ASTM E2232-21

Standard Guide for Selection and Use of Mathematical Methods for Calculating Absorbed Dose in Radiation Processing Applications

SIGNIFICANCE AND USE
4.1 Use as an Analytical Tool—Mathematical methods provide an analytical tool to be employed for many applications related to absorbed dose determinations in radiation processing. Mathematical calculations may not be used as a substitute for routine dosimetry in some applications (for example, medical device sterilization, food irradiation).  
4.2 Dose Calculation—Absorbed-dose calculations may be performed for a variety of photon/electron environments and irradiator geometries.  
4.3 Evaluate Process Effectiveness—Mathematical models may be used to evaluate the impact of changes in product composition, loading configuration, and irradiator design on dose distribution.  
4.4 Complement or Supplement to Dosimetry—Dose calculations may be used to establish a detailed understanding of dose distribution, providing a spatial resolution not obtainable through measurement. Calculations may be used to reduce the number of dosimeters required to characterize a procedure or process (for example, dose mapping).  
4.5 Alternative to Dosimetry—Dose calculations may be used when dosimetry is impractical (for example, granular materials, materials with complex geometries, material contained in a package where dosimetry is not practical or possible).  
4.6 Facility Design—Dose calculations are often used in the design of a new irradiator and can be used to help optimize dose distribution in an existing facility or radiation process. The use of modeling in irradiator design can be found in Refs (2-7).  
4.7 Validation—The validation of the model should be done through comparison with reliable and traceable dosimetric measurements. The purpose of validation is to demonstrate that the mathematical method makes reliable predictions of dose and other transport quantities. Validation compares predictions or theory to the results of an appropriate experiment. The degree of validation is commensurate with the application. Guidance is given in the documents referenced in Annex A2.  
...
SCOPE
1.1 This guide describes different mathematical methods that may be used to calculate absorbed dose and criteria for their selection. Absorbed-dose calculations can determine the effectiveness of the radiation process, estimate the absorbed-dose distribution in product, or supplement or complement, or both, the measurement of absorbed dose.  
1.2 Radiation processing is an evolving field and annotated examples are provided in Annex A6 to illustrate the applications where mathematical methods have been successfully applied. While not limited by the applications cited in these examples, applications specific to neutron transport, radiation therapy and shielding design are not addressed in this document.  
1.3 This guide covers the calculation of radiation transport of electrons and photons with energies up to 25 MeV.  
1.4 The mathematical methods described include Monte Carlo, point kernel, discrete ordinate, semi-empirical and empirical methods.  
1.5 This guide is limited to the use of general purpose software packages for the calculation of the transport of charged or uncharged particles and photons, or both, from various types of sources of ionizing radiation. This standard is limited to the use of these software packages or other mathematical methods for the determination of spatial dose distributions for photons emitted following the decay of 137Cs or 60Co, for energetic electrons from particle accelerators, or for X-rays generated by electron accelerators.  
1.6 This guide assists the user in determining if mathematical methods are a useful tool. This guide may assist the user in selecting an appropriate method for calculating absorbed dose. The user must determine whether any of these mathematical methods are appropriate for the solution to their specific application and what, if any, software to apply.
Note 1: The user is urged to apply these predictive techniques while being aware of the need for experien...

General Information

• Status: Published
• Publication Date: 14-Jun-2021
• ICS: 07.020 - Mathematics
• ICS: 17.240 - Radiation measurements
• Technical Committee: E61 - Radiation Processing
• Drafting Committee: E61.04 - Specialty Application

Relations

• Refers: ASTM E3083-17 - Standard Terminology Relating to Radiation Processing: Dosimetry and Applications
• Effective Date: 01-Oct-2017
• Refers: ASTM E482-11e1 - Standard Guide for Application of Neutron Transport Methods for Reactor Vessel Surveillance, E706 (IID)
• Effective Date: 01-Jun-2011
• Refers: ASTM E482-11 - Standard Guide for Application of Neutron Transport Methods for Reactor Vessel Surveillance, E706 (IID)
• Effective Date: 01-Jun-2011
• Refers: ASTM E482-07 - Standard Guide for Application of Neutron Transport Methods for Reactor Vessel Surveillance, E706 (IID)
• Effective Date: 01-Jun-2007
• Refers: ASTM E482-01 - Standard Guide for Application of Neutron Transport Methods for Reactor Vessel Surveillance, E706 (IID)
• Effective Date: 10-Jun-2001
• Refers: ASTM E482-89(1996) - Standard Guide for Application of Neutron Transport Methods for Reactor Vessel Surveillance, E706 (IID)
• Effective Date: 10-Jun-2001

Overview

ASTM E2232-21 is the Standard Guide for Selection and Use of Mathematical Methods for Calculating Absorbed Dose in Radiation Processing Applications. Developed by ASTM International, this guide assists users in selecting appropriate mathematical methods for calculating absorbed dose when applying ionizing radiation in various processing environments. This standard is widely used in industries such as medical device manufacturing, food irradiation, and materials processing to better understand, predict, and validate absorbed dose distributions.

The guide provides comprehensive direction for method selection, implementation, and validation. It covers electron and photon radiation transport modeling up to 25 MeV, supporting improved design, process optimization, and dose mapping. The standard specifically excludes detailed guidance for neutron transport, radiation therapy, and shielding design, focusing instead on absorbed dose calculation relevant to industrial and research-based radiation processing.

Key Topics

• Mathematical Methods: The guide outlines four main types of mathematical approaches:

  • Monte Carlo methods
  • Deterministic methods, including discrete ordinate and point kernel approaches
  • Empirical methods
  • Semi-empirical methods

• Criteria for Use: It distinguishes cases where mathematical dose calculation can supplement or complement routine dosimetry, or where it can serve as an alternative when physical dosimetry is impractical.

• Validation & Verification: Emphasizes the importance of validating models through comparison with experimentally measured and traceable dosimetric data to ensure reliability.

• Uncertainty Estimation: Recommends accompanying absorbed dose calculations with uncertainty estimates, as done for physical measurements, to support robust process control and compliance.

• Prerequisites for Application:

  • Documented facility and product geometry
  • Characterization of radiation sources (energy, type, output)
  • Comprehensive recordkeeping of software, equipment, and data used in calculations
  • Competency of personnel in modeling and radiation physics

• Method Selection: Selection of the appropriate mathematical method depends on source type (electrons or photons), complexity of the material geometry, need for spatial detail and resolution, and available software capabilities.

Applications

ASTM E2232-21 provides practical value in a variety of industrial and scientific applications:

• Process Design & Optimization: Mathematical dose calculation aids in the design and modification of irradiation facilities or processes, helping optimize dose distribution and minimize under- or over-exposure.

• Dose Mapping: Enables detailed, high-resolution spatial mapping of dose without an extensive array of physical dosimeters, ideal for complex products or packaging.

• Process Qualification & Control: Supports qualification activities such as Operational Qualification (OQ) and Performance Qualification (PQ), essential for regulatory compliance, especially in healthcare product sterilization or food irradiation.

• Product and Packaging Assessment: Useful for assessing dose uniformity in materials with complex geometry or in bulk, granular forms where dosimeter placement is difficult.

• Supplemental Insight: Provides refined insights into dose distribution, complementing routine dosimetry and identifying potential process variations or inefficiencies.

Related Standards

Several key international and industry standards relate to or complement ASTM E2232-21:

• ASTM E3083: Terminology Relating to Radiation Processing-Dosimetry and Applications
• ISO/ASTM 51707: Guide for Estimating Uncertainties in Dosimetry for Radiation Processing
• ISO/ASTM 52628: Practice for Dosimetry in Radiation Processing
• ISO 12749-4: Nuclear energy, nuclear technologies, and radiological protection-Vocabulary-Part 4: Dosimetry for radiation processing
• ICRU Report 85a: Fundamental Quantities and Units for Ionizing Radiation
• NIST Technical Note 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results
• JCGM 100:2008 & 200:2012: Guides for the Expression of Uncertainty in Measurement and International Vocabulary of Metrology

