Imputing radiobiological parameters of the linear-quadratic dose-response model from a radiotherapy fractionation plan

The objective in cancer radiotherapy is to maximize tumor-kill while limiting toxic effects of radiation dose on nearby organs-at-risk (OAR). Given a fixed number of treatment sessions, planners thus face the problem of finding a dosing sequence that achieves this goal. This is called the fractionation problem, and has received steady attention over a long history in the clinical literature. Mathematical formulations of the resulting optimization problem utilize the linear-quadratic (LQ) framework to characterize radiation dose-response of tumors and OAR. This yields a nonconvex quadratically constrained quadratic program. The optimal dosing plan in this forward problem crucially depends on the parameters of the LQ model. Unfortunately, these parameters are difficult to estimate via in vitro or in vivo studies, and as such, their values are unknown to treatment planners. The clinical literature is thus replete with debates about what parameter values will make specific dosing plans effective. This paper formulates this as an inverse optimization problem. The LQ dose-response parameters appear in the objective function, the left hand side, and the right hand side of the forward problem, and none of the existing generic methods can provide an exact solution of the inverse problem. This paper exploits the structure of the problem and identifies all possible parameter values that render the given dosing plan optimal, in closed-form. This closed-form formula is applied to dosing-plans from three clinical studies published within the last two years.