Convex and Robust Optimization Methods for Modality Selection in External Beam Radiotherapy
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The goal in external beam radiotherapy (EBRT) for cancer is to maximize damage to the tumor while limiting toxic effects of radiation dose on the organs-at-risk (OAR). EBRT can be delivered via different modalities such as photons, protons, and neutrons. The choice of an optimal modality depends on the anatomy of the irradiated area, and the relative physical and biological properties of the modalities under consideration. There is no single universally dominant modality. The research objective of this dissertation is to apply convex and robust optimization methods to facilitate modality selection and corresponding dosing decisions in EBRT. The organization of this dissertation is outlined here. Chapter 1: Optimal modality selection The first chapter presents the first-ever math- ematical formulation of the optimal modality selection problem. This formulation employs the well-known linear-quadratic (LQ) dose-response framework to model the effect of radia- tion on the tumor and the OAR. The chapter proves that this formulation can be tackled by solving the Karush-Kuhn-Tucker conditions of optimality, which reduce to an analytically tractable quartic equation. Extensive numerical experiments are performed to gain insights into the effect of biological and physical properties on the choice of an optimal modality or combination of modalities. Chapter 2: Robust modality selection The feasible region and optimal solutions for the nominal modality selection problem studied in the first chapter depend on the param- eters of the LQ dose-response model. Unfortunately, “true” values of these parameters are unknown. The second chapter addresses this issue by proposing a robust counterpart of the nominal formulation. As is common in the theoretical literature on robust optimization, unknown parameter values are assumed to belong to intervals. These intervals are called uncertainty sets. The chapter shows that a robust solution can be derived by solving a finite number of nominal subproblems via a KKT approach similar to the first chapter. Again, numerical experiments are performed to gain insight into the optimal choice of modality as well as the price of robustness. Chapter 3: Spatiotemporally integrated modality selection The models in the first two chapters may be viewed as “spatiotemporally separated.” In particular, base-case radi- ation intensity profiles that deliver base-case doses for each modality are implicitly assumed to be available to the treatment planner. Any dose different from the base-case can be ad- ministered simply by appropriately scaling the base-case intensity profiles. Consequently, the decision variables in the first two chapters correspond to the dose administered by each modality. As shown in the first two chapters, this simplification leads to analytically tractable formulations. However, recent literature on dose optimization for the single modality case has shown that such a simplification may result in some loss of optimality. The third chapter therefore provides a spatiotemporally integrated formulation of the modality selection problem. This formulation is also based in the LQ model of dose-response. The decision variables here correspond to the fluence-maps (vectors) for each modality. The resulting model is inevitably of a larger scale and computationally more difficult than the ones in the first two chapters. Specifically, the model is a nonconvex quadratically constrained quadratic program (QCQP). An efficient method rooted in convex programming is explored for its approximate solution. Numerical experiments are performed to obtain insight into the optimal choice of modality and its biological effect on tumor. This dissertation establishes a mathematically rigorous foundation for modality selection and dosing in EBRT that is rooted in the clinically well-accepted LQ model of dose-response. The hope is that this foundation and associated insights via numerical experiments will help practitioners make judicious decisions while treating their patients.