Page 62 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

HEAT TRANSFER APPLICATIONS IN BIOLOGICAL SYSTEMS 39

Weinbaum-Jiji Bioheat Equation. Since 1980, researchers (Chato, 1980; Chen and Holmes, 1980;
Weinbaum et al., 1984) have begun to question the validity of the Pennes bioheat equation. Later,
Weinbaum and Jiji (1985) developed a new equation for microvascular blood tissue heat transfer,
based on an anatomic analysis (Weinbaum et al., 1984) to illustrate that the predominant mode of
heat transfer in the tissue was the countercurrent heat exchange between a thermally significant
artery and vein pair. The near-perfect countercurrent heat exchange mechanism implies that most of
the heat leaving the artery is transferred to its countercurrent vein rather than released to the sur-
rounding tissue. Once there is a tissue temperature gradient along the countercurrent vessel axes, the
artery and vein will transfer a different amount of energy across a plane perpendicular to their axes
even if there is no net mass flow. This gives rise to a net energy transfer that is equivalent to an
enhancement in tissue conductivity in the axial direction of the vessels. In the Weinbaum-Jiji bioheat
equation, the thermal effect of the blood perfusion is described by an enhancement in thermal con-
ductivity k , appearing in the traditional heat conduction equation,
eff

∂ T 2
ρC t = k ∇ T + q m k eff = k [1 + f ( ω ] ) (2.6)
eff
t
t
t ∂
It was shown that k eff is a function of the local blood perfusion rate and local vascular geometry.
The main limitations of the Weinbaum-Jiji bioheat equation are associated with the importance
of the countercurrent heat exchange. It was derived to describe heat transfer in peripheral tissue only,
where its fundamental assumptions are most applicable. In tissue area containing a big blood vessel
(>200 μm in diameter), the assumption that most of the heat leaving the artery is recaptured by its
countercurrent vein could be violated; thus, it is not an accurate model to predict the temperature
field. In addition, this theory was primarily developed for closely paired microvessels in muscle
tissue, which may not always be the main vascular structure in other tissues, such as the renal cor-
tex. Furthermore, unlike the Pennes bioheat equation, which requires only the value of local blood
perfusion rate, the estimation of the enhancement in thermal conductivity requires that detailed
anatomical studies be performed to estimate the vessel number density, size, and artery-vein spacing
for each vessel generation, as well as the blood perfusion rate (Zhu et al., 1995). These anatomic data
are normally not available for most blood vessels in the thermally significant range.

A New Modified Bioheat Equation. The Pennes and Weinbaum-Jiji models represent two extreme
situations of blood-vessel thermal interaction. In the original Pennes model, the arterial blood releases
all of its heat to the surrounding tissue in the capillaries and there is no venous rewarming. Pennes
did not realize that thermal equilibration was achieved in vessels at least an order of magnitude larger
than the capillaries. In contrast, in the Weinbaum-Jiji model the partial countercurrent rewarming is
assumed to be the main mechanism for blood-tissue heat transfer. The derivation of the Weinbaum-
Jiji equation is based on the assumption that heat transfer between the artery and the vein does not
depart significantly from a perfect countercurrent heat exchanger. In other words, most of the heat
lost by the artery is recaptured by its countercurrent vein rather than lost to the surrounding tissue.
Subsequent theoretical and experimental studies have shown that this is a valid assumption only for
vessels less than 200 μm diameter (Charny et al., 1990; Zhu et al., 1996a).
Several theoretical studies have suggested that one way to overcome the shortcomings of both
models was to introduce a “correction coefficient” in the Pennes perfusion term (Chato, 1980; Baish,
1994; Brinck and Werner, 1994; Weinbaum et al., 1997; Zhu et al., 2002). In 1997, Weinbaum and
coworkers (Weinbaum et al., 1997) modified the Pennes source term on the basis of the thermal
analysis of a basic heat transfer unit of muscle tissue, a 1-mm-diameter tissue cylinder containing
blood vessels smaller than 200 μm in diameter, as shown in Fig. 2.2. The countercurrent heat
exchange between the s artery and vein defined in the anatomical studies of Myrhage and Eriksson
(1984) led to the estimation of the heat loss recaptured by the s vein. The strength of the source
term was then rederived taking into account the rewarming of the countercurrent venous blood in the
s tissue cylinder. The thermal equilibration analysis on the countercurrent s artery and vein in the tissue