138
Dynamics
of
the
Vascular
System
where a and b are constants. The exponent b is normally negative.
Thus, an inverse relationship is established between arterial compliance
and blood pressure; with increasing blood pressure arterial compliance
decreases.
R
Fig.
4.5.9:
Nonlinear arterial system model proposed by Li et al.
(1990),
incorporating a
pressure-dependent compliance.
Z,
is characteristic impedance
of
the ascending aorta,
R,
is total peripheral resistance,
C(P)
is the pressure-dependent compliance, represented by
a
variable capacitor. Q
is
aortic
flow.
Figure 4.5.9 shows that the flow through the compliance branch
of
the nonlinear model is given by
where P(t) and Q(t) are the pressure and flow through the compliance
branch, respectively. This flow can also be expressed
as
Q,
(t)
=
C(
P)
.
dP(
t)/dt
(4.5.66)
Equate these
two
equations, resulting in
This equation defines the dynamic relationship between pressure and
flow for a nonlinear compliance element. Numerical methods can be
employed to solve this equation.
Using difference representations, we have
At
=
ti,,
-
ti
=
dt
(4.5.68)