Analytical linear energy transfer calculations for proton therapy

As the relative biological effectiveness of protons depends on the linear energy transfer (LET), simple methods for LET calculations are desired for the optimization of proton therapy. This work provides an analytical model for the LET on the central axis of broad proton beams in water, which can also be applied to spread-out Bragg peaks. For realistic treatment situations with polyenergetic beams, the LET is here defined as a local mean of the proton stopping power, weighted by the local energy spectrum. The proposed model considers only Coulomb interactions and neglects nonelastic nuclear interactions. By assuming a Gaussian shape for the energy spectrum and by using a suitable parametrization of the stopping power, analytical expressions for the track averaged and the dose averaged LET are derived, which account for range straggling as well as for the initial width of the energy spectrum. The analytical model was evaluated by Monte Carlo simulations with GEANT 3.21. Local energy spectra were simulated to obtain LET distributions for several cases, using clinical energies between 70 and 250 MeV and varying widths of the initial energy spectrum. Good agreement was found between the analytical model and the Monte Carlo simulations (with maximum deviations of 0.5 keV per micrometer), which justifies the assumptions used in the derivation of the analytical model.