An analytical approximation of depth-dose distributions for therapeutic proton beams

A simple analytical approximation for the proximal part of the depth-dose distribution of a spread-out-Bragg-peak (SOBP) proton beam is derived. The basic assumption made is that the range-energy relationship follows a power law. The analytical formula depends only on the ratio of the distance from the proximal edge of the SOBP and the width of the SOBP. A comparison with measurements and with exact numerical calculations reveals a maximum deviation of 5% and a typical deviation of less than 2% of the target dose. The model comprises an analytical expression for the weighting of individual Bragg peaks as a function of the range, which is necessary to obtain an SOBP of the desired width and depth. Finally, an analytical expression for the total energy released in normal tissue is found by integrating the depth-dose distribution.