Summary

ASTM E2232-21 ensures that radiation processing professionals have a rigorous, validated approach to absorbed dose calculation. By guiding users through model selection, application, verification, and uncertainty analysis, the standard improves the reliability, safety, and consistency of radiation processing across diverse industries.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E2232 − 21
Standard Guide for
Selection and Use of Mathematical Methods for Calculating
Absorbed Dose in Radiation Processing Applications
This standard is issued under the fixed designation E2232; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
NOTE 1—The user is urged to apply these predictive techniques while
1. Scope
beingawareoftheneedforexperienceandalsotheinherentlimitationsof
1.1 This guide describes different mathematical methods
both the method and the available software. Information pertaining to
that may be used to calculate absorbed dose and criteria for
availabilityandupdatestocodesformodelingradiationtransport,courses,
their selection. Absorbed-dose calculations can determine the
workshops and meetings can be found in Annex A1. For a basic
understanding of radiation physics and a brief overview of method
effectiveness of the radiation process, estimate the absorbed-
selection, refer to Annex A3.
dose distribution in product, or supplement or complement, or
both, the measurement of absorbed dose.
1.7 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the
1.2 Radiation processing is an evolving field and annotated
responsibility of the user of this standard to establish appro-
examples are provided in Annex A6 to illustrate the applica-
tions where mathematical methods have been successfully priate safety, health, and environmental practices and deter-
applied. While not limited by the applications cited in these mine the applicability of regulatory limitations prior to use.
examples, applications specific to neutron transport, radiation
1.8 This international standard was developed in accor-
therapy and shielding design are not addressed in this docu-
dance with internationally recognized principles on standard-
ment.
ization established in the Decision on Principles for the
1.3 This guide covers the calculation of radiation transport Development of International Standards, Guides and Recom-
of electrons and photons with energies up to 25 MeV.
mendations issued by the World Trade Organization Technical
Barriers to Trade (TBT) Committee.
1.4 The mathematical methods described include Monte
Carlo, point kernel, discrete ordinate, semi-empirical and
2. Referenced Documents
empirical methods.
1.5 This guide is limited to the use of general purpose 2.1 ASTM Standards:
software packages for the calculation of the transport of
E482Guide for Application of Neutron Transport Methods
charged or uncharged particles and photons, or both, from
for Reactor Vessel Surveillance
various types of sources of ionizing radiation. This standard is
E3083Terminology Relating to Radiation Processing: Do-
limited to the use of these software packages or other math-
simetry and Applications
ematical methods for the determination of spatial dose distri-
2.2 ISO/ASTM Standards:
butions for photons emitted following the decay of Cs or
51707Guide for Estimating Uncertainties in Dosimetry for
Co, for energetic electrons from particle accelerators, or for
Radiation Processing
X-rays generated by electron accelerators.
52628Practice for Dosimetry in Radiation Processing
1.6 This guide assists the user in determining if mathemati-
2.3 ISO Standard:
cal methods are a useful tool.This guide may assist the user in
ISO 12749-4Nuclear energy, nuclear technologies, and
selecting an appropriate method for calculating absorbed dose.
radiological protection — Vocabulary — Part 4: Dosim-
The user must determine whether any of these mathematical
etry for radiation processing
methods are appropriate for the solution to their specific
application and what, if any, software to apply.
For referenced ASTM and ISO/ASTM standards, visit the ASTM website,
This guide is under the jurisdiction of ASTM Committee E61 on Radiation www.astm.org, or contact ASTM Customer Service at [e-mail]. For
Processing and is the direct responsibility of Subcommittee E61.04 on Specialty Annual Book of ASTM Standards volume information, refer to the standard’s
Application. Document Summary page on the ASTM website.
Current edition approved June 15, 2021. Published July 2021. Originally Available from International Organization for Standardization (ISO), ISO
approved in 2002. Last previous edition approved in 2020 as E2232-20. DOI: Central Secretariat, Chemin de Blandonnet 8, CP 401, 1214 Vernier, Geneva,
10.1520/E2232-21. Switzerland, https://www.iso.org.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2232 − 21
2.4 International Commission on Radiation Units and Mea- 3.1.6 discrete ordinate method—a deterministic method for
surements Reports: approximate numerical solution of the transport equation in
ICRU Report 85aFundamental Quantities and Units for whichthedirectionofmotionisdividedintoafinitenumberof
Ionizing Radiation discrete ordinate angles.
2.5 JCGM Documents: 3.1.6.1 Discussion—In the discrete ordinates
JCGM 100:2008GUM 1995, with minor corrections Evalu- approximation,thetransportequationbecomesasetofcoupled
ation of measurement date - Guide to the Expression of equations, one for each discrete ordinate. Particle behaviors
Uncertainty in Measurement along paths intermediate to described paths are approximated
JCGM 200:2012VIM International Vocabulary of Metrol- byaweightedaverage(numericalquadrature)ofadjacentpaths
ogy - Basic and General Concepts and Associated Terms (1). Themethodisusefulforbothelectronandphotonsources
when appropriate assumptions can be made.
3. Terminology
3.1.7 empirical method—a method derived from fitting an
3.1 Definitions:
approximating function to experimental data or Monte Carlo
3.1.1 accuracy (VIM)—closeness of agreement between a
calculation result.
measured quantity value and a true quantity value of a
3.1.7.1 Discussion—Empirical models are generally devel-
measurand.
oped by fitting equations (for example, polynomial) to experi-
3.1.2 benchmarking—comparing model predictions to inde- mental data or simulation output derived from another math-
pendentmeasurementsorcalculationsundersimilarconditions
ematical method.
using defined criteria of uncertainty.
3.1.8 history (of a particle)—record of all simulated inter-
3.1.2.1 Discussion—Benchmarking is a prerequisite before
actions along particle’s track as used in stochastic simulations
routine use of a mathematical model. Refer to 8.1 and Annex
(for example, Monte Carlo).
A5.
3.1.8.1 Discussion—Aparticle history begins with the start-
3.1.3 biasing (in a Monte Carlo simulation)—adjustment of ing position, energy and direction of a particle, follows all its
the source particle selection or the transported particle weight,
interactions,andterminatesinoneofseveraloutcomessuchas
or both, in a statistically valid manner so as to increase the absorption, escape from the boundary of the problem, or
particles in a region where the detector response is most
reaching a cut-off limit (such as a cut-off energy). A particle
important. history is the systematic generation of a random, simulated
3.1.3.1 Discussion—Biasing is a method used to reduce the
particle track that is obtained according to the known physical
estimated uncertainty or computer run times of Monte Carlo interactions of either electrons or photons with the material
simulations. Monte Carlo simulations using the natural prob-
being traversed. History and particle history are considered
abilities of physical events may require unacceptably long run synonymous.
times to accumulate statistics for rare events. The simulated
3.1.9 mathematical method—a method of solution of an
probabilitiesmaybealteredtoachievetheuncertaintygoalsfor
electron or photon transport problem, or both, using algebraic
the simulation in acceptable run times by biasing the sampling
relations and mathematical operations to represent the system
from the probability distributions. The number of particles
and its dynamics.
tracked and the particle weights may be adjusted so as to
3.1.10 mathematical model—a mathematical description of
ensure a statistically valid sample from the probability distri-
a physical problem based on physical laws or empirical
butions. Appropriate biasing requires a detailed knowledge of
correlation, or both.
the model and the influence of rare events. As with all
3.1.11 Monte Carlo method—a simulation method used for
simulations, results should be compared with benchmark
calculating absorbed dose, energy spectra, charge, fluence and
measurements or simulation results originated by a different
fluence rate in a volume of interest using a statistical summary
code.
of the radiation interactions.
3.1.4 build-up factor—ratio of the total value of a specified
3.1.11.1 Discussion—AMonte Carlo calculation consists of
radiation quantity (such as absorbed dose) at any point in that
running a large number of particle histories (simulations) until
medium to the contribution to that quantity from the incident
someacceptablestatisticaluncertaintyinthedesiredcalculated
un-collided radiation reaching that point.
quantity (such as dose) has been reached. This calculation
3.1.4.1 Discussion—The concept of build-up applies to the
method is suitable for problems involving either electrons or
transport of photons.
photons or both. This technique produces a probabilistic
3.1.5 deterministic method—a mathematical method using
approximation to the solution of a problem by using statistical
transport equations to directly calculate the radiation field over
sampling techniques. See also stochastic and history.
all space as a function of radiation source and boundary
3.1.12 numerical convergence—process in which the itera-
conditions.
tive solution of an equation or set of equations changes by less
3.1.5.1 Discussion—The point kernel and discrete ordinate
than some defined value.
methods are examples of deterministic methods.
Available from International Commission on Radiation Units and
Measurements, 7910 Woodmont Ave., Suite 800, Bethesda, MD 20815 USA. Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
Available from JCGM-BIPM website at: http://www.bipm.org. this standard.
E2232 − 21
3.1.12.1 Discussion—The mathematical equations describ- 3.1.20 non-statistical component of uncertainty—
ing a problem are often so complex that an analytical (alge- component of uncertainty evaluated by means other than
braic) solution is not possible. The solution of the equations statistical analysis of a series of calculated values.
can be estimated by an iterative process of progressively
3.1.20.1 Discussion—There are non-statistical components
refiningapproximatesolutionsatagridofdiscretelocations.A
of uncertainties associated with the necessary simplifying
consistent set of solutions arrived at by this method achieves
assumptions needed to approximate the physical paths of
numerical convergence. Convergence may not be obtained if
electrons in the model and uncertainties in the cross-sections
the discrete locations are too widely separated (that is, the grid
for the different interactions. These uncertainties can be esti-
is too coarse).
mated by analytical techniques.Anon-statistical component of
uncertainty could result from the difference in geometry and
3.1.13 point kernel method—a deterministic method for
material composition of the modelled irradiator versus the
