A Unified Framework From Boltzmann Transport to Proton Treatment Planning

Source abstract

ABSTRACT We develop a unified stochastic–deterministic framework for proton transport in radiotherapy. The deterministic formulation is based on a Boltzmann–Fokker–Planck (BFP) equation with continuous slowing‐down, angular diffusion, and scattering terms, posed in a kinetic variational setting. The stochastic formulation describes proton trajectories by a track‐length parameterized Markov process with energy loss, angular diffusion, and jump scattering. The main result is an operator‐theoretic connection between these two descriptions. We identify the backward generator of the stochastic process with the adjoint BFP operator and show that the associated resolvent represents the Green's function of the deterministic transport equation. This gives a rigorous interpretation of deterministic fluence as an expected occupation density of stochastic particle paths. We then apply this duality to dose computation and treatment planning. Dose is represented equivalently as a deterministic fluence functional and as an expected accumulated stopping‐power functional along stochastic tracks. Finally, we formulate a hybrid SDE–PDE optimization framework in which stochastic forward simulations are combined with deterministic adjoint equations for gradient computation. We prove existence and differentiability of the resulting objective functional and derive the corresponding first‐order optimality system.