Biomedical Engineering Reference
In-Depth Information
on some anatomical understanding, considering the countercurrent arterio-
venous vessels. As pointed out by Roetzel and Xuan (1998), the model may
be useful in describing a temperature field in a single organ, but would not be
convenient to apply to the whole thermoregulation system. Excellent reviews
on these bioheat transfer equations may be found in Chato (1980) and Charny
(1992).
Khaled and Vafai (2003) and Khanafer and Vafai (2006) stress that the
theory of porous media is most appropriate for treating heat transfer in bio-
logical tissues since it contains fewer assumptions as compared to different
bioheat transfer equations. Roetzel and Xuan (1998) and Xuan and Roetzel
(1997) exploited the volume averaging theory (VAT) previously established
for the study of porous media (e.g., Cheng 1978, Nakayama 1995), to formu-
late a two-energy equation model accounting for the thermal nonequilibrium
between the blood and peripheral tissue. In their model, the perfusion term
is replaced by the interfacial convective heat transfer term. This point should
be examined since the interfacial convective heat transfer is different from
perfusion heat transfer. Naturally, the former takes place even in the absence
of the latter.
In this chapter, we present a rigorous mathematical development based
on VAT so as to achieve a complete set of the volume averaged governing
equations for bioheat transfer and blood flow. Most shortcomings in existing
models will be overcome. We start with the case of isolated blood vessels
and the surrounding tissue, to establish a two-energy equation model for the
blood and tissue temperatures. We shall identify the terms describing the
blood perfusion and dispersion in the resulting equation and revisit the Pennes
model, the Wulff model, and their modifications.
Subsequently, the two-energy equation model is extended to the three-
energy equation model, so as to account for the effect of countercurrent heat
transfer between closely spaced arteries and veins in the blood circulatory
system. In this model, three distinctive energy equations are derived for the
arterial blood phase, venous blood phase, and tissue phase with three individ-
ual temperatures. Capillaries providing a continuous connection between the
countercurrent terminal arteries and veins are modeled introducing the perfu-
sion bleed-off rate. It will be shown that the resulting model, under appropri-
ate conditions, naturally reduces to those introduced by Chato (1980), Bejan
(1979), Weinbaum and Jiji (1985), and others for countercurrent heat transfer
for the case of closely aligned pairs of vessels. A useful expression for the lon-
gitudinal effective thermal conductivity for the tissue can be obtained without
dropping the perfusion source terms. The expression turns out to be quite sim-
ilar to Bejan's and Wienbaum and Jiji's expressions. Furthermore, the effect
of spatial distribution of perfusion bleed-off rate on total countercurrent heat
transfer is discussed in depth exploiting the present bioheat transfer model.
As for an application of a bioheat equation, the freezing process within
a tumor during cryoablation therapy is investigated both analytically and
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