Using Stable Distributions to Characterize Proton Pencil Beams
  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. 
Frank Van den Heuvel, Francesca Fiorini, Niek Schreuder, Ben George
physics.med-ph