Spatiotemporally optimal fractionation in radiotherapy

We present a spatiotemporally integrated formulation of the optimal fractionation problem using the standard log-linear-quadratic survival model. Our objective is to choose a fluence map and a number of fractions to maximize the biological effect of tumor dose averaged over its voxels subject to maximum dose, mean dose, and dose-volume constraints for various normal tissues. Constraints are expressed in biologically effective dose equivalents.We propose an efficient convex programming method to approximately solve the resulting computationally difficult model. Through extensive computer simulations on 10 head-and-neck and prostate cancer test cases with a broad range of radiobiological parameters, we compare the biological effect on tumors obtained by our integrated approach relative to that from two other models. The first is a traditional intensity modulated radiation therapy (IMRT) fluence map optimization model that does not optimize the number of fractions. The second assumes that a fluence map is available a priori from a traditional IMRT optimization model and then optimizes the number of fractions, thus separating the spatial and temporal components. The improvements in tumor biological effect over IMRT were 9%-52%, with an average of 22% for head-and-neck, and 53%-108%, with an average of 69% for prostate. The improvements in tumor biological effect over the spatiotemporally separated model were 15%-45%, with an average of 27%, and 17%-23%, with an average of 21%, for head-andneck and prostate, respectively. This suggests that integrated optimization of the fluence map and the number of fractions could improve treatment efficacy, as measured within the linear-quadratic framework.