A Theoretical Framework for Learning Tumor Dose-Response Uncertainty in Individualized Spatiobiologically Integrated Radiotherapy

Recent theoretical research has employed the linear-quadratic model of dose-response in stochastic control formulations for spatiobiologically integrated radiotherapy. The goal is to maximize the expected tumor kill while limiting the biologically effective dose administered to nearby organs at risk under tolerable limits. This is attempted by adapting fluence maps to the uncertain evolution of tumor-cell densities observed in functional images acquired at the beginning of each treatment session. One limitation of this research is that the treatment planner is assumed to know the probability distribution of a crucial dose-response parameter in the linear-quadratic model. This paper proposes a Bayesian stochastic control framework to relax this assumption. An algorithm rooted in certainty-equivalent control is devised to simultaneously learn this probability distribution while adapting fluence maps based on dose-response data collected from functional images over the treatment course. This algorithm’s performance is compared via numerical simulations with two other solution procedures that are also rooted in certainty equivalent control. The first one is a clairvoyant method. This assumes that the treatment planner knows the probability distribution, and hence serves as an idealized gold standard. The other one uses a fixed value of the dose-response parameter as available from the literature, and hence provides a natural benchmark without learning. The tumor kill achieved by the learning algorithm is statistically indistinguishable from the clairvoyant approach, whereas it can be about 20% higher than the no-learning benchmark. Both these conclusions bode well for individualized spatiobiologically integrated radiotherapy using functional images, at least in theory.