Adaptive Deterministic Proton Dose Calculation Using a Semi-Analytical Linear Boltzmann Transport Solver
Abstract
Purpose
Monte Carlo (MC) methods are widely used for proton therapy dose calculation due to their high physical accuracy, but their stochastic nature results in statistical noise and long computation times, particularly for high-resolution dose calculations and iterative optimization workflows. Deterministic solvers of the linear Boltzmann transport equation (LBTE) have achieved clinical success in photon dose calculation by providing noise-free solutions with improved efficiency; however, comparable approaches for proton transport remain limited due to the increased dimensionality and complexity of energy loss and angular scattering. This work aims to develop an efficient deterministic solver for proton dose calculation based on a semi-analytical formulation of the LBTE.
Methods
Proton phase-space density is obtained through a hybrid formulation combining a numerical solution of a two-dimensional Fokker–Planck equation with an analytical Fermi–Eyges solution for angular transport. A corrected, unconditionally stable Crank–Nicolson scheme is employed for depth discretization, while a local discontinuous Galerkin method is used for energy discretization. Adaptive refinement is applied in both the depth and energy domains to concentrate computational effort in regions with steep dose gradients and rapidly varying phase-space structure.
Results
Dose distributions calculated with the proposed deterministic solver were compared against Monte Carlo simulations using large particle histories. The deterministic solution converged toward the MC reference dose with systematic mesh refinement in both spatial and energy dimensions. The adaptive refinement strategy substantially reduced computational cost, yielding approximately a 75% reduction in computation time relative to uniformly refined meshes while maintaining comparable dose accuracy.
Conclusion
An adaptive, deterministic semi-analytical solver for proton beam dose calculation based on the LBTE was developed and validated. By combining numerical and analytical transport formulations with adaptive discretization, the proposed approach achieves Monte Carlo–level accuracy with significantly improved computational efficiency, supporting its potential utility for high-resolution and iterative proton therapy dose calculation workflows.