A counter current vascular network model of heat transfer in tissues

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^jy model. These results also show that the ' 'perfusion'' value ( W) in the Pennes' BHTE is not necessarily equal to the ' 'true'' tissue perfusion (P) 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. Introduction The Pennes' (1948) bio-heat transfer equation (BHTE) de­veloped to model temperatures in living tissues, and the more recently developed alternative equations having the same goal (e.g., Chen and Holmes, 1980; Jiji, et al., 1984; Lagendijk and Mooibroek, 1986; Weinbaum, et al., 1984), are all approxima­tions, with each attempting to formulate a single, general field equation that can predict the overall characteristics of the tem­perature distributions in tissues. The field equations so devel­oped approximate the physics of the actual fully conjugated blood/tissue heat transfer processes associated with the pres­ence of the common, frequently occurring blood vessels. That is, they approximate the sum of all the convective heat transfer rates between those blood vessels (whose effects the approxi­mate equations are intended to replace) and the solid tissue matrix in which those vessels are imbedded. These approxima­tions are don e by replacing the actual convective blood vessel/ tissue heat transfer term in the conservation of energy equation with a new term or with an altered parameter in the tissue energy balance equation. Evaluating the validity of these approximate equations is a difficult task since their predictions ultimately must be compared to difficult to obtain experimental in vivo anatomical, physiological, and temperature data. The most ex­tensive such combined experimental and modeling work, is re-