A Counter Current Vascular Network Model of Heat Transfer in Tissues

Abstract

A fully conjugated blood vessel network model (FCBVNM) for calculating tissue temperatures has been developed, tested, and studied. This type of model represents a more fundamental approach to modeling temperatures in tissues than do the generally used approximate equations such as the Pennes’ BHTE or effective thermal conductivity equations. As such, this type of model can be used to study many important questions at a more basic level. For example, in the particular hyperthermia application studied herein, a simple vessel network model predicts that the role of counter current veins is minimal and that their presence does not significantly affect the tissue temperature profiles: the arteries, however, removed a significant fraction of the power deposited in the tissue. These more fundamental models can also be used to check the validity of approximate equations. For example, using the present simple model, when the temperatures calculated by the FCBVNM are used for comparing predictions from two approximation equations (a simple effective thermal conductivity and a simple Pennes’ bio-heat transfer equation formulation of the same problem) it is found that the Pennes’ equation better approximates the FCBVNM temperatures than does the k eff model. These results also show that the “perfusion” value (Ẇ) in the Pennes’ BHTE is not necessarily equal to the “true” tissue perfusion (Ṗ) as calculated from mass flow rate considerations, but can be greater than, equal to, or less than that value depending on (1) how many vessel levels are modeled by the BHTE, and (2) the “true” tissue perfusion magnitude. This study uses a simple, generic vessel network model to demonstrate the potential usefulness of such fully conjugated vessel network models, and the associated need for developing and applying more complicated and realistic vascular network models. As more realistic vascular models (vessel sizes, orientations, and flow rates) are developed, the predictions of the fully conjugated models should more closely model and approach the true tissue temperature distributions, thus making these fully conjugated models more accurate and valuable tools for studying tissue heat transfer processes.