Proton stopping powers averaged over beam energy spectra

Stopping powers averaged over the proton energy spectra at various depths in water and a tissue-like material were calculated for proton beams with initial energies between 50 and 250 MeV. The analysis was made for proton beams with a spread-out Bragg peak (SOBP) to assess in particular the variations of stopping power with depth in the SOBP plateau region. Nuclear interactions were accounted for in determination of the proton energy spectra. In modulated beams, stopping power varies considerably with depth along the plateau region of the spread-out peak. Moreover, stopping powers at the same water depth may differ from each other by up to 20% if referring to modulated or non-modulated beams. Calculations of water-air mass stopping power ratios were also performed, with and without the inclusion of nuclear interactions, for modulated and non-modulated beams. The stopping power ratios do not depend significantly on proton energy, and this makes the effect of accounting for nuclear interactions in proton fluence calculation negligible. For the same reason the stopping power ratios for modulated beams do not appreciably differ from those referred to non-modulated beams with the same Emax at the same depth in water. The present results also show that the dependence of stopping power on energy spread and spatial divergence for proton beams is not negligible in some conditions. Some conclusions are drawn on the possibility of using the residual range as a descriptor of the proton beam quality for different experimental beam conditions.