A Mathematical Model for Cell Killing by Heat Applied to a C3H Mammary Carcinoma in Vivo

Abstract

Jung (1986) has proposed a mathematical model for cell killing by hyperthermia which assumes that heat killing involves two steps: the production (p) of non-lethal lesions at random and a subsequent conversion (c) into lethal lesions. The p & c model has been shown to predict the survival of CHO cells heated in vitro even when complicated biological phenomena such as thermotolerance and step-down heating (SDH) are involved (Jung 1986, 1991). In the present study the objective was to test the p & c model's ability to describe the effect of single heating and SDH in an experimental tumour in vivo. The endpoint was tumour growth delay (GD). The doubling times (DT) for untreated and heated tumours were similar, and the surviving fraction (SF) could therefore be estimated using: SF = −in(2)·GD/DT. SF was fitted to the model by non-linear regression. The p & c model adequately described the GD obtained by SDH (39–44·5°C) and single heating above 42·5°C. Multiple linear regression showed that the residuals for single heating and SDH were independent of both heating time and temperature. However, the residuals for single heating (41–44·5°C) were significantly correlated to heating time when analysed separately. The GD obtained by the use of extended single heating times at or below 42·5°C was therefore overestimated by the model. Development of chronic thermotolerance during heating may account for the observed divergence. The Arrhenius plots for both p and c were log-linear with activation energies of 678 and 311 kJ/mol, respectively. Jung (1986) has previously reported similar p and c activation energies above 42·5°C for CHO cells in vitro.