  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.
