Biomedical Engineering Reference
In-Depth Information
vessel as it bulges with each heartbeat. For a segment of artery, the continuity equation
becomes
dQ
dx
¼
GP
þ
C
dP
ð
4
:
70
Þ
dt
where
is leakage through the blood vessel wall. This pair of hydraulic equations—
Eqs. (4.69) and (4.70)—was used to describe each of 125 segments of the arterial system
and was the first model sufficiently detailed to explain arterial pressure and flow wave
reflection. Arterial branching leads to reflected pressure and flow waves that interact in this
pulsatile system. Physical R-L-C circuits were constructed and built into large transmission
line networks with measured voltages and currents corresponding to hydraulic pressures
and flows, respectively. If distributed arterial properties such as pulse wave reflection are
not of interest, the arterial system load seen by the heart can be much reduced, as an elec-
trical network may be reduced to an equivalent circuit.
The most widely used arterial load is the three-element model shown in Figure 4.37. The
model appears as an electrical circuit due to its origin prior to the advent of the digital
computer.
G
Z
0
is the characteristic impedance of the aorta, in essence the aorta's flow resis-
tance.
is transverse arterial compliance, the inverse of elastance, and describes stretch
of the arterial system in the radial direction.
C
s
is the peripheral resistance, describing
the systemic arteries' flow resistance downstream of the aorta. This simple network may
be used to represent the systemic arterial load seen by the left ventricle. The following
ordinary differential equation relates pressure at the left-hand side,
R
s
p
(
t
), to flow,
Q
(
t
):
þ
Z
0
C
s
dp
1
R
s
p
ð
t
Þ¼
Q
ð
t
Þ
þ
Z
C
s
dQ
dt
0
R
s
dt
þ
1
ð
4
:
71
Þ
Z
0
p
Q
C
s
R
s
FIGURE 4.37
Equivalent systemic arterial load. Circuit elements are described in the text.
EXAMPLE PROBLEM 4.14
Using basic circuit theory, derive the differential equation (Eq. (4.71)) from Figure 4.37.
Solution
Define node 1 as shown in Figure 4.38. By Kirchoff's current law, the flow
Q
going into node 1
is equal to the sum of the flows
Q
1
and
Q
2
coming out of the node:
Q
¼
Q
1
þ
Q
2
