Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characteristic Sweeps and Source Iteration for Charged-Particle Transport with Continuous Slowing-Down and Angular Scattering

Ben S. Ashby, Alex Lukyanov, Tristan Pryer

TL;DR

This work introduces a semi-analytic, deterministic framework for charged-particle transport that blends method-of-characteristics energy-advection with a fixed-point scattering iteration, enabling explicit directional sweeps while controlling angular coupling. The authors establish a robust variational setting with coercivity and a posteriori error bounds, derive contraction criteria for the continuum and discrete source iterations, and decompose angular errors into quadrature, cone truncation, and iteration effects. They provide detailed angular discretisation theory, conservation properties, and practical stopping rules, then validate the approach through proton and carbon ion simulations, including a data-informed stopping power for heavy ions and a reduced multi-species model for secondary production. The numerical results demonstrate ballistic benchmarks, forward-peaked scattering behavior, and distal tails in carbon transport, underscoring the method’s accuracy, stability, and suitability for planning and uncertainty studies in particle therapy. Overall, the framework offers a scalable, error-controlled alternative to Monte Carlo for forward-propagation problems in therapy-relevant geometries with explicit angular information.

Abstract

We develop a semi-analytic deterministic framework for charged-particle transport with continuous slowing-down in energy and angular scattering. Directed transport and energy advection are treated by method-of-characteristics integration, yielding explicit directional sweeps defined by characteristic maps and inflow data. Scattering is incorporated through a fixed-point (source-iteration) scheme in which the angular gain is lagged, yielding a sequence of decoupled directional solves coupled only through angular sums. The method is formulated variationally in a transport graph space adapted to the charged particle drift. Under standard monotonicity and positivity assumptions on the stopping power and boundedness assumptions on cross sections, we establish coercivity and boundedness of the transport bilinear form, prove contraction of the source iteration under a subcriticality condition and derive a rigorous a posteriori bound for the iteration error, providing an efficient stopping criterion. We further analyse an elastic discrete-ordinates approximation, including conservation properties and a decomposition of angular error into quadrature, cone truncation and finite iteration effects. Numerical experiments for proton transport validate the characteristic sweep against an exact ballistic benchmark and demonstrate the predicted fixed-point convergence under forward-peaked scattering. Carbon-ion simulations with tabulated stopping powers and a reduced multi-species coupling illustrate Bragg peak localisation and distal tail formation driven by secondary charged fragments.

Characteristic Sweeps and Source Iteration for Charged-Particle Transport with Continuous Slowing-Down and Angular Scattering

TL;DR

This work introduces a semi-analytic, deterministic framework for charged-particle transport that blends method-of-characteristics energy-advection with a fixed-point scattering iteration, enabling explicit directional sweeps while controlling angular coupling. The authors establish a robust variational setting with coercivity and a posteriori error bounds, derive contraction criteria for the continuum and discrete source iterations, and decompose angular errors into quadrature, cone truncation, and iteration effects. They provide detailed angular discretisation theory, conservation properties, and practical stopping rules, then validate the approach through proton and carbon ion simulations, including a data-informed stopping power for heavy ions and a reduced multi-species model for secondary production. The numerical results demonstrate ballistic benchmarks, forward-peaked scattering behavior, and distal tails in carbon transport, underscoring the method’s accuracy, stability, and suitability for planning and uncertainty studies in particle therapy. Overall, the framework offers a scalable, error-controlled alternative to Monte Carlo for forward-propagation problems in therapy-relevant geometries with explicit angular information.

Abstract

We develop a semi-analytic deterministic framework for charged-particle transport with continuous slowing-down in energy and angular scattering. Directed transport and energy advection are treated by method-of-characteristics integration, yielding explicit directional sweeps defined by characteristic maps and inflow data. Scattering is incorporated through a fixed-point (source-iteration) scheme in which the angular gain is lagged, yielding a sequence of decoupled directional solves coupled only through angular sums. The method is formulated variationally in a transport graph space adapted to the charged particle drift. Under standard monotonicity and positivity assumptions on the stopping power and boundedness assumptions on cross sections, we establish coercivity and boundedness of the transport bilinear form, prove contraction of the source iteration under a subcriticality condition and derive a rigorous a posteriori bound for the iteration error, providing an efficient stopping criterion. We further analyse an elastic discrete-ordinates approximation, including conservation properties and a decomposition of angular error into quadrature, cone truncation and finite iteration effects. Numerical experiments for proton transport validate the characteristic sweep against an exact ballistic benchmark and demonstrate the predicted fixed-point convergence under forward-peaked scattering. Carbon-ion simulations with tabulated stopping powers and a reduced multi-species coupling illustrate Bragg peak localisation and distal tail formation driven by secondary charged fragments.
Paper Structure (25 sections, 8 theorems, 114 equations, 16 figures, 1 algorithm)

