Title:
Simultaneous optimization of dose distributions and fractionation schemes in particle radiotherapy.

Abstract:
The paper considers the fractionation problem in intensity modulated proton therapy (IMPT). Conventionally, IMPT fields are optimized independently of the fractionation scheme. In this work, we discuss the simultaneous optimization of fractionation scheme and pencil beam intensities.
This is performed by allowing for distinct pencil beam intensities in each fraction, which are optimized using objective and constraint functions based on biologically equivalent dose (BED). The paper presents a model that mimics an IMPT treatment with a single incident beam direction for which the optimal fractionation scheme can be determined despite the nonconvexity of the BED-based treatment planning problem.
For this model, it is shown that a small α∕β ratio in the tumor gives rise to a hypofractionated treatment, whereas a large α∕β ratio gives rise to hyperfractionation. It is further demonstrated that, for intermediate α∕β ratios in the tumor, a nonuniform fractionation scheme emerges, in which it is optimal to deliver different dose distributions in subsequent fractions. The intuitive explanation for this phenomenon is as follows: By varying the dose distribution in the tumor between fractions, the same total BED can be achieved with a lower physical dose. If it is possible to achieve this dose variation in the tumor without varying the dose in the normal tissue (which would have an adverse effect), the reduction in physical dose may lead to a net reduction of the normal tissue BED. For proton therapy, this is indeed possible to some degree because the entrance dose is mostly independent of the range of the proton pencil beam.
The paper provides conceptual insight into the interdependence of optimal fractionation schemes and the spatial optimization of dose distributions. It demonstrates the emergence of nonuniform fractionation schemes that arise from the standard BED model when IMPT fields and fractionation scheme are optimized simultaneously. Although the projected benefits are likely to be small, the approach may give rise to an improved therapeutic ratio for tumors treated with stereotactic techniques to high doses per fraction.