Applications of Nonlinear Programming to the Optimization of Fractionated Protocols in Cancer Radiotherapy

The present work of review collects and evidences the main results of our previous papers on the optimization of fractionated radiotherapy protocols. The problem under investigation is presented here in a unitary framework as a nonlinear programming application that aims to determine the optimal schemes of dose fractionation commonly used in external beam radiotherapy. The radiation responses of tumor and normal tissues are described by means of the linear quadratic model. We formulate a nonlinear, non-convex optimization problem including two quadratic constraints to limit the collateral normal tissue damages and linear box constraints on the fractional dose sizes. The general problem is decomposed into two subproblems: (1) analytical determination of the optimal fraction dose sizes as a function of the model parameters for arbitrarily fixed treatment lengths; and (2) numerical determination of the optimal fraction number, and of the optimal treatment time, in different parameter settings. After establishing the boundedness of the optimal number of fractions, we investigate by numerical simulation the optimal solution behavior for experimentally meaningful parameter ranges, recognizing the crucial role of some parameters, such as the radiosensitivity ratio, in determining the optimality of hypo- or equi-fractionated treatments. Our results agree with findings of the theoretical and clinical literature.