Improving Accuracy in Arrhenius Models of Cell Death: Adding a Temperature-Dependent Time Delay

The Arrhenius formulation for single-step irreversible unimolecular reactions has been used for many decades to describe the thermal damage and cell death processes. Arrhenius predictions are acceptably accurate for structural proteins, for some cell death assays, and for cell death at higher temperatures in most cell lines, above about 55 °C. However, in many cases—and particularly at hyperthermic temperatures, between about 43 and 55 °C—the particular intrinsic cell death or damage process under study exhibits a significant “shoulder” region that constant-rate Arrhenius models are unable to represent with acceptable accuracy. The primary limitation is that Arrhenius calculations always overestimate the cell death fraction, which leads to severely overoptimistic predictions of heating effectiveness in tumor treatment. Several more sophisticated mathematical model approaches have been suggested and show much-improved performance. But simpler models that have adequate accuracy would provide useful and practical alternatives to intricate biochemical analyses. Typical transient intrinsic cell death processes at hyperthermic temperatures consist of a slowly developing shoulder region followed by an essentially constant-rate region. The shoulder regions have been demonstrated to arise chiefly from complex functional protein signaling cascades that generate delays in the onset of the constant-rate region, but may involve heat shock protein activity as well. This paper shows that acceptably accurate and much-improved predictions in the simpler Arrhenius models can be obtained by adding a temperature-dependent time delay. Kinetic coefficients and the appropriate time delay are obtained from the constant-rate regions of the measured survival curves. The resulting predictions are seen to provide acceptably accurate results while not overestimating cell death. The method can be relatively easily incorporated into numerical models. Additionally, evidence is presented to support the application of compensation law behavior to the cell death processes—that is, the strong correlation between the kinetic coefficients, ln{A} and Ea, is confirmed.