48
Dynamics
of
the
Vascular
System
one that
is
infinite in the longitudinal direction along the blood vessel
axis and the other is in the radial direction. Thus, Laplace’s law for an
artery can be written as:
T=p.r
(3.1
.
1
8)
This assumes the artery has a thin-wall or that the ratio of arterial wall
thickness (h) to arterial lumen radius (r) is small, or h/r
i
1/10. Here p is
the intramural-extramural pressure difference, or the transmural pressure.
When the arterial wall thickness is taken into account, the Lame equation
becomes relevant:
(3.1.1
9)
Arteries have been assumed to be incompressible. Although not
exactly
so,
this is in general a good approximation.
To assess the
compressibility
of
a material, the Poisson ratio is defined. It is the ratio
of
We obtain from the above
definitions, the Poisson ratio as:
E,
Ar
Ir
All1
fJ=-=-
(3.1.20)
When radial strain is half that of longitudinal strain, or when
CJ
=
0.5,
the material is said to be incompressible. This means that when
a
cylindrical material
is
stretched, its volume remains unchanged.
Or, in
the case of an artery, when it is stretched, its lumenal volume remains
unchanged.
Experimental measurements
to
obtain the Poisson ratio for
arteries have shown
CT
to be close
to
0.5.
Arteries, therefore, can be
considered to be close to being incompressible.
A
purely elastic material differs from a viscoelastic material. The
former depends only on strain (eqns. 3.1.16 and 3.1.17) while the latter
depends on the rate of change of strain, or strain rate (dddt) also. The
artery as a viscoelastic material exhibits stress-relaxation, creep, and
hysteresis phenomena (Fig. 3.1.1).