A structural model of heat transfer due to blood vessels in living tissue

Abstract

Numerical investigations of heat transfer in single and multiple thick-walled pipes and countercurrent pairs are used to deduce relationships between fluid and solid temperatures needed to develop more accurate thermal models of living tissue in the extremities. A structural model of heat transfer in living tissue is developed using currently available anatomical and physiological data for the extremities. In order to improve the heat transfer basis of thermal modeling under in vivo conditions, four heat transfer problems based on structures found in the extremities are solved using in vivo parameters-the thick-walled pipe and countercurrent pair, and the multiple thick-walled pipes and countercurrent pairs-are studied. Low resolution numerical models are devised to approximate the thick-walled pipe and the non-concentric thick-walled countercurrent pair in square geometries. A constant heat transfer coefficient at the fluid-solid interface adequately approximates the fluid and solid temperatures for moderate flow conditions (Peclet number of 10 < °Pe < 1000). In the thick-walled countercurrent pair, countercurrent exchange and fluid-solid thermal interaction are found to act simultaneously, giving rise to imperfect countercurrent exchange. Fluid and solid temperatures in the multiple thick-walled pipes and pairs near the outer boundary resemble those of the single thick-walled pipes and pairs. The countercurrent pairs near the center also exhibit imperfect countercurrent exchange. In cylinders with L* > 1 containing multiple countercurrent pairs, the shapes of the temperature profiles cannot be distinguished from the temperature profile shapes of cylinders containing multiple thick-walled pipes. Fluid and solid temperatures in multiple parallel pipes may be approximated with a field equation which has the same form as the Pennes' bioheat equation. Unlike Pennes' equation, the coefficients for the blood thermal energy term quantify the dependence of the amount of thermal energy transferred between blood and tissue with the geometry of the blood the flow rate through the dimensionless axial length L*, and the dimensionless axial coordinate x* . Comparisons of structural model temperatures with available in vivo temperature studies show that blood and tissue temperatures are consistent with fluid and solid temperatures of either multiple unpaired pipes or multiple countercurrent pairs embedded in a solid cylinder. Further improvements of the basis for in vivo heat transfer modeling are crucially dependent upon more extensive comparison with three-dimensional in vivo studies.