Biomedical Engineering Reference
In-Depth Information
FIGURE 5.11
Figure associated with
example problem.
Lamina 1
r
L
=
5 mm
r
S
=
2 mm
v
1
=
10 cm/s
Lamina 2
r
L
=
4.6 mm
r
S
=
1.6 mm
v
2
=
35 cm/s
Lamina 3
r
L
=
4.2 mm
r
S
=
1.2 mm
v
3
=
50 cm/s
Lamina 4
r
L
=
3.8 mm
r
S
=
0.8 mm
v
4
=
80 cm/s
Lamina 5
r
L
3.4 mm
r
S
=
0.4 mm
v
5
=
105 cm/s
=
5.7
WAVE PROPAGATION IN ARTERIAL CIRCULAT
ION
Wave propagation in arteries is primarily concerned with the displacement of the arte-
rial wall in response to the progression of pressure/velocity waves through the blood
vessel (due to elasticity of the vessel wall). To fully represent this condition, a fairly com-
plex set of mathematical equations must be solved simultaneously. This solution would
describe an instance of fluid structure interactions, where the fluid forces impact the
wall and the wall has some response (solutions that make few assumptions are discussed
in Chapter 13). To begin to understand the concepts of wave propagation, let us first
consider a blood vessel that is cylindrical and straight with mechanical properties that
are homogenous and elastic. Also, we will first make the assumption that the fluid is
incompressible, inviscid, and relatively slow (in the main direction of flow). A pressure
pulse at one end of the blood vessel will cause a geometric change in the vessel, which
propagates along the tube length at a particular speed. In most instances, the wavelength
is much larger than the wave amplitude, which allows us to assume that the flow veloc-
ity is still one-dimensional. If the amplitude of the wave were large, fluid would need to
enter/leave the space created/removed when the wave passed through the blood vessel
wall, and this typically would not be in the same direction of the majority of the fluid
flow. Therefore, a coupled Navier-Stokes solution would be required to determine the
velocity profile.
Combining all of these assumptions and using the continuity equation and the Navier-
Stokes equation, one can determine the speed of wave propagation in an artery. First we
should consider the forces acting on the blood vessel during the pressure pulse through
the vessel (
Figure 5.12
). We will assume that the forces balance in the y-direction and
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