1、Designation: E2232 16Standard Guide forSelection and Use of Mathematical Methods for CalculatingAbsorbed Dose in Radiation Processing Applications1This standard is issued under the fixed designation E2232; the number immediately following the designation indicates the year oforiginal adoption or, in
2、 the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This guide describes different mathematical methodsthat may be used to calculate ab
3、sorbed dose and criteria fortheir selection. Absorbed-dose calculations can determine theeffectiveness of the radiation process, estimate the absorbed-dose distribution in product, or supplement or complement, orboth, the measurement of absorbed dose.1.2 Radiation processing is an evolving field and
4、 annotatedexamples are provided in Annex A6 to illustrate the applica-tions where mathematical methods have been successfullyapplied. While not limited by the applications cited in theseexamples, applications specific to neutron transport, radiationtherapy and shielding design are not addressed in t
5、his docu-ment.1.3 This guide covers the calculation of radiation transportof electrons and photons with energies up to 25 MeV.1.4 The mathematical methods described include MonteCarlo, point kernel, discrete ordinate, semi-empirical andempirical methods.1.5 This guide is limited to the use of genera
6、l purposesoftware packages for the calculation of the transport ofcharged or uncharged particles and photons, or both, fromvarious types of sources of ionizing radiation. This standard islimited to the use of these software packages or other math-ematical methods for the determination of spatial dos
7、e distri-butions for photons emitted following the decay of137Cs or60Co, for energetic electrons from particle accelerators, or forX-rays generated by electron accelerators.1.6 This guide assists the user in determining if mathemati-cal methods are a useful tool. This guide may assist the user insel
8、ecting an appropriate method for calculating absorbed dose.The user must determine whether any of these mathematicalmethods are appropriate for the solution to their specificapplication and what, if any, software to apply.NOTE 1The user is urged to apply these predictive techniques whilebeing aware
9、of the need for experience and also the inherent limitations ofboth the method and the available software. Information pertaining toavailability and updates to codes for modeling radiation transport, courses,workshops and meetings can be found in Annex A1. For a basicunderstanding of radiation physi
10、cs and a brief overview of methodselection, refer to Annex A3.1.7 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica
11、-bility of regulatory requirements prior to use.2. Referenced Documents2.1 ASTM Standards:2E170 Terminology Relating to Radiation Measurements andDosimetryE482 Guide for Application of Neutron Transport Methodsfor Reactor Vessel Surveillance2.2 ISO/ASTM Standards:251707 Guide for Estimating Uncertai
12、nties in Dosimetry forRadiation Processing2.3 International Commission on Radiation Units and Mea-surements Reports:3ICRU Report 85a Fundamental Quantities and Units forIonizing Radiation2.4 United States National Institute of Standards and Tech-nology:4NIST Technical Note 1297 (1994 edition) Guidel
13、ines forEvaluating and Expressing the Uncertainty of NIST Mea-surement Results3. Terminology3.1 Definitions:1This guide is under the jurisdiction of ASTM Committee E61 on RadiationProcessing and is the direct responsibility of Subcommittee E61.04 on SpecialtyApplication.Current edition approved Dec.
