ON ONE MATHEMATICAL MODEL OF FREEZING LIVING BIOLOGICAL TISSUE WITH A CYLINDRICAL CRYOPROBE

Abstract

When cooling living biological tissue (active, non-inert medium), cryoinstruments with various forms of cooling surface are used. Cryoinstruments can be located both on the surface of biological tissue, and completely penetrate into it. With a decrease in the temperature of the cooling surface, an unsteady temperature field appears in the tissue, which in the general case depends on three spatial coordinates and time. Of interest are both the distribution of the temperature field in the tissue, and the size of the zones of cryophase, freezing and the influence of cold, as well as the time of stabilization of the temperature field. Today there are a large number of scientific publications that consider mathematical models of cryodestruction of biological tissue. However, in the overwhelming majority of them, the Pennes equation (or some of its modifications) is taken as the basis of the mathematical model, from which the linear nature of the dependence of heat sources of biological tissue on the desired temperature field is visible. This character of the dependence does not allow one to describe the actually observed spatial localization of heat. In addition, Pennes' model does not take into account the fact that the freezing of the intercellular fluid occurs much earlier than the freezing of the intracellular fluid and the heat corresponding to these two processes is released at different times. In the proposed work, a new mathematical model of cooling and freezing of living biological tissue by a sufficiently extended cryoprobe with a cylindrical cooling surface is constructed. The model takes into account the above features and has applications in cryobiology and cryomedicine. A method is proposed for the numerical study of the problem posed, based on the application of the through counting technique without explicitly identifying the boundary of the influence of cold and the boundaries of the phase transition. The method was successfully tested earlier by the author in solving a number of two-dimensional problems arising in cryomedicine. Some numerical computer calculations are presented.