Biomedical Engineering Reference
In-Depth Information
If one integrates the velocity over the cross-sectional area of the blood vessel, this results
in the well-known Poiseuille law for steady, incompressible, axisymmetric flow in a cylin-
drical vessel
R
4
l
where Q is the volume flow rate (cardiac output for blood), R is the vessel radius,
Q
¼ p
r
P
=
8 m
r
P is the
pressure gradient,
m
is the blood viscosity
,
and
l
is the length of the blood vessel over which
the pressure gradient is acting.
Poiseuille's law (sometimes called the Hagen-Poiseuille law) states that assuming lami-
nar flow in a tube, flow (Q) is proportional to its radius to the fourth power (R
4
) and
inversely proportional to its length (
). The general principle
of this law is that small changes in the internal diameter of a blood vessel lumen can make
a big difference to the rate of flow. The flow in the aorta is large—in part because the radius
of the aorta is large. The flow in smaller vessels is less, not only because the branching
blood vessels segment the available flow but also because the vessel radius is smaller.
As already stated, when the hematocrit is smaller, the blood viscosity is as well. As a result,
an anemic patient has a smaller blood viscosity, and therefore, according to the Poiseuille
law, the flow rate is larger. The resistance to blood flow (Poiseuille) corresponds to the rela-
tionship between volume flow rate (Q) and the pressure gradient (
l
) and viscosity of the fluid (
m
r
P) by the equation
R
s
where R
s
is the resistance to flow. If one uses the Poiseuille flow equation, then the flow
resistance takes the form of
Q
¼r
P
=
R
4
Thus, the resistance to flow inside a given blood vessel is proportional to the fluid viscosity
and inversely related to the vessel radius to the fourth power.
Throughout the analysis of fluid flow in a vessel, the driving mechanism has been the
pressure gradient (
R
s
¼
8
m
l
=p
P). An example of the relationship between pressure and flow is the
relationship between left ventricular pressure in the heart, left atrial pressure, and aortic
pressure. The heart valves associated with the left heart (mitral valve between the left
atrium and left ventricle, and aortic valve between the left ventricle and the aorta) open
and close via the difference in pressure across each valve. This is evident in Figure 14.33,
which shows the pressure and timing relationship across the two left heart valves.
Another manifestation of pressure drop producing a fluid flow is the relationship
between pressure across the systemic circulation, resulting from flow across the aortic
valve, through the aorta, arteries, arterioles, capillaries, venules, veins, and the vena cava.
The pressure across the systemic circulation begins as a pulsatile pressure in the aorta
(at a mean pressure of about 100 mm Hg) and ends at the vena cava with a mean pressure
of about 5 mm Hg. This is shown in Figure 14.34, along with the smaller pressures asso-
ciated with the right heart and the pulmonary circulation.
r
14.2.5 Boundary Layers
The Poiseuille assumption of fully developed flow (
0) refers to the fluid bound-
ary layer, the region of the flow field where viscous forces predominate. What is “fully
@
u
z
=@
z
¼
