Mathematics and Statistics Vol. 9(1), pp. 65 - 70
DOI: 10.13189/ms.2021.090111
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On One Mathematical Model of Cooling Living Biological Tissue

B. K. Buzdov *
Institute for Computer Science and Problems of Regional Management KBSC Russian Academy of Sciences, Nalchik, Russia


When cooling living biological tissue (active, non-inert medium), cryomedicine uses cryo-instruments with various forms of cooling surface. Cryoinstruments are located on the surface of biological tissue or 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. To date, 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 are built with a flat rectangular applicator located on its surface. The model takes into account the above features and is a three-dimensional boundary-value problem of the Stefan type with nonlinear heat sources of a special type and has applications in cryosurgery. A method is proposed for the numerical study of the problem posed, based on the use of locally one-dimensional difference schemes without explicitly separating the boundary of the influence of cold and the boundaries of the phase transition. The method was previously successfully tested by the author in solving other two-dimensional problems arising in cryomedicine.

Mathematical Modeling in Cryobiology and Cryomedicine, Stefan Type Problems in Cryomedicine

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] B. K. Buzdov , "On One Mathematical Model of Cooling Living Biological Tissue," Mathematics and Statistics, Vol. 9, No. 1, pp. 65 - 70, 2021. DOI: 10.13189/ms.2021.090111.

(b). APA Format:
B. K. Buzdov (2021). On One Mathematical Model of Cooling Living Biological Tissue. Mathematics and Statistics, 9(1), 65 - 70. DOI: 10.13189/ms.2021.090111.