Two boundary model for freezing front propagation in biological tissue

The response of the living tissue to the effects of strong heating or cooling can cause the blood flow rate to vary by an order of magnitude. A mathematical model for the freezing of living tissue is formulated which takes into account the nonlocal temperature dependence of the blood flow rate when the temperature distribution in the tissue is substantially nonuniform, as in cryosurgery. INTRODUCTION Mathematical analysis and prediction of temperature distribution in living tissue during the process of freezing has been used in the study and optimization of cryosurgical procedures. In the last years a number of different approaches to describing the heat transfer process in living tissue have been proposed (for a review see[1]–[15]). Within the framework of these approaches the obtained bioheat equation is of the cρ ∂T ∂t = ∇(k∇T )− cbρbJf(T − Ta) + S , (1) where T is the tissue temperature, c and ρ denote the specific heat and density of the tissue, cb and ρb are the specific heat and density of blood, J is the blood flow rate per unit tissue evolume, (i.e. the volume of blood flowing through a unit volume of tissue per unit time), k is the thermal conductivity of the tissue [14],[15], Ta is the systemic arterial blood temperature and S is the rate of metabolic heat generation. Cofactor f , ranging from 0 to 1 is due to heat exchange between arterial and venous blood flowing through the nearest vessels[5]-[10]. Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3b Naukova str. L’viv, 290601, Ukraine Department of Physics, Moscow State University, Vavilova str., 46–92, 117333 Moscow, Russia Department of Mathematics and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA