Oblique incidence for broad monoenergetic proton beams.

Purpose: The depth dose of a monoenergetic broad parallel proton beam has been modeled in a number of ways, but evidently not yet for oblique incidence. The purpose of this investigation is to find an accurate analytic formula for this case, which can then be used to model the depth dose of a broad beam with an initial Gaussian angular distribution.

Methods: The Bortfeld model of depth dose in a broad normally incident proton beam has been extended to the case of oblique incidence. This extension uses an empirically determined Gaussian parameter sigma(x) which (roughly) characterizes the off-axis dose of a proton pencil beam. As with Bortfeld's work, the modeling is done in terms of parabolic cylinder functions. To obtain the depth dose for an initial angular distribution, the result is integrated over the angle of incidence, weighted by a Gaussian probability function. The predictions of the theory have been compared to MCNPX Monte Carlo calculations for four phantom materials (water, bone, aluminum, and copper) and for initial proton energies of 50, 100, 150, 200, and 250 MeV.

Results: Comparisons of the depth dose predicted by this theory with Monte Carlo calculations have established that with very good accuracy, sigma(x) can be taken to be independent both of the depth and of the angle of incidence. As a function of initial proton range or of initial proton energy, sigma(x) has been found to obey a power law to very high accuracy. Good fits to Monte Carlo calculations have also been found for an initial Gaussian angular distribution.

Conclusions: This investigation is the first step in the accurate modeling of a proton pencil beam with initial Gaussian angular distribution. It provides the longitudinal factor, with its Bragg peak buildup and sharp distal falloff. A transverse factor must still be incorporated into this theory and this will give the lateral penumbra of a collimated proton beam. Also, it will be necessary to model the dose of product particles from nuclear interactions of the proton beam. With the accurate modeling of a pencil beam, it will be possible to accurately take into account the effect of localized tissue inhomogeneities.