SDE models for proton transport and associated Monte Carlo methods

As part of modern cancer therapy, protons are fired into the body where they decelerate and deposit energy, killing cancerous cells. We are interested to know how to model the evolution of a proton through inhomogeneous material as the solution to an SDE with path discontinuities (in energy and velocity). Given this mathematical structure, we are interested in optimal ways to build exact Monte Carlo approximations to the proton transport equation, taking account of modern interacting particle methodology. These questions form part of a larger series of mathematical questions around building efficient numerical models for treatment planning, which interact Monte Carlo methods with numerical optimisation.