A NOVEL RADIATION TRANSPORT ALGORITHM FOR RADIOGRAPHY SIMULATIONS

Feyzi Inanc

Iowa State University, Center for Nondestructive Evaluation, Ames, IA 50011

ABSTRACT. The simulations used in the NDE community are becoming more realistic with the introduction of more physics. In this work, we have developed a new algorithm that is capable of representing photon and charged particle fluxes through spherical harmonic expansions in a manner similar to well known discrete ordinates method with the exception that Boltzmann operator is treated through exact integration rather than conventional Legendre expansions. This approach provides a mean to include radiation interactions for higher energy regimes where there are additional physical mechanisms for photons and charged particles.

INTRODUCTION As with a variety of scientific disciplines, modeling various x-ray physics and simulating radiography procedures have been gaining momentum for some time. Till recently, most of the radiographic modeling has been limited to low energy ranges specific to the x-ray tubes. In that energy range, photoelectric absorption forms most of the interactions. Most of the available radiography simulators neglect other interactions without significant loss of accuracy. When the photon energy window is pushed to levels higher than x-ray tube ranges, limiting the physics to photoelectric absorption starts to loose its adequacy since other interaction mechanisms start to play a larger role than the photoelectric absorption. Along with this, necessity of modeling other physics becomes important. One other reason why this study has been undertaken is that secondary radiation sources become important in some cases. With the increasing energy levels and increasing size of the illuminated object, the secondary radiation source starts to play an important role in the end result. In addition to this, the primary radiation source may also have a volumetric distribution in contrast with the conventional radiography implementations. For example, nuclear fuel material in a reactor core or a radioactive waste drum will have volumetric radiation source distributed throughout the object and that radiation will act as volumetric radiation source. This is in contrast with the point source approximation commonly used in the conventional radiography. The scenarios involving high energy primary radiation source and/or primary/secondary volumetric sources require an algorithm more sophisticated than currently used ray tracing algorithms. Since more sophisticated algorithms come with a computational overhead, any new algorithm gearing up to tackle such problems should be able to utilize the potential offered by the platforms using multiple processors. In this work, we will detail a new algorithm that can handle most of the challenges described above.

GOVERNING EQUATIONS The algorithm is based on a deterministic approach that employs the integral form of the transport equation. Since the high energy cases include charged particle transport coupled to the photon transport, this system can be described by Equation (1). (1)

In Equation (1), γ

ψ stand for the angular flux for photons, e

ψ for electrons, and p

ψ for

positrons. ),,( OrG Eψ describes the boundary condition imposed by an external source

such as an x-ray tube illuminating the object. γq represents a primary photon source distributed through the problem domain. If e and p are used instead ofγ , same term represents volumetric electron and positron primary source. γγ →q represents the secondary volumetric photon source induced by photons. The second index represents the type of the volumetric source and the first one represents the type that drives the source. All three integral transport equations are coupled to each other through these secondary volumetric sources. The boundary conditions, ),,( OrG Eψ , in Equation (1) are to be obtained by using the primary radiation sources external to the problem domain such as a x-ray tube source. The primary volumetric source terms, peq //γ , are to be supplied as input in the same fashion with the boundary conditions. A typical example for such a source can be radioactive isotopes distributed through the object. In contrast with these two terms, the secondary volumetric source terms need to be calculated. The general form of these secondary volumetric sources can be explained through Boltzmann and Fokker-Planck operators as given in Equation (2) [1,2,3].

(2) While the first term in Equation (2) is the Boltzmann operator, the second term is Fokker-Planck operator. The second term is mainly for the charged particles rather than the photons.

