On the scattering power of radiotherapy protons

Abstract

Purpose: First, to show that accurate formulas for scattering power T must take into account the competition between the Gaussian core and the single scattering tail of the angular distribution, which affects the rate of change in the Gaussian width and leads to the single scattering correction (SSC). Second, to show that the SSC requires that T ( x ) be nonlocal: Besides material properties and energy at the point of interest, it must depend in some fashion on how much multiple scattering has already taken place. Third, after reviewing five previous formulas (three local and two nonlocal), to derive an improved “differential Molière” formula T dM . Last, to investigate, by studying some practical cases, when an accurate formula for T is actually needed. Methods: We first take the numerical derivative of the Molière/Fano/Hanson ⟨ θ 2 ⟩ in order to find the true SSC. We simplify the formula for T IC (ICRU Report 35) for protons, introducing a new material dependent property, the “scattering length” X S , analogous to radiation length X 0 . We then use T IC as a basis for T dM by including a nonlocal correction factor f dM which, by virtue of the Øverås approximation, parametrizes the single scattering correction. Results: The improved scattering power is T dM ≡ f dM ( p v , p 1 v 1 ) × ( E s / p v ) 2 1 / X S where f dM ≡ 0.5244 + 0.1975 lg ( 1 − ( p v / p 1 v 1 ) 2 ) + 0.2320 lg ( p v ) − 0.0098 lg ( p v ) lg ( 1 − ( p v / p 1 v 1 ) 2 ) , p 1 v 1 (MeV) is the initial product of proton momentum and speed, p v is the same at the point of interest, and E s = 15.0 MeV . T dM is easily computed and generalizes readily to mixed slabs because f dM is not material dependent. Conclusions: Whether an accurate formula for T is required depends very much on the problem at hand. For beam spreading in water, five of the six formulas for T give almost identical results, suggesting that patient dose calculations are insensitive to T . That is not true, however, of beam spreading in Pb. At the opposite extreme, the projected rms beam width at the end of a Pb/Lexan/air stack, analogous to the upstream modulator in a passive beam spreading system, is sensitive to T . In this case a simple experiment would discriminate between all but two of the six formulas discussed. Scattering power applies as much to Monte Carlo as to deterministic transport calculations. Using T in any of its forms will avoid step size dependence. Using the best available T could be important in general purpose Monte Carlo codes, which are expected to give the correct answer to many different problems.