14、 1, 2016. Published January 2017. Originallyapproved in 2002. Last previous edition approved in 2010 as E2232-10. DOI:10.1520/E2232-16.2For referenced ASTM and ISO/ASTM standards, visit the ASTM website,www.astm.org, or contact ASTM Customer Service at serviceastm.org. ForAnnual Book of ASTM Standar
15、ds volume information, refer to the standardsDocument Summary page on the ASTM website.3Available from International Commission on Radiation Units andMeasurements, 7910 Woodmont Ave., Suite 800, Bethesda, MD 20815 USA.4Available as a download from the NIST web site at: http:/physics.nist.gov/Pubs/gu
16、idelines/TN1297/tn1297s.pdf.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles
17、 for theDevelopment of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.13.1.1 accuracy (VIM)closeness of agreement between ameasured quantity value and a true quantity value of ameasurand.3.1.2 benchmarkingcompari
18、ng model predictions to inde-pendent measurements or calculations under similar conditionsusing defined criteria of uncertainty.3.1.2.1 DiscussionBenchmarking is a prerequisite beforeroutine use of a mathematical model. Refer to 8.1 and AnnexA5.3.1.3 biasing (in a Monte Carlo simulation)adjustment o
19、fthe source particle selection or the transported particle weight,or both, in a statistically valid manner so as to increase theparticles in a region where the detector response is mostimportant.3.1.3.1 DiscussionBiasing is a method used to reduce theestimated uncertainty or computer run times of Mo
20、nte Carlosimulations. Monte Carlo simulations using the natural prob-abilities of physical events may require unacceptably long runtimes to accumulate statistics for rare events. The simulatedprobabilities may be altered to achieve the uncertainty goals forthe simulation in acceptable run times by b
21、iasing the samplingfrom the probability distributions. The number of particlestracked and the particle weights may be adjusted so as toensure a statistically valid sample from the probability distri-butions. Appropriate biasing requires a detailed knowledge ofthe model and the influence of rare even
22、ts. As with allsimulations, results should be compared with benchmarkmeasurements or simulation results originated by a differentcode.3.1.4 build-up factorratio of the total value of a specifiedradiation quantity (such as absorbed dose) at any point in thatmedium to the contribution to that quantity
23、 from the incidentun-collided radiation reaching that point.3.1.4.1 DiscussionThe concept of build-up applies to thetransport of photons.3.1.5 deterministic methoda mathematical method usingtransport equations to directly calculate the radiation field overall space as a function of radiation source
24、and boundaryconditions.3.1.5.1 DiscussionThe point kernel and discrete ordinatemethods are examples of deterministic methods.3.1.6 discrete ordinate methoda deterministic method forapproximate numerical solution of the transport equation inwhich the direction of motion is divided into a finite numbe
25、r ofdiscrete ordinate angles.3.1.6.1 DiscussionIn the discrete ordinatesapproximation, the transport equation becomes a set of coupledequations, one for each discrete ordinate. Particle behaviorsalong paths intermediate to described paths are approximatedby a weighted average (numerical quadrature)
26、of adjacent paths(1).5The method is useful for both electron and photon sourceswhen appropriate assumptions can be made.3.1.7 empirical methoda method derived from fitting anapproximating function to experimental data or Monte Carlocalculation result.3.1.7.1 DiscussionEmpirical models are generally
27、devel-oped by fitting equations (for example, polynomial) to experi-mental data or simulation output derived from another math-ematical method.3.1.8 history (of a particle)record of all simulated inter-actions along particles track as used in stochastic simulations(for example, Monte Carlo).3.1.8.1
28、DiscussionA particle history begins with the start-ing position, energy and direction of a particle, follows all itsinteractions, and terminates in one of several outcomes such asabsorption, escape from the boundary of the problem, orreaching a cut-off limit (such as a cut-off energy). A particlehis
29、tory is the systematic generation of a random, simulatedparticle track that is obtained according to the known physicalinteractions of either electrons or photons with the materialbeing traversed. History and particle history are consideredsynonymous.3.1.9 mathematical methoda method of solution of
30、anelectron or photon transport problem, or both, using algebraicrelations and mathematical operations to represent the systemand its dynamics.3.1.10 mathematical modela mathematical description ofa physical problem based on physical laws or empiricalcorrelation, or both.3.1.11 Monte Carlo methoda si
31、mulation method used forcalculating absorbed dose, energy spectra, charge, fluence andfluence rate in a volume of interest using a statistical summaryof the radiation interactions.3.1.11.1 DiscussionA Monte Carlo calculation consists ofrunning a large number of particle histories (simulations) until
32、some acceptable statistical uncertainty in the desired calculatedquantity (such as dose) has been reached. This calculationmethod is suitable for problems involving either electrons orphotons or both. This technique produces a probabilisticapproximation to the solution of a problem by using statisti
