Edward W. Larsen

A method for incorporating organ motion due to breathing into 3D dose calculations

A method is proposed that incorporates the effects of intratreatment organ motion due to breathing on the dose calculations for the treatment of liver disease. Our method is based on the convolution of a static dose distribution with a probability distribution function (PDF) which describes the nature of the motion. The organ motion due to breathing is assumed here to be one-dimensional (in the superior-inferior direction), and is modeled using a periodic but asymmetric function (more time spent at exhale versus inhale). The dose distribution calculated using convolution-based methods is compared to the static dose distribution using dose difference displays and the effective volume (Veff) of the uninvolved liver, as per a liver dose escalation protocol in use at our institution. The convolution-based calculation is also compared to direct simulations that model individual fractions of a treatment. Analysis shows that incorporation of the organ motion could lead to changes in the dose prescribed for a treatment based on the Veff of the uninvolved liver. Comparison of convolution-based calculations and direct simulation of various worst-case scenarios indicates that a single convolution-based calculation is sufficient to predict the dose distribution for the example treatment plan given.

Fast iterative methods for discrete-ordinates particle transport calculations

Abstract In discrete-ordinates (S N ) simulations of large problems involving linear interactions between radiation and matter, the underlying linear Boltzmann problem is discretized and the resulting system of algebraic equations is solved iteratively. If the physical system contains subregions that are optically thick with small absorption, the simplest iterative process, Source Iteration, is inefficient and costly. During the past 40 years, significant progress has been achieved in the development of acceleration methods that speed up the iterative convergence of these problems. This progress consists of ( i ) a theory to derive the acceleration strategies, ( ii ) a theory to predict the convergence properties of the new strategies, and ( iii ) the implementation of these concepts in production computer codes. In this Review we discuss the theoretical foundations of this work, the important results that have been accomplished, and remaining open questions.

Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II

In a recent article (Larsen, Morel, and Miller, J. Comput. Phys.69, 283 (1987)), a theoretical method is described for assessing the accuracy of transport differencing schemes in highly scattering media with optically thick spatial meshes. In the present article, this method is extended to enable one to determine the accuracy of such schemes in the presence of numerically unresolved boundary layers. Numerical results are presented that demonstrate the validity and accuracy of our analysis.

UNCONDITIONALLY STABLE DIFFUSION-SYNTHETIC ACCELERATION METHODS FOR THE SLAB GEOMETRY DISCRETE ORDINATES EQUATIONS. PART I: THEORY.

The authors consider the slab geometry discrete ordinates equations, with the weighted diamond, linear characteristic, linear discontinuous, and linear moments spatial differencing schemes. For each differencing scheme we derive a diffusion-synthetic, source-correction acceleration method which, for model (infinite medium, isotropic scattering, constant cross section, constant mesh spacing) problems, unconditionally reduces the spectral radius of the iteration method from the unaccelerated value of c (the scattering ratio) to less than or equal to c/3.

Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering

The multigroup and P{sub 1} and Simplified P{sub N} equations are shown to be a family of asymptotic approximation to the multigroup transport equation with anisotropic scattering. The physical assumptions are that the material system is optically thick, the probability of absorption is small, and the mean scattering angle {anti {mu}}{sub o} is not close to unity.

Benchmark results for particle transport in a binary Markov statistical medium

Abstract We give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process. A Monte Carlo procedure is used to generate a physical realization of the statistics, and a discrete ordinate numerical transport solution is generated for this realization. The ensemble averaged solution, as well as the variance, is obtained by averaging a large number of such calculations. Reflection and transmission results are given for several problems in both rod and planar geometry. In a separate development, two coupled transport equations are derived which formally described transport in a random binary mixture for arbitrary mixing statistics. Closing these equations by approximating their coupling terms in a low order and intuitive way leads to a model for stochastic transport previously obtained via the master equation. The present derivation, based upon approximating exact equations, allows in principle the opportunity to develop more accurate models by making higher order approximations in the coupling terms.

The simplified P3 approximation

Abstract The simplified P3 (SP3) approximation to the multigroup neutron transport equation in arbitrary geometries is derived using a variational analysis. This derivation yields the SP3 equations along with material interface and Marshak-like boundary conditions. The multigroup SP3 approximation is reformulated as a system of within-group problems that can be solved iteratively. An “explicit” iterative algorithm for solving the within-group problem is described, Fourier analyzed, and shown to be more efficient than the traditional FLIP implicit algorithm. Numerical results compare diffusion (P1), simplified P2 (SP2), and simplified P3 calculations of a mixed-oxide (MOX) fuel benchmark problem to a reference transport calculation. The SP3 approximation can eliminate much of the inaccuracy in the diffusion and SP2 calculations of MOX fuel problems.

Computational Efficiency of Numerical Methods for the Multigroup, Discrete-Ordinates Neutron Transport Equations: The Slab Geometry Case

A study of spatial discretization schemes for the multigroup discrete-ordinates transport equations in slab geometry is described. The purpose of the study is to determine the most computationally efficient method, defined as the one that produces the minimum error for a given cost. Cost is defined as the total amount of computer time required to complete one inner iteration, given a limit on storage, and three error norms are used to measure the accuracies of edge fluxes, cell average fluxes, and integral parameters. Three test problems are studied: the first is a model one-group problem examined in detail, while the second and third are more realistic multigroup problems. One conclusion is that a new method, labeled linear characteristic, significantly outperforms all other methods that have been implemented up to the present time. 15 references.

Asymptotic analysis of radiative transfer problems

The equations of radiative transfer are systematically analyzed by asymptotic methods. To lowest order, the classical equilibrium diffusion approximation is recovered. The next order analysis leads to the equilibrium diffusion differential equations and initial condition, but with a boundary condition containing a linear extrapolation distance α. This quantity is related tothe solution of a canonical halfspace problem and is computed by deriving an appropriate variational principle. For the case of no scattering, an exact Wiener-Hopf solution is available. The FN solution technique is also applied to the problem of obtaining α with good results. Higher order asymptotic radiative transfer descriptions are discussed and, while not immediately constituting practical calculational techniques, do have implications for computing the parameters in the multiband treatment of the frequency variable.

Convergence rates of spatial difference equations for the discrete-ordinates neutron transport equations in slab geometry

AbstractThe order of convergence, as the spatial cell widths tend to zero, is derived for six numerical methods that have been proposed for the slab geometry, multigroup, discrete-ordinates neutron transport equations. Our results, which in two cases differ from earlier experimental results, are illustrated by means of a simple test problem.