Main Article Content

Beslan Buzdov


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.


Download data is not yet available.


Metrics Loading ...

Article Details

How to Cite
Novel computational tools and databases


Baissalov R. A semi-empirical treatment planning model for optimization of multiprobe cryosurgery/ Baissalov R., G.A. Sandison, B.J. Donnelly J.C.et al. // Phys. Med. Biol. – 2000. – 45. – P. 1085-1098.

Rossi M.R. Experimental verification of numerical simulations of cryosurgery with application to computerized planning / Rossi M.R. and Rabin Y. // Phys. Med. Biol. – 2007. – 52. – P. 4553-4567.

Rabin Y. Numerical solution of the multidimensional freezing problem during cryosurgery / Rabin Y. and Shitzer A. // ASME J. Biomech. Eng.-1998.- 120(1).-P. 32-37.

Pennes H.H. Analysis of tissue and arterial blood temperature in the resting human forearm/. Pennes H.H. // J. Appl. Physiol. – 1948. – Vol. 1. – P. 93-102.

Berezovskiy A.A. Odnomernaya lokal’naya zadacha Stefana plosko-parallel’noy kriodestruktsii biologicheskoy tkani [One-dimensional local Stefan problem for plane cryodestruction of biological tissue] / Berezovskiy A.A. // Zadachi teploprovodnosti s podvizhnimi granitsami [Heat conduction problems with moving boundaries]. – Kiyev . – 1985. – P. 3-8.(Prepr./AN USSR. In-t matematiki:85.2) [in Russian].

Berezovskiy A.A. Nestatsionarnyye zadachi sfericheski-simmetrichnoy gipotermii biotkani [Unsteady problems for spherically symmetric hypothermia of biotissue] / Berezovskiy A.A., Zhurayev K.O., Yurtin I.I. // Zadachi Stefana so svobodnymi granitsami [Stefan’s problems with free boundaries]. – Kiyev. – 1990. – P. 9-20. (Prepr./AN USSR. In-t matematiki:90.27) [in Russian].

Berezovskiy A.A. Matematicheskoye prognozirovaniye kriovozdeystviya na biologicheskiye tkani [Mathematical prediction of cryotherapy on biological tissues] / Berezovskiy A.A., Leontyev Yu.V. // Kriobiologiya [Cryobiology]. – Kiyev. – 1989. – Naukova Dumka. – №3. – P. 7-13 [in Russian].

Budak B.M. Raznostnyy metod so sglagivaniyem koeffitsientovdlya resheniya zadachi Stefana [Difference method with smoothing factors for solving the Stefan’s problem] / Budak B.M., Solovyeva E.N., Uspenskiy A.B. // GVMMF [Journal of computational mathematics and mathematical physics]. – 1965. – Vol.5. – № 5. – P.828-840 [in Russian].

Budak B.M. Raznostnyye metody resheniya nekotorykh krayevykh zadach tipa Stefana [Difference methods for solving some boundary value problems of Stefan’s type] // Budak B.M., Vasilyev F.P., Uspenskiy A.B. // V sb Chislennyye metody v gazovoy dinamike [Computational methods in gas dynamics]. – Iss.4. – M. Izd-vo MGU. – 1965. – P.139-183 [in Russian].

Samarskiy A.A. Ekonomichnaya schema skvoznogo scheta dlya mnogomernoy zadachi Stefana [Efficient through calculation scheme for the multidimensional Stefan’s problem] / Samarskiy A.A., Moiseenko B.D. // GVMMF [Journal of computational mathematics and mathematical physics]. – 1965. – Vol.5. – № 5. – P.816-827 [in Russian].

Buzdov B.K. Mathematical modeling of biological tissue cryodestruction/ Buzdov B.K. // Applied Mathematical Sciences. – 2014. – Vol. 8. – no. 57. – P. 2823 – 2831.

Buzdov B.K. Two-dimensional boundary problems of Stefan’s type in cryomedicine/ Buzdov B.K. // Applied Mathematical Sciences. – 2014. – Vol. 8. – no. 137. – P. 6841-6848.

Buzdov B.K. Modelirovaniye kriodestruktsii biologicheskoy tkani/ Buzdov B.K. // Matematicheskoye modelirovaniye [Mathematical modeling]. – 2011. – Vol. 23. – №3. – P. 27-37[in Russian].

Buzdov B.K. Ob odnoy dvumernoy krayevoy zadache tipa Stefana, voznikayushchey v kriokhirurdii [On one two-dimensional boundary value problem of the Stefan type arising in cryosurgery] / Buzdov B.K. // Itogi nauki I tekhniki. Seriya: Sovremennaya matematika i yeye prilozheniya. Tematicheskiye obzory [Results of Science and Technology. Contemporary mathematics and its applications. Thematic reviews]. – 2019. – Vol.167. – P.20-26 [in Russian].

Buzdov B.K. Numerical study of two-dimensional mathematical model with variable heat exchange coefficient which arises in cryosergary/ Buzdov B.K. // Journal of Applied and Indastrial Mathematics. – 2017. – v. 11. – № 4. – P. 494-499.

Buzdov B.K.Dvumernaya krayevaya zadacha tipa Stefana dlya polukol’tsa [Two-dimensional boundary value problem of Stefan type for a semiring]/. Buzdov B.K. // Izvestiya vuzov. Severo-Kavkazskiy region. Yestestvennyye nauki [Proceedings of higher educational institutions. North Caucasian region. Series: Natural Sciences]. – 2007. – №1. – P. 30-33 [in Russian].

Buzdov B.K. On One Mathematical Model of Cooling Living Biological Tissue / Buzdov B.K.// Mathematics and Statistics. – 2021. – Vol.9. – No.1.-P.65-70.