Page 59 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

36 BIOMECHANICS OF THE HUMAN BODY

muscle were closely juxtaposed artery-vein pairs (Weinbaum et al., 1984). Thermal equilibration in
the artery (approximately 50 to 300 μm in diameter) in a countercurrent pair was estimated based on
a simple heat conduction analysis in the cross-sectional plane. It was noted that the thermal equili-
bration length in the countercurrent artery was at least 3 times shorter than that in a single vessel
of the same size embedded in a tissue cylinder (Weinbaum et al., 1984). Significantly, short thermal
equilibration length in comparison with that of a single vessel suggests that the primary blood tissue
heat exchange mechanism for vessels larger than 50 μm in the deep layer is the incomplete counter-
current heat exchange. Therefore, for modeling heat transfer in these tissue regions, reasonable
assumptions related to the countercurrent heat exchange mechanism can be made to simplify the
mathematical formulation.
Theoretical analysis of the thermal equilibration in a large vessel in the cutaneous layer (Chato,
1980; Weinbaum et al., 1984) demonstrated that its thermal equilibration length was much longer
than its physical length during normal and hyperemic conditions despite the close distance from the
skin surface. It was suggested that the large vessels in the cutaneous layer can be far from thermal
equilibration and are, therefore, capable of delivering warm blood from the deep tissue to the skin
layer. This superficial warm blood shunting is very important in increasing the normal temperature
gradient at the skin surface and, therefore, plays an important role in losing heat during heavy exer-
cise. On the contrary, during surface cooling there is rapid cutaneous vasoconstriction in the skin.
The minimally perfused skin, along with the underlying subcutaneous fat, provides a layer of insu-
lation, and the temperature gradient from the skin surface into the muscle becomes almost linear
(Bazett, 1941) yielding the lowest possible heat transfer from the body.

2.3 BIOHEAT TRANSFER MODELING

The effects of blood flow on heat transfer in living tissue have been examined for more than a cen-
tury, dating back to the experimental studies of Bernard in 1876. Since then, mathematical
modeling of the complex thermal interaction between the vasculature and tissue has been a topic
of interest for numerous physiologists, physicians, and engineers. A major problem for theoretical
prediction of temperature distribution in tissue is the assessment of the effect of blood circulation,
which is the dominant mode of heat removal and an important cause of tissue temperature
inhomogeneity.
Because of the complexity of the vascular geometry, there are two theoretical approaches describing
the effect of blood flow in a biological system. Each approach represents two length scales over
which temperature variations may occur.
• Continuum models, in which the effect of blood flow in the region of interest is averaged over
a control volume. Thus, in the considered tissue region, there is no blood vessel present; how-
ever, its effect is treated by either adding an additional term in the conduction equation for the
tissue or changing some of the thermophysical parameters in the conduction equation. The
continuum models are simple to use since the detailed vascular geometry of the considered
tissue region need not be known as long as one or two representative parameters related to the
blood flow are available. The shortcoming of the continuum model is that since the blood
vessels disappear, no point-by-point variation in the blood temperature is available. Another
shortcoming is associated with the assumptions introduced when the continuum model was
derived. For different tissue regions and physiological conditions, these assumptions may not
be valid.
• Vascular models, in which blood vessels are represented as tubes buried in tissue. Because of the
complicate vascular geometry one may only consider several blood vessels and neglect the others.
Recent studies (Dorr and Hynynen, 1992; Crezee and Lagendijk, 1990; Roemer, 1990) have
demonstrated that blood flow in large, thermally unequilibrated vessels is the main cause for tem-
perature nonhomogeneity during hyperthermia treatment. Large blood vessels may significantly
cool tissue volumes around them, making it very difficult to cover the whole tumor volume with