calculating dose based on integrating the contributions from
point sources. actual irradiator. Other sources of non-statistical component of
uncertainty are the inadequate description of the problem and
3.1.13.1 Discussion—The point kernel method is typically
approximations to actual physics.
used for photon transport applications. The radiation source is
modeled as a large set of point sources. The absorbed dose,
3.1.21 transport equation—an integro-differential equation
dose equivalent or exposure is estimated at a dose point by
describing the motion of particles or radiation through a
integrating the contribution from each of the point sources. A
medium.
multiplicative value (the semi-empirical build-up factor) is
3.1.21.1 Discussion—The transport equation contains vari-
used to account for the contribution from scattered (indirect)
ous terms corresponding to sources of particles, particle
radiationfromregionsnotinthedirectpathbetweenthesource
streaming and particle scattering in and out of an infinitesimal
point and field point.
volume of phase space.
3.1.14 radiation field—a function describing the particle
3.1.22 uncertainty of calculation result—non-negative pa-
density and the distributions of energy, direction and particle
rameter associated with the result of a calculation that charac-
type at any point.
terizes the spread of values that could reasonably be attributed
3.1.15 radiation transport theory—an analytical description
to the derived quantity.
ofthepropagationofaradiationfieldaccordingtothephysical
3.1.22.1 Discussion—Like absorbed-dose measurement, the
laws governing the interaction of radiation with matter.
absorbed-dose calculation should also be accompanied by an
3.1.15.1 Discussion—In its most general form, transport
estimate of uncertainty.
theory is a special branch of statistical mechanics, which deals
3.1.23 validation—accumulation of documented experi-
with the interaction of the radiation field with matter.
mental evidence, used to demonstrate that the mathematical
3.1.16 semi-empirical model—an empirical model in which
method is a reliable prediction technique.
thefittingparametersareconstrainedsothatthemodelsatisfies
3.1.23.1 Discussion—Validation compares a code or theory
one or more physical laws or rules.
with results of an appropriate experiment.
3.1.16.1 Discussion—Thesatisfactionofsuchphysicalrules
3.1.24 verification—confirmation by examination of evi-
may enable the model to be applicable over a wide range of
dence that the mathematical method has been properly and
energies and materials.
successfully applied to the problem.
3.1.17 spatial mesh—subdivision of the radiation interac-
3.1.24.1 Discussion—It is important to know the type of
tion volume of interest into a grid of discrete spatial elements
radiation sources, geometries, energies, etc. for which a code
for performing a transport calculation.
has been validated. The calculated results will also depend on
3.1.18 statistical component of uncertainty—component of
quantities at the user’s disposal such as cut-off energy (for
uncertainty evaluated by statistical analysis of a series of
Monte Carlo) or mesh size (for discrete ordinate methods).
calculated values.
Verification demonstrates that theory was implemented in the
3.1.18.1 Discussion—The inherent sampling uncertainty of
way intended, and that the simulation was performed in
the Monte Carlo method can be estimated as a statistical
accordance with its requirements and specifications.
uncertainty by applying statistical sampling techniques to the
3.1.25 zoning—The geometric description used to break up
number of simulated histories. For calculations without
a larger region into smaller segments in which to calculate the
biasing,thestatisticaluncertaintyscalesasthereciprocalofthe
dose.
square root of the number of histories.
3.1.25.1 Discussion—Partitioning a zone into smaller seg-
3.1.19 stochastic methods—methods using mathematical
ments is referred to as subzoning.
equations containing random variables to describe or summa-
rize the physical processes in the system being studied. A
3.2 Definitions of other terms used in this standard that
random variable is a variable whose value is a function of a
pertain to radiation measurement and dosimetry may be found
statistical distribution of random values.
in ISO/ASTM Practice 52628. Other terms that pertain to
3.1.19.1 Discussion—The Monte Carlo method is the only radiation measurement and dosimetry may be found in Termi-
stochastic method discussed in this guide. See also Monte nology E3083 and ISO Terminology 12749-4. Where
Carlo and history. appropriate, definitions used in these standards have been
E2232 − 21
derived from, and are consistent with definitions in ICRU 4.9 Uncertainty—An absorbed dose prediction should be
Report 85a, and general metrological definitions given in the accompaniedbyanestimateofoveralluncertainty,asitiswith
VIM. absorbed-dose measurement (refer to ISO/ASTM 51707 and
JCGM100:2008 and JCGM200:2012). In many cases,
absorbed-dose measurement helps to establish the uncertainty
4. Significance and Use
in the dose calculation.
4.1 Use as an Analytical Tool—Mathematical methods pro-
4.10 This guide should not be used as the only reference in
vide an analytical tool to be employed for many applications
the selection and use of mathematical models. The user is
related to absorbed dose determinations in radiation process-
encouraged to contact individuals who are experienced in
ing. Mathematical calculations may not be used as a substitute
mathematical modelling and to read the relevant publications
for routine dosimetry in some applications (for example,
in order to select the best tool for their application. Radiation
medical device sterilization, food irradiation).
processing is an evolving field and the references cited in the
4.2 Dose Calculation—Absorbed-dose calculations may be
annotated examples of Annex A6 are representative of the
performed for a variety of photon/electron environments and
various published applications. Where a method is validated
irradiator geometries.
with dosimetry, it becomes a benchmark for that particular
application.
4.3 Evaluate Process Effectiveness—Mathematical models
may be used to evaluate the impact of changes in product
composition, loading configuration, and irradiator design on 5. Classification of Mathematical Methods and General
dose distribution. Application
5.1 Mathematical methods for radiation transport can be
4.4 Complement or Supplement to Dosimetry—Dose calcu-
used to estimate the absorbed dose to a small volume or point.
lations may be used to establish a detailed understanding of
The dose distribution within the entire product can be deter-
dose distribution, providing a spatial resolution not obtainable
mined by calculations at different points within the product.
through measurement. Calculations may be used to reduce the
number of dosimeters required to characterize a procedure or
5.2 Types of Methods—Fourgeneraltypesofmethodsarein
process (for example, dose mapping).
use: Monte Carlo, deterministic, semi-empirical and empirical.
Both Monte Carlo and deterministic methods are based on the
4.5 Alternative to Dosimetry—Dose calculations may be
detailed physics of the interaction of radiation with matter.
used when dosimetry is impractical (for example, granular
materials, materials with complex geometries, material con- 5.2.1 Monte Carlo methods involve simulating paths of a
finite number of photons or electrons and estimating dose by
tained in a package where dosimetry is not practical or
possible). summing and averaging the histories of many energy deposi-
tion events.
4.6 Facility Design—Dosecalculationsareoftenusedinthe
5.2.2 Deterministic methods use equations describing the
design of a new irradiator and can be used to help optimize
transport of radiation in matter to perform a direct estimate of
dose distribution in an existing facility or radiation process.
the total radiation field, absorbed dose and other responses.
The use of modeling in irradiator design can be found in Refs
5.2.3 Empirical and semi-empirical methods are based on
(2-7).
statistical relationships of measurements or calculations for a
4.7 Validation—The validation of the model should be done
particular system.
through comparison with reliable and traceable dosimetric
5.3 Monte Carlo Method—The Monte Carlo method simu-
measurements.Thepurposeofvalidationistodemonstratethat
lates the paths of particles such as electrons and photons from
the mathematical method makes reliable predictions of dose
the source to the dose volume. See Note 1, Refs (8-19) and
and other transport quantities.Validation compares predictions
Annex A1 for examples and codes. See also A3.3 and A3.4.4
or theory to the results of an appropriate experiment. The
for brief discussions of the physics of electron and photon
degree of validation is commensurate with the application.
transport and the Monte Carlo method respectively.
Guidance is given in the documents referenced in Annex A2.
5.3.1 Advantages—Unlike other methods, the Monte Carlo
4.8 Verification—Verification is the confirmation of the
method can, in principle, account for all interactions and
mathematical correctness of a computer implementation of a
providearealisticsimulationofactualallscatteringandenergy
mathematical method. This can be done, for example, by
lossevents.Allcontributionstotheabsorbeddosecanbetaken
comparing numerical results with known analytic solutions or
into account including electron and photon scattering from
with other computer codes that have been previously verified.
nearby objects. (See Note 3.) In addition, the Monte Carlo
Verification should be done to ensure that the simulation is
method has the great advantage of being the method most
appropriate for the intended application. Refer to 3.1.24.
capable of simulating the actual radiation transport in complex
NOTE 2—Certain applications of the mathematical model deal with
three-dimensional geometry.
Operational Qualification (OQ), Performance Qualification (PQ) and
process control in radiation processing such as the sterilization of NOTE 3—Such objects could be structures outside the system of
healthcareproducts.Theapplicationanduseofthemathematicalmodelin irradiated material(s) for which the dose distribution is to be calculated.
these applications may have to meet regulatory requirements. Refer to For example, these might include shielding layers, photon beam
Section 6 for prerequisites for application of a mathematical method and collimators, e-beam accelerator heads, or walls of concrete or lead
Section8forrequirementsbeforeroutineuseofthemathematicalmethod. surrounding a Co radiation source.
E2232 − 21
5.3.2 Disadvantages—Because electrons (including those methodshavebeendevelopedtosolvetheseequations (24).All
generated by photons) in the energy range of 50 keV to 10 of these methods place limits on the angular variable such that
MeV undergo large numbers of scattering events, exact simu- the incident radiation is represented as streaming only along a
lation of all photon and electron paths is not feasible or finitenumberofdirectionsratherthanallpossibledirectionsas
practical.Instead,approximateelectronpathsareemployed,as containedinthetransportequation.Extensionofthistechnique
in the so-called “condensed history Monte Carlo method” (20 to 2-D and 3-D has been done by several authors (25).
and 21).Forelectrons,approximateartificialtrajectoriesusing 5.4.2 Point Kernel Methods—Pointkernelmethodsareused
