Chapter 3 summarizes major equations of statistical and stochastic mechanics, important for description of molecular transport.

This chapter studies a robust counterpart with interval range uncertainty for LQ dose-response parameters in the fractionation problem with two modalities. As in previous chapters, the robust formulation is inevitably infinite-dimensional. The chapter describes its equivalent reformulation that calls for solving a finite set of subproblems. Each of these subproblems can be solved using the two analytical solution methods described in the previous chapter. Clinical insights into the resulting treatment plans are derived via numerical experiments.

This chapter extends the formulation from the previous chapter by including multiple organs-at-risk. The four analytical solution methods from the previous chapter are generalized to this case. Again, the structure of the solution depends on the relative values of the LQ dose-response parameters of the tumor and the organs-at-risk. The four solution methods are only adequate in two of the three possibilities that can arise regarding these relative parameter values. While these two possibilities are natural extensions of similar scenarios from the previous chapter, the third one is unique to this chapter. The chapter thus includes a fifth solution method to tackle this third possibility. That approach relies on reformulating the quadratically constrained quadratic program as an equivalent linear program with two variables. Clinical insights are obtained via numerical experiments.

This chapter accommodates uncertainty in the LQ dose-response parameters of the spatiotemporally integrated formulation of the fractionation problem from the previous chapter. It pursues a robust optimization approach, where the uncertainty is modeled using interval ranges for possible parameter values. The resulting robust formulation is infinite-dimensional. The chapter describes an approach rooted in linear programming duality to equivalently reformulate this as a finite collection of finite-dimensional convex linear-quadratic programs. The effect of the size of interval ranges on the treatment plans is studied via computational experiments.

The number of treatment sessions and the doses administered in each of these sessions were the decision variables in all chapters thus far. In practice, these doses are in turn determined by the intensity profile of the radiation field. In this chapter, we study a new, more general class of formulations where these intensity profiles are optimized directly along with the number of treatment sessions. These formulations fall within the realm of spatiotemporally integrated fractionation. They take the form of mixed integer nonconvex nonlinear programs and are computationally intractable to solve exactly. The chapter describes an approximate solution method rooted in solving a sequence of convex linear-quadratic problems. Clinical insights into the resulting treatment plans are derived via computational experiments.

This chapter continues with the theme of tackling uncertainty in the LQ dose-response parameters within formulations of the optimal fractionation problem. However, it pursues an approach called inverse optimization, which is quite different from the earlier robust optimization methodology. Unlike in usual (forward) optimization, the goal in inverse optimization is to determine parameters of a model that would render given values of decision variables optimal. In our context, this reduces to finding parameters of the LQ dose-response model that would render a given dosing plan optimal to the fractionation problem. The chapter relies on the structure of optimal dosing plans as derived in earlier chapters to solve this inverse problem in closed-form. In fact, the chapter calculates simple formulas for all possible values of LQ parameters that would make a given dosing plan optimal. The calculation procedure is illustrated through a numerical example.

This chapter uses the linear quadratic (LQ) dose-response model to present a mathematical formulation of the optimal fractionation problem, assuming that there is a single healthy tissue (organ-at-risk) nearby. The decision variables in this formulation are the number of sessions in the treatment course and the radiation doses administered in each of these sessions. The chapter first studies this formulation by fixing the number of sessions at an arbitrary positive integer. The resulting model is a nonconvex quadratically constrained quadratic program in the dosing decisions. A closed-form solution to this model is derived via four different analytical methods. The form of this solution depends on the relative values of the LQ dose-response parameters of the tumor and the organ-at-risk. In particular, the chapter shows that it is optimal to administer either a positive dose in a single session and no dose in other sessions, or an identical positive dose in each session. This solution is then substituted back into the original formulation and an optimal number of sessions is determined using calculus. Clinical insights are obtained via numerical experiments.

This chapter is written for a scientifically literate reader without assuming any specific background in medicine, physics, or mathematics. The chapter emphasizes that while radiation kills tumor, it also damages healthy tissue nearby. Thus, the fundamental challenge is to maximize tumor-kill while maintaining toxic effects on healthy tissue within tolerable limits. Three approaches to manage this tradeoff are described: spatial localization of radiation dose; temporal dispersion of radiation dose; and selection of an appropriate radiation modality. The optimal fractionation problem is then introduced within the context of these three approaches. The monograph relies upon the linear quadratic (LQ) framework of radiation dose response to build and solve mathematical formulations of the optimal fractionation problem. This LQ framework is recalled briefly. This chapter provides motivation for all subsequent material in the monograph. An outline of the remainder of the monograph is included at the end.

This chapter outlines some questions that remain open in the optimal fractionation literature within the context of the LQ dose-response framework. Opportunities for theoretical and algorithmic research are outlined.

All previous chapters considered the optimal fractionation problem with a single radiation modality to administer dose. This chapter includes two competing modalities in a formulation of the fractionation problem using the LQ dose-response framework. The decision variables are thus the number of treatment sessions administered with each modality and the dose administered in each of these sessions via the corresponding modality. Two analytical methods for exact solution of this formulation are described. The effect of relative radiobiological powers and physical dose-deposition profiles of the two competing modalities on optimal treatment plans is explored via numerical experiments.

The previous chapter demonstrated that an optimal dosing plan for the fractionation problem depends on the values of the LQ dose-response parameters for the tumor and the organs-at-risk. Unfortunately, these parameter values are unknown and difficult to estimate accurately. The literature often instead reports estimated interval ranges for these values. This chapter therefore pursues a robust optimization approach to the fractionation problem. The goal is to find a dosing plan that would not violate toxicity limits for the organs-at-risk as long as the “true” values of the unknown parameters belong to estimated interval ranges. These ranges are called uncertainty intervals. In fact, among all such robust plans, the treatment planner is interested in finding one that maximizes tumor-kill. The chapter provides a formulation for this problem, which is inevitably infinite-dimensional. Structural insights from the previous two chapters are utilized to reformulate this problem such that it can be instead tackled by solving a finite set of linear programs with two variables. The effect of the size of the uncertainty interval on the dosing plans is studied via numerical experiments.