2 - Fractionation with a Single Organ-at-Risk

Summary

This chapter uses the linear quadratic (LQ) dose-response model to present a mathematical formulation of the optimal fractionation problem, assuming that there is a single healthy tissue (organ-at-risk) nearby. The decision variables in this formulation are the number of sessions in the treatment course and the radiation doses administered in each of these sessions. The chapter first studies this formulation by fixing the number of sessions at an arbitrary positive integer. The resulting model is a nonconvex quadratically constrained quadratic program in the dosing decisions. A closed-form solution to this model is derived via four different analytical methods. The form of this solution depends on the relative values of the LQ dose-response parameters of the tumor and the organ-at-risk. In particular, the chapter shows that it is optimal to administer either a positive dose in a single session and no dose in other sessions, or an identical positive dose in each session. This solution is then substituted back into the original formulation and an optimal number of sessions is determined using calculus. Clinical insights are obtained via numerical experiments.