Page 63 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

40 BIOMECHANICS OF THE HUMAN BODY

C
t t t S
50–100 μm dia.
t P

S. S

20–50 μm dia.
S. P P
t
1 mm dia. S
muscle
cylinder
c

BF P 100–300 μm dia.

SAV

300–1000 μm dia.
0.5 m
FIGURE 2.2 Macro- and microvascular arrangement in skeletal muscle. The blood supply for the
muscle tissue cylinder comes from a branching countercurrent network of supply vessels. The primary
(P) vessels originating from the SAV vessels, run obliquely across the muscle tissue cylinders and then
branch into the long secondary (s) vessels. [From Myrhage and Eriksson (1984), with permission.]

cylinder led to the following bioheat transfer equation:
∂ T 2
T +
(
k
ρC t =∇ T + εωρ C T − ) q m (2.7)
b b
t
t
a0
t
t ∂
Note that the only modification to the Pennes model is a correction coefficient in the Pennes source
term. This correction coefficient can be viewed as a weighting function to correct the overestimation
of the original Pennes perfusion term. An easy-to-use closed-form analytic expression was derived
for this coefficient that depends on the vessel spacing and radius. From the anatomic studies of the
vascular arrangements of various skeletal muscles, the correction coefficient was found to vary from
0.6 to 0.8 under normal physiological conditions, indicating that there is a 20 to 40 percent rewarming
of the countercurrent vein. Note that it is close to neither unity (the Pennes model) nor zero (the
Weinbaum-Jiji model). Thus, both the Pennes and Weinbaum-Jiji bioheat equations are not valid for
most muscle tissue. Furthermore, as shown in Zhu et al. (2002), the arterial temperature T may not
a0
be approximated as the body core temperature either, unless the local blood perfusion is very high.
In most physiological conditions, it is a function of the tissue depth, blood vessel bifurcation pattern,
and the local blood perfusion rate.
2.3.2 Experimental and Theoretical Studies to Validate the Models
Pennes (1948) performed a series of experimental studies to validate his model. He inserted and
pulled thermocouples through the arms of nine male subjects to measure the radial temperature