Analytical solution to the radiotherapy fractionation problem including dose bound constraints

TL;DR

This paper provides analytical solutions to the radiotherapy dose fractionation problem with dose bounds, optimizing treatment strategies for maximizing tumor effect or minimizing organ damage under the linear-quadratic model.

Contribution

It introduces the first analytical solutions considering minimum and maximum dose constraints in radiotherapy fractionation planning.

Findings

Hypofractionation is preferable in certain cases.

Hyperfractionation is optimal under different conditions.

Solutions are simple to compute with basic tools.

Abstract

This paper deals with the classic radiotherapy dose fractionation problem for cancer tumors concerning the following goals: a) To maximize the effect of radiation on the tumor, restricting the effect produced to the organs at risk (healing approach). b) To minimize the effect of radiation on the organs at risk, while maintaining enough effect of radiation on the tumor (palliative approach). We will assume the linear-quadratic model to characterize the radiation effect and consider the stationary case (that is, without taking into account the timing of doses and the tumor growth between them). The main novelty with respect to previous works concerns the presence of minimum and maximum dose fractions, to achieve the minimum effect and to avoid undesirable side effects, respectively. We have characterized in which situations is more convenient the hypofractionated protocol (deliver few fractions with high dose per fraction) and in which ones the hyperfractionated regimen (deliver a large number of lower doses of radiation) is the optimal strategy. In all cases, analytical solutions to the problem are obtained in terms of the data. In addition, the calculations to implement these solutions are elementary and can be carried out using a pocket calculator.