MULTIDIMENSIONAL COUPLED PHOTON-ELECTRON TRANSPORT SIMULATIONS USING NEUTRAL PARTICLE S N CODES

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1 Computational Medical Physics Workin Group Workshop II ep 30 Oct University of Florida (UF) Gainesville Florida UA on CD-ROM American Nuclear ociety LaGrane Park IL (007) MULTIDIMNIONAL COUPLD PHOTON-LCTRON TRANPORT IMULATION UING NUTRAL PARTICL N COD Dan Ilas Mark L. Williams Doulas. Peplow and Bernadette L. Kirk Oak Ride National Laboratory ilasd@ornl.ov; williamsml@ornl.ov; peplowde@ornl.ov; kirkbl@ornl.ov ABTRACT Durin the past two years a study was underway at ORNL to assess the suitability of the popular N neutral particle codes ANIN DORT and TORT for coupled photon-electron calculations specific to external beam therapy of medical physics applications. The CPX-BFP code was used to enerate the cross sections. The computational tests were performed on phantoms typical of those used in medical physics for external beam therapy with materials simulated by water at different densities and the comparisons were made aainst Monte Carlo simulations that served as benchmarks. Althouh the results for one-dimensional calculations were encourain it appeared that the hiher dimensional transport codes had fundamental difficulties in handlin the electron transport. The results of two-dimensional simulations usin the code DORT with an 6 fully symmetric quadrature set aree fairly with the reference Monte Carlo results but not well enouh for clinical applications. While the photon fluxes are in better areement (enerally within less than 5% from the reference) the discrepancy increases sometimes very sinificantly for the electron fluxes. The paper however focuses on the results obtained with the three-dimensional code TORT which had converence difficulties for the electron roups. Numerical instabilities occurred in these roups. These instabilities were more pronounced with the deree of anisotropy of the problem. KYWORD: discrete ordinates electron transport Monte Carlo. INTRODUCTION The use of standard neutral particle discrete ordinates codes to solve the transport of chared particles is hihly desirable because of the wide availability and maturity of such codes. The Boltzmann equation: ˆ ( r ˆ ) ( r ) ( r ˆ ) d d ˆ ( r ˆ ˆ ) ( r ˆ Ω ψ Ω Ω Ω Ω ψ Ω ) Q( r ψ ) () Ω t 0 4π can be used (with appropriate boundary conditions) to find the anular flux distribution throuhout a domain of interest. However in situations typical of electron transport the cross sections are extremely anisotropic with straiht-ahead (close to θ 0 or μ ˆ ˆ 0 Ω Ω ) scatterin cross sections bein many orders of manitude larer than the cross sections at wider s

