Analytical range evaluation for therapeutic protons in stack of materials

‎As one of the most clinically relevant parameters in proton radiotherapy‎, ‎the range of incident particles can be measured either by counting the number of protons or through depth-dose evaluation in the target‎. ‎In the latter‎, ‎the range is defined as the depth in the target at the distal 80% point of the Bragg peak‎. ‎In this work‎, ‎a highly accurate analytical model was employed to predict depth-dose distribution‎, ‎and hence the range‎, ‎in a desired target‎. ‎Aiming to study the effect of energy spread on the range‎, ‎proton beams with initial Gaussian distributions have been considered‎. ‎For our arbitrary tested energies‎, ‎the results show that the more the width of energy distribution increases‎, ‎the more the Bragg peaks shift in depth‎, ‎by about‎ -‎0.25% to‎ -25%, ‎compared with those of monoenergetic beams‎. ‎Furthermore‎, ‎it was found that for different widths of initial energy spectrum‎, ‎keeping the mean energy the same‎, ‎the range remains unchanged‎. ‎It was also shown that the results corresponding to utilizing analytical range determination for proton beams of different incident energies in stack of materials deviate from those of Monte Carlo simulations by less than 1.7%‎. ‎The results are encouraging‎, ‎although accurate modeling of analytical proton dose distribution in the presence of tissue inhomogeneities is still an unsolved problem‎.