ISSUES WITH THE HIGH ENERGY RADIOGRAPHYSIMULATIONS

FeyziInane

Iowa State University Center for Nondestructive Evaluation Ames, IA 50010, USA

ABSTRACT. Although most of the industrial radiography relies on x-ray tubes, in some cases,gamma rays or linear accelerators form the interrogating radiation beam. Due to high energylevels of photons in such cases, new physics start to accompany radiation absorption, scatteringevents. Most of these new physics are interactions of charged particles produced by high-energyphotons. Charged particles with sufficient energy can produce photons. High-energyradiography simulations need to account for both photon and charged particle transport forproper computation of photon fluxes. This work will outline various physical mechanisms thatneed to be taken into consideration and challenges that come with designing deterministicalgorithms that can handle the mathematics related to these new physical mechanisms.

INTRODUCTION

Although industrial radiography and medical diagnostic radiography employ x-raytubes as the interrogating beam sources, there are other cases where the radiation source isnot a x-ray tube but either a radioisotope emitting high-energy photons or a linearaccelerator tube. In such cases, the photon beam will be composed of either multiplehigh-energy photon lines or a spectrum that can go from a few keV to many MeV levels. Inthe x-ray tube based implementations, the photon energy levels dictate that only photonprobable interaction mechanisms are photoelectric absorption, coherent scattering andincoherent scattering. Since the primary photons are already low energy photons, anysecondary photon resulting from such interactions will not be instrumental in the outcomeof the radiography because of their very low energy levels and relatively small quantities.With the increase in the primary photon energy levels, another photon interactionmechanism, pair production, becomes possible and photoelectric absorption loses itsdominance over the other interaction mechanisms. One byproduct of this trend isincreasing amount of energy transferred to charged particles through scattering, pairproduction and photoelectric absorption interactions. Since amount of energy transferredto the charged particles in terms of kinetic energy is instrumental in production ofsecondary photons, secondary photons start to increase both in quantity and importance.The major difference between the primary photons and secondary photons is thatsecondary photons are less energetic compared to the primary photons. Since the photondetector usually are more sensitive to lower energy photons than the higher energy ones,estimation of secondary photon fluxes incident upon the detectors becomes an importantissue in the high-energy radiography. In this article, we will provide some insight into theissues that are important in developing a deterministic algorithm that can be used insimulating high-energy radiography procedures.

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LITERATURE REVIEW

Early Monte Carlo charged particle transport computations started 1960's [1].Monte Carlo computations were used for photon transport even before that date [2]. Afterthose initial attempts, Monte Carlo based transport calculations became widely availablewith the advent of computers. Today, there are many general purpose Monte Carlo codesthat can be used for various forms of transport calculations. Some of the well-knowngeneral-purpose Monte Carlo codes are MCNP [3,4] and EGS4 [5]. These codes or theirderivatives are routinely used in various transport calculations and dosimetry. In contrastwith the Monte Carlo methods, research for development of deterministic transport toolsfor charged particle transport started much later and gained speed in the recent years. In1967, Zerby et. al. provided an extensive review of the electron transport theory and someexperimental and computational results [6]. Most of the deterministic methods use discreteordinates method. The first one has been reported by Bartine et.al. in early 1970's [7,8]and they implemented an extended transport correction for handling forward peakingscattering cross section. This has been followed up by a series of articles reportingimplementation of Fokker-Planck operators [9-15]. Following these initial reports, therehave been many reports studying various aspects of the Boltzmann-Fokker-Planckequations used in the charged-particle transport. Some of the later ones have beenimplemented to solve the pencil beam or collimated beam problems that are typical tomedical radiotherapy implementations [16-18].

MOTIVATION

Detection efficiency is quite important in radiography for improving spatialresolution. Regardless of the detector type, detection efficiency is a strong function of thephoton energy. Since higher energy levels result in lower interaction rates, detectors willdetect the incoming photons with lower efficiency levels. As a result, photon detectionefficiency is much better for lower energy levels because of higher interaction coefficients.Very energetic photons transfer significant amounts of energy to charged particles emittedas a result of photon object interactions. These charged particles, in turn, generatesecondary photons. These photons are likely to have energy levels low compared to theprimary photons that caused emission of charged particles. Introduction of thesesecondary photons into the overall photon spectrum is likely to cause distortions in thelower portions of the overall photon spectrum. A typical example of such a case isprovided by Takeuchi and et. al. [19]. One of the problems they worked on is made up ofa 10 cm. thick lead plate with 6.2 MeV photons incident on one side. They provide thephoton flux spectrum on the other side of the lead plate with and without bremsstrahlungphotons. According to their finding, two cases provide similar results above 3 MeV.Around 1 MeV, case with bremsstrahlung photons provides a spectrum that is about 60%larger than the case without bremsstrahlung. Those numbers are about 40, 30, 15 and 10 %for 1.5, 2, 2.5 and 3 MeV respectively. In another problem, they study the photon fluxinside the lead shield. In this case they analyze the flux at 4 and 10 mean free path (mfp)distances from the boundary with a 8 MeV photon source. The differences at 4 mfp are500, 350, 220, 100 and 50 % at 1, 1.5, 2, 2.5 and 3 MeV respectively. There is not muchdifference above 5 MeV. For 10 mfp case, these numbers are about 160, 100, 50, 25 and10 % for the same energy levels. As seen from these numbers, introduction of chargedparticle transport is likely to introduce significant amounts of secondary photons at energylevels where the detectors are more sensitive. For lower atomic number materials, levelsof the secondary photon fluxes are likely to be much lower than the ones given for lead.

