Page 58 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

HEAT TRANSFER APPLICATIONS IN BIOLOGICAL SYSTEMS 35

Chato (1980) first theoretically investigated the heat transfer from individual blood vessels in
three configurations: a single vessel, two vessels in counterflow, and a single vessel near the skin sur-
face. It was shown that the Graetz number, proportional to the blood flow velocity and radius, is the
controlling parameter determining the thermal equilibration between the blood and tissue. For blood
vessels with very low Graetz number, blood quickly reaches the tissue temperature. It was also
demonstrated that heat transfer between the countercurrent artery and vein is affected by the vessel
center-to-center spacing and mass transport between them.
In an anatomic study performed on rabbit limbs, Weinbaum et al. (1984) identified three vascular
layers (deep, intermediate, and cutaneous) in the outer 1-cm tissue layer. Subsequently, three fun-
damental vascular structures were derived from the anatomic observation: (1) an isolated vessel
embedded in a tissue cylinder, as shown by the intermediate tissue layer; (2) a large artery and its
countercurrent vein oriented obliquely to the skin surface, as shown in the deep tissue layer; and
(3) a vessel or vessel pair running parallel to the skin surface in the cutaneous plexus. These three
vascular structures served as the basic heat transfer units in the thermal equilibration analysis in
Weinbaum et al. (1984).
As shown in Weinbaum et al. (1984), 99 percent thermal equilibration length of a single blood
vessel embedded in a tissue cylinder was derived as
x = 1.15aPrRe[0.75 + Kln(R/a)] (2.1)
cr
where a and R are the blood vessel and tissue cylinder radii, respectively; Pr and Re are the blood
flow Prandtl number and Reynolds number, respectively; and K is the ratio of blood conductivity to
tissue conductivity. It is evident that x is proportional to the blood vessel size and its blood flow
cr
velocity. Substituting the measured vascular geometry and the corresponding blood flow rate number
for different blood vessel generations (sizes) from a 13-kg dog (Whitmore, 1968), one could calcu-
late the thermal equilibration length as listed in Table 2.1.
Several conclusions were drawn from the comparison between x cr and L. In contrast to
previous assumptions that heat transfer occurs in the capillary bed, for blood vessels smaller than
50 μm in diameter, blood quickly reaches the tissue temperature; thus, all blood-tissue heat transfer
must have already occurred before entering into these vessels. For blood vessels larger than 300 μm
in diameter, there is little change in blood temperature in the axial direction because of their
much longer thermal equilibration length compared with the vessel length. The medium-sized
vessels between 50 and 300 μm in diameter are considered thermally significant because of their
comparable thermal equilibration length and physical length. Those blood vessels are primary
contributors to tissue heat transfer. Note that the conclusions are similar to that drawn by Chato
(1980).
The most important aspect of the bioheat transfer analysis by Weinbaum and coinvestigators was
the identification of the importance of countercurrent heat transfer between closely spaced, paired
arteries and veins. The countercurrent heat exchange mechanism, if dominant, was suggested as an
energy conservation means since it provides a direct heat transfer path between the vessels. It was
observed that virtually all the thermally significant vessels (>50 μm in diameter) in the skeletal

TABLE 2.1 Thermal Equilibration Length in a Single
Vessel Embedded in a Tissue Cylinder

Vessel radius Vessel length
a, mm L, cm R/a x , cm
cr
300 1.0 30 9.5
100 0.5 20 0.207
50 0.2 10 0.014
20 0.1 7 0.0006