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ALGORITHM One of the main features of the algorithm is that three coupled integral equations are to be solved iteratively rather than simultaneously. In that scheme, the photon fluxes are computed by using the photon equation. These photon fluxes are then used to compute volumetric secondary charged particle sources. Once these sources are known, the charged particle transport equations are solved for charged particle fluxes. These fluxes, in turn, will be used to compute the secondary volumetric photon sources that will be substituted into the photon equation. At that point, photon equation will be solved for a second time and the results from that step will be added to the photon fluxes obtained in the first iteration. Since the photon flux diminishes very rapidly with each iteration, charged particle sources will diminish as well. Therefore, we will not solve the charged particle equations for a second time. Another major feature is the transport equations are written for discrete directions rather than continuous angular domain. This is shown in Equation (3) for the generic form of the integral transport. The energy domain is discretized as well. (3) Flux Expansions Photon and charged particle fluxes are to be approximated by spherical harmonics approximation [4, 5]. In that approximation, expansion moments are functions of energy and space. The spherical harmonics add the angular dimension as shown in Equation (4). (4) In this approximation, the moments are determined through a weighted residual scheme utilizing the orthogonality of the spherical harmonics and this is shown in Equation (5). (5) Due to discretization in the angular domain, Equation (5) will use numerical quadratures. While spherical harmonics approximation reduces amount of memory required for storing information, it is also very beneficial in manipulating the source terms into a form where the source terms are expressed as a dot product of two vectors. Parallelization In this work, we adapted an angular decomposition approach for parallelization. Since the overall domain is discretized into finite number of directions, certain groups of directions are assigned to each processor in the platform as shown in figure 1. When each processor is done with its share of directions, it performs its share of the integration described by Equation (5). This is followed by a communication step where each processor trades its part with all processors and those parts are brought together to obtain the moments of the spherical harmonics representation of the fluxes.

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FIGURE 1. Schematic diagram of angular decomposition in two dimensional cylindrical coordinates. Sweeping Algorithm Equation (3) is well positioned for using ray tracing algorithms. The source terms in that equation are to be subjected to line integration. The boundary term is to be attenuated along the same line. Our ray tracing follows the idea used in the discrete ordinates method with some modifications [4]. The sweeping algorithm provides the unknown flux values at the vertices of the voxels shown in figure 2. Line integrals are performed along the direction vectors that end at the vertex point for which the flux is to be computed. Since all the voxels have the same sizes, coordinates of the starting points of the direction vectors and the magnitudes are the same for a given sweep. The boundary flux value at the starting point is computed by exponential integration that uses the flux values at the four vertices of the facet containing the starting point. That value is attenuated all the way to the vertex at which the flux value is sought. Each voxel is assumed to have homogenous material properties and constant source distribution. Since the source term is treated to be constant, it can be integrated analytically to provide the local contribution to the flux. Another assumption used is that the flux is accepted to have a three dimensional exponential behavior in the voxel. The sweeping algorithm provides the flux values at all vertices of the voxels. By using flux values at the vertices, an average flux is computed for each voxel. This value is then inserted into discretized version of Equation (5) to obtain the moment values for the given voxel. Each processor then trades the local component of the moment values with the other processors to obtain the total value of the moments. One important is sue in the sweeping algorithm is to ensure that sweeping direction is consistent with the direction vector. The overall sweeping algorithm is shown in figure 3. FIGURE 2. Voxel and coordinates system used in the computations.

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FIGURE 3. A 2D sweeping diagram for a direction with positive x&y components. Source Computations As seen in Equation (2), secondary source is made up of two terms. While one uses the Boltzmann operator and the other involves Fokker-Planck operator. Due to the spherical harmonics approximation, Fokker-Planck operator term can be manipulated quite easily. The real challenge is with the Boltzmann operator. For this operator, we use a direct integration approach [6,7] that uses spherical harmonics approximation to reduce the source term into a dot product of two vectors. Since there are different types of physical interactions forming the source terms, type of the interaction has an influence on how the direction integration is formulated for the specific interaction. Whether or not there is a correlation between angular distribution and energy spectrum of the photon/particle after an interaction plays an important role in the formulations. This has an influence on the mathematical treatment of the source term. One of these extreme cases is Compton scattering. In that case, angular deflection determines the energy loss. Therefore, scattering kernel includes a delta function as seen in Equation (6). After a coordinate transformation where the final direction is polar direction, we obtain Equation (7). With the integration over polar angle, we obtain Equation (8). At that point, spherical harmonics form of the flux is substituted into Equation (8) to obtain Equation (9).

