Mechanistic model of radiation-induced cancer after fractionated radiotherapy using the linear-quadratic formula

A simple mechanistic model for predicting cancer induction after fractionated radiotherapy is developed. The model is based upon the linear-quadratic model. The inductions of carcinomas and sarcomas are modeled separately. The linear-quadratic model of cell kill is applied to normal tissues which are unintentionally irradiated during a cancer treatment with radiotherapy. Tumor induction is modeled such that each transformation process results in a tumor cell. The microscopic transformation parameter was chosen such that in the limit of low dose and acute exposure, the parameters of the linear-no-threshold model for tumor induction were approached. The differential equations describing carcinoma and sarcoma inductions can be solved analytically. Cancer induction in this model is a function of treatment dose, the cell kill parameters (α,β), the tumor induction variable (μ), and the repopulation parameter (ξ). Carcinoma induction shows a bell shaped behavior as long as cell repopulation is small. Assuming large cell repopulation rates, a plateaulike function is approached. In contrast, sarcoma induction is negligible for low doses and increases with increasing dose up to a constant value. The proposed model describes carcinoma and sarcoma inductions after fractionated radiotherapy as an analytical function of four parameters. In the limit of low dose and for an instant irradiation it reproduces the results of the linear-no-threshold model. The obtained dose-response curves for cancer induction can be implemented with other models such as the organ-equivalent dose model to predict second cancers after radiotherapy.