large path length steps and a multiple-scattering approach to mainly for photon transport problems (26). In point kernel
particle deflections are employed in standard Monte Carlo methods, the radiation source volume is approximated by a
codes.(SeeAnnexA1.)ThestandardMonteCarlocodeslisted number of isotropic source points. The absorbed dose at each
in Annex A1 and Refs (8-19) use this condensed history dose point is obtained by summing the dose contribution from
approach. However, such approximate paths may lead to all source points. The calculation takes into account the
significant errors, particularly at locations where transport distance between the dose point and the source point and
across surfaces or material interfaces is important. See Note 4. approximates the scatter within the intervening product
through the use of a build-up factor. Build-up factors are
NOTE 4—In some Monte Carlo codes (17), improved accuracy near
theoretically calculated and sometimes fitted to empirical
material boundaries has been obtained using shorter paths near interfaces
functions. These factors provide an approximation for the
between different materials.
NOTE 5—To reduce computational time, limits to the problem may be contribution of scattered photons from surrounding material.
specified, such as physical boundaries and energy cut-offs, when the
Approximations are also required to account for the energy
contributions to the problem made outside of these boundaries are no
spectrum and variations in the atomic number in different
longer expected to be significant. Variance reduction techniques help to
intervening or scattering materials.
improve the rate of numerical convergence but require a sophisticated
understanding of probability distributions.
NOTE 6—There are a number of general databases available for the
photon buildup factors needed for these codes (Annex A1).
5.3.2.1 One of the main difficulties with this method is that
when applied to geometries where reductions in fluence
5.4.3 Advantages—Deterministic methods may be faster
spanning several orders of magnitude might occur, or where
than Monte Carlo, and can be benchmarked against dosimetry.
the absorbed dose in very small volumes is required, the
5.4.4 Disadvantages—Deterministicmethodsgivenoinnate
statistical component of the uncertainty will often be large.
estimate of statistical uncertainty. Iterative solution methods
Thiscanbealleviatedusingvariancereductiontechniques.See
may be susceptible to numerical convergence errors and
Note 5.
oscillatory solutions.
5.3.2.2 Calculations of dose should provide a range of dose
5.4.5 Uncertainties—There are three sources of uncertain-
values over a region near where the dose is to be measured.
ties in deterministic models. These are (1) the approximations
This is to permit estimation of the effect of variations in the
usedtocreatephysicalmodelsandcross-sections(forexample,
location/orientation of a dosimeter in that region. This deter-
energy straggling is neglected in deterministic methods), (2)
mines the dose sensitivity associated with placement of the
the effect of representing a continuous problem in space, angle
dosimeter and allows determination of this type of error.
and energy with a finite mesh in all these variables and (3)
5.3.3 Statistical Uncertainty—The inherent uncertainty in
truncation error due to a finite number of discrete ordinates.
the calculated value of dose due to sampling in the Monte
5.4.6 The accuracy of the point kernel treatment may be
Carlomethodcanbeestimatedbyapplyingstatisticalsampling
comparable to that of a Monte Carlo calculation for configu-
techniques to the number of histories. For calculations without
rations where the point kernel approximation is valid (27).
biasing,thestatisticaluncertaintyscalesasthereciprocalofthe
5.5 Empirical and Semi-empirical Methods:
square root of the number of histories run.
5.5.1 Empirical—Empirical methods typically involve fit-
5.3.3.1 Special care must be taken when using variance
ting analytical functions to experimental measurements (or to
reduction techniques which are used to increase statistics in an
calculations using other methods). The model equations are
otherwise poorly populated phase space (for example, shield-
typically specific to a particular radiation facility and their
ing calculation where only high energy photons are tracked
predictive capabilities are not generally transferable to other
through the shield). This is accomplished by introducing
facilities or products. Some simple equations exist for calcu-
sampling probabilities which may be highly varying and have
latingtherangeofelectronsincondensedmatter (28),electron
an adverse effect on the convergence of Monte Carlo calcula-
energy loss (29) and depth-dose relationships in various
tions.
materials (30).
5.4 Deterministic Methods—These methods use analytical 5.5.2 Semi-Empirical—These are empirical methods in
equations to summarize radiation fluence rate through target
which the fitting parameters are constrained so that the model
materials. Such complex equations cannot be solved directly satisfies one or more physical laws or rules. These methods
but must be solved iteratively in the computer calculations.
provide a more generally applicable mathematical model than
5.4.1 Discrete Ordinates Methods—These methods have theempiricalmethodandareadjustabletophysicalparameters
been used for both electron and photon sources (22 and 23). ofthefacility,sourceandproducts,suchasenergy,densityand
This name is given to several closely related techniques for composition. In general, these are software-based programs
obtaining approximate solutions to the transport equations that withvariableparameterinputs.Equations,codesanddatabases
contain both integral and partial derivative terms. Various are available (31-34).
E2232 − 21
source shroud have been removed. The tote irradiator uses a shuffle-and-
5.5.3 Advantages—Empirical and semi-empirical models
dwell concept. Each product tote is irradiated for a defined period of time
are fast and do not require cross-sections, build-up factors and
before it is moved to the next irradiation position. The source rack
zoning since they are implicitly included in the coefficients of
containing the radiation sources is shown (35).
the model. No special knowledge, such as needed for Monte
NOTE9—Fig.2showsaphotographontheleftofaresearchcarrierand
Carlo or deterministic methods, is required. Semi-empirical
the graphical user interface window of a mathematical model shown on
the right photograph.All product is contained in aluminum totes. For the
models may be applicable to multiple facilities.
research carrier, product is brought into the radiation chamber and
5.5.4 Disadvantages—Empirical methods are likely to be
irradiated for a defined period of time, and then leaves the irradiation
verylimitedintheirapplication.Generally,empiricallyderived
chamber. The graphical user interface shows ray tracing between the
equations cannot be transferred to other sites or irradiation
radiation source (1) and the dose volume (2) (36).
applications,orboth,thatwerenotpartoftheoriginaldatabase
6.1.2 Detailed drawings of materials to be irradiated
used to generate the model. These methods may be difficult to
(products, targets) and their associated geometries, with physi-
implement for systems with complicated geometry.
cal verification of the same (composition of constituents,
NOTE 7—Although empirical or semi-empirical codes may give some
densities) should be collected and documented.
useful guidance, modern Monte Carlo codes on modern platforms are
6.1.3 The type of source(s) present (electrons, photons),
often very fast in these types of applications.
source energy spectrum, source output angular distribution,
5.5.5 Uncertainties—Uncertainty in both methods is influ-
source size (point or distributed, diffuse source with variable
enced by factors such as lack of homogeneity in the product,
activityetc.)andthenumberofsourcesshouldbespecifiedand
dosimeter location and uncertainty associated with dose mea-
documented.
surements.
NOTE 10—In the case of gamma-ray sources (for example, Co
sources), the photon energy spectrum may be difficult to obtain experi-
6. Prerequisites for Application of a Mathematical
mentally or estimate theoretically. In general, for photons with energies
Method
200 keV and above, a broad low energy contribution to the spectrum is
6.1 Facility and Related Geometry Considerations: created via Compton scattering.
6.1.1 Detailed drawings of irradiation facility equipment,
6.2 Personnel—Trained personnel should be involved in all
source-relatedequipmentandassociatedgeometries,shouldbe
aspectsofmodeldevelopment,programexecution,datareduc-
obtained, physically verified, and documented. Examples of
tion and the evaluation of results. There is no standard set of
gamma irradiation facilities are given in Figs. 1 and 2.
qualifications that can be recommended. Interaction of person-
nel with all phases of the modeling exercise should be
NOTE 8—Fig. 1 shows a physical model of a typical gamma irradiator
with product in aluminum totes. For clarity, eight totes and part of the documented according to the end-user’s policy and procedural
FIG. 1 Solid Model of a modified Nordion JS9600 Irradiator with a two layer roller conveyor, showing the product totes (1) and the ra-
diation source (2). The model was developed using EGSPP (35)
E2232 − 21
FIG. 2 Picture and Simulation of a Gamma Production Irradiator and Research Loop (36)
plans. The individual developing or using the selected model 7.1.2 Specificationoffacility(transportmechanism,support
should be actively involved in the verification experiment(s). structures, biological shield as per 6.1) where required for the
See Section 8 concerning the verification and validation particular calculation.
experiments.
7.1.3 Specification of target materials and geometries as per
6.2.1 All personnel involved in modelling using MC and 6.1.
other techniques should be knowledgeable in radiation physics
7.1.4 Declaration of personnel as per 6.2.
and should have received training for the code(s) that they are
7.1.5 Specification of computer hardware and software as
using.
per 6.3 (see also 7.2).
6.2.2 All training and significant experience of personnel
7.2 Criteria for Selection—Most problems are rarely mod-
involved in the modeling effort should be documented.
eledexactlyastheyappearinreality;majorapproximationsfor
6.3 Computer Equipment and Software—Requirements
simplification may be required to reduce the amount of effort
should be reviewed and documented.
required to build the model description and run times. These
6.3.1 Sufficient information regarding the equipment and
assumptions should be documented. Method selection will be
software that was used to run the calculations, including
primarily determined by the following criteria:
version numbers and compilers as appropriate, should be
7.2.1 Source Description—For a photon source, any of the
recorded to enable the calculations to be reproduced.
four methods may be chosen. For an electron source, the point
6.3.2 All operating system software, modeling software,
kernel method is not recommended since the point kernel
compilers and commercial products such as spreadsheets and
method assumes that the energy of the interacting particle is
dataanalysistoolsshouldhavetheirtitlesandversionnumbers
deliveredatapointandthendistributedstatisticallyaroundthat
recorded.
reaction point, as in the case of photons. On the contrary,
6.4 Allrelevantdosimetrydata,reportsofmeasurementand