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GOVERNING EQUATIONS AND ISSUES

One of the major differences between x-ray tube and high energy radiographymathematical formulation is that high-energy radiography requires a set of simultaneoustransport equations representing photon transport and charged particle transport rather thanonly photon transport. If we use the integral transport equation notation, we can writethese equations for a high-energy photon source as given below.

R" KTRT -l^(r-R"Q,E)clR" -JZ* (r-R"Q,E)dR"

Ik(r,E,G)=\qk(r-KQ,E,Q)e0 dR+Ik(rr,E,O)e ° k = 1,2,3 (1)

where Ik(r,E,Q) is photon, electron and positron flux respectively for k=l,2,3. Since theinterrogating radiation in radiography is usually an external photon source, the second termin the right hand side of equation (1) will be nonzero only for k=l. The source term inequation (1) is given in equation (2).

(2)

The first term in the RHS of equation (2) is Boltzmann operator, the second is continuesangular deflection term and the third is continues slowing down operator. The fourth termis the source term induced by the other types of particles. The second and third terms areusually known as Fokker-Planck operators used for cases where the scattering is stronglyforward peaked and energy loss per interaction is very small. Photon transport equation inequation (1) does not have Fokker-Planck operators. The fourth term in the RHS ofequation set (1) provides the coupling among three equations. Photons contribute toelectron flux through photoelectric absorption, incoherent scattering and pair production.In the photoelectric absorption, photoelectron carries out the photon energy minus bindingenergy. In the incoherent scattering, recoiled electron receives the energy that scatteredphoton looses. In the pair production interaction, electron and positron receive their sharesof energy that is left over after the pair production energy is taken off from the incidentphoton. This is the only mechanism that causes a positron flux. Electrons contribute tophoton flux through bremsstrahlung and fluorescence radiation following an ionizationevent. Positrons contribute to photon flux though bremsstrahlung, fluorescence andannihilation processes. All these mechanisms form the source term in the RHS of equation(2). While photons and charged particles interact with the matter and contribute the fluxesof other types, some of the interaction mechanisms that were not listed here contribute tothe flux of its own particle or photon type.

Photon Transport Issues

The major issue with the high-energy photon transport is severity of the anisotropyseen in the incoherent scattering. With the level of anisotropy shown in figure 1, thesampling of the scattering kernel becomes quite challenging. The conventional Legendrepolynomials approximation used in most of the deterministic methods face convergencedifficulties for scattering kernels that are highly anisotropic for such cases.

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Scattering Angle

FIGURE 1. Behavior of incoherent scattering kernel with incident photon energy levels.

Issues With Charged Particles

One of the issues that makes charged particle transport different than the photontransport is amount of energy lost through charged particle interactions. While photonscan slow down to photoelectric absorption dominant energy levels through a few scatteringevents with relatively large energy losses, charged particles slow down MeV levels to a fewkeV energy levels only after thousands of interactions. Since energy groups can not bechosen that small with the current computational power levels, conventional multigroupapproach is insufficient to represent this type of energy loss. The continuous slowingdown operator in equation (1) is meant to handle this energy loss process. Through thatoperator, charged particle is assumed to loose energy continuously. Continuous slowingdown term requires scattering kernels to be decomposed into two components where oneof the components represent the cases where the energy loss is relatively large enough tocause ionization and the other component represents the cases where the energy loss issmall. The slowing down operator is constructed on the basis of the component where theenergy losses are assumed to be small. The other component of the scattering kernel iskept with the Boltzmann integral operator. The stopping power term, Sk(E), in equation(2) is given below [20].

