Fast SDE-based Monte Carlo dose calculation for proton therapy validated against Geant4

TL;DR

The paper presents a stochastic differential equation (SDE) framework for proton dose calculation that replaces frequent small-angle Coulomb interactions with a diffusion term while keeping probabilistic Monte Carlo fidelity. By anchoring cross sections and mean excitation energies to established physics (Bethe–Bloch, energy straggling, Molière, Kalbach–Moore) and employing Bragg-additivity, the model can adapt to diverse materials without free parameter tuning. Validations against Geant4 in both homogeneous and heterogeneous phantoms show range differences within $\sim$0.4 mm and 3D gamma pass rates $>98\%$ under $3\%/2~\mathrm{mm}$, with about a $5\times$ speed-up, highlighting the method’s potential for rapid plan verification and adaptive planning. The results demonstrate that the SDE approach can achieve high fidelity dose predictions at a fraction of the computational cost, and its parallelizability suggests substantial practical impact for clinical workflows.

Abstract

Objective: To validate a newly proposed stochastic differential equation (SDE)-based model for proton beam energy deposition by comparing its predictions with those from Geant4 in simplified phantom scenarios. Approach: Building on previous work in Crossley et al. (2025), where energy deposition from a proton beam was modelled using an SDE framework, we implemented the model with standard approximations to interaction cross sections and mean excitation energies, which makes simulations easily adaptable to new materials and configurations. The model was benchmarked against Geant4 in homogeneous and heterogeneous phantoms. Main results: The SDE-based dose distributions agreed well with Geant4, showing range differences within 0.4 mm and 3D gamma pass rates exceeding 98% under 3%/2 mm criteria with a 1% dose threshold. The model achieved a computational speed-up of approximately fivefold relative to Geant4, consistent across different Geant4 physics lists. Significance: These results demonstrate that the SDE approach can reproduce accuracy comparable to high-fidelity Monte Carlo for proton therapy at a fraction of the computational cost, highlighting its potential for accelerating dose calculations and treatment planning.

Figures (12)

• Figure 1: The three main interactions of a proton with matter. An elastic scattering (top) with the nucleus, a proton-nucleus collision which may be elastic or inelastic (centre), and an inelastic Coulomb interaction with atomic electrons (bottom).
• Figure 2: A typical Bragg Peak curve in a homogeneous medium. Depth is normalised to the proton range R90, and dose is normalised to its maximum value.
• Figure 3: 2D central-axis dose distribution comparison using a monoenergetic 100 MeV proton beam in a homogeneous water phantom.
• Figure 4: 1D comparison between SDE and Geant4 for two monoenergetic proton beams (100 and 150 MeV) in a homogeneous water phantom, including pointwise calculations for percentage dose differences in the lower subplots.
• Figure 5: Lateral profile comparison between SDE (marker points) and Geant4 (solid lines) for two proton energies in a homogeneous water phantom.