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

Charged particles such as protons and carbon ions are an increasingly important tool in radiotherapy. There are however unresolved physics issues impeding optimal implementation, including estimation of dose deposition in non-homogeneous tissue, an essential aspect of treatment optimization. Monte Carlo (MC) methods can be employed to estimate radiation profile, and whilst powerful, these are computationally expensive, limiting practicality. In this work, we start from fundamental physics in the form of the Bethe equation to yield a novel approximate analytical solution for particle range, energy and linear energy transfer (LET). The solution is given in terms of the exponential integral function with relativistic co-ordinate transform, allowing application at radiotherapeutic energy levels (50-350 MeV protons, 100-600 Mev/a.m.u carbon ions). Model results agreed closely for protons and carbon-ions (mean error within approximate to 1%) of literature values. Agreement was high along particle track, with some discrepancy manifesting at track-end. The model presented has applications within a charged particle radiotherapy optimization framework as a rapid method for dose and LET estimation, capable of accounting for heterogeneity in electron density and ionization potential.