An approximate analytical formula for 3D dose deposition in water by a Gaussian proton beam

A proton linear accelerator named TOP-IMPLART, developed as a prototype of a full linear machine for proton therapy, is currently in operation at the ENEA laboratories of the Frascati Research Center in Italy. In anticipation of test irradiations of volumes or, more specifically, simulated treatment plans to check the linac's performance, it is desirable to have a streamlined tool for calculating dose deposition in three spatial dimensions and for designing volumetric spread-out Bragg peaks. To this end, a preliminary quick approach to 3D dose calculation that avoids lengthy Monte Carlo simulations would be quite convenient. Here, we present the derivation of an analytical formula for the approximate calculation of the dose volume deposited in water by a Gaussian proton beam. The formula combines Bortfeld's Bragg curve model with Preston–Koehler's model for the widening of pencil beams. Example tests are shown.