162
Dynamics
of
the Vascular
System
approaches infinity, or
a,+m
,
though the minimum reflection is only a
few percent. This is shown in Fig. 5.3.2. The area ratios for the aortic
junctions are usually between 1.15 and 1.25 (table 4. l), which represent
a very small amount of reflections (Li et al., 1984). These same small
amount of local reflections were found on the analog model of the
systemic arterial tree by Westerhof et
al.
(1969).
In terms of fluid
dynamics, optimal energy transfer is achieved when the area ratio is
close to one.
The relationship of local reflection coefficient, area ratio and junction
vessel characteristic impedances can be easily appreciated from the
following analysis. We first relate the characteristic impedance of the
blood vessel to its geometric and elastic properties. From the water-
hammer formula above for the characteristic impedance, we have
pc
2,
=-
m2
(5.3.16)
Knowing the Moens-Korteweg relation and substituting for pulse wave
velocity, we have:
(5.3.1
7)
Now with a bifurcation, the resultant characteristic impedance of the
daughter vessel branch impedances,
Z1
and
Z2,
is:
111
-=-+-
Zd
z, z2
(5.3.18)
For an equi-bifurcation, we have
(5.3.19)
or for n equal daughter branches:
