Optimal number and sizes of the doses in fractionated radiotherapy

We address the problem of finding the optimal dose fractionation for cancer radiotherapy schedules of the kind one fraction/day, five fractions/week. Using the LQ model with exponential repopulation to represent the radiation response of tumour and normal tissues, we formulate a constrained nonlinear programming optimization problem in terms of the dose fraction sizes and of the total number of dose fractions. Constraints are imposed to guarantee that the damages to the early and late reacting normal tissues do not exceed maximal tolerable levels, as well as to limit the size of the daily dose fractions. The optimal solutions are found in two consecutive steps. The first step is the analytical determination of the optimal dose sizes for a fixed, but arbitrary number of fractions. The optimal protocols are classified according to the values of the tumour radiosensitivity ratio, ?/?, and of the daily dose upper bound, while the optimal fraction sizes are expressed as a function of the normal tissue parameters. The second step of the optimization consists in the numerical computation of the optimal number of dose fractions, and then of the optimal overall treatment time, considering specific tumour classes identified by the values of ?/?. We prove that the optimal number of fractions is finite so that it can be determined by a limited number of direct comparisons among the cost function values obtained for the sequence of optima with fixed length. While the radiosensitivity and repopulation parameters of the early and late responding tissues are set according to the literature, we investigate the behaviour of the optimal solution, even in comparison with standard clinical protocols, for wide variations of the tumour parameters and of the daily dose upper bound, evidencing the influence of the model parameters on the optima. In particular, we recognize that the value of the tumour ?/? ratio compared to the the normal tissue radiosensitivity determines the hypo- or equi-fractionation of the treatment scheme. The crucial role of the product of the radiosensitivity coefficient ? and the tumour cell doubling time TP on the optimal duration of the whole treatment has been highlighted