Optimal number and sizes of the doses in fractionated radiotherapy according to the LQ model

We address a non-linear programming problem to find the optimal scheme of dose fractionation in cancerradiotherapy. Using the LQ model to represent the response to radiation of tumour and normal tissues,we formulate a constrained non-linear optimization problem in terms of the variables number and sizesof the dose fractions. Quadratic constraints are imposed to guarantee that the damages to the early andlate responding normal tissues do not exceed assigned tolerable levels. Linear constraints are set to limitthe size of the daily doses. The optimal solutions are found in two steps: i) analytical determination ofthe optimal sizes of the fractional doses for a fixed, but arbitrary number of fractions n; ii) numericalsimulation of a sequence of the previous optima for n increasing, and for specific tumour classes. Weprove the existence of a finite upper bound for the optimal number of fractions. So, the optimum withrespect to n is found by means of a finite number of comparisons amongst the optimal values of theobjective function at the first step. In the numerical simulations, the radiosensitivity and repopulationparameters of the normal tissue are fixed, while we investigate the behaviour of the optimal solution forwide variations of the tumour parameters, relating our optima to real clinical protocols. We recognizethat the optimality of hypo or equi-fractionated treatment schemes depends on the value of the tumourradiosensitivity ratio compared to the normal tissue radiosensitivity. Fast growing, radioresistant tumoursmay require particularly short optimal treatments.