33、calsampling techniques. See also stochastic and history.3.1.12 numerical convergenceprocess in which the itera-tive solution of an equation or set of equations changes by lessthan some defined value.3.1.12.1 DiscussionThe mathematical equations describ-ing a problem are often so complex that an anal
34、ytical (alge-braic) solution is not possible. The solution of the equationscan be estimated by an iterative process of progressivelyrefining approximate solutions at a grid of discrete locations.Aconsistent set of solutions arrived at by this method achievesnumerical convergence. Convergence may not
35、 be obtained ifthe discrete locations are too widely separated (that is, the gridis too coarse).3.1.13 point kernel methoda deterministic method forcalculating dose based on integrating the contributions frompoint sources.3.1.13.1 DiscussionThe point kernel method is typicallyused for photon transpo
36、rt applications. The radiation source ismodeled as a large set of point sources. The absorbed dose,5The boldface numbers in parentheses refer to the list of references at the end ofthis standard.E2232 162dose equivalent or exposure is estimated at a dose point byintegrating the contribution from eac
37、h of the point sources. Amultiplicative value (the semi-empirical build-up factor) isused to account for the contribution from scattered (indirect)radiation from regions not in the direct path between the sourcepoint and field point.3.1.14 radiation fielda function describing the particledensity and
38、 the distributions of energy, direction and particletype at any point.3.1.15 radiation transport theoryan analytical descriptionof the propagation of a radiation field according to the physicallaws governing the interaction of radiation with matter.3.1.15.1 DiscussionIn its most general form, transp
39、orttheory is a special branch of statistical mechanics, which dealswith the interaction of the radiation field with matter.3.1.16 semi-empirical modelan empirical model in whichthe fitting parameters are constrained so that the model satisfiesone or more physical laws or rules.3.1.16.1 DiscussionThe
40、 satisfaction of such physical rulesmay enable the model to be applicable over a wide range ofenergies and materials.3.1.17 spatial meshsubdivision of the radiation interac-tion volume of interest into a grid of discrete spatial elementsfor performing a transport calculation.3.1.18 statistical compo
41、nent of uncertaintycomponent ofuncertainty evaluated by statistical analysis of a series ofcalculated values.3.1.18.1 DiscussionThe inherent sampling uncertainty ofthe Monte Carlo method can be estimated as a statisticaluncertainty by applying statistical sampling techniques to thenumber of simulate
42、d histories. For calculations withoutbiasing, the statistical uncertainty scales as the reciprocal of thesquare root of the number of histories.3.1.19 stochastic methodsmethods using mathematicalequations containing random variables to describe or summa-rize the physical processes in the system bein
43、g studied. Arandom variable is a variable whose value is a function of astatistical distribution of random values.3.1.19.1 DiscussionThe Monte Carlo method is the onlystochastic method discussed in this guide. See also MonteCarlo and history.3.1.20 non-statistical component of uncertaintycomponent o
44、f uncertainty evaluated by means other thanstatistical analysis of a series of calculated values.3.1.20.1 DiscussionThere are non-statistical componentsof uncertainties associated with the necessary simplifyingassumptions needed to approximate the physical paths ofelectrons in the model and uncertai
45、nties in the cross-sectionsfor the different interactions. These uncertainties can be esti-mated by analytical techniques.Anon-statistical component ofuncertainty could result from the difference in geometry andmaterial composition of the modelled irradiator versus theactual irradiator. Other source
46、s of non-statistical component ofuncertainty are the inadequate description of the problem andapproximations to actual physics.3.1.21 transport equationan integro-differential equationdescribing the motion of particles or radiation through amedium.3.1.21.1 DiscussionThe transport equation contains v
47、ari-ous terms corresponding to sources of particles, particlestreaming and particle scattering in and out of an infinitesimalvolume of phase space.3.1.22 uncertainty of calculation resultnon-negative pa-rameter associated with the result of a calculation that charac-terizes the spread of values that
48、 could reasonably be attributedto the derived quantity.3.1.22.1 DiscussionLike absorbed-dose measurement, theabsorbed-dose calculation should also be accompanied by anestimate of uncertainty.3.1.23 validationaccumulation of documented experi-mental evidence, used to demonstrate that the mathematical
49、method is a reliable prediction technique.3.1.23.1 DiscussionValidation compares a code or theorywith results of an appropriate experiment.3.1.24 verificationconfirmation by examination of evi-dence that the mathematical method has been properly andsuccessfully applied to the problem.3.1.24.1 DiscussionIt is important to know the type ofradiation sources, geometries, energies, etc. for which a codehas been validated. The calculated results will also depend onquantities at the users disposal such as cut-off energy (forMonte Carlo) or mesh
