A Unified Framework from Boltzmann Transport to Proton Treatment Planning

Abstract: This work develops a rigorous mathematical formulation of proton transport by integrating both deterministic and stochastic perspectives. The deterministic framework is based on the Boltzmann-Fokker-Planck equation, formulated as an operator equation in a suitable functional setting. The stochastic approach models proton evolution via a track-length parameterised diffusion process, whose infinitesimal generator provides an alternative description of transport. A key result is the duality between the stochastic and deterministic formulations, established through the adjoint relationship between the transport operator and the stochastic generator. We prove that the resolvent of the stochastic process corresponds to the Green's function of the deterministic equation, providing a natural link between fluence-based and particle-based transport descriptions. The theory is applied to dose computation, where we show that the classical relation: dose = (fluence * mass stopping power) arises consistently in both approaches. Building on this foundation, we formulate a hybrid optimisation framework for treatment planning, in which dose is computed using a stochastic model while optimisation proceeds via adjoint-based PDE methods. We prove existence and differentiability of the objective functional and derive the first-order optimality system. This framework bridges stochastic simulation with deterministic control theory and provides a foundation for future work in constrained, adaptive and uncertainty-aware optimisation in proton therapy.