electronsinteractcontinuouslywithmatteralongtheirpathand
other physical evidence should be collected and filed or because of this the point kernel method is not appropriate.
referenced for use in validation of model performance. See
7.2.2 Level of Detail—The level of detail to be included in
Section 8 concerning validation experiments.
the model, or the granularity of the problem, will influence the
methodselection.Iftheproblemcanbedescribedasregionsof
7. Specification of Modeling Strategy and Method
homogeneous material, the point kernel method may be most
Selection
appropriateforphotonproblems,ifspeedandspatialresolution
areimportant.Iftheproblemmustbefurtherbrokendowninto
7.1 Specification of the Modeling Effort—All modeling
smaller regions of different material (composition or density)
approaches should be described in the form of a written
in order to achieve accuracy, more complex input files will be
protocol detailing the requirements for successful execution
needed.
andsubsequentcompletionoftheexercise(s)relativetowritten
criteria for success. The protocol should, at a minimum, 7.2.2.1 Available software may have geometry replication
include: and tiling features that are very useful for this purpose. If the
7.1.1 Specification of the source type and geometry as per target size is small relative to geometry or source description,
6.1. Monte Carlo may require biasing or modification to include a
E2232 − 21
larger volume wherein the dose will be an average value over 8.1.1 Model Benchmarking—Model benchmarking is used
a larger volume than desired. The Monte Carlo method can be both to verify a mathematical method and to validate the
used to provide a refinement of the point kernel build-up
overall model construction and underlying physics of the
calculation used with the point kernel method. This can then
method to produce reliable results. Comparing current model
helppointkernelcalculationsachievearequiredaccuracywith
resultswithpreviouslywell-characterizedsystemsispartofthe
optimized efficiency (time, resolution) (27, 37, 38).
modeltesting.Comparingmodelresultswithdosimetryforthe
7.2.3 Set-up Time—The complexity of three-dimensional
specific problem being modeled is strongly recommended
problem descriptions in the input files and manipulation of the
whenever possible. Differences between measurement and
output files is where most of the effort is concentrated and can
calculationsshouldbeconsistentwithuncertaintyestimatesfor
be very time consuming. It may also be necessary to make
both the measurements and the calculations.
modificationstothecodetoaccommodatethespecificproblem
8.1.1.1 Therearealimitednumberofreferencedbenchmark
to be solved. If modifications to the code are necessary,
examples in the literature and these may be inadequate in
revalidation will be required, particularly if the physics mod-
number to validate a method and inadequate in detail for
eled in the code has been changed.
comparison with the model under consideration. The model of
7.3 Selection of Method Type: the application of interest should be as nearly the same as
possibletothebenchmarkexample.Benchmarkexamplesmay
7.3.1 The criteria for selection of a method type require
input from various sources. Such sources include in-house and be found in Annex A5. An example comparing the results of
several methods (Monte Carlo, deterministic and semi-
outside modeling expertise, model-based testing history and
availability of verified and validated modeling code(s). These empirical) with dosimetry can be found in Ref (39).
criteria should be documented as per 7.1.
NOTE 13—One or more well-defined problems may be run through the
7.3.2 Evaluation of the impact of the code on those items
model on the user’s hardware and software platform(s) and compared to
stated in 7.1.1 – 7.1.5 will typically be geared towards
accepted results for execution of the model generated by one or more
minimization of time for model set-up, execution and evalua- organizations (typically, this includes, at a minimum, the firm issuing the
modeling software). Input and output are compared, and the modeling
tion in exchange for exactness of solution set(s).
package’s performance is deemed verified upon successful completion of
7.3.3 There are currently no written methods available for
the test(s).
determining the optimum code to use. However, some general
NOTE 14—Formal software testing is not addressed in this guide. It is
guidelines are as follows:
desirabletoperformcalculationswithamodelingcodethathasundergone
7.3.3.1 Empirical equations can be developed, evaluated
a formal software validation program.The level of validation is commen-
suratewiththeapplication,andmustbejustifiedbytheuser.Theintended
against experimental results and, when found to satisfy written
use of software may also have GMPor ISO implications. Refer to Annex
criteria within the limits established in the documentation,
A2 for references and Guide E482 for further guidance on software
accepted and applied.
validation. Validation of computer modeling software is a complex issue.
7.3.3.2 If empirical equations are unsatisfactory as deter-
In many cases, validation of all aspects of operation of the code under all
mined by the user’s criteria, deterministic or stochastic
proposed modeling conditions is not feasible. The user is advised of the
possibility that none of the software packages referenced in Annex A1
solutions, or both, may be sought.
may be validated to national or international standards. The user is also
NOTE 11—Deterministic or stochastic approaches, or both, may be
advisedtocomparethecalculationresultswiththeexperimentalresults.If
utilized for the express purpose of supplementing a sparse measurement
this is not possible it would be convenient to use, at least, two different
database so that empirical relationships can be established and employed.
computer-modeling codes.
NOTE12—BecauseofthemorerigorousphysicalmodelsusedinMonte
8.2 Particulars of Three-Dimensional Model
Carlo codes, these may be considered for the purpose of verifying or
Construction—Procedures for building and using a three-
validating performance of a proposed deterministic or empirical solution.
dimensional model to integrate code results with dosimetry
7.3.3.3 Variousoptionsareavailabletotheend-userseeking
(verification) are discussed in Annex A5.
deterministic or stochastic solutions, or both. Software pack-
ages related to these modeling techniques are listed in Annex
8.3 Precautions and Implementation—It is important to test
A1. Refer to Table A3.1 in Annex A3 for guidance.
all assumptions for validity and to compare the results against
7.3.3.4 Inallcases,validationofmodelperformanceshould
dosimetry whenever possible.
bedoneusingacomprehensivemeasurementdatabase(dosim-
8.3.1 Dosimetry may be used to “fine tune” the model for
etry results). See Section 8 concerning validation.
the current system. This is an acceptable and recommended
practice when performed by qualified personnel.
8. Verification and Validation of Model Performance
8.4 The verification and validation procedure should be
8.1 ModelVerificationandValidation—Validationcompares
adhered to and documented.
the code output to results of an appropriate experiment.
8.5 Validation and Verification of New Computer Code
Verification confirms that the theory was implemented in a
Releases—Revisions of mathematical models are intended to
mathematically correct manner. Both verification and valida-
improve the physics or software functionality, or both. At a
tion of a model require the use of a comprehensive measure-
ment database of dosimetry results and other accepted calcu- minimum, verification of output from the updated software
with output from previously run input files should be per-
lations. In practice, verification and validation efforts often
overlap during model testing. formed.
E2232 − 21
9. Uncertainty in Model/Method Prediction 9.4.3.1 Discussion—Limited ability of a 1-D code to match
the number of layers in the problem forces combination of
9.1 Similar to dosimetric measurement, an estimate of
different materials, limited angle of incidence or inability to
uncertainty should accompany dose calculations. As a
run Monte Carlo in adjoint mode.
minimum, accuracy of the calculated dose value may be
9.4.4 Other Factors Limiting Appropriate Benchmarking—
expressed as the ratio of the calculated absorbed dose to the
Ability to run dosimetry or lack of access to the problem;
measured absorbed dose. The accepted degree of agreement
failure to perform a good measurement; lack of traceability or
between calculation and measurement will depend on the
inability to perform dosimetry in critical elements of the
user’s requirements.
problem.
NOTE 15—Measured values of absorbed dose are usually expressed in
9.4.5 Information about the Problem—Insufficient commu-
terms of dose to water. Dose in materials in a measurement or resulting
nication between the experimentalist and the model builder
from a model calculation might be expressed in terms of other materials
may obscure important details. Whenever possible, the model
which might then exhibit a numerical difference, related to differences in
stoppingpowersorabsorptioncoefficients(seeNoteA4.2andNoteA4.3).
buildershouldbewitnesstotheexperimentandinvolvedinall
measurements.
9.1.1 Refer to 2.1 forASTM standards on dosimetry meth-
ods and uncerta
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E2232 − 20 E2232 − 21
Standard Guide for
Selection and Use of Mathematical Methods for Calculating
Absorbed Dose in Radiation Processing Applications
This standard is issued under the fixed designation E2232; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This guide describes different mathematical methods that may be used to calculate absorbed dose and criteria for their
selection. Absorbed-dose calculations can determine the effectiveness of the radiation process, estimate the absorbed-dose
distribution in product, or supplement or complement, or both, the measurement of absorbed dose.
1.2 Radiation processing is an evolving field and annotated examples are provided in Annex A6 to illustrate the applications where
mathematical methods have been successfully applied. While not limited by the applications cited in these examples, applications
specific to neutron transport, radiation therapy and shielding design are not addressed in this document.
1.3 This guide covers the calculation of radiation transport of electrons and photons with energies up to 25 MeV.
1.4 The mathematical methods described include Monte Carlo, point kernel, discrete ordinate, semi-empirical and empirical
methods.
1.5 This guide is limited to the use of general purpose software packages for the calculation of the transport of charged or
uncharged particles and photons, or both, from various types of sources of ionizing radiation. This standard is limited to the use
of these software packages or other mathematical methods for the determination of spatial dose distributions for photons emitted
137 60
following the decay of Cs or Co, for energetic electrons from particle accelerators, or for X-rays generated by electron
accelerators.
1.6 This guide assists the user in determining if mathematical methods are a useful tool. This guide may assist the user in selecting
an appropriate method for calculating absorbed dose. The user must determine whether any of these mathematical methods are
appropriate for the solution to their specific application and what, if any, software to apply.
