An approximate analytical solution of the Bethe equation for charged particles in the range of radiotherapy energy

Charged particles such as protons and carbon ions are an increasing tool in radiation therapy. However, unresolved physical problems prevent optimal performance, including estimating the deposited dose in non-homogeneous tissue, is an essential aspect of optimizing treatment. The Monte Carlo (MC) method can be used to estimate the amount of radiation, but, this powerful computing operation is very expensive, and has the ability to restrict it. In this work, we use basic physics in the form of the Bethe equation to provide a new analytical solution for range, energy and LET of particles. This solution is presented in terms of the functional integral by converting the relativistic harmonics, which allows it to be used at the level of radiotherapy energy (protons 50-350 MeV, carbon ions of 100-600 Mev / a.m.u). The agreement along the path of the particles, with some differences in reaching the path is high. The model presented in an optimization framework for radiation particle radiation is estimated as a rapid method for dose and LET, which is able to account for heterogeneity in electron density and ionization potential.