2 anles (for example for μ 0. 8 ). This creates major difficulties in usin the Leendre expansion formalism typical of N codes because the number of Leendre terms needed to adequately represent the cross section would be hue (~000). A preferred approach to deal with the hihly forward-peaked transport is to use the Boltzmann- Fokker-Planck (BFP) equation which is based on the decomposition of the scatterin kernel into reular (wide anle scatterin) and sinular (forward-peaked scatterin) components: s ( Ωˆ Ω ˆ ) ( Ωˆ Ω ˆ r r ) ( r Ωˆ Ω ˆ ) () s re s sin The reular part is treated by the usual Leendre expansion while the sinular part incurs a different treatment. The BFP equation is thus obtained as Ωˆ ψ T ( r ˆ Ω) [ ( r ) ψ ( r Ωˆ )] ( ) ( ) ( r ˆ ) ( r ˆ ψ Ω ψ Ω) r μ ( r ) ψ ( r Ωˆ ) d 0 4π μ dω ˆ s re μ μ ( r ˆ ˆ ) ( r ˆ Ω Ω ψ Ω ) Q( r ) ϕ Where ( r ) and T ( r ) are the restricted stoppin power and the restricted momentum transfer respectively. They are defined based on the sinular part of the scatterin kernel. The term containin the stoppin power second from left hand side of q. (3) is termed the continuous slowin down (CD) term and the term containin the momentum transfer third from left hand side of q. (3) is the continuous scatterin (C) term. The use of the BFP equation assumes modifications to the existin standard discrete ordinates codes to include the Fokker-Planck terms (enery and anular derivatives in q. 3) and will not be analyzed here. Returnin to the standard discrete ordinates codes one way to reduce the manitude of the scatterin cross section is to separate the sharp peak at μ by modelin it with a Dirac-delta function δ ( μ ). If the Leendre expansion of the cross section is cut at the N th order and the cross section moments are corrected by subtractin the (N) th moment the P N transportcorrected P N expansion is obtained []. The moments so obtained are much smaller in manitude and the corrected scatterin cross section is less forward-peaked. The method used in conjunction with a Gauss quadrature in -D calculations allows the use of a much smaller expansion order (N~0) with very reasonable results. A literature survey on the deterministic approach for electron or coupled photon-electron transport with existin neutral particle Boltzmann solvers evidenced that while a rich literature exists for dealin with one-dimensional cases little work was done for the multidimensional calculations. The seed paper for the one-dimensional calculations can be considered Morel s 98 paper []. Reardin the multidimensional cases one exception is C.R. Drumm s 997 t (3) Computational Medical Physics Workin Group Workshop II ep 30 Oct /8

3 paper [3] which uses a Goudsmit-aunderson approach to prepare multiroup Leendre cross sections appropriate for the standard discrete ordinates codes. The Goudsmit-aunderson is limited to infinite-medium problems and nelects the hard-inelastic (wide anle) scatterin. The present study focuses on evaluatin the TORT [4] discrete ordinates code for electron transport with cross-sections prepared by CPX-BFP [5] a modified version of the CPX code [6].. CRO CTION PRPARATION The CPX-BFP code [5] was contributed to the Radiation afety Information Center (RICC) at ORNL by the Russian Academy of ciences Keldysh Institute of Applied Mathematics. This code has the capability of producin data needed for explicit Fokker-Planck treatment as well as indirect treatments such as in the oriinal CPX (the CPX-G method of ref. [3] is not available). The followin three options are available in CPX-BFP: () N -CD Produces restricted stoppin powers at the multiroup enery boundaries for transport codes that explicitly treat the CD term usin diamond differencin but represents the C anular operator indirectly by Leendre expansion. () N -BFP Produces restricted stoppin powers and restricted momentum transfer coefficients for transport codes that explicitly treat both CD and C differential operators usin finite difference. (3) N -Indirect Produces data for transport codes that indirectly treat both the CD and C operators by includin modified cross sections in the standard multiroup library. This approach is nearly identical to the CPX methodoloy but the results are in a different format. The CPX-BFP code was used to produce electron interaction data for the ORNL transport codes in this project. In the N -indirect option the total and scatterin cross sections for use in the standard discrete ordinates codes are redefined for a P L expansion as: ~ T / t t BFP L( L ) (4) Δ ( ) ( ) ~ T / s l s l BFP [ L( L ) l( l ) ] / δ for (5) Δ where the BFP-labeled terms are the terms that would be used in a BFP treatment (in case of the scatterin kernel it represents the Leendre moment for the reular part of the scatterin crosssection). The sum of the first two terms in qs. (4) and (5) represent the CD cross sections that are enerated with the N -CD option which explicitly uses the stoppin power. The preferred approach in this study however was to use the N -CD cross sections and embed the stoppin power term by post processin in the ARV.5 code [7]. This code uses a Computational Medical Physics Workin Group Workshop II ep 30 Oct /8

