Energy optimization procedure for treatment planning with laser-accelerated protons

A simple analytical model is found that predicts the exact proton spectrum needed to obtain a spread-out-Bragg peak (SOBP) distribution for laser-accelerated proton beams. The theory is based on the solution to the Boltzmann kinetic equation for the proton distribution function. The resulting analytical expression allows one to calculate the SOBP proton energy spectra for the different beamlet sizes and modulation depths that can be readily implemented in the calculation of energy and intensity modulated proton dose distributions. Since the practical implementation of energy modulation for proton beams is realized through the discrete superposition of individual Bragg peaks, it is shown that there exists an optimal relationship between the energy sampling size and the width of the initial proton energy distribution.