## Robust, Non-stationary, and Adaptive Fractionation in Radiotherapy

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Ajdari, Ali

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In external beam radiotherapy for cancer, high-energy radiation is passed through the pa- tient’s body from an outside source to kill tumor cells. The challenge is that radiation also damages healthy tissue and organs-at-risk (OAR) in its path. The objective therefore is to devise treatment plans that maximize tumor-damage while protecting healthy anatomies. Treatment planners attempt two separate methods to attain this goal: spatial and biological. The spatial side focuses on the geometry and physics of the problem. The key consider- ation here is the location of the tumor relative to the nearby healthy regions as seen in an anatomical image, and the dose (energy absorbed per unit mass) deposition properties of the radiation beam. The treatment planner prescribes a high dose to the tumor and puts upper limits on the doses delivered to the healthy regions. Intensity Modulated Radiation Therapy (IMRT) technology is then employed to tune the profile (fluence-map) of the radiation beam to administer a dose that is as close as possible to this tumor-conforming prescription. Sev- eral mathematical optimization models and solution algorithms for this problem have been developed and embedded into treatment planning systems. The biological side of planning exploits the difference between the dose-response charac- teristics of tumors and healthy anatomies. For example, healthy cells are believed to possess better damage repair capabilities than tumor cells. Thus, treatment is delivered over mul- tiple sessions to give healthy tissue some time to recover between sessions. This is called fractionation. Fractionation also gives the tumor some time to re-oxygenate, which increases its sensitivity to radiation. Tumors, however, proliferate during the treatment course, and hence, too long a treatment course may not be ideal. One key question on this biological side is to determine the optimal number of treatment sessions. This is called the fraction- ation problem. Existing optimization research on the fractionation problem relies on the linear-quadratic (LQ) model of dose-response with tumor- and OAR-specific parameters to approximately capture the behavior of the complex biological system involved. Recent studies have suggested that an integrated approach that simultaneously tackles the spatial and biological sides of the problem may lead to a higher tumor-damage as com- pared to tackling the two aspects separately. The goal in such integrated formulations is to simultaneously find the fluence-map and the number of sessions that maximize tumor- damage while limiting toxic effects of dose on the healthy anatomies. Emerging advances in quantitative functional imaging technologies are enabling planners to observe the tumor’s actual dose-response over the treatment course. This provides additional opportunities for better-utilizing the LQ model by dynamically adapting treatment plans to further improve outcomes. The challenge, however, is that spatiobiologically integrated formulations based on the LQ model typically yield nonconvex quadratically constrained quadratic programming problems, which are computationally difficult to solve exactly. The research objective of this dissertation is to develop efficient convex, robust, and dynamic optimization methods to formulate and approximately solve different nonadaptive and adaptive versions of the spatiobiologically integrated fractionation problem within the LQ framework. Chapter 1 briefly describes state-of-the-art literature on spatiobiologically integrated fractionation. Each subsequent chapter is motivated by a distinct limitation of an existing formulation of the spatiobiologically integrated fractionation problem from this literature. Chapter 2: The solutions offered by existing formulations of the fractionation problem crucially depend on the assumed values of the dose-response parameters of the LQ model. Unfortunately, “true” values of these parameters are unknown. Consequently, a solution of the fractionation problem may not be feasible in practice. This concern is addressed in Chapter 2 via a robust formulation, whose solution remains feasible as long as the dose- response parameters belong to an interval. An efficient solution method rooted in a convex, finite-dimensional reformulation of the resulting infinite-dimensional problem is proposed. The price of robustness and feasibility properties of the robust solution are quantified via numerical experiments. Chapter 3: Existing spatiobiologically integrated formulations of the fractionation problem assume that the fluence-map is not changed across treatment sessions. From a computational viewpoint, this simplifies the problem significantly. Chapter 3 relaxes this assumption, and proposes an efficient solution method that allows the fluence-maps to vary across sessions. The quality of the time-variant solutions produced by this method is compared against traditional time-invariant solutions via numerical experiments. Chapter 4: Adaptive spatiobiologically integrated fractionation attempts to alter fluence- maps according to the observed evolution of tumor cell density in functional images. Chapter 4 proposes a formulation and solution method that also determine the length of the remaining treatment course adaptively. Potential benefits of such adaptive treatment-length planning are investigated through numerical experiments. Chapter 5: Adaptive fluence-map planning methods assume that the treatment planner knows the probability distribution of the uncertainty in the tumor’s dose-response parame- ters. In contrast to this “clairvoyant” approach, Chapter 5 proposes an alternative formula- tion, where the treatment planner learns this distribution from tumor-response information observed in functional images over the treatment course while also adaptively optimizing fluence-maps. This yields a Bayesian stochastic control formulation whose exact solution is impossible to derive. The chapter proposes a simple approximate solution method rooted in certainty equivalent control, and compares its performance agains a clairvoyant certainty equivalent control scheme and a “no learning” approach via numerical experiments. Finally, Chapter 6 outlines limitations of this dissertation work and describes two direc- tions for future research.