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

The claim that Monte Carlo is the most accurate method is a case of misattributed credit. This claim is based on experience with advanced systems MC-NPX, Geant4 and EGS. These systems achieve remarkable performance because they use most accurate physics, not because they use random numbers. The latter simplifies algorithms, but contaminates the solution with random noise. Currently prevalent fast Monte Carlo algorithms retain this worst part while achieving high computing speed at the expense of the best part - accurate physics. We employ an opposite strategy. We develop a Boltzmann solver for protons that retains unchanged the physics of most advanced Monte Carlo systems. We eliminate random noise, because our solution method is deterministic. Our method is also applicable to heavier ions, helium and carbon, for example.

To develop a fast and accurate deterministic Boltzmann solver for protons. It calculates dose distributions and fluence spectra. The spectra are needed for biological modelling. The main application is treatment planning of proton beam therapy.

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.

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 $\gamma$ -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.

Results of the study provide a foundation for achieving a high computing speed with uncompromised accuracy in proton treatment planning systems.