NOTE 1—The user is urged to apply these predictive techniques while being aware of the need for experience and also the inherent limitations of both
the method and the available software. Information pertaining to availability and updates to codes for modeling radiation transport, courses, workshops
and meetings can be found in Annex A1. For a basic understanding of radiation physics and a brief overview of method selection, refer to Annex A3.
1.7 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
1.8 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
This guide is under the jurisdiction of ASTM Committee E61 on Radiation Processing and is the direct responsibility of Subcommittee E61.04 on Specialty Application.
Current edition approved Dec. 1, 2020June 15, 2021. Published March 2021July 2021. Originally approved in 2002. Last previous edition approved in 20162020 as
E2232-16.E2232-20. DOI: 10.1520/E2232-20.10.1520/E2232-21.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2232 − 21
2. Referenced Documents
2.1 ASTM Standards:
E482 Guide for Application of Neutron Transport Methods for Reactor Vessel Surveillance
E3083 Terminology Relating to Radiation Processing: Dosimetry and Applications
2.2 ISO/ASTM Standards:
51707 Guide for Estimating Uncertainties in Dosimetry for Radiation Processing
52628 Practice for Dosimetry in Radiation Processing
2.3 ISO Standard:
ISO 12749-4 Nuclear energy, nuclear technologies, and radiological protection — Vocabulary — Part 4: Dosimetry for radiation
processing
2.4 International Commission on Radiation Units and Measurements Reports:
ICRU Report 85a Fundamental Quantities and Units for Ionizing Radiation
2.5 United States National Institute of Standards and Technology:
NIST Technical Note 1297 (1994 edition) Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement
Results
2.5 JCGM Documents:
JCGM 100:2008 GUM 1995, with minor corrections Evaluation of measurement date - Guide to the Expression of Uncertainty
in Measurement
JCGM 200:2012 VIM International Vocabulary of Metrology - Basic and General Concepts and Associated Terms
3. Terminology
3.1 Definitions:
3.1.1 accuracy (VIM)—closeness of agreement between a measured quantity value and a true quantity value of a measurand.
3.1.2 benchmarking—comparing model predictions to independent measurements or calculations under similar conditions using
defined criteria of uncertainty.
3.1.2.1 Discussion—
Benchmarking is a prerequisite before routine use of a mathematical model. Refer to 8.1 and Annex A5.
3.1.3 biasing (in a Monte Carlo simulation)—adjustment of the source particle selection or the transported particle weight, or both,
in a statistically valid manner so as to increase the particles in a region where the detector response is most important.
3.1.3.1 Discussion—
Biasing is a method used to reduce the estimated uncertainty or computer run times of Monte Carlo simulations. Monte Carlo
simulations using the natural probabilities of physical events may require unacceptably long run times to accumulate statistics for
rare events. The simulated probabilities may be altered to achieve the uncertainty goals for the simulation in acceptable run times
by biasing the sampling from the probability distributions. The number of particles tracked and the particle weights may be
adjusted so as to ensure a statistically valid sample from the probability distributions. Appropriate biasing requires a detailed
knowledge of the model and the influence of rare events. As with all simulations, results should be compared with benchmark
measurements or simulation results originated by a different code.
3.1.4 build-up factor—ratio of the total value of a specified radiation quantity (such as absorbed dose) at any point in that medium
to the contribution to that quantity from the incident un-collided radiation reaching that point.
3.1.4.1 Discussion—
The concept of build-up applies to the transport of photons.
3.1.5 deterministic method—a mathematical method using transport equations to directly calculate the radiation field over all space
as a function of radiation source and boundary conditions.
For referenced ASTM and ISO/ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book
of ASTM Standards volume information, refer to the standard’s Document Summary page on the ASTM website.
Available from International Organization for Standardization (ISO), ISO Central Secretariat, Chemin de Blandonnet 8, CP 401, 1214 Vernier, Geneva, Switzerland,
https://www.iso.org.
Available from International Commission on Radiation Units and Measurements, 7910 Woodmont Ave., Suite 800, Bethesda, MD 20815 USA.
Available as a download from the NIST web site at: http://physics.nist.gov/Pubs/guidelines/TN1297/tn1297s.pdf.
Available from JCGM-BIPM website at: http://www.bipm.org.
E2232 − 21
3.1.5.1 Discussion—
The point kernel and discrete ordinate methods are examples of deterministic methods.
3.1.6 discrete ordinate method—a deterministic method for approximate numerical solution of the transport equation in which the
direction of motion is divided into a finite number of discrete ordinate angles.
3.1.6.1 Discussion—
In the discrete ordinates approximation, the transport equation becomes a set of coupled equations, one for each discrete ordinate.
Particle behaviors along paths intermediate to described paths are approximated by a weighted average (numerical quadrature) of
adjacent paths (1). The method is useful for both electron and photon sources when appropriate assumptions can be made.
3.1.7 empirical method—a method derived from fitting an approximating function to experimental data or Monte Carlo calculation
result.
3.1.7.1 Discussion—
Empirical models are generally developed by fitting equations (for example, polynomial) to experimental data or simulation output
derived from another mathematical method.
3.1.8 history (of a particle)—record of all simulated interactions along particle’s track as used in stochastic simulations (for
example, Monte Carlo).
3.1.8.1 Discussion—
A particle history begins with the starting position, energy and direction of a particle, follows all its interactions, and terminates
in one of several outcomes such as absorption, escape from the boundary of the problem, or reaching a cut-off limit (such as a
cut-off energy). A particle history is the systematic generation of a random, simulated particle track that is obtained according to
the known physical interactions of either electrons or photons with the material being traversed. History and particle history are
considered synonymous.
3.1.9 mathematical method—a method of solution of an electron or photon transport problem, or both, using algebraic relations
and mathematical operations to represent the system and its dynamics.
3.1.10 mathematical model—a mathematical description of a physical problem based on physical laws or empirical correlation,
or both.
3.1.11 Monte Carlo method—a simulation method used for calculating absorbed dose, energy spectra, charge, fluence and fluence
rate in a volume of interest using a statistical summary of the radiation interactions.
3.1.11.1 Discussion—
A Monte Carlo calculation consists of running a large number of particle histories (simulations) until some acceptable statistical
uncertainty in the desired calculated quantity (such as dose) has been reached. This calculation method is suitable for problems
involving either electrons or photons or both. This technique produces a probabilistic approximation to the solution of a problem
by using statistical sampling techniques. See also stochastic and history.
3.1.12 numerical convergence—process in which the iterative solution of an equation or set of equations changes by less than
some defined value.
3.1.12.1 Discussion—
The mathematical equations describing a problem are often so complex that an analytical (algebraic) solution is not possible. The
solution of the equations can be estimated by an iterative process of progressively refining approximate solutions at a grid of
discrete locations. A consistent set of solutions arrived at by this method achieves numerical convergence. Convergence may not
be obtained if the discrete locations are too widely separated (that is, the grid is too coarse).
3.1.13 point kernel method—a deterministic method for calculating dose based on integrating the contributions from point sources.
3.1.13.1 Discussion—
The point kernel method is typically used for photon transport applications. The radiation source is modeled as a large set of point
sources. The absorbed dose, dose equivalent or exposure is estimated at a dose point by integrating the contribution from each of
the point sources. A multiplicative value (the semi-empirical build-up factor) is used to account for the contribution from scattered
(indirect) radiation from regions not in the direct path between the source point and field point.
The boldface numbers in parentheses refer to the list of references at the end of this standard.
E2232 − 21
3.1.14 radiation field—a function describing the particle density and the distributions of energy, direction and particle type at any
point.
3.1.15 radiation transport theory—an analytical description of the propagation of a radiation field according to the physical laws
governing the interaction of radiation with matter.
3.1.15.1 Discussion—
In its most general form, transport theory is a special branch of statistical mechanics, which deals with the interaction of the
radiation field with matter.
3.1.16 semi-empirical model—an empirical model in which the fitting parameters are constrained so that the model satisfies one
or more physical laws or rules.
3.1.16.1 Discussion—
The satisfaction of such physical rules may enable the model to be applicable over a wide range of energies and materials.
3.1.17 spatial mesh—subdivision of the radiation interaction volume of interest into a grid of discrete spatial elements for
performing a transport calculation.
3.1.18 statistical component of uncertainty—component of uncertainty evaluated by statistical analysis of a series of calculated
values.
3.1.18.1 Discussion—
The inherent sampling uncertainty of the Monte Carlo method can be estimated as a statistical uncertainty by applying statistical
sampling techniques to the number of simulated histories. For calculations without biasing, the statistical uncertainty scales as the
reciprocal of the square root of the number of histories.
3.1.19 stochastic methods—methods using mathematical equations containing random variables to describe or summarize the
physical processes in the system being studied. A random variable is a variable whose value is a function of a statistical distribution
of random values.
3.1.19.1 Discussion—
The Monte Carlo method is the only stochastic method discussed in this guide. See also Monte Carlo and history.
3.1.20 non-statistical component of uncertainty—component of uncertainty evaluated by means other than statistical analysis of
a series of calculated values.
3.1.20.1 Discussion—
There are non-statistical components of uncertainties associated with the necessary simplifying assumptions needed to approximate
the physical paths of electrons in the model and uncertainties in the cross-sections for the different interactions. These uncertainties
can be estimated by analytical techniques. A non-statistical component of uncertainty could result from the difference in geometry
and material composition of the modelled irradiator versus the actual irradiator. Other sources of non-statistical component of
