A mathematical framework for spatiotemporal optimality in radiation therapy

Success in radiotherapy depends on a complicated radiobiological interaction between radiation and malignant and healthy tissues. The current practice of increasing tumor control while keeping normal-tissue complication below a threshold level focuses on controlling the spatial and temporal distribution of radiation. Delivering a highly concentrated dose to tumors by Intensity Modulated Radiation Therapy (IMRT) minimizes the dose to normal tissues. This spatially optimized dose distribution is delivered over several weeks, typically one fraction per day, to allow normal tissues to recover from radiation damage between treatment sessions. The current standard is to deliver the same dose every fraction, ignoring the possible biological change in tumors and normal tissues over time during the treatment course. This thesis begins by designing a multi-objective method to achieve a spatially optimal dose distribution. Then a Markov decision process approach is developed to adaptively optimize the prescription dose in each fraction. Moreover, I propose a synthesis of spatial and temporal optimization strategies by constructing a discrete-time, stochastic control framework, which I name Dynamic Biologically Conformal Radiation Therapy (DBCRT).;The spatial optimization problem is tackled by designing a novel, two-level hierarchical process. The upper-level optimization is done with a multi-objective genetic algorithm (multi-GA) to ensure a lower organs-at-risk (OAR) dose whenever possible, and the lower-level optimization employs efficient quadratic programming to achieve adequate target coverage. The goal is to find a small set of Pareto-optimal plans that exhibit diverse dosimetriccharacteristics, and that can he easily filtered by a decision maker. My numerical results indicate that the set of plans obtained this way is more diverse with lower OAR-dose than the more common single-objective approach of adjusting weighting factors. I then optimize the prescribed dose in each fraction using a Markov decision process without assuming a constant dose per fraction. This model incorporates some key biological features of the problem through intuitive choices of state and action spaces, as well as transition probability and reward functions. Numerical simulations exhibit a clinically intuitive, monotone optimal prescription-dose policy.;I then propose DBCRT, a mathematical framework, where the dose distribution is spatiotemporally optimized. DBCRT possesses the unique ability to simultaneously balance the spatial and temporal aspects of dose optimization. Specifically, DBCRT is a general, discrete-time, stochastic control formalism whose system states correspond to the patient's biological condition and whose controls relate to radiation beam intensities to be employed in each treatment session. For illustration purposes, I focus on special cases with cell-based states for tumors and dose-based states for normal tissues. As exact solution of this problem is computationally demanding, three approximate control schemes called certainty equivalence control, open-loop feedback control, and open-loop control are investigated to solve the problem approximately. I present DBCRT in two different settings: (i) finite horizon, where the total number of treatment sessions is fixed a priori, and (ii) infinite horizon, where the optimal stopping point is found adaptively in addition to the beamlet intensities to be used if the treatment continues. Numerical simulations show that DBCRT achieves a significantly higher tumor cell-kill at the end of the treatment course as compared to the current static, deterministic approaches.