Proceedings of COBEM 2005 18th International Congress of Mechanical Engineering Copyright © 2005 by ABCM November 6-11, 2005, Ouro Preto, MG HYBRID ANALYTICAL NUMERICAL SOLUTION TO THE BIOHEAT TRANSFER EQUATION A. V. Presgrave, R. O. C. Guedes, F. Scofano Neto Departamento de Engenharia Mecânica e de Materiais Instituto Militar de Engenharia Praça General Tibúrcio 80 Rio de Janeiro, RJ, 22290-270 scofano@ime.eb.br Abstract. The bioheat transfer problem can be defined as the effect of blood flow on heat transfer in living tissue. Such physical situations have been examined for a long time but since direct measurements are extremely difficult to carry, there is a natural preference for mathematical models to study this phenomenon. The most commonly employed model for heat transfer within an organic tissue taking into account blood flow effects is the so-called Pennes bioheat equation. From a mathematical point of view, the Pennes model can be regarded as a standard heat diffusion equation together with an extra term that accounts for blood perfusion within the tissue. The main contribution of this work is to establish an analytical-numerical solution of a generalized bioheat transfer transient problem that accounts for a temperature dependent perfusion effect. This general solution is then applied to a specific situation related to the thermal balloon endometrial ablation treatment which is a modern and efficient medical procedure used in menorraghia therapy. The Generalized Integral Transform Technique is employed to analytically tackle the bioheat transfer equation. Finally, results are presented and compared to previously published data in the literature . Keywords: bioheat, Pennes model, blood perfusion effect, eigenfuction expansion. 1. Introduction A series of modern procedures in medicine such as cancer treatments by hyperthermia, refractive surgeries employing lasers and cryosurgery based interventions for the cure of benign prostate hyperplasia, are fundamentally dependent on a previous knowledge of the temperature distribution of the affected tissues. Due to the difficulties of a direct temperature measurement in living tissues, the determination and the solution of the mathematical models for such bioheat transfer problems are desirable. A distinguishing feature of this particular form of heat transfer process is to assess the influence of the local blood perfusion through the vascular network on the temperature distribution. In many physiological processes, such as human thermoregulation and inflammation, a significant temperature difference is found between the tissue and the blood through which it flows and thus convective heat transfer occurs altering the temperature of both the tissue and the blood. As pointed out by Diller and Ryan (1998), this interaction relies on various parameters including the rate of perfusion and the vascular anatomy and pathology. The accurate determination of the rate of blood perfusion in a certain organic tissue is quite an involved matter as it is affected by many a series of effects such as physical activity, physiological stimulus and environmental conditions and over the years, numerous techniques have been established in order to carefully evaluate this parameter. It is commonly agreed that the first work that describes the thermal interaction between an organic perfused tissue was advanced almost 60 years ago by Pennes (1948). Its main objective was to experimentally determine the radial temperature distribution in the forearm of nine unanesthetized human subjects. In his observations, Pennes noticed a temperature difference of three to four degrees between the skin and the interior of the arm, which he attributed to the effects of metabolism and to the heat transfer with the arterial blood perfused through the microvasculature. He then proposed a simple model to describe these two effects which were incorporated into the standard thermal diffusion equation. Over the years, alternative and more elaborate models that account for effects of vessel size, countercurrent heat exchange, combination of partial countercurrent exchange and bleed-off perfusion were proposed to describe the blood and tissue thermal interaction, Hartnett and Irvine (1992). However, it is interesting to notice while these more sophisticated models provide a larger depth in the analysis, they lack the generality and simplicity of the Pennes model. Therefore, it is no surprise that up to this day, this model is widely employed in bioheat transfer predictions (Azevedo, 2004, Presgrave, 2005, Chan, 1991). 2. Analysis The Pennes bioheat equation mentioned earlier, can be described as an energy balance that takes into account the effects of the metabolism and the blood perfusion in an organic tissue. Thus, the transient temperature field of the tissue can be generally described as: