Monitoring the Distance and Velocity of Protons in a Medium for Biomedical Applications Using a Straightforward Mathematical Approach

Radiotherapy is one of the major cancer treatments that uses controlled doses of ionizing radiation to damage tumor cells. The monitoring of charged particles within a medium is of tremendous importance in radiotherapy. Monte Carlo methods can be used to estimate radiation profiles; however, despite their effectiveness, they are computationally expensive, which limits their practicality. To simplify the analysis, approximate analytical methods focused on estimating the range of charged particles and their velocity function within a medium have been previously derived. Previous solutions include non-elementary functions, such as the exponential integral function with relativistic coordinate transformations, or the use of regular perturbation methods accounting for small relativistic effects. In this paper, a much simpler approach is presented to assist practitioners in the field of radiotherapy. Using the proposed method, the particles’ range and velocities are calculated exclusively with elementary functions. The main advantage of the proposed approach, aside from its straightforward application, is its suitability for relativistic velocities. The equations derived in this paper were successfully tested at the radiotherapy level, accommodating protons with energies of up to 350 MeV.