Using Stable Distributions to Characterize Proton Pencil Beams

TL;DR

This paper introduces alpha-stable distributions as an efficient and accurate method for characterizing proton pencil beams in treatment planning, enabling better interpolation and implementation in existing systems.

Contribution

The study demonstrates that alpha-stable distributions can effectively model proton pencil beams with fewer parameters and are compatible with current treatment planning systems.

Findings

Stable distributions fit proton beam data as well as standard models.

Fewer parameters are needed for the stable distribution representation.

The method allows interpolation of non-measured dose data.

Abstract

Purpose: To introduce and evaluate the use of stable distributions as a means of describing the behavior of charged particle pencil beams in a medium, with specific emphasis on proton beam scanning (PBS). Methods: The proton pencil beams of a clinically commissioned proton treatment facility are replicated in a Monte Carlo simulation system (FLUKA). For each available energy the beam deposition in water medium is characterized by the dose deposition. Using an alpha--stable distribution methodology each beam with a nominal energy $E$ is characterized by the lateral spread at depth $z$: $S(z;\alpha,\gamma,E)$ and a total energy deposition $I_D(z)$. The beams are then described as a function of the variation of the parameters at depth. Finally, an implementation in a freely available open source dose calculation suite (matRad, DKFZ, Heidelberg, Germany) is proposed. Results: Quantitatively, the fit of the stable distributions, compared to those implemented in standard treatment planning systems, are equivalent. The efficiency of the representation is better (2 compared to 3 and more parameters needed). The meta--parametrization (i.e. the description of the dose deposition by only providing the fitted parameters) allows for interpolation of non--measured data. In the case of the clinical data used in this paper, it was possible to only commission 1 out of 5 nominal energies to obtain a viable data set. Conclusions: Alpha--stable distributions are intrinsically suited to describe charged particle pencil beams in a medium and can be easily implemented in existing treatment planning systems. The use of alpha-distributions can easily be extended to other particles.