Page 34 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

MODELING OF BIOMEDICAL SYSTEMS 11

Q2 Q3 Q4
Arteries Arterioles Capillaries Venules
R1 L P2 R2 P3 R3 P5 R4

Q1 C1 C2 C3
Q5

P ith P ith P ith

P1 P6

Right ventricle Left atrium

Pulmonary circulation model
FIGURE 1.5 A model of pulmonary circulation. P ith is the intrathoracic pressure which is the exter-
nal pressure on the pulmonary blood vessels.

P − P = (1/C )∫(Q − Q )dt (1.22)
5 ith 3 4 5
Q = Q (1.23)
3 4
P − P = R Q (1.24)
5 6 4 5
The capacitance is due to distensibility of the vessel. The capillaries are stiffer and less distensible,
and therefore have minimal capacitance.
Electrical analog models have been used in the study of cardiovascular, pulmonary, intestinal, and
urinary system dynamics. Recently, Barnea and Gillon (2001) have used an electrical analog model
to simulate flow through the urethra. Their model consisted of a simple L, R, C circuit with a vari-
able capacitor. The time varying capacitor simulated the time-dependent relaxation of the urethra.
They used two types of resistance: a constant resistance to simulate Poiseouille-type viscous pres-
sure drop and a flow-dependent resistance to simulate Bernoulli-type pressure loss. With real-time
pressure-flow data sets, Barnea and Gillon (2001) have used the model to estimate urethral resistance
and changes in urethral compliance during voiding, and have suggested that the urethral elastance
(inverse of compliance) estimated by the model provides a new diagnostic tool. Ventricular and atri-
al pumping can be modeled using similar techniques. The actual pump (pressure source) can be mod-
eled as a variable capacitor. Figure 1.6 shows a model of the left heart with a multisegment
representation of the ventricle (Rideout, 1991). Kerckhoffs et al. (2007) have coupled an electrical
analog model of systemic circulation with a finite element model of cardiac ventricular mechanics.

1.3 MECHANICAL MODELS

Mechanical models consisting of combinations of springs and dashpots are very popular in numer-
ous disciplines. Spring dashpot models have been used to model the mechanical behavior of vis-
coelastic materials and can be used to represent the one dimensional behavior of tissue and other
biological materials. In a linear spring, the force is proportional to the change in length or the strain.