Jump stochastic differential equations for the characterisation of the Bragg peak in proton beam radiotherapy

TL;DR

This work introduces a first-principles stochastic framework for proton transport in tissue based on a jump stochastic differential equation indexed by track length, capturing energy loss, elastic and non-elastic interactions, and angular diffusion. It extends the classical Bragg peak concept to 2D/3D via the Bragg manifold and Bragg surface, defined through a path functional U and a density for the resolvent, and links the model to Bethe–Bloch stopping power. The authors show one-dimensional consistency with the Bortfeld Bragg-curve formulation, derive a diffusive approximation, and demonstrate Monte Carlo-style 3D simulations that reproduce Bragg-peak behavior and its higher-dimensional generalizations. The framework provides a robust mathematical basis for calibration against nuclear data and for solving inverse problems in treatment planning and dose verification, potentially enabling integration with Geant4 and inverse Bayesian methods. Overall, the paper lays a theoretical foundation for high-dimensional, probabilistic proton energy-deposition modelling with practical pathways to data-driven calibration and clinical application.

Abstract

Proton beam radiotherapy stands at the forefront of precision cancer treatment, leveraging the unique physical interactions of proton beams with human tissue to deliver minimal dose upon entry and deposit the therapeutic dose precisely at the so-called Bragg peak, with no residual dose beyond this point. The Bragg peak is the characteristic maximum that occurs when plotting the curve describing the rate of energy deposition along the length of the proton beam. Moreover, as a natural phenomenon, it is caused by an increase in the rate of nuclear interactions of protons as their energy decreases. From an analytical perspective, Bortfeld proposed a parametric family of curves that can be accurately calibrated to data replicating the Bragg peak in one dimension. We build, from first principles, the very first mathematical model describing the energy deposition of protons. Our approach uses stochastic differential equations and affords us the luxury of defining the natural analogue of the Bragg curve in two or three dimensions. This work is purely theoretical and provides a new mathematical framework which is capable of encompassing models built using Geant4 Monte Carlo, at one extreme, to pencil beam calculations with Bortfeld curves at the other.

Figures (4)

• Figure 1: The three main interactions of a proton with matter. An elastic Coulomb scattering (top) with the nucleus, a non-elastic proton-nucleus collision (centre), and an inelastic Coulomb interaction with atomic electrons (bottom).
• Figure 2: An illustration of the Bragg Peak, $D(z)$, which plots energy deposition per unit cm against depth of proton beam into patient's tissue. In clinical settings, typical proton beams of energies range from 200 to 250 MeV are required to achieve the range of the order of 20-30 cm in tissue for treating deep-seated tumors.
• Figure 3: The numerical functional we use for $10^3\sigma_{\rm ne}$ used in the simulations.
• Figure 4: (left) Realisation of $10^6$ proton paths in 3D projected in to 2D with the height map, with height on the log scale in base 10, representing the resulting value of the Bragg surface $(x,y)\mapsto\int\mathtt{B}(x,y,z){\mathrm d} z$. (centre) a 2D slice through the 3D height map of the stopping power $(x,y)\mapsto\mathtt{B}(x,y, 0)$ for the same simulated protons beam, with again with height on the log scale (base 10), added according to stopping power. (right) The projection of the Bragg surface onto the $x$-axis (this time on the linear scale), $x\mapsto \int\int\mathtt{B}(x,y, z){\mathrm d} y{\mathrm d} z$, giving a classical rate of energy deposition resembling a Bragg peak.

Theorems & Definitions (4)

• proof
• proof
• proof : Proof of Theorem \ref{['diffusivethrm']}
• proof : Proof of Theorem \ref{['density']}