160
Dynamics
of
the
Vascular System
Pc
2,
=-
m2
(5.3.12)
This gives the characteristic impedance of a uniform cylindrical vessel
that is independent of wave reflections. Of course, the blood flow is
assumed to be Newtonian and that the viscosity of blood and the vessels
wall are neglected in this formulation.
5.3.2
Area
Ratio
Concept
From
a
simple geometric perspective, the cross-sectional area
of
the
adjoining vessels should provide some quantitative estimates
of
the
mismatching characteristics of pulse transmission characteristics. Thus,
the branching vessel lumen areas come into play. This concept of area
ratio
has been
examined by several
investigators.
For
instance,
Karreman (1952) used area ratio in his mathematical formulation of
wave reflection at an arterial junction. By assuming both the wall and
fluid are non-viscous, and wall thickness remains the same for an
infinitely long tube, he arrived at a value of area ratio (the ratio of the
sum of the areas of daughter vessels to that of the mother vessel) for a
reflectionless bifurcation of about 1.15,
With a modification by considering tethered elastic tubes containing
viscous fluid, Womersley
(1958)
later arrived at a similar result with a
correcting factor
q,
for a bifurcation, assuming the daughter vessels have
identical characteristics (rd, cd):
(5.3.13)
The local reflection coefficient due
to
the equi-bifurcation is then given
by:
(5.3.14)
