Optimal fractionation in radiotherapy with multiple normal tissues.

The goal in radiotherapy is to maximize the biological effect (BE) of radiation on the tumour while limiting its toxic effects on healthy anatomies. Treatment is administered over several sessions to give the normal tissue time to recover as it has better damage-repair capabilities than tumour cells. This is termed fractionation. A key problem in radiotherapy involves finding an optimal number of treatment sessions (fractions) and the corresponding dosing schedule. A major limitation of existing mathematically rigorous work on this problem is that it includes only a single normal tissue. Since essentially no anatomical region of interest includes only one normal tissue, these models may incorrectly identify the optimal number of fractions and the corresponding dosing schedule. We present a formulation of the optimal fractionation problem that includes multiple normal tissues. Our model can tackle any combination of maximum dose, mean dose and dose-volume type constraints for serial and parallel normal tissues as this is characteristic of most treatment protocols. We also allow for a spatially heterogeneous dose distribution within each normal tissue. Furthermore, we do not a priori assume that the doses are invariant across fractions. Finally, our model uses a spatially optimized treatment plan as input and hence can be seamlessly combined with any treatment planning system. Our formulation is a mixed-integer, non-convex, quadratically constrained quadratic programming problem. In order to simplify this computationally challenging problem without loss of optimality, we establish sufficient conditions under which equal-dosage or single-dosage fractionation is optimal. Based on the prevalent estimates of tumour and normal tissue model parameters, these conditions are expected to hold in many types of commonly studied tumours, such as those similar to head-and-neck and prostate cancers. This motivates a simple reformulation of our problem that leads to a closed-form formula for the dose per fraction. We then establish that the tumour-BE is quasiconcave in the number of fractions; this ultimately helps in identifying the optimal number of fractions. We perform extensive numerical experiments using 10 head-and-neck and prostate test cases to uncover several clinically relevant insights.