Based on physicochemical mechanics, Chapter 8 discusses complicated problems of multicomponent and thermodiffusion, and gives the generalizedFick’s law and generalized Onsagedr’s reciprocal relations.

Chapter 11 summarizes the major ideas of the book, and discusses their possible applications in more complex processes, such as biological evolution and social and economical phenomena.

Chapter 7 gives derivation of major laws of mechanochemstry, colloid chemistry, chemical kinetics of mono- and bimolecular reactions and electrochemistry.

Chapter 5 summarizes the basics of nonequilibrium thermodynamicsand shows its limitations.

Chapter 9 describes fluctuations-based energy and heat transport, including thermoelectric and thermomagnetic phenomena.

Chapter 4 reminds the basics of equilibrium chemical thermodynamics and reformulates it for transport in the presence of external fields.

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.