Exact Solution to the One-Dimensional Inverse-Stefan Problem in Nonideal Biological Tissues

Abstract

A new analytic solution of the inverse-Stefan problem in biological tissues is presented. The solution, which is based on the enthalpy method, assumes that phase change occurs over a temperature range and includes the thermal effects of metabolic heat generation, blood perfusion, and density changes. As a first stage a quasi-steady-state solution is derived, defined by uniform velocities of the freezing fronts and thus by constant cooling rates at those interfaces. Next, the fixed boundary condition leading to the quasi-steady state is calculated. It is shown that the inverse-Stefan problem may not be solved exactly for a uniform initial condition, but rather for a very closely approximating exponential initial condition. Very good agreement is obtained between the new solution and an earlier one assuming biological tissues to behave as pure materials in which phase change occurs at a single temperature. A parametric study of the new solution is presented taking into account property values of biological tissues at low freezing rates typical of cryosurgical treatments.