A new model of steady-state heat transport in perfused tissue is presented. The key elements of the model are as follows: (1) a physiologically-based algorithm for simulating the geometry of a realistic vascular tree containing all thermally significant vessels in a tissue; (2) a means of solving the conjugate heat transfer problem of convection by the blood coupled to three-dimensional conduction in the extravascular tissue, and (3) a statistical interpretation of the calculated temperature field. This formulation is radically different from the widely used Pennes and Weinbaum-Jiji bio-heat transfer equations that predict a loosely defined local average tissue temperature from a local perfusion rate and a minimal representation of the vascular geometry. Instead, a probability density function for the tissue temperature is predicted, which carries information on the most probable temperature at a point and uncertainty in that temperature due to the proximity of thermally significant blood vessels. A sample implementation illustrates the dependence of the temperature distribution on the flow rate of the blood and the vascular geometry. The results show that the Pennes formulation of the bio-heat transfer equation accurately predicts the mean tissue temperature except when the arteries and veins are in closely spaced pairs. The model is useful for fundamental studies of tissue heat transport, and should extend readily to other forms of tissue transport including oxygen, nutrient, and drug transport.
Asymptotic Periodicity for a Class of Partial Integrodifferential Equations