Jump stochastic differential equations for the characterisation of the Bragg peak in proton beam radiotherapy

Abstract

Proton beam radiotherapy stands at the forefront of precision cancer treatment, leveraging the unique physical interactions of proton beams with human tissue to deliver minimal dose upon entry and deposit the therapeutic dose precisely at the so-called Bragg peak, with no residual dose beyond this point. The Bragg peak is the characteristic maximum that occurs when plotting the curve describing the rate of energy deposition along the length of the proton beam. Moreover, as a natural phenomenon, it is caused by an increase in the rate of nuclear interactions of protons as their energy decreases. From an analytical perspective, Bortfeld proposed a parametric family of curves that can be accurately calibrated to data replicating the Bragg peak in one dimension. We build, from first principles, the very first mathematical model describing the energy deposition of protons. Our approach uses stochastic differential equations and affords us the luxury of defining the natural analogue of the Bragg curve in two or three dimensions. This work is purely theoretical and provides a new mathematical framework which is capable of encompassing models built using Geant4 Monte Carlo, at one extreme, to pencil beam calculations with Bortfeld curves at the other.