4 weihted -step scheme in enery to incorporate the stoppin power. The resultin cross sections that are to be used in the standard codes are defined as: Δ ~ / CD t t P (6) / / / where for ~ Δ Δ CD l s l s P P P (7) The cross sections were enerated on 40 electron roups (and 40 photon roups). 3. RULT 3.. Problem description Calculations were performed on a fully 3-D realistic human phantom provided by the University of North Carolina. The full depth of the phantom (6 voxels 0.4 cm in size) was used alon the y direction but only a reion 0x0 voxels wide was used for the cross sectional x-z plane (Fi. ). Fiure. Cross sectional views throuh the computational phantom To capture the scatterin of the electrons which is assumed to be very forward peaked we used a quadrature with such forward peakin features. This forward biased quadrature had a total of 560 directions pointin forward and 70 directions pointin backwards. The photon source is modeled as a surface source (at y9.576cm on the oriinal phantom i.e. minimum y value) with a spectrum typical of a radiotherapy source and is assumed to emit Computational Medical Physics Workin Group Workshop II ep 30 Oct /8

5 photons within a 30 o cone of directions (5 o with respect to the y axis) in which a lare number of quadrature set directions lay. The source s spatial extension is constrained to a 0 0 voxel square central on the plane perpendicular to the depth (xz plane for the oriinal phantom). This set up allows for testin the behavior of TORT on a more realistic case with a non-isotropic source and a sharp spatial radient in the penumbra reion. 3.. Three dimensional results The Monte-Carlo code Gnrc [8] was used as reference and we compared the ratio TORT/G for the total photon flux and the total electron flux throuhout the phantom. Fiure shows these ratios for a cut plane z5cm. The photon source is at the bottom and impines upwards. Ideally the TORT/G flux ratio should be constant (i.e. solid color). With appropriate normalization between TORT and Gnrc this ratio should be throuhout. However because of the numerical problems in TORT (and to a lesser extent because of the statistical errors in Gnrc) the ratio of the fluxes computed by TORT and Gnrc is not constant. For photons the variation is about 5% in most of the phantom with an underestimation by TORT in reions close to the source s axis and an overestimation towards the periphery of the source beam reion. For the electron fluxes ratio the variation is still around 5% in most of the phantom (with appropriate rescalin the variation within one unique color on Fiure riht is about ±4%) with an underestimation in the central reion. The electron flux ratio decreases in reions which are further away from the source. The most fundamental problem however was that while the ratio for photons was reasonably close to the ratio for electron fluxes was orders of manitude different. Fiure. The ratio TORT/G for the total photon (left) and electron fluxes at z5cm The above scopin results were obtained usin inner iterations per enery roup. The computational time was about half of that used for Gnrc to achieve a reasonable relative error (below 5%). Difficulties in the electron roup s converence were found when we attempted to increase the number of iterations for each enery roup. The solution may divere after a taret number of iterations. Fiure 3 shows that ratio TORT/G for the photon fluxes alon a central depth line. It can be noticed that the curves for and 4 iterations per enery roup practically overlap. Over the whole phantom raisin the number of iterations to 4 for the photon roups leads to chanes Computational Medical Physics Workin Group Workshop II ep 30 Oct /8

6 smaller than % in the flux distribution. When the quadrature set was chaned to the 6 fully symmetric difficulties bean to show up even for the photons because there were too few directions within the 30 o cone (only 4 non-zero weihted directions compared to 440 directions for the biased set) to adequately capture the hihly forward-peaked fluxes. The disareement for the 6 fully symmetric quadrature starts only after a few centimeters from the source. TORT/G ratio iterations iterations 6F ( iterations) y (cm) Fiure 3. The TORT/G ratio for photon fluxes alon a central depth line for three calculations Fundamental converence difficulties are observed for the electron roups when the number of iterations is increased. Fiure 4 exposes a resonance-like increase in the electron flux at some point alon the central depth axis. The 6 fully symmetric set also has difficulties in handlin the electrons TORT/G ratio iterations 4 iterations 6F ( iterations) y (cm) Fiure 4. The TORT/G ratio for electron fluxes alon a central depth line for three calculations Fiure 5 shows how the numerical instability has propaated throuhout the phantom when the number of inner iterations is increased. The red color shows a diverence in electron flux errors probably due to the difficulties in handlin lare spatial radients. Computational Medical Physics Workin Group Workshop II ep 30 Oct /8

