Using stable distributions to characterize proton pencil beams.

Purpose: To introduce and evaluate the use of stable distributions as a methodology to quantify the behavior of proton pencil beams in a medium.

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 a stable distribution methodology, each beam with a nominal energy E is characterized by the lateral spread at depth z: S(z; α, γ, E) and a total energy deposition I (z, E). The parameter α describes the tailedness of the distributions, while γ is used to scale the size of the function. The beams can then be described completely by a function of the variation of the parameters with depth.

Results: Quantitatively, the fit of the stable distributions, compared to those implemented in some standard treatment planning systems, are equivalent for all but the highest energies (i.e., 230 MeV/u). The decrease in goodness of fit makes this methodology comparable to a double Gaussian approach. The introduction of restricted linear combinations of stable distributions also resolves that particular case. More importantly, the meta-parameterization (i.e., the description of the dose deposition by only providing the fitted parameters) allows for interpolation of nonmeasured data. In the case of the clinical commissioning data used in this paper, it was possible to only commission one out of five nominal energies to obtain a viable dataset, valid for all energies. An additional parameter β allows to describe asymmetric beam profiles as well.

Conclusions: Stable distributions are intrinsically suited to describe proton pencil beams in a medium and provide a tool to quantify the propagation of proton beams in a medium.