Analysis of the Inverse Problem of Freezing and Thawing of a Binary Solution During Cryosurgical Processes

Abstract

An integral solution for a one-dimensional inverse Stefan problem is presented. Both the freezing and subsequent thawing processes are considered. The medium depicting biological tissues, is a nonideal binary solution wherein phase change occurs over a range of temperatures rather than at a single one. A constant cooling, or warming, rate is imposed at the lower temperature boundary of the freezing/thawing front. This condition is believed to be essential for maximizing cell destruction rate. The integral solution yields a temperature forcing function which is applied at the surface of the cryoprobe. An average thermal conductivity, on both sides of the freezing front, is used to improve the solution. A two-dimensional, axisymmetric finite element code is used to calculate cooling/warming rates at positions in the medium away from the axis of symmetry of the cryoprobe. It was shown that these cooling/warming rates were always lower than the prescribed rate assumed in the one-dimensional solution. Thus, similar, or even higher, cell destruction rates may be expected in the medium consistent with existing in vitro data. Certain problems associated with the control of the warming rate during the melting stage are discussed.