arXiv subjects

Christian Bäumer

Publications and source records attributed to Christian Bäumer.

3 recordsLinked to original sources

arXiv2026

Realizability-preserving finite element discretizations of the $M_1$ model for dose calculation in proton therapy

We present a deterministic framework for proton therapy dose calculation based on finite element discretizations of the energy-dependent $M_1$ moment model. The nonlinear $M_1$ system is derived from the Fokker--Planck equation for charged particles and closed using an entropy-based approximation of the second moment. Energy is treated as a pseudo-time coordinate. The zeroth and first moments of the proton fluence are evolved backward in energy. To ensure hyperbolicity and physical admissibility, we employ a monolithic convex limiting (MCL) strategy. Representing the standard continuous Galerkin discretization in terms of auxiliary `bar' states, we construct a nonlinear scheme that is provably invariant domain preserving (IDP) w.r.t. convex realizable sets consisting of all admissible states. The realizability of the bar states is enforced using the MCL technology for homogeneous hyperbolic systems. The forcing induced by stiff scattering is incorporated using Strang-type operator splitting. We use an explicit strong-stability-preserving Runge--Kutta method for the radiation transport subproblem and exact integration in the forcing steps, which guarantees the IDP property. The deposited dose is defined as the integral of a weighted zeroth moment over a bounded energy range. It is accumulated during the backward-in-energy evolution. Numerical experiments demonstrate that the proposed Strang-MCL method produces accurate and physically consistent dose distributions.

math.NA↗

arXiv2025

Realizability-preserving monolithic convex limiting in continuous Galerkin discretizations of the M1 model of radiative transfer

We discretize the $M_1$ model of radiative transfer using continuous finite elements and propose a tailor-made monolithic convex limiting (MCL) procedure for enforcing physical realizability. The $M_1$ system of nonlinear balance laws for the zeroth and first moments of a probability distribution function is derived from the linear Boltzmann equation and equipped with an entropy-based closure for the second moment. To ensure hyperbolicity and physical admissibility, evolving moments must stay in an invariant domain representing a convex set of realizable states. We first construct a low-order method that is provably invariant domain preserving (IDP). Introducing intermediate states that represent spatially averaged exact solutions of homogeneous Riemann problems, we prove that these so-called bar states are realizable in any number of space dimensions. This key auxiliary result enables us to show the IDP property of a fully discrete scheme with a diagonally implicit treatment of reactive terms. To achieve high resolution, we add nonlinear correction terms that are constrained using a two-step MCL algorithm. In the first limiting step, local bounds are imposed on each conserved variable to avoid spurious oscillations and maintain positivity of the scalar-valued zeroth moment (particle density). The second limiting step constrains the magnitude of the vector-valued first moment to be realizable. The flux-corrected finite element scheme is provably IDP. Its ability to prevent nonphysical behavior while attaining high-order accuracy in smooth regions is verified in a series of numerical tests. The developed methodology provides a robust simulation tool for dose calculation in radiotherapy.

math.NA↗

arXiv2022

Characterization of pixelated silicon detectors for daily quality assurance measurements in proton therapy

The advanced imaging and delivery techniques in proton therapy allow conformal high-dose irradiation of the target volume with high accuracy using pencil beam scanning or beam shaping apertures. These irradiation methods increasingly include small radiation fields with large dose gradients, which require detector systems with high spatial resolution for quality assurance. In addition the measurement of all success parameters for daily quality assurance with only one proton field and one simple detector system would save a lot of time in clinical usage. Based on their good spatial resolution and high rate compatibility, pixelated silicon detectors could meet the new requirements. To assess their applicability in proton therapy, ATLAS pixelated silicon detectors are used to measure the lateral beam profile with high spatial resolution. Furthermore, a dose dependent detector calibration is presented to allow the measurement of the requested output constancy. A strategy to verify the proton energy during the daily quality assurance is under study. Promising results from proof-of-principle measurements at the West German Proton Therapy Centre in Essen, Germany, have been obtained.

physics.med-ph↗