This paper contains 25 sections, 8 theorems, 114 equations, 16 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Charged Particle Transport Models
  3. Stopping power and the Bragg--Kleeman approximation
  4. Data-informed stopping for heavy ions
  5. Boundary conditions
  6. Variational Formulation and Functional Setting
  7. Source iteration at the continuum level
  8. Angular approximation and error control
  9. Angular discretisation
  10. Source iteration
  11. Error control for the angular approximation
  12. Setting and error decomposition
  13. Quadrature consistency on $\mathcal{A}$
  14. Cone truncation
  15. Source-iteration error and stopping
  16. ...and 10 more sections

Key Result

Theorem 3.1

Assume $S\in W^{1,\infty}(I)$ with $S(E)>0$ and $S'(E)\le 0$ on $I$, $\sigma_T\in L^\infty(I)$ with $\sigma_T(E)\ge 0$, and $\sigma_S\in L^\infty(\mathbb{S}^{d-1}\times\mathbb{S}^{d-1}\times I\times I)$ with $\sigma_S\ge 0$. Define the coercivity norm and the stronger norm Assume moreover that so that $\|w\|_{L^2(\Omega)} \le \mu^{-1/2}\|(-S')^{1/2}w\|_{L^2(\Omega)} \le \mu^{-1/2}\|w\|_{\mathca

Figures (16)

  • Figure 1: The three main interactions of a proton with matter. A nonelastic proton--nucleus collision, an inelastic Coulomb interaction with atomic electrons and elastic Coulomb scattering with the nucleus.
  • Figure 2: Left: proton range--energy data in water from PSTAR with Bragg--Kleeman fit $R_{\mathrm{BK}}(E)=\alpha E^p$. Right: stopping power reconstructed from the same range data via $\widehat{S}(E)=(\mathrm dR/\mathrm dE)^{-1}$, obtained by differentiating a cubic spline fit of $R(E)$, together with the Bragg--Kleeman stopping power $S_{\mathrm{BK}}(E)=(\alpha p E^{p-1})^{-1}$ and a Bethe--Bloch stopping power $S_{\mathrm{BB}}(E)$. Data taken from PSTAR berger2005star.
  • Figure 3: Left: carbon range--energy data from ICRU with Bragg--Kleeman fit $R_{\mathrm{BK}}(E)=\alpha E^p$ (energy in MeV/u). Right: stopping power reconstructed from the same range data via $\widehat{S}(E)=(\mathrm dR/\mathrm dE)^{-1}$, obtained by differentiating a cubic spline fit of $R(E)$, together with the Bragg--Kleeman stopping power $S_{\mathrm{BK}}(E)=(\alpha p E^{p-1})^{-1}$ and a Bethe--Bloch-shaped surrogate $S_{\mathrm{BB}}(E)$. Data taken from ICRU icru2005report73.
  • Figure 4: Angular computational domain restricted to a cone centred on $\boldsymbol{\omega}_\star$.
  • Figure 5: Ballistic (degenerate scattering) benchmark. Left: pointwise relative error of the numerical transport solution against the exact characteristic solution \ref{['eq:exact_dirac_solution']}. Right: computed dose field for the same test case.
  • ...and 11 more figures

Theorems & Definitions (19)

  • Theorem 3.1: Coercivity and boundedness of the continuous bilinear forms
  • proof
  • Proposition 3.2: Convergence of source iteration
  • proof
  • Remark 4.1: Continuum conservation
  • Proposition 4.2: Discrete conservation under exact angular quadrature
  • proof
  • Remark 4.3
  • Proposition 4.4: Convergence of discrete-ordinates source iteration
  • proof
  • ...and 9 more