Fast proton transport and neutron production in proton therapy using Fourier neural operators

TL;DR

A surrogate model based on Fourier Neural Operators (FNO) for fast prediction of angle- and energy-resolved proton transport and neutron production within proton therapy and demonstrates robust generalization with respect to irradiated geometry and beam characteristics.

Abstract

Objective: Real-time adaptive proton range verification systems based on produced neutrons require accurate information on their non-isotropic momentum distributions within short times, for which Monte Carlo (MC) methods are too computationally expensive. We present a surrogate model based on Fourier Neural Operators (FNO) for fast prediction of angle- and energy-resolved proton transport and neutron production within proton therapy. Approach: We treat the irradiated phantom and the proton beam's state as depth-evolving series, respectively of different materials, and of spatial, angular and energy phase space density distributions. The task is solved auto-regressively by learning changes in the distributions of protons and those of produced neutrons. For training and evaluation, two datasets of 47 MC simulations featuring different primary intensities were produced. Simulated geometries were extracted from a thoracic CT scan as series of laterally homogeneous materials. Main Results: An average relative $L^2$ discrepancy of 0.067 and 0.137 was achieved by the predicted proton and neutron distributions, respectively. This corresponded to an average gamma passing rate in the spatial distributions of 99.95$\%$ and 99.40$\%$. Training with higher primary intensities led to improvements between 12$\%$ and 30$\%$ in density metrics. Inference over depths of 40 cm at a resolution of 0.5 mm required on average 23.17 s per beam. Significance: The proposed proton beam surrogate generates accurate spatial and momentum distributions of neutrons at MC-level accuracy within seconds, while demonstrating robust generalization with respect to irradiated geometry and beam characteristics. This approach can be used for prototyping and operation of range verification systems, other tasks such as neutron dose estimation, and can be extended to include other kinds of secondary emissions.

Figures (12)

• Figure 1: Schematic of the NOVO range verification system. Detection is performed for both prompt gamma rays and neutrons produced during treatment.
• Figure 2: Visualization of the relevant dimensions in the proposed surrogate: (a) the spatial dimensions; (b) the directional and energy dimensions. Retained dimensions ($R$, $E$ and $\theta$) are colored in blue, while those assumed as independently distributed (meaning $\alpha$ and $\phi$) are shown in orange. The discretized depth position $z_{k}$ is instead colored in pink as it is handled only indirectly by the surrogate's operators.
• Figure 3: Details of the performed simulations: (a) the phantom utilized to sample candidate material sequences by casting randomly-directed lines (left) and corresponding material profiles per depth in Houndfield Units (HU) (right); (b) initial energy spread function as a function of the average energy.
• Figure 4: Visualization of the dataset: (a) the selected geometry function per depth shown for each simulated energy within the dataset; (b) examples of MC-binned densities $\psi^{p}_{k}$ and $\psi^{n}_{k}$ taken from the simulation highlighted in (a), showing only bins having a probability mass greater than $0.1\%$ and with projections of the distributions along each pair of dimensions plotted on the domain boundaries. The evolution over each step $k$ is also animated within this figure.
• Figure 5: Details of the proposed surrogate: (a) the FNO architecture employed to implement $G^{p}$ and $G^{n}$ is shown, along with the shape of the data and network weights through its depth; (b) inter-operation between proton data and the multi-step surrogate components $G^{p}$ and $F^{p}$, where $f_{w}$ represents the mapping from the proton beam energy and CT number to the water equivalent thickness $w_k$; (c) inter-operation between the predicted proton phase space and the $G^{n}$ and $F^{n}$ components producing the neutron phase space prediction.