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.

The general premise of this chapter is to address thermodynamic behaviors and structure of charged macromolecules in non-dilute conditions, such as semidilute and concentrated solutions. After a summary of uncharged macromolecules in concentrated solutions, the coupling between the electrostatic and topological correlations is treated. Five regimes of polymer concentrations are outlined accompanied by a collection of experimental data. Spontaneous formation of large aggregates formed by similarly charged macromolecules is described in detail.

The scope of the book is outlined with specific examples of phenomenology that are outlined and explained in subsequent chapters. The necessity of bridging electrostatic and topological correlations to understand the behavior of charged macromolecules is addressed.

This chapter introduces important concepts such as Gouy-Chapman length, double-layer, Manning condensation, and regularization of the charge of a geometrical object in electrolyte solutions. A clear description of counterion distribution around charged objects is presented.

This brief epilogue emphasizes that this book is a gateway to new avenues in addressing living matter and new water-based materials.