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(9) If integration over energy in Equation (9) is discretized through a numerical quadrature and then order of summations in that equation is rearranged, we obtain Equation (10). (10) Some terms in Equation (10) are given by the following expressions. Integrations in the above expressions contain the Jacobians of the coordinate transformations [8]. We do not have an open form for expressing the original coordinates in the new coordinates. Therefore, we carry out that integration by using a quadrature. This quadrature can easily be obtained by using the original coordinates rather than the coordinate system where the final direction is the polar direction. Another case where there is no correlation between the angular deflection and the energy loss is pair production. In this case, incident photon gives rise to the birth of a pair of charged particles. In this case, angular distribution of the charged particles and their energy distribution are not correlated to each other. Electron source term driven by pair production is given by Equation (11). If the charged particle production cross section is substituted by the full expression, we obtain Equation (12). Following a coordinate transport similar to the one in the previous case and discretization of the integrations, we end up with Equation (13). As seen in this equation, its form is quite similar to Compton scattering case where there are two vectors. While the column vector is computed only once before each sweep, the row vector contains the moments of the spherical harmonics expansion. (11)

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RESULTS In this article, we outlined the new algorithm that we have developed for solving photon-charged particle radiation problems that we come across in the radiography simulations, shielding problems and dosimetry where the energy range is higher than the x-ray tube based radiography and there might be volumetric radiation sources inside the object. Although the computer codes based on the algorithm outlined above are not complete yet, a significant portion of the coding has been completed. The first part of the coding that involves photon transport has been completed. The part that will be used for computing volumetric distribution of the charged particle sources has been completed as well. The next step in the coding efforts is to implement the radiation transport part for the charged particles. The algorithm for that part will be basically the same fo r the photon radiation part. While the algorithm for the charged particles is similar to the photon transport, charged particle part will need a iteration convergence scheme in place to speed up the convergence. Since the charged particles go through very small energy losses through interactions, a typical source iteration to solve such a problem would not be sufficient. In addition, we should also device an approach that will be used for decomposing charged particle scattering cross sections into two parts where one will be used by the Boltzmann term while the other is used by the Fokker-Planck term. Once we are done with the charged particle part, the next step is to compute volumetric photon source driven by the charged particle fluxes. ACKNOWLEDGMENTS This manuscript has been authored by Iowa State University of Science and Technology under Contract No. W-7405-ENG-82. REFERENCES 1. Franke B.C., Larsen E.W., “Radial Moment Calculations of Coupled Electron-

Photon Beams”, Nucl. Sci. Eng., 140, pp.1-22, 2002. 2. Börgers C., Larsen E.W., “On the Accuracy of the Fokker-Planck and Fermi Pencil

Beam Equations for Charged Particle Transport”, Med. Phys., 23, 1996. 3. Morel J. E., “Fokker-Planck Calculations Using Standard Discrete Ordinates

Transport Codes”, Nucl. Sci. Eng., 79, pp. 340-356, 1981. 4. Lewis, E.E. and Miller, W.F. Jr., Computational Methods of Neutron Transport,

American Nuclear Society, 1993, Illinois, pp. 116-203. 5. Bell, G.I. and Glasstone, S., Nuclear Reactor Theory, Van Nostrand Reinhold

Company, 1970, New York, pp.129-243. 6. Inanc F. and Gray J.N., “Scattering Simulations in Radiography” Int. J. Appl.

Radiat. Iso., 48 (10),1997. 7. Inanc F., “Analysis of X-Ray and Gamma Ray Scattering Through Computational

Experiments”, Journal of Nondestructive Evaluation, 18 (2), 1999. 8. Kaplan W., Advanced Calculus, Addison-Wesley Publishing, 1984, Massachusetts,

pp.238-243. 9. Wylie, C.R. and Barrett, L. C., Advanced Engineering Mathematics, McGraw-Hill

Book Company, 1982, New York, pp. 625-638.