uncertainty are the inadequate description of the problem and approximations to actual physics.
3.1.21 transport equation—an integro-differential equation describing the motion of particles or radiation through a medium.
3.1.21.1 Discussion—
The transport equation contains various terms corresponding to sources of particles, particle streaming and particle scattering in
and out of an infinitesimal volume of phase space.
3.1.22 uncertainty of calculation result—non-negative parameter associated with the result of a calculation that characterizes the
spread of values that could reasonably be attributed to the derived quantity.
3.1.22.1 Discussion—
Like absorbed-dose measurement, the absorbed-dose calculation should also be accompanied by an estimate of uncertainty.
3.1.23 validation—accumulation of documented experimental evidence, used to demonstrate that the mathematical method is a
reliable prediction technique.
3.1.23.1 Discussion—
Validation compares a code or theory with results of an appropriate experiment.
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3.1.24 verification—confirmation by examination of evidence that the mathematical method has been properly and successfully
applied to the problem.
3.1.24.1 Discussion—
It is important to know the type of radiation sources, geometries, energies, etc. for which a code has been validated. The calculated
results will also depend on quantities at the user’s disposal such as cut-off energy (for Monte Carlo) or mesh size (for discrete
ordinate methods). Verification demonstrates that theory was implemented in the way intended, and that the simulation was
performed in accordance with its requirements and specifications.
3.1.25 zoning—The geometric description used to break up a larger region into smaller segments in which to calculate the dose.
3.1.25.1 Discussion—
Partitioning a zone into smaller segments is referred to as subzoning.
3.2 Definitions of other terms used in this standard that pertain to radiation measurement and dosimetry may be found in
ISO/ASTM Practice 52628. Other terms that pertain to radiation measurement and dosimetry may be found in Terminology E3083
and ISO Terminology 12749-4. Where appropriate, definitions used in these standards have been derived from, and are consistent
with definitions in ICRU Report 85a, and general metrological definitions given in the VIM.
4. Significance and Use
4.1 Use as an Analytical Tool—Mathematical methods provide an analytical tool to be employed for many applications related to
absorbed dose determinations in radiation processing. Mathematical calculations may not be used as a substitute for routine
dosimetry in some applications (for example, medical device sterilization, food irradiation).
4.2 Dose Calculation—Absorbed-dose calculations may be performed for a variety of photon/electron environments and irradiator
geometries.
4.3 Evaluate Process Effectiveness—Mathematical models may be used to evaluate the impact of changes in product composition,
loading configuration, and irradiator design on dose distribution.
4.4 Complement or Supplement to Dosimetry—Dose calculations may be used to establish a detailed understanding of dose
distribution, providing a spatial resolution not obtainable through measurement. Calculations may be used to reduce the number
of dosimeters required to characterize a procedure or process (for example, dose mapping).
4.5 Alternative to Dosimetry—Dose calculations may be used when dosimetry is impractical (for example, granular materials,
materials with complex geometries, material contained in a package where dosimetry is not practical or possible).
4.6 Facility Design—Dose calculations are often used in the design of a new irradiator and can be used to help optimize dose
distribution in an existing facility or radiation process. The use of modeling in irradiator design can be found in Refs (2-7).
4.7 Validation—The validation of the model should be done through comparison with reliable and traceable dosimetric
measurements. The purpose of validation is to demonstrate that the mathematical method makes reliable predictions of dose and
other transport quantities. Validation compares predictions or theory to the results of an appropriate experiment. The degree of
validation is commensurate with the application. Guidance is given in the documents referenced in Annex A2.
4.8 Verification—Verification is the confirmation of the mathematical correctness of a computer implementation of a mathematical
method. This can be done, for example, by comparing numerical results with known analytic solutions or with other computer
codes that have been previously verified. Verification should be done to ensure that the simulation is appropriate for the intended
application. Refer to 3.1.24.
NOTE 2—Certain applications of the mathematical model deal with Operational Qualification (OQ), Performance Qualification (PQ) and process control
in radiation processing such as the sterilization of healthcare products. The application and use of the mathematical model in these applications may have
to meet regulatory requirements. Refer to Section 6 for prerequisites for application of a mathematical method and Section 8 for requirements before
routine use of the mathematical method.
4.9 Uncertainty—An absorbed dose prediction should be accompanied by an estimate of overall uncertainty, as it is with
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absorbed-dose measurement (refer to ISO/ASTM 51707 and NIST Technical Note 1297). JCGM100:2008 and JCGM200:2012).
In many cases, absorbed-dose measurement helps to establish the uncertainty in the dose calculation.
4.10 This guide should not be used as the only reference in the selection and use of mathematical models. The user is encouraged
to contact individuals who are experienced in mathematical modelling and to read the relevant publications in order to select the
best tool for their application. Radiation processing is an evolving field and the references cited in the annotated examples of Annex
A6 are representative of the various published applications. Where a method is validated with dosimetry, it becomes a benchmark
for that particular application.
5. Classification of Mathematical Methods and General Application
5.1 Mathematical methods for radiation transport can be used to estimate the absorbed dose to a small volume or point. The dose
distribution within the entire product can be determined by calculations at different points within the product.
5.2 Types of Methods—Four general types of methods are in use: Monte Carlo, deterministic, semi-empirical and empirical. Both
Monte Carlo and deterministic methods are based on the detailed physics of the interaction of radiation with matter.
5.2.1 Monte Carlo methods involve simulating paths of a finite number of photons or electrons and estimating dose by summing
and averaging the histories of many energy deposition events.
5.2.2 Deterministic methods use equations describing the transport of radiation in matter to perform a direct estimate of the total
radiation field, absorbed dose and other responses.
5.2.3 Empirical and semi-empirical methods are based on statistical relationships of measurements or calculations for a particular
system.
5.3 Monte Carlo Method—The Monte Carlo method simulates the paths of particles such as electrons and photons from the source
to the dose volume. See Note 1, Refs (8-19) and Annex A1 for examples and codes. See also A3.3 and A3.4.4 for brief discussions
of the physics of electron and photon transport and the Monte Carlo method respectively.
5.3.1 Advantages—Unlike other methods, the Monte Carlo method can, in principle, account for all interactions and provide a
realistic simulation of actual all scattering and energy loss events. All contributions to the absorbed dose can be taken into account
including electron and photon scattering from nearby objects. (See Note 3.) In addition, the Monte Carlo method has the great
advantage of being the method most capable of simulating the actual radiation transport in complex three-dimensional geometry.
NOTE 3—Such objects could be structures outside the system of irradiated material(s) for which the dose distribution is to be calculated. For example,
these might include shielding layers, photon beam collimators, e-beam accelerator heads, or walls of concrete or lead surrounding a Co radiation source.
5.3.2 Disadvantages—Because electrons (including those generated by photons) in the energy range of 50 keV to 10 MeV undergo
large numbers of scattering events, exact simulation of all photon and electron paths is not feasible or practical. Instead,
approximate electron paths are employed, as in the so-called “condensed history Monte Carlo method” (20 and 21). For electrons,
approximate artificial trajectories using large path length steps and a multiple-scattering approach to particle deflections are
employed in standard Monte Carlo codes. (See Annex A1.) The standard Monte Carlo codes listed in Annex A1 and Refs (8-19)
use this condensed history approach. However, such approximate paths may lead to significant errors, particularly at locations
where transport across surfaces or material interfaces is important. See Note 4.
NOTE 4—In some Monte Carlo codes (17), improved accuracy near material boundaries has been obtained using shorter paths near interfaces between
different materials.
NOTE 5—To reduce computational time, limits to the problem may be specified, such as physical boundaries and energy cut-offs, when the contributions
to the problem made outside of these boundaries are no longer expected to be significant. Variance reduction techniques help to improve the rate of
numerical convergence but require a sophisticated understanding of probability distributions.
5.3.2.1 One of the main difficulties with this method is that when applied to geometries where reductions in fluence spanning
several orders of magnitude might occur, or where the absorbed dose in very small volumes is required, the statistical component
of the uncertainty will often be large. This can be alleviated using variance reduction techniques. See Note 5.
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5.3.2.2 Calculations of dose should provide a range of dose values over a region near where the dose is to be measured. This is
to permit estimation of the effect of variations in the location/orientation of a dosimeter in that region. This determines the dose
sensitivity associated with placement of the dosimeter and allows determination of this type of error.
5.3.3 Statistical Uncertainty—The inherent uncertainty in the calculated value of dose due to sampling in the Monte Carlo method
can be estimated by applying statistical sampling techniques to the number of histories. For calculations without biasing, the
statistical uncertainty scales as the reciprocal of the square root of the number of histories run.
5.3.3.1 Special care must be taken when using variance reduction techniques which are used to increase statistics in an otherwise
poorly populated phase space (for example, shielding calculation where only high energy photons are tracked through the shield).
This is accomplished by introducing sampling probabilities which may be highly varying and have an adverse effect on the
convergence of Monte Carlo calculations.
5.3.4 Non-statistical Component of Uncertainty—These uncertainties can be estimated by analytical techniques, which may
include sensitivity analysis (changing a value of a parameter by an amount related to its uncertainty and rerunning the calculation
