynamic viscosity. In the absolute metric system, the basic unit of viscosity is the poise [1 poise = 1 g/(cm · s)] and in the System International of Units (SI) it is
F F
t0 t1 t2
t0<t1<t2 solid
material fluid
plate
a b plate
Solid Fluid
Fig. 1 Behavior of a solid ( a ) and fluid ( b ) under the action of a constant shear force
Pseudoplastic Bingham plastic
Shear stress
Shear rate
Newtonian Dilatant Fig. 2 Shear stress as a function of the shear
rate for one-dimensional flow
91 Hemodynamics
kg/(m · s) or Pa · s [1 Pa · s = 1 N · s/m 2 ]. The dynamic viscosity of water is 1.0 centipoise (cP) at 20°C, and that of blood with a normal hematocrit is 3.5–4.0 cP at body temperature.
Many common fluids exhibit non-Newtonian behavior. Two familiar examples are toothpaste and Lucite paint. The latter is very thick when in the can, but becomes thin when sheared by brushing.
This shear-thinning characteristic can be seen in the blood flow; in the low-wall-shear-stress region around bifurcations, the viscosity of blood increases through red blood cells aggrega- tion, whereas in capillaries with diameters less than 400 m m, the viscosity decreases via the redistribution of red blood cells at the centerline of the vessel (Fahraeus–Lindqvist effect).
Numerous empirical equations have been proposed to describe non-Newtonian fluid flow [7] . The power law model, one of the simplest, can adequately represent the relationship between shear stress and the shear rate:
n yx
= k du , t dy
(2)
where the exponent n is the flow behavior index and the coefficient k is the consistency index;
both are determined empirically.
Governing Equations (Poiseuille’s Law)
General fluid flow can be described mathematically by the Navier–Stoke equation, which can be reduced to the following for steady pipe flow (Fig. 3 ) [8]
(P1 P )2 2 2 u = (R r ),
4 L
- -
m (3)
where P 1 and P 2 are the pressures at the ends of the length ( L ), R is the internal radius, and r is the radius of the pipe. This is the equation for a parabola, where u = 0 when r = R and is a maximum when r = 0 at the centerline of the pipe. The total volume rate of flow Q can be defined as
1 2 2 2 4 1 2
0 0
2 ( ) ( )
d 2 d ( )d ,
4 8
R R R
o
P P R P P
Q u A ur r r R r r
L L
p p
p m m
− −
=
∫
=∫
=∫
− = (4)where d A is an element of area. This is commonly referred to as the Poiseuille equation. It is important to note that the volume flow is directly related to the fourth power of the radius, and
0 R
P1 P2
u u(r)
r
-R Pipe wall
Pipe length, L
Fig. 3 Parabolic velocity profile of pipe flow driven by a pressure difference
92 J.C. Kim and E.B. Shim that the flow increases exponentially with the radius of the pipe. The mean velocity is obtained by dividing the volume flow Q by the cross-sectional area p R 2 , so that from (4) we get
2 1 2
max
( ) 1
8 2 .
R P P
u Q u
A mL
= = − = (5)
Finally, the wall shear stress is given by
d 1 1 2 4
d 2 .
w
P P
u u
r R L R
t =m = − = m (6)
Although the wall shear stress is proportional to the mean velocity in laminar flow, it is constant in turbulent flow. Recently, numerous studies have reported on the role of hemodynamic wall shear stress in atherosclerosis formed around the outer edge of vessel bifurcations by disturbed flow [9] . Arterial level shear stress (>15 dynes/cm 2 ) induces endothelial quiescence and an atheroprotective gene expression profile, whereas low shear stress (< 4 dynes/cm 2 ), which is prevalent at atherosclerosis-prone sites, stimulates an atherogenic phenotype.
Laminar and Turbulent Flows
Viscous flow regimes are classified as laminar or turbulent based on the flow structure. In the lamina regime, the flow structure is characterized by smooth motion in laminae, or layers. The flow structure in the turbulent regime is characterized by random, three-dimensional motions of fluid particles in addition to the mean motion (Fig. 4 ). In laminar flow, no macroscopic mixing of adjacent fluid layers occurs. A thin filament of dye injected into a laminar flow appears as a single line, except for the slow diffusion due to molecular motion. By contrast, a dye filament injected into turbulent flow disperses quickly throughout the flow field; the line of dye breaks up into myriad entangled threads of dye. This behavior of turbulent flow is caused by velocity fluctuations. For steady laminar flow, the velocity at a point remains constant with time. In turbulent flow, the velocity trace indicates random fluctuations of the instantaneous velocity about the temporal mean velocity. We can consider the instantaneous velocity to be the sum of the temporal mean velocity and the fluctuating component.
Poiseuille’s law relating steady flow and the pressure gradient in a pipe is no longer valid when the flow velocity exceeds a certain limit. The critical point is dependent on the diameter of the pipe, the mean velocity of the flow, and the density and viscosity of the liquid. This is expressed as a dimensionless quantity known as the Reynolds number (Re):
Re ruD.
= m (7)
The critical Reynolds number is usually about 2,000. This value, however, is determined experimentally and is highly dependent on the experimental conditions. In the human body, the maximum Reynolds number in large arteries exceeds the critical value, although the average Reynolds number is below the critical value. Furthermore, pulsatile flow becomes unstable at Reynolds numbers below 2,000 [10] . Turbulent blood flow has been suggested to have a direct effect on the arterial wall. Post-stenotic dilation has been attributed to progressive weakening of the wall by turbulence, which has a potential role in atherogenesis.
93 Hemodynamics