7 Fiure 5. The ratio TORT/G for the total electron flux at z5cm In conclusion the tests show that the ORNL s three-dimensional discrete ordinates transport code TORT has fundamental converence difficulties in transportin the electrons with the electron cross sections derived accordin to the CPX methodoloy. Because of this it is not possible to make meaninful comparisons for the 3D electron transport case. 4. CONCLUION Althouh reasonable ood areement was obtained for one-dimensional calculations for both photon and electron transport it appears that the hiher dimensional standard discrete ordinates transport codes have fundamental difficulties in handlin the electron transport. The three dimensional code TORT had fundamental converence difficulties for electrons. On confiuration with an isotropic source the two dimensional code DORT [9] ave results (not shown in this paper) close to the reference Monte Carlo but not enouh for clinical applications. An 6 fully symmetric quadrature set was used for the DORT calculations. It appears that the difficulties stem from the cross section preparation. A deeper insiht into this issue is necessary. The computational time for the discrete ordinates calculations compared well aainst Monte Carlo for the problems considered. It should be noticed that the codes used are eneral purpose codes not specialized for voxelized eometries. pecializin the codes for this type of eometry could lead to sinificant ains in computational speeds. As a final conclusion the explicit (direct) treatment of the Fokker-Planck terms is recommended when dealin with the electron transport. Computational Medical Physics Workin Group Workshop II ep 30 Oct /8

8 ACKNOWLDGMNT The study was partially funded by the National Cancer Institute (NCI) of the National Institutes of Health (NIH) under Grant Number R CA464. The authors want to acknowlede the help of Dr. Charles O. later in settin up the TORT input files. RFRNC. J.. Morel On the Validity of the xtended Transport Cross-ection Correction for Low- nery lectron Transport Nuclear cience and nineerin 7 pp.64-7 (979).. J.. Morel Fokker-Planck Calculations Usin tandard Discrete Ordinates Transport Codes Nuclear cience and nineerin 79 pp (98). 3. C.R. Drumm Multidimensional lectron-photon Transport with tandard Discrete Ordinates Codes Nuclear cience and nineerin 7 pp.- (997). 4. W. A. Rhoades and R. L. Childs The TORT Three-Dimensional Discrete Ordinates Neutron/Photon Transport Code ORNL-668 Oak Ride National Laboratory (987). 5. A. M. Voloschenko CPX-BFP: Version of Multiroup Coupled lectron-photon Cross- ection Generation Code CPX Adapted for olvin the Chared Particle Transport in the Boltzmann-Fokker-Plank Formulation with the Use of Discrete Ordinate Method User s Guide Report No Keldysh Inst. of Appl. Math. Russian Ac. of cience Moscow (004). 6. L. J. Lorence J.. Morel and G. D. Valdez Physics Guide to CPX/ONLD: A Multiroup Coupled lectron-photon Cross ection Generatin Code AND andia National Laboratories (989). 7. A. M. Voloschenko. V. Gukov and A. V. hwetsov ARV-.5: Preprocessor for the Workin Macroscopic Cross-ection FMAC-M Format for Transport Calculations User s Guide Rept. No Keldysh Inst. of Appl. Math. Russian Ac. of cience Moscow (004). 8. I. Kawrakow Accurate Condensed History Monte Carlo imulation of lectron Transport. I. Gnrc the new G4 version Medical Physics (000) 9. W. A. Rhoades and R. L. Childs The DORT Two-Dimensional Discrete Ordinates Transport Code Nuclear cience and nineerin (May 988). Computational Medical Physics Workin Group Workshop II ep 30 Oct /8