Robust and probabilistic optimization of dose schedules in radiotherapy

We consider the effects of parameter uncertainty on the optimal radiation schedule in the context of the linear-quadratic model. Our interest arises from the observation that if interpatient variations in normal tissues and tumor sensitivities to radiation or sparing factor of the organ at risk (OAR) are not accounted for during radiation scheduling, the performance of the therapy may be strongly degraded or the OAR may receive a substantially larger dose than the maximum threshold. This paper proposes two radiation scheduling concepts to incorporate inter-patient variability into the scheduling optimization problem. The first approach is a robust formulation that formulates the problem as a conservative model that optimizes the worst case dose scheduling that may occur, assuming that the parameters vary within given intervals. The second method is a probabilistic approach, where the model parameters are given by a set of random variables. Our probabilistic formulation insures that our constraints are satisfied with a given probability, and that our objective function achieves a desired level with a stated probability. We used the same transformation as [37] to reduce the resulting optimization problem to two dimensions. We showed that the optimal solution in the absence of uncertainty in the tumor radio-sensitivity parameters (α and β) occurs at one of the corners of the feasible region. However if we incorporate uncertainty in α and β into the optimization problem, this result does not hold anymore. In this case, we showed that the optimal solution lies on the boundary of the feasible region and we implemented a branch and bound algorithm to find the global optimal solution. We demonstrated how the configuration of optimal schedules in the presence of uncertainty compares to optimal schedules in the absence of uncertainty (conventional schedule). We observed that if the number of fractions in the optimal conventional schedule is the same as the robust and stochastic solutions, it is preferable to administer equal or smaller total dose. In addition if there exist more (fewer) treatment sessions in the probabilistic or robust solution compared to the conventional schedule, a reduction in total dose squared (total dose) will be expected. Finally, we performed numerical experiments in the setting of head-and-neck tumors including several normal tissues to reveal the effect of parameter uncertainty on optimal schedules and to evaluate the sensitivity of the model to the choice of key model parameters.