12.4 Propagation of the Pressure Pulse
385
12.4 Propagation of the Pressure Pulse
At any location in the blood vessel, the motion of blood is driven by the local pressure gradient. This gradient is determined by the propagation of the
pressure pulse. Description of the full complexity of pulse-wave propagation
is beyond the scope of this book. We will consider here some idealized cases
which, however, illustrate the basic principles of pulsative blood flow. Let us
assume an infinitely long, straight cylindrical tube of circular cross-section. The
excess pressure due to the heart action, p, and the tube cross-sectional area,
A, are functions of distance, x, and time, t. Moreover, we will assume that
the amplitude of pressure is small and the flow is essentially one-dimensional
with a longitudinal component u(x, t). Thus, the continuity principle (see Appendix C.2) requires:
aA a
-+-(uA)=O.
at ax
(12.28)
The momentum equation for one-dimensional motion in a tube becomes (see
Eq. C.30):
au
au 1 ap
-+u-+--=O.
at
ax pax
(12.29)
For simplicity, we assume that in the elastic tube there is a single-valued relationship between the cross-sectional area, A, and pressure p:
p = P(A).
(12.30)
In reality, this relationship is much more complex due to viscoelasticity of the
tube wall. However, in the case of an elastic tube, the tube radius, a, is linearly
proportional to the blood pressure in the vessel:
a = ao + ap,
(12.31)
in which ao is the radius, when p = O. In the case when velocity is small, the
term uau/ax in Eq. (12.29) can be ignored; thus:
au 1 ap
-+--=0.
at pax
Because A = rra 2 , Eq. (12.28) can be rewritten as follows:
au 2aa
-+--=0
ax a at
'
(12.32 )
(12.33)
385
12.4 Propagation of the Pressure Pulse
At any location in the blood vessel, the motion of blood is driven by the local pressure gradient. This gradient is determined by the propagation of the
pressure pulse. Description of the full complexity of pulse-wave propagation
is beyond the scope of this book. We will consider here some idealized cases
which, however, illustrate the basic principles of pulsative blood flow. Let us
assume an infinitely long, straight cylindrical tube of circular cross-section. The
excess pressure due to the heart action, p, and the tube cross-sectional area,
A, are functions of distance, x, and time, t. Moreover, we will assume that
the amplitude of pressure is small and the flow is essentially one-dimensional
with a longitudinal component u(x, t). Thus, the continuity principle (see Appendix C.2) requires:
aA a
-+-(uA)=O.
at ax
(12.28)
The momentum equation for one-dimensional motion in a tube becomes (see
Eq. C.30):
au
au 1 ap
-+u-+--=O.
at
ax pax
(12.29)
For simplicity, we assume that in the elastic tube there is a single-valued relationship between the cross-sectional area, A, and pressure p:
p = P(A).
(12.30)
In reality, this relationship is much more complex due to viscoelasticity of the
tube wall. However, in the case of an elastic tube, the tube radius, a, is linearly
proportional to the blood pressure in the vessel:
a = ao + ap,
(12.31)
in which ao is the radius, when p = O. In the case when velocity is small, the
term uau/ax in Eq. (12.29) can be ignored; thus:
au 1 ap
-+--=0.
at pax
Because A = rra 2 , Eq. (12.28) can be rewritten as follows:
au 2aa
-+--=0
ax a at
'
(12.32 )
(12.33)