Biomedical Engineering Reference
In-Depth Information
could be quite significant owing to a large surface to volume ratio. These mod-
els, however, were not able to take account of either metabolic reaction or per-
fusion bleed-off from the artery to vein. Keller and Seiler (1971) established
a one-dimensional bioheat transfer model to include the countercurrent heat
transfer for the subcutaneous tissue region with arteries, veins, and capillaries.
Weinbaum and Jiji (1979, 1985) proposed a model, which is based on some
anatomical understanding, considering the countercurrent arterio-venous ves-
sels. Roetzel and Xuan (1998) pointed out that 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. The foregoing survey prompts
us to establish a multidimensional model that can be applied to the regions
of extremity, where the countercurrent heat transfer happens between closely
spaced arteries and veins in the blood circulatory system. Excellent reviews
on these bioheat transfer equations may be found in Chato (1980), Charny
(1992), and Khaled and Vafai (2003).
In this section, we shall extend the volume averaging procedure described
for the heat transfer between the isolated vessels and the surrounding tissue
to the case of countercurrent bioheat transfer in a blood circulatory system.
The set of macroscopic governing equations consists of continuity and momen-
tum equations for both arterial and venous blood phases and three individual
energy equations for the two blood phases and the surrounding tissue phase.
It will be shown that most shortcomings in existing models are overcome in
the present model. Capillaries providing a continuous connection between the
countercurrent terminal arteries and veins are modeled introducing the per-
fusion bleed-off rate, originally introduced in the pioneering paper by Pennes
(1948). It has been found that the resulting model under certain conditions
reduces to existing models for countercurrent heat transfer such as Chato
(1980), Bejan (1979), Keller and Seiler (1971), Roetzel and Xuan (1998), and
Weinbaum and Jiji (1985) for the case of closely aligned pairs of artery and
vein. A general expression has been presented for the longitudinal effective
thermal conductivity in the energy equation for the tissue. To examine the
present model, we shall apply it to the countercurrent blood vessel config-
uration examined by Chato (1980). While Chato assumed the constancy of
the perfusion bleed-off rate, we shall allow the spatial distribution of perfu-
sion bleed-off rate and investigate its effect on the total countercurrent heat
transfer.
1.5.2 Three-Energy Equation Model Based on the
Volume Averaging Theory
A schematic view of the tissue layer close to the skin surface is shown in
Figure 1.4, in which the arteries and veins are paired, such that the coun-
tercurrent heat transfer takes place. Thus, we assign individual dependent
variables such as temperature to the arterial blood, venous blood, and tissue,
which leads us to propose a three-energy equation model.
Search WWH ::




Custom Search