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

This paper introduces a Fourier Neural Operator-based surrogate model that rapidly and accurately predicts angle- and energy-resolved proton transport and neutron production in proton therapy, achieving Monte Carlo-level accuracy within seconds to enable real-time adaptive range verification and neutron dose estimation.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: The "Crystal Ball" for Cancer Treatment

Imagine Proton Therapy as a super-precise sniper rifle used to shoot cancer cells. Unlike regular radiation (which is like a shotgun blast that hurts everything in its path), proton beams are smart. They travel through the body, stop exactly at the tumor, and release all their energy right there, sparing the healthy organs behind it. This stopping point is called the Bragg Peak.

However, there's a catch: Anatomy is messy. Patients breathe, their organs shift, and their tissues vary in density (like air in lungs vs. bone). If the sniper misjudges the distance by even a few millimeters, they might miss the tumor or hit healthy tissue.

To fix this, doctors want a real-time "range verification" system. They want to know exactly where the beam stopped while the patient is still on the table.

The Problem: The "Slow Motion" Camera

To figure out where the beam stopped, scientists usually look at neutrons (tiny particles) that are knocked loose when the proton beam hits the body. By tracking these neutrons, they can reconstruct the path of the beam.

The problem is that calculating how these neutrons behave is incredibly hard. The current gold standard method is called Monte Carlo simulation.

• The Analogy: Imagine trying to predict the path of a single raindrop by simulating every single molecule of wind, every bump on the ground, and every other raindrop it might hit. To get a clear picture, you have to simulate billions of raindrops.
• The Result: It takes a supercomputer hours or days to run one simulation. In a hospital, you need the answer in seconds to adjust the treatment while the patient is there. It's too slow.

The Solution: The "AI Crystal Ball"

The authors of this paper built an AI model (specifically using something called Fourier Neural Operators) that acts as a "crystal ball." Instead of simulating every single particle interaction from scratch, the AI has learned the rules of how protons and neutrons behave.

• The Analogy: Think of the Monte Carlo method as a student who has to do every math problem from scratch to solve an equation. The AI is like a genius who has seen enough problems to instantly recognize the pattern and give you the answer immediately.

How It Works (The "Video Game" Approach)

The researchers broke the problem down into a step-by-step video game:

1. The Map: They took a CT scan of a patient's chest (specifically looking at the lungs, which are very tricky because they are full of air and tissue).
2. The Levels: They sliced the patient's body into 800 tiny layers (like pages in a book).
3. The Players:
   • The Proton: The main character moving through the layers.
   • The Neutron: A sidekick that gets created when the proton hits something.
4. The AI's Job: The AI looks at the proton in Layer 1, figures out what happens in Layer 2, then Layer 3, and so on. It predicts two things at every step:
   • Where the protons are going next.
   • How many neutrons are being created and in what direction.

The Results: Fast and Accurate

The team tested their AI against the slow "supercomputer" method.

• Speed: The supercomputer took years of total computing time to generate the data. The AI model took about 23 seconds to do the same job for a whole treatment beam. That's a speed-up of millions of times!
• Accuracy: The AI was incredibly accurate.
  • For the protons, it was 99.95% accurate compared to the supercomputer.
  • For the neutrons, it was 99.40% accurate.
  • The Analogy: If the supercomputer's prediction was a perfect map, the AI's prediction was a map where you couldn't tell the difference unless you were looking through a microscope.

Why This Matters

This isn't just about being fast; it's about safety and adaptability.

• Real-Time Adjustments: Because the AI is so fast, doctors could potentially use it during the treatment. If the patient moves or breathes differently than expected, the system could instantly recalculate where the beam is going and adjust the dose on the fly.
• Better Protection: By understanding exactly where the neutrons are going, doctors can better protect healthy organs from unnecessary radiation damage.

The Catch (Limitations)

The AI is currently trained on "flat" slices of the body (like looking at a loaf of bread slice by slice). It assumes the left and right sides of that slice are the same. In reality, a human body is 3D and lumpy.

• The Analogy: The AI is great at predicting the weather in a straight line down a hallway, but it's still learning how to predict the weather in a complex, twisting cave. The researchers plan to upgrade the AI to handle full 3D complexity next.

Summary

This paper introduces a super-fast AI assistant for cancer doctors. It replaces a slow, heavy calculation with a quick, smart prediction, allowing for safer, more precise proton therapy that can adapt to the patient's body in real-time. It turns a process that used to take days into one that takes seconds.

Technical Summary

Here is a detailed technical summary of the paper "Fast Proton Transport and Neutron Production in Proton Therapy Using Fourier Neural Operators" by Blangiardi et al.

1. Problem Statement

Proton Therapy (PT) offers superior dose conformity due to the Bragg peak, but its efficacy relies on precise range verification to account for anatomical changes and uncertainties in stopping power.

• The Challenge: Real-time adaptive PT requires accurate, rapid prediction of secondary particle emissions (specifically fast neutrons and prompt gammas) to verify the proton range during treatment.
• Current Limitations:
  • Monte Carlo (MC) Methods: While considered the gold standard for accuracy, MC simulations (e.g., PHITS, TOPAS) are computationally prohibitive for online, real-time clinical workflows.
  • Analytical/Approximate Methods: Traditional fast algorithms often fail in highly heterogeneous geometries or lack the detailed angular and energy resolution required for neutron-based range verification.
  • Existing AI Solutions: Most deep learning approaches focus on spatial dose maps, neglecting the critical joint spatial, energy, and angular distributions of secondary particles necessary for multi-particle range verification systems (like the NOVO project).

