130
Dynamics
of
the
Vascular
System
5
=
(X
-
jY)(l-
jk,
tan@
C
(4.5.44)
It has been shown that the viscous component is relatively small in large
arteries, such as the aorta.
The viscous
loss
represented by the
magnitude of tan4 is less than 10%.
Although in adequate, the Voigt model and the Maxwell model
continues to be popular choices when taking into account of viscoelastic
properties of the arterial wall. Womersley, as well as Morgan and Kiely
(1954), and Jager et al. (1965) employed the Voigt model to describe the
arterial wall properties. Jager et al. (1965) also assumed a thick-walled
model, when the arterial wall thickness is a large fraction of the radius. A
linearized Navier Stokes equation and dynamic deformation of the wall
were also incorporated to arrive at a complex wave velocity:
Eh 2r+h
3p
(r
+
h)2
(1
-
40)
c=-
(4.5.45)
In general, pressure and
flow
are obtained as periodic solutions with
spatial and temporal dependences as:
(4.5.47)
Linear theories are based on certain assumptions as discussed in the
previous section, they are mathematically tractable and allow solutions
for pressure and flow to be expressed in closed forms.
4.5.3
The Lumped Model of the Arterial System: The Windkessel
The idea of a lumped model of the arterial circulation was first described
by Hales in 1733.
Albeit largely qualitative, he did emphasize the
storage properties of large arteries and the dissipative nature of small
peripheral resistance vessels. In his description, the blood ejected by the
