Robust optimization with time-dependent uncertainty in radiation therapy

ABSTRACT In the recent past, robust optimization methods have been developed and successfully applied to a variety of single-stage problems. More recently, some of these approaches have been extended to multi-stage settings with fixed uncertainties. However, in many real-world applications, uncertainties evolve over time, rendering the robust solutions suboptimal. This issue is particularly prevalent in medical decision making, where a patient’s condition can change during the course of the treatment. In the context of radiation therapy, changes in cell oxygenation directly affect the response to radiation. To address such uncertain changes, we provide a general robust optimization framework that incorporates time-dependent uncertainty sets in a tractable fashion. Temporal changes reside within a cone, whose projection at each step yields the current uncertainty set. We develop conic robust two-stage linear problems and provide their robust counterparts for uncertain constraint parameters, covering the range of radiation therapy problems. For a clinical prostate cancer case, the time-dependent robust approach improves the tumor control throughout the treatment, as opposed to current methods that lose efficacy at some stage. We show that this advantage does not bear additional risks compared to current clinical methods. For intermediate diagnostics, we provide the optimal observation timing that maximizes the value of information. While these findings are relevant to clinical settings, they are also general and can be applied to a broad range of applications; e.g., in maintenance scheduling.