A study of heat transfer during cryosurgery of lung cancer.

In this study, a mathematical model describing two-dimensional bio-heat transfer during cryosurgery of lung cancer is developed. The lung tissue is cooled by a cryoprobe by imposing its surface at a constant temperature or a constant heat flux or a constant heat transfer coefficient. The freezing starts and the domain is distributed into three stages namely: unfrozen, mushy and frozen regions. In stage I where the only unfrozen region is formed, our problem is an initial-boundary value problem of the hyperbolic partial differential equation. In stage II where mushy and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations and in stage III where frozen, mushy, and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations. The solution consists of the three-step procedure: (i) transformation of problem in non-dimensional form, (ii) by using finite differences, the problem converted into ordinary matrix differential equation and moving boundary problem of ordinary matrix differential equations, (iii) applying Legendre wavelet Galerkin method the problem is transferred into the generalized system of Sylvester equations which are solved by applying Bartels-Stewart algorithm of generalized inverse. The complete analysis is presented in the non-dimensional form. The consequence of the imposition of boundary conditions on moving layer thickness and temperature distribution are studied in detail. The consequence of Stefan number, Kirchoff number and Biot number on moving layer thickness are also studied in specific.

A study of cryosurgery of lung cancer using Modified Legendre wavelet Galerkin method.

In this paper, we have developed a new mathematical model describing bio-heat transfer during cryosurgery of lung cancer. The lung tissue cooled by a flat probe whose temperature decreases linearly with time. The freezing process occurs in three stages and the whole region is divided into solid, mushy and liquid region. The heat released in the mushy region is considered as discontinuous heat generation. The model is an initial-boundary value problem of the hyperbolic partial differential equation in stage 1 and moving boundary value problem of parabolic partial differential equations in stage 2 and 3. The method of the solution consists of four-step procedure as transformation of problem in dimensionless form, the problem of hyperbolic partial differential equation converted into ordinary matrix differential equation and the moving boundary problem of parabolic partial differential equations converted into moving boundary problem of ordinary matrix differential equations by using finite differences in space, transferring the problem into the generalized system of Sylvester equations by using Legendre wavelet Galerkin method and the solution of the generalized system of Sylvester equation are solved by using Bartels-Stewart algorithm of generalized inverse. The whole analysis is presented in dimensionless form. The effect of cryoprobe rate on temperature distribution and the effect of Stefan number on moving layer thickness is discussed in detail.