£,

S(E) = fa (3)

where ass(E—>E',jio) represents scattering interactions with small energy losses. Thedecomposition of charged particle scattering kernels into two components is not a uniqueprocess and it is an issue that needs to be dealt carefully. Momentum transfer, a^E), incontinues angular deflection term in equation (2) is given by equation (4) [20].

cKE) = (4)

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Computational Issues

The most important issue in the computational aspect of the photon-chargedparticle transport is how to solve three integral equations simultaneously. Solving these ina simultaneous manner with the current computational resources poses a major difficulty.Luckily, it is possible to be able separate these three equations from each other andimplement an iterative algorithm that would provide the answer sought in thecomputations. Usually, the nature of the problem dictates the character of the iterativealgorithm. In this specific case, an interrogating photon beam induces charged particlefluxes. Those charged particle fluxes, in return, end up contributing to the photon flux.This secondary photon flux contributed by the charged particle interactions is quite smallcompared to the primary photon flux. Therefore, any charged-particle flux resulting fromthe interactions of the secondary flux will be negligibly small. Therefore, all we need tocompute is the primary photon flux, primary charged particle fluxes and then secondaryphoton flux. Therefore, the iteration will start with solving the photon transport equationby using the external source. Once the photon fluxes are computed, electron and positronsource distributions through the problem domain will be calculated. Electron and positroncharged particle transport equations will be solved by using these source distributions.Once the charged particle fluxes are known, they will be used for calculatingbremsstrahlung, annihilation and fluorescence photon source distributions. Secondaryphoton flux will be done based on these secondary photon source distributions. Theresulting photon fluxes will be obtained by summing the primary and secondary photonfluxes.

Another issue with the photon problem is sampling of the incoherent scatteringkernel. As it is seen in figure 1, mean scattering angle at high energy levels is very small.To avoid any potential sampling problems, we plan to use a dynamic approach to theapproximation of scattering integral by quadratures. The scattering angle will be dividedinto intervals and a different quadrature will be used in each interval. Partitioning of thescattering angle domain will vary with the energy levels. Since a direct integrationapproach will be used for solving the integral transport equations, this approach shouldprove to be efficient enough to sample scattering kernel appropriately.

Another issue that needs is attention is presence of slowing down terms in the RHSof the charged particle transport equations. Photons do not gain energy through scattering.Therefore, the usual approach in the photon transport is to start with the highest energygroup equation and then work down to the lowest energy group equation. Because of theslowing down terms in the charged particle transport, this approach is not feasible. In themultigroup approximation, the slowing down term and the flux is to be expressed in termsof the slowing down and flux terms in the higher and lower energy groups. This generatesa strong coupling between adjacent group equations. Therefore, instead of a top downapproach, either a simultaneous solution technique or an iterative technique should beadapted for solving the energy group equations.

Another issue that is not very visible with the photon transport is the convergenceof the scattering source iterations. The scattering source iteration is a very popularapproach in deterministic transport based algorithms. This approach is very efficient insolving the photon transport equation. Because high-energy photons can escape the objectboundaries or slow down to photoelectric absorption dominant energy levels very rapidlythrough large energy loss incoherent scattering, scattering source iteration converges veryrapidly. In contrast with the photon transport, energy loss in the charged particle transportis usually very small. In addition, charged particles do not travel far and escape from theboundaries is minimal. Therefore, the convergence of scattering source surfaces as a majorissue in the computations. This issue has been studied for discrete ordinates methodsexhaustively and various algorithms developed for enhancing the convergence rate of

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scattering source iterations are reviewed in an article [21]. Although that article addressesthe issue for discrete ordinates method, facing similar difficulties with integral transportbased methods should not be surprising.

Data Issues

Gathering and arranging data for high-energy radiography computations is animportant issue. Although data for all types of interactions are available in the literature,they need to be organized into proper forms for utilization in the computations. In thephoton transport, the most crucial data are scattering and pair production cross sectiondata. Since coherent scattering and binding effects are negligible for such high energylevels, Klein-Nishina formula can provide incoherent scattering cross sections. Pairproduction and photoelectric absorption cross sections are readily available in the literature[22].

Most of the charged-particle interaction cross sections are described in a report byLorence et al. [23]. We hope to make use of that document in generation of the crosssection to be used in the computations.

CONCLUSIONS

As seen from the discussions in the previous sections, high-energy radiography isradically different than the x-ray tube based radiography. Therefore, it is quite difficult toadapt the x-ray tube based radiography simulation codes to high-energy radiography caseswith small modifications. One reason for such a radical shift is that number of physicalinteractions in the high-energy regimes is much larger than the physical interactions seenin the lower energy levels. Therefore, simulation codes should be able to accommodate thephysics introduced at higher energy levels. In addition, charged particles play a role in thehigher energy levels. Although bremsstrahlung generation starts to dominate charged-particle physics at higher levels, it may still have enough contribution to the overall photonflux to distort the flux in the lower end of the spectrum where the detectors are moresensitive to the incoming radiation. One other thing that needs to be kept in mind thatthere are three transport equation in high energy radiography simulations in contrast withthe one transport equation in the x-ray tube radiography cases. This increasescomputational efforts significantly. Therefore, quick and easy solutions should not beexpected for high-energy radiography simulations. Such simulations can be performedonly with considerable computational resources that may include multiple processorplatforms. Such computational resources may not be available to typical users who areplanning to simulate high-energy radiography scenarios.

ACKNOWLEDGMENTS

This manuscript has been authored by Iowa State University of Science andTechnology under Contract No. W-7405-ENG-82.

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