Numerical Solution of the Multidimensional Freezing Problem During Cryosurgery

Abstract

A multidimensional, finite difference numerical scheme for the freezing process of biological tissues during cryosurgery is presented, which is a modification of an earlier numerical solution for inanimate materials. The tissues are treated as nonideal materials, freezing over a temperature range and possessing temperature-dependent thermophysical properties, blood perfusion, and metabolic heat generation. The numerical scheme is based on the application of an effective specific heat, substituting the intrinsic property, to include the latent heat effect within the phase transition temperature range. Results of the numerical solution were verified against an existing exact solution of a one-dimensional inverse Stefan problem in Cartesian coordinates. Results were further validated against experimental data available from the literature. The utility of the numerical solution for the design and application of cryodevices is demonstrated by parametric studies of the freezing processes around spherical and cylindrical cryoprobes. The parameters studied are the cryoprobe cooling power and the dimensions of the frozen region. Results are calculated for typical thermophysical properties of soft biological tissues, for angioma and for water.

Experimental verification of numerical simulations of cryosurgery with application to computerized planning

“…Since this comparison is performed on gelatin, the blood perfusion term and metabolic heat generation are set to zero in the current study. Note that heating due to blood perfusion during cryosurgery is a second order effect and metabolic heat generation is negligible (Rabin and Shitzer 1998). While the blood perfusion rate and metabolic heat generation are set to zero for the purpose of the current study, the numerical scheme presented above accounts for these parameters.…”

“…Verification of the uniform grid numerical scheme presented by Rabin and Shitzer (1998) is the objective of the current study, by comparison with experimental data. Since this comparison is performed on gelatin, the blood perfusion term and metabolic heat generation are set to zero in the current study.…”

“…(1), using a finite difference scheme (Rabin and Shitzer 1998), and is presented here in brief for the completeness of presentation. For this purpose, Eq.…”

“…For the purpose of computerized planning, it is assumed in previous studies , Rabin and Shitzer 1998, Tanaka et al 2006) that the classical bioheat equation (Pennes 1948) can be used to model heat transfer during cryosurgery of the prostate:…”

“…Here, heat transfer simulations can be calibrated with temperature measurements in the target tissue for the purpose of parametric estimation of tissue properties (Rabin et al 1996), assist in evaluating the certainty in temperature measurements (Rabin 1998), and augment imaging in real time by identifying specific temperature thresholds within the frozen region (Gilbert et al 1997). However, the quality of those simulations depends on the certainty in value of the thermophysical properties of the tissue (Rabin 2003), and the accuracy of the numerical scheme used.…”