2. Methodology

The authors propose a surrogate modeling framework using Fourier Neural Operators (FNO) and Gradient Boosted Trees to predict the evolution of proton phase space and the subsequent production of neutrons.

A. Theoretical Formulation

• Physics Basis: The problem is modeled using the Linear Boltzmann Transport Equation (LBTE) under the Continuous Slowing Down Approximation (CSDA) and Fokker-Planck approximations.
• Reformulation: Instead of solving the full integro-differential equation directly, the authors treat the phantom depth ($z$) as a pseudo-time dimension.
• Operators: Two main operators are learned:
  1. Proton Transport Operator ($G_p$): Maps the incident proton phase space density ($\psi_p$) and geometry ($a$) at depth $k$ to the density at $k+1$.
  2. Neutron Production Operator ($G_n$): Maps the local proton density and geometry to the produced neutron phase space density ($\psi_n$).
• Dimensionality Reduction: To make the problem tractable, the model assumes lateral homogeneity (cylindrical symmetry) within the phantom. The phase space is reduced to 3 dimensions: Radial distance ($R$), Energy ($E$), and Polar angle ($\theta$). Azimuthal angles are treated as statistically independent.

B. Data Generation

• Simulation Engine: PHITS (Particle and Heavy Ion Transport code System) version 3.341.
• Dataset:
  • Derived from a thoracic CT scan (Lung CT Segmentation Challenge).
  • 47 unique material sequences (rays cast through the CT) representing diverse tissue compositions (lung, bone, soft tissue).
  • Primary Energies: 70–250 MeV.
  • Two Noise Levels:
    • ES8: $10^8$ primary protons (higher statistical noise).
    • ES9: $10^9$ primary protons (lower noise, higher fidelity).
• Discretization:
  • Protons: Log-spaced in $R$ and $\theta$ to capture high-density regions; linear in $E$.
  • Neutrons: Coarser binning due to lower particle counts and spread-out distributions.

C. Model Architecture

• Fourier Neural Operators (FNO):
  • Used for $G_p$ and $G_n$ to learn the solution operator of the PDEs.
  • Utilizes Fast Fourier Transforms (FFT) to perform global convolutions in Fourier space, capturing long-range dependencies efficiently.
  • Architecture: 4 Fourier layers with soft-gating skip connections and MLPs.
  • Recursive Strategy: The model predicts step-by-step ($k \to k+1$) but is trained to handle variable step counts ($q$) to mitigate error accumulation over long depths (up to 40 cm).
• Intensity Functionals ($F^p, F^n$):
  • Implemented using Gradient Boosted Trees (XGBoost).
  • Predict the relative reduction in proton intensity and the absolute number of produced neutrons, which are multiplied by the phase space densities to reconstruct absolute particle counts.

3. Key Contributions

1. First FNO Application in Proton Therapy: This is the first study to apply Fourier Neural Operators for fast surrogate modeling of proton transport and secondary neutron production.
2. Joint Distribution Prediction: Unlike previous works focusing solely on dose, this model predicts the full 4D phase space (3D spatial + energy + angle) of both protons and neutrons, which is critical for neutron-based range verification.
3. Auto-regressive Depth Evolution: The framework treats depth as a sequential process, allowing the model to generalize to arbitrary phantom depths without retraining.
4. Robustness to Heterogeneity: The model demonstrates the ability to generalize across complex, patient-derived material sequences (lung, bone, soft tissue) without specific retraining for each geometry.

4. Results

The models were evaluated against MC ground truth using metrics including Probability Density Function-Mean Absolute Error (PDF-MAE), Wasserstein Distance, Gamma Passing Rate ($\gamma_{2mm,2\%}$), and Relative L2 Discrepancy.

• Accuracy:
  • Protons: Achieved an average relative L2 discrepancy of 0.067 and a spatial gamma passing rate of 99.95%.
  • Neutrons: Achieved an average relative L2 discrepancy of 0.137 (on ES8) and 0.121 (on ES9), with a gamma passing rate of 99.40%.
  • Wasserstein Distance: Both tasks showed distances below 1.0% of the domain size, indicating that the "work" required to transport the predicted distribution to the ground truth is less than one bin width.
• Impact of Data Quality: Training on the higher-statistics dataset (ES9, $10^9$ protons) improved neutron prediction metrics by 12–30% compared to ES8, highlighting the sensitivity of neutron modeling to statistical noise.
• Speed:
  • MC Simulation Time: ~14–27 CPU-years for the full dataset.
  • Surrogate Inference Time: ~23.17 seconds per beam (11.15s for protons, 6.67s for neutrons) on a single NVIDIA A100 GPU.
  • Speedup: Several orders of magnitude faster than MC, making it viable for clinical workflows.

5. Significance and Future Work

• Clinical Impact: The proposed surrogate enables the prototyping and operation of real-time range verification systems based on neutron detection, a critical step toward adaptive proton therapy.
• Generalizability: The approach is not limited to neutrons; it can be extended to other secondary emissions (prompt gammas, positron emitters) and tasks like neutron dose estimation.
• Limitations:
  • Lateral Homogeneity: The current model assumes lateral homogeneity within the phantom. Future work aims to extend this to full 3D heterogeneous geometries.
  • Data Cost: Generating high-fidelity training data (ES9) is computationally expensive. Future strategies may involve variance reduction techniques or semi-analytical definitions.
  • Zero-Shot Super-Resolution: The current discretization prevents zero-shot inference at different resolutions, a capability inherent to FNOs that was not explored due to data constraints.

In conclusion, this work bridges the gap between the high accuracy of Monte Carlo simulations and the speed requirements of clinical proton therapy, providing a robust, data-driven tool for next-generation range verification.