Biomedical Engineering Reference
In-Depth Information
and as a result, the branching is area preserving, which leads to a constant blood
velocity. In small vessels (capillaries, arterioles) the pulse is strongly damped
because Poiseuille flow dominates and significant energy is dissipated. This
leads to area-increasing branching, so blood flow slows down, almost ceasing in
the capillaries.
A detailed treatment of pulsatile flow is complicated. Here, we present a
condensed version that contains the features pertinent to the scaling problem. In
contrast to non-pulsatile, Poiseuille flow, blood vessels cannot be considered
rigid for pulsatile flow because vessels expand and contract as the pulse wave
generated by the contraction of the heart propagates along them. The classic
Poiseuille resistance of the rigid tube, relating the fluid volume flow rate to the
driving pressure gradient, is thereby generalized to a complex impedance, signi-
fying attenuated wave propagation (32-34).
The equation of motion governing fluid flow is the Navier-Stokes equation
(35). Neglecting nonlinear terms responsible for turbulence, this is:
s
v
S
=
N
2
vp
.
[1]
s
t
Here, the vector
v
is the local fluid velocity at some time
t
,
p
is the local pres-
sure, and S is the fluid density. If the fluid is incompressible, then local conser-
vation of fluid requires /&
v
= 0. When combined with Eq. [1], this gives the
subsidiary condition
/
2
p
= 0.
[2]
The analogous equation governing the elastic motion of the tube is the Navier
equation. Neglecting nonlinear terms, this is given by:
2
s
[
S
=
E
2
Y
p
,
[3]
w
s
t
2
where the vector [ is the local displacement of the tube wall, S
w
is its density,
and
E
is its modulus of elasticity. These three coupled equations, [1], [2], and
[3], must be solved subject to boundary conditions that require the continuity of
velocity and force at the tube wall interfaces.
In the approximation where the vessel wall thickness,
h
, is small compared
to the static equilibrium value of the vessel radius,
r
, i.e.,
h
<<
r
, the problem
can be solved analytically, as first shown by Womersley (32), to give
2
--
3/2
2
0
c
c
Ji
(
B
B
)
c
S
®
2
and
Z
,
[4]
3/2
2
J
(
i
)
Q
rc
0
0
