Mathematical model for the thermal enhancement of radiation response: thermodynamic approach

Abstract

Radiotherapy can effectively kill malignant cells, but the doses required to cure cancer patients may inflict severe collateral damage to adjacent healthy tissues. Recent technological advances in the clinical application has revitalized hyperthermia treatment (HT) as an option to improve radiotherapy (RT) outcomes. Understanding the synergistic effect of simultaneous thermoradiotherapy via mathematical modelling is essential for treatment planning. We here propose a theoretical model in which the thermal enhancement ratio (TER) relates to the cell fraction being radiosensitised by the infliction of sublethal damage through HT. Further damage finally kills the cell or abrogates its proliferative capacity in a non-reversible process. We suggest the TER to be proportional to the energy invested in the sensitisation, which is modelled as a simple rate process. Assuming protein denaturation as the main driver of HT-induced sublethal damage and considering the temperature dependence of the heat capacity of cellular proteins, the sensitisation rates were found to depend exponentially on temperature; in agreement with previous empirical observations. Our findings point towards an improved definition of thermal dose in concordance with the thermodynamics of protein denaturation. Our predictions well reproduce experimental in vitro and in vivo data, explaining the thermal modulation of cellular radioresponse for simultaneous thermoradiotherapy.

Figure 1

Left: Schematic survival probabilities for the three cases depicted on the right. (a) Cell killing as a single rate process with transition rate from alive (A) to dead (D) $\alpha$. (b) Two-step cell killing process in the LQ-model for radiation. A cell transits from the alive state (A) to the dead state (D) through two possible paths: $\alpha$ for direct killing (a single hit suffices to kill), and $\beta$ for indirect killing (when two hits are required to kill). (c) Combined HT+RT: HT-induced damage elevates cells from state (A) to an activated state ($A^{\prime }$), effectively reducing the $\alpha /\beta$ ratio. Since $\beta$ is more efficiently reduced, the direct path $\alpha$ dominates the killing process and consequently reduces the survival probability.

Figure 2

(a) and (c) show the linear dependency of the thermal enhancement ratio (TER) on time of exposure for CHO cells in vitro and C3H mammary carcinoma cells in mice tumours in vivo, respectively. The slope of the linear fitting clearly depends on the temperature of the hyperthermia treatment, and the natural logarithm of the slope was plotted as a function of temperature for both datasets in (b) and (d). The linear trend lines show the exponential behaviour of the temperature dependent rate k(T) according to Eq. (6). The data for CHO cells (a, b) and C3H mammary carcinoma (c, d) was extracted from and, respectively.

Figure 3

Thermal enhancement $(TER-1)$ as function of the relative temperature $(T-T_g)$ for M8013 mouse mammary carcinoma cells in vitro. (a) Thermotolerant modification of the cell line and (b) Non-thermotolerant cells. Vertical axes displayed in logarithmic scale. The lines are exponential fittings of the $TE{R}_{\alpha }-1$ and $TE{R}_{\beta }-1$ points together.

Figure 4

Survival curves of cells after HT treatment. Data extracted from reference (a) $-ln(S)$ as function of HT treatment time for different temperatures. The lines correspond to quadratic fittings. (b) $-ln(S)$ as function of the thermal dose as defined in Eq. (20). (c) $-ln(S)$ as function of the thermal dose as defined in Eq. (21).

Figure 5

Thermal enhancement TER as function of the thermal dose as defined by Eq. (20) ((a) and (c)) and Eq. (21) ((b) and (d)). (a) and (b) show the performance of both dosimetry concepts for CHO cells in vitro. (c) and (d) show the performance for C3H mammary carcinoma cells in mice tumours in vivo. The filled (orange) symbols in (b) correspond to the TERs obtained at 43 °C.