Page 32 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

MODELING OF BIOMEDICAL SYSTEMS 9

where V is the volume. However, volume is a function of pressure. Momentum balance for the vessel
wall can be expressed as
P = P + (h/a )σ (1.10)
ext 0
where P is the external pressure on the outside of the vessel wall, h is the wall thickness, and σ is
ext
the hoop stress in the wall. The hoop stress is a function of wall radius a and modulus of elasticity
E of the wall, and can be expressed as
2
σ= (E/2)[(a/a ) − 1] (1.11)
0
where a is the unstretched radius. Since the length of the segment does not change, the above equa-
0
tion can be expressed as
σ= (E/2)[(V/V ) − 1] (1.12)
0
where V is the volume of the vessel segment and V is the unstretched volume. Equations (1.10),
0
(1.11), and (1.12) can be combined as
dV/dt = CdP/dt (1.13)

where C = (2V a /hE) (1.14)
0 0
C is often referred to as the compliance or capacitance.
Substituting Eq. (1.13) in Eq. (1.9) results in
CdP/dt = Q − Q (1.15)
in out
Equation (1.15) can be expressed in terms of an electrical equivalent as follows:
−
E = (1/C)∫idt (1.16)
Equations (1.7) and (1.16) can be used to simulate either a segment of a blood vessel or the entire blood
vessel itself. In small blood vessels, the inductance L is very low when compared to the resistance term
R, and therefore, the inductance term can be neglected in small arteries, arterioles, and capillaries. Since
there is no oscillation of pressure in the capillaries, the inductance term can be neglected in vessels
downstream of the capillary including venules, veins, vena cava, etc. (Chu and Reddy, 1992).
An electrical analog model of the circulation in the leg is illustrated in Fig. 1.4. Let us consider
the flow from the femoral artery into the small leg arteries. There is no inductance in small leg arteries,
and there is only the resistance. Since the small arteries are distensible, they have capacitance (com-
pliance). The muscular pressure (P MP ) acts as the external pressure on the majority of small leg arter-
ies. Consequently, P MP is used as the reference pressure across the capacitor. The arterioles do not
have inductance, but have a variable resistance which is controlled by neurogenic and metabolic fac-
tors. In this model, the precapillary sphincters and the capillaries are lumped together. Since the cap-
illaries are rather rigid, they do not have any capacitance (compliance), but the combined resistance
of the sphincters and capillaries is variable subject to metabolic control. For instance, precapillary
sphincters dilate in the presence of lactic acid and other end products of metabolism. Venules have
resistance and a variable capacitance. This capacitance is subject to neurogenic control since the
diameter of the venule is under neurogenic control. From the venules, the flow goes into leg small
veins which have a resistance and a variable capacitance subject to neurogenic control. In addition,
the venules have valves which only permit unidirectional flow. These valves can be modeled as
diodes. Again, the reference pressure for the capacitor is the muscle pressure P MP . It is well known
that the blood flow in the legs is aided by the muscle pump which is essentially the external pressure
oscillations on the blood vessel wall due to periodic skeletal muscle contractions during walking, etc.
The muscle pump is absent in bedridden patients. Extremity pumps are used on such patients to
enhance blood flow to the legs. These extremity pumps provide a periodic a graded sequential exter-
nal compression of the leg. The electrical analog model shown in Fig. 1.4 can be easily modified to
simulate the effect of these extremity pumps.