Modelling the transport of ionizing radiation using the finite element method

Boman, Eeva; Tervo, Jani; Vauhkonen, Marko

doi:10.1088/0031-9155/50/2/006

Cited by 24 publications

(34 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Finite element methods have been investigated in the angular domain to yield improved approximations and mitigation of ray-effects (Miller et al 1973, Briggs et al 1975, Martin and Duderstadt 1977, Martin et al 1981, Boman et al 2005, Merton et al 2008, Kophazi and Lathouwers 2015. There exists an important interdependence between discontinuous angular and spatial finite elements.…”

Section: Introductionmentioning

confidence: 99%

A novel upwind stabilized discontinuous finite element angular framework for deterministic dose calculations in magnetic fields

et al. 2018

View full text Add to dashboard Cite

Angular discretization impacts nearly every aspect of a deterministic solution to the linear Boltzmann transport equation, especially in the presence of magnetic fields, as modeled by a streaming operator in angle. In this work a novel stabilization treatment of the magnetic field term is developed for an angular finite element discretization on the unit sphere, specifically involving piecewise partitioning of path integrals along curved element edges into uninterrupted segments of incoming and outgoing flux, with outgoing components updated iteratively. Correct order-of-accuracy for this angular framework is verified using the method of manufactured solutions for linear, quadratic, and cubic basis functions in angle. Higher order basis functions were found to reduce the error especially in strong magnetic fields and low density media. We combine an angular finite element mesh respecting octant boundaries on the unit sphere to spatial Cartesian voxel elements to guarantee an unambiguous transport sweep ordering in space. Accuracy for a dosimetrically challenging scenario involving bone and air in the presence of a 1.5 T parallel magnetic field is validated against the Monte Carlo package GEANT4. Accuracy and relative computational efficiency were investigated for various angular discretization parameters. 32 angular elements with quadratic basis functions yielded a reasonable compromise, with gamma passing rates of 99.96% (96.22%) for a 2%/2 mm (1%/1 mm) criterion. A rotational transformation of the spatial calculation geometry is performed to orient an arbitrary magnetic field vector to be along the z-axis, a requirement for a constant azimuthal angular sweep ordering. Working on the unit sphere, we apply the same rotational transformation to the angular domain to align its octants with the rotated Cartesian mesh. Simulating an oblique 1.5 T magnetic field against GEANT4 yielded gamma passing rates of 99.42% (95.45%) for a 2%/2 mm (1%/1 mm) criterion.

show abstract

Section: Introductionmentioning

confidence: 99%

A novel upwind stabilized discontinuous finite element angular framework for deterministic dose calculations in magnetic fields

et al. 2018

View full text Add to dashboard Cite

show abstract

“…Hence there exists c > 0 (namely c = min{κ, 1 2 }) such that B(ψ, ψ) c ψ 2 We have the following sufficient algebraic criterion for the coercitivity assumption which has a physical background ( [4,Appendix], cf. also [10] for one species of particles).…”

Section: Variational Form Of Equations Existence Of Solutionsmentioning

confidence: 99%

On coupled Boltzmann transport equation related to radiation therapy

Tervo

2007

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We consider a system of Boltzmann transport equations which models the charged particle evolution in media. The system is related to the dose calculation in radiation therapy. Although only one species of particles, say photons is invasing these particles mobilize other type of particles (electrons and positrons). Hence in realistic modelling of particle transport one needs a coupled system of three Boltzmann transport equations. The solution of this system must satisfy the inflow boundary condition. We show existence and uniqueness result of the solution applying generalized Lax-Milgram Theorem. In addition, we verify that (in the case of external therapy) under certain assumptions the "incoming flux to dose operator" D 1 is compact. Also the adjoint D * 1 is analyzed. Finally we consider the inverse planning problem as an optimal control problem. Its solution can be used as an initial solution of the actual inverse planning problem.

show abstract

“…Results with similar accuracy as Monte Carlo calculations were achieved in promising computation times. A finite-element discretization of the transport equation was proposed in [8].…”

Section: Introductionmentioning

confidence: 99%

Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation

Berthon

Frank

Sarazin

et al. 2011

Commun. comput. phys.

View full text Add to dashboard Cite

The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method.

show abstract

Modelling the transport of ionizing radiation using the finite element method

Cited by 24 publications

References 21 publications

A novel upwind stabilized discontinuous finite element angular framework for deterministic dose calculations in magnetic fields

A novel upwind stabilized discontinuous finite element angular framework for deterministic dose calculations in magnetic fields

On coupled Boltzmann transport equation related to radiation therapy

Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation

Contact Info

Product

Resources

About