Page 64 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

HEAT TRANSFER APPLICATIONS IN BIOLOGICAL SYSTEMS 41

profiles. He also measured the skin temperature distributions along the axis of the upper limb, as
well as around the circumference of the forearm. Pennes then modeled the arm as a long cylinder
and calculated the steady-state radial temperature profile. In this theoretical prediction, since the
blood perfusion rate ω could not be directly measured, Pennes adjusted this parameter in his model
to fit the solution to his experimental data for a fixed, representative ambient temperature and meta-
bolic heating rate. The fitted value of blood perfusion rate ω was found to be between 1.2 and 1.8 mL
blood/min/100 g tissue, which is a typical range of values for resting human skeletal muscle.
Recently, Wissler (1998) reevaluated Pennes’ original paper and analyzed his data. He found that the
theoretical prediction agrees very well with Pennes’ experimental results if the data were analyzed
in a more rigorous manner.
Profound understanding of the heat transfer site and the countercurrent heat exchange between
paired significant vessels was gained through the experiments (Lemons et al., 1987) performed on
rabbit thigh to measure the transverse tissue temperature profiles in the rabbit thigh using fine ther-
mocouples. The experimental study was designed to achieve two objectives. The first is to examine
whether there exists detectable temperature difference between tissue and blood for different-size
vessels. Existing detectable blood-tissue temperature difference implies that blood has not reached
thermal equilibration with the surrounding tissue. The second is to examine the temperature differ-
ence between the countercurrent artery and vein. If the countercurrent heat exchange is dominant in
blood tissue heat transfer, the vein must recapture most of the heat leaving the artery; thus, the tem-
perature difference between the countercurrent artery and vein should not vary significantly in the
axial direction.
Experimental measurements (Lemons et al., 1987) revealed small temperature fluctuations of up
to 0.5°C in the deep tissue. The irregularities in the tissue temperature profiles were closely associ-
ated with the existence of blood vessels in the vicinity of the thermocouple wire. It was shown that
temperature fluctuation was observed in all the blood vessels larger than 500 μm, in 67 percent of
the vessels between 300 and 500 μm, and in 9 percent of the vessels between 100 and 300 μm. No
temperature fluctuation was observed in blood vessels less than 100 μm in diameter. This finding
indicates that the assumption in the Pennes model that arterial blood reaches the capillary circu-
lation without significant prior thermal equilibration is inaccurate for this vascular architecture, and
thus most of the significant blood-tissue heat transfer occurs in the larger vessels upstream. It was
also observed that the temperature field rarely exceeded 0.2°C in any countercurrent pair, even when
the difference in temperature between the skin and the central part of the rabbit thigh exceeded 10°C.
This implies the effectiveness of the countercurrent heat exchange process throughout the vascular
tree.
Similar experiments were performed by He et al. (2002, 2003) to measure directly the tempera-
ture decays along the femoral arteries and veins and their subsequent branches in rats. The experi-
mental results have demonstrated that the venous blood in mid-size blood veins recaptured up to
41 percent of the total heat released from their countercurrent arteries under normal conditions. As
expected, the contribution of countercurrent rewarming is reduced significantly to less than 15 percent
for hyperemic conditions.
In a series of experiments with an isolated perfused bovine kidney, Crezee and Lagendijk (1990)
inserted a small plastic tube into the tissue of a bovine kidney and measured the resulting tempera-
ture fields in a plane perpendicular to the tube while heated water was circulated through it, with the
kidney cortex perfused at different rates. They also used thermocouples to map the temperature dis-
tribution in the tissue of isolated bovine tongues perfused at various perfusion rates (Crezee et al.,
1991). By examining the effect of increased perfusion on the amplitude and width of the thermal pro-
file, they demonstrated that the temperature measurements agreed better with a perfusion-enhanced
k as opposed to the perfusion source term in the Pennes equation.
eff
Charny (Charny et al., 1990) developed a detailed one-dimensional three-equation model. Since
this model was based on energy conservation and no other assumptions were introduced to simplify
the analysis of the blood flow effect, it was viewed as a relatively more accurate model than both the
Pennes and Weinbaum-Jiji equation. The validity of the assumptions inherent in the formulation of
the Weinbaum-Jiji equation was tested numerically under different physiological conditions. In addi-
tion, the temperature profile predicted by the Pennes model was compared with that by the three-
equation model and the difference between them was evaluated. The numerical simulation of the