A novel Boltzmann equation solver for calculation of dose and fluence spectra distributions for proton beam therapy

TL;DR

This work introduces a novel deterministic Boltzmann transport solver (DBS) for proton beam therapy that solves the Boltzmann transport equation via a line-integrated, iterative approach, delivering dose, fluence spectra, and LET/RBE-relevant data without statistical noise. DBS uses the same physical models as leading Monte Carlo codes, including multiple Coulomb scattering and nuclear interactions, and leverages Vavilov and Molière formalisms with adaptive step and discretization schemes to balance speed and accuracy. The authors validate DBS in water across 40–220 MeV, reporting 1%–1 mm level agreement with Geant4 and extreme speed improvements (milliseconds versus hours), while also providing detailed fluence spectra for radiobiological modelling. The practical impact is substantial: a fast, accurate, spectrally rich dose-calculation tool that supports advanced radiobiological models and can be extended to heavier ions, offering a foundation for improved treatment planning and biological optimization in hadron therapy.$

Abstract

Approach. We solve the Boltzmann transport equation using an iterative procedure. Our algorithm accounts for Coulomb scattering and nuclear reactions. It uses the same physical models, as do the most rigorous Monte Carlo systems. Thereby it achieves the same low level of systematic errors. Our solver does not involve random sampling. The solution is not contaminated by statistical noise. This means that the overall uncertainties of our solver are lower than those realistically achievable with Monte Carlo. Furthermore, our solver is orders of magnitude faster. Its another advantage is that it calculates fluence spectra. They are needed for calculation of relative biological effectiveness, especially when advanced radiobiological models are used that may present a challenge for other algorithms. Main results. We have developed a novel Boltzmann equation solver, have written prototype software, and completed its testing for calculations in water. For 40-220 MeV protons we calculated fluence spectra, depth doses, three-dimensional dose distributions for narrow Gaussian beams. The CPU time was 5-11 ms for depth doses and fluence spectra at multiple depths. Gaussian beam calculations took 31-78 ms. All the calculations were run on a single Intel i7 2.9 GHz CPU. Comparison of our solver with Geant4 showed good agreement for all energies and depths. For the 1\%/1 mm $γ$-test the pass rate was 0.95-0.99. In this test, 1\% was the difference between our and Geant4 doses at the same point. The test included low dose regions down to 1\% of the maximum dose.