to compare the results. Various elements of the calculation can be validated with dosimetry.
NOTE 6—There is great potential for large discrepancies in results because there is no estimate of non-statistical component of uncertainty resulting from
software. Refer to Section 9. Construction of an uncertainty budget is recommended.
5.4 Deterministic Methods—These methods use analytical equations to summarize radiation fluence rate through target materials.
Such complex equations cannot be solved directly but must be solved iteratively in the computer calculations.
5.4.1 Discrete Ordinates Methods—These methods have been used for both electron and photon sources (22 and 23). This name
is given to several closely related techniques for obtaining approximate solutions to the transport equations that contain both
integral and partial derivative terms. Various methods have been developed to solve these equations (24). All of these methods
place limits on the angular variable such that the incident radiation is represented as streaming only along a finite number of
directions rather than all possible directions as contained in the transport equation. Extension of this technique to 2-D and 3-D has
been done by several authors (25).
5.4.2 Point Kernel Methods—Point kernel methods are used mainly for photon transport problems (26). In point kernel methods,
the radiation source volume is approximated by a number of isotropic source points. The absorbed dose at each dose point is
obtained by summing the dose contribution from all source points. The calculation takes into account the distance between the dose
point and the source point and approximates the scatter within the intervening product through the use of a build-up factor.
Build-up factors are theoretically calculated and sometimes fitted to empirical functions. These factors provide an approximation
for the contribution of scattered photons from surrounding material. Approximations are also required to account for the energy
spectrum and variations in the atomic number in different intervening or scattering materials.
NOTE 6—There are a number of general databases available for the photon buildup factors needed for these codes (Annex A1).
5.4.3 Advantages—Deterministic methods may be faster than Monte Carlo, and can be benchmarked against dosimetry.
5.4.4 Disadvantages—Deterministic methods give no innate estimate of statistical uncertainty. Iterative solution methods may be
susceptible to numerical convergence errors and oscillatory solutions.
5.4.5 Uncertainties—There are three sources of uncertainties in deterministic models. These are (1) the approximations used to
create physical models and cross-sections (for example, energy straggling is neglected in deterministic methods), (2) the effect of
representing a continuous problem in space, angle and energy with a finite mesh in all these variables and (3) truncation error due
to a finite number of discrete ordinates.
5.4.6 The accuracy of the point kernel treatment may be comparable to that of a Monte Carlo calculation for configurations where
the point kernel approximation is valid (27).
5.5 Empirical and Semi-empirical Methods:
5.5.1 Empirical—Empirical methods typically involve fitting analytical functions to experimental measurements (or to
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calculations using other methods). The model equations are typically specific to a particular radiation facility and their predictive
capabilities are not generally transferable to other facilities or products. Some simple equations exist for calculating the range of
electrons in condensed matter (28), electron energy loss (29) and depth-dose relationships in various materials (30).
5.5.2 Semi-Empirical—These are empirical methods in which the fitting parameters are constrained so that the model satisfies one
or more physical laws or rules. These methods provide a more generally applicable mathematical model than the empirical method
and are adjustable to physical parameters of the facility, source and products, such as energy, density and composition. In general,
these are software-based programs with variable parameter inputs. Equations, codes and databases are available (31-34).
5.5.3 Advantages—Empirical and semi-empirical models are fast and do not require cross-sections, build-up factors and zoning
since they are implicitly included in the coefficients of the model. No special knowledge, such as needed for Monte Carlo or
deterministic methods, is required. Semi-empirical models may be applicable to multiple facilities.
5.5.4 Disadvantages—Empirical methods are likely to be very limited in their application. Generally, empirically derived
equations cannot be transferred to other sites or irradiation applications, or both, that were not part of the original database used
to generate the model. These methods may be difficult to implement for systems with complicated geometry.
NOTE 7—Although empirical or semi-empirical codes may give some useful guidance, modern Monte Carlo codes on modern platforms are often very
fast in these types of applications.
5.5.5 Uncertainties—Uncertainty in both methods is influenced by factors such as lack of homogeneity in the product, dosimeter
location and uncertainty associated with dose measurements.
6. Prerequisites for Application of a Mathematical Method
6.1 Facility and Related Geometry Considerations:
6.1.1 Detailed drawings of irradiation facility equipment, source-related equipment and associated geometries, should be obtained,
physically verified, and documented. Examples of gamma irradiation facilities are given in Figs. 1 and 2.
NOTE 8—Fig. 1 shows a physical model of a typical gamma irradiator with product in aluminum totes. For clarity, eight totes and part of the source shroud
FIG. 1 Solid Model of a modified Nordion JS9600 Irradiator with a two layer roller conveyor, showing the product totes (1) and the ra-
diation source (2). The model was developed using EGSPP (35)
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FIG. 2 Picture and Simulation of a Gamma Production Irradiator and Research Loop (36)
have been removed. The tote irradiator uses a shuffle-and-dwell concept. Each product tote is irradiated for a defined period of time before it is moved
to the next irradiation position. The source rack containing the radiation sources is shown (35).
NOTE 9—Fig. 2 shows a photograph on the left of a research carrier and the graphical user interface window of a mathematical model shown on the right
photograph. All product is contained in aluminum totes. For the research carrier, product is brought into the radiation chamber and irradiated for a defined
period of time, and then leaves the irradiation chamber. The graphical user interface shows ray tracing between the radiation source (1) and the dose
volume (2) (36).
6.1.2 Detailed drawings of materials to be irradiated (products, targets) and their associated geometries, with physical verification
of the same (composition of constituents, densities) should be collected and documented.
6.1.3 The type of source(s) present (electrons, photons), source energy spectrum, source output angular distribution, source size
(point or distributed, diffuse source with variable activity etc.) and the number of sources should be specified and documented.
NOTE 10—In the case of gamma-ray sources (for example, Co sources), the photon energy spectrum may be difficult to obtain experimentally or estimate
theoretically. In general, for photons with energies 200 keV and above, a broad low energy contribution to the spectrum is created via Compton scattering.
6.2 Personnel—Trained personnel should be involved in all aspects of model development, program execution, data reduction and
the evaluation of results. There is no standard set of qualifications that can be recommended. Interaction of personnel with all
phases of the modeling exercise should be documented according to the end-user’s policy and procedural plans. The individual
developing or using the selected model should be actively involved in the verification experiment(s). See Section 8 concerning the
verification and validation experiments.
6.2.1 All personnel involved in modelling using MC and other techniques should be knowledgeable in radiation physics and
should have received training for the code(s) that they are using.
6.2.2 All training and significant experience of personnel involved in the modeling effort should be documented.
6.3 Computer Equipment and Software—Requirements should be reviewed and documented.
6.3.1 Sufficient information regarding the equipment and software that was used to run the calculations, including version numbers
and compilers as appropriate, should be recorded to enable the calculations to be reproduced.
6.3.2 All operating system software, modeling software, compilers and commercial products such as spreadsheets and data
analysis tools should have their titles and version numbers recorded.
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6.4 All relevant dosimetry data, reports of measurement and other physical evidence should be collected and filed or referenced
for use in validation of model performance. See Section 8 concerning validation experiments.
7. Specification of Modeling Strategy and Method Selection
7.1 Specification of the Modeling Effort—All modeling approaches should be described in the form of a written protocol detailing
the requirements for successful execution and subsequent completion of the exercise(s) relative to written criteria for success. The
protocol should, at a minimum, include:
7.1.1 Specification of the source type and geometry as per 6.1.
7.1.2 Specification of facility (transport mechanism, support structures, biological shield as per 6.1) where required for the
particular calculation.
7.1.3 Specification of target materials and geometries as per 6.1.
7.1.4 Declaration of personnel as per 6.2.
7.1.5 Specification of computer hardware and software as per 6.3 (see also 7.2).
7.2 Criteria for Selection—Most problems are rarely modeled exactly as they appear in reality; major approximations for
simplification may be required to reduce the amount of effort required to build the model description and run times. These
assumptions should be documented. Method selection will be primarily determined by the following criteria:
7.2.1 Source Description—For a photon source, any of the four methods may be chosen. For an electron source, the point kernel
method is not recommended since the point kernel method assumes that the energy of the interacting particle is delivered at a point
and then distributed statistically around that reaction point, as in the case of photons. On the contrary, electrons interact
continuously with matter along their path and because of this the point kernel method is not appropriate.
7.2.2 Level of Detail—The level of detail to be included in the model, or the granularity of the problem, will influence the method
selection. If the problem can be described as regions of homogeneous material, the point kernel method may be most appropriate
for photon problems, if speed and spatial resolution are important. If the problem must be further broken down into smaller regions
of different material (composition or density) in order to achieve accuracy, more complex input files will be needed.
7.2.2.1 Available software may have geometry replication and tiling features that are very useful for this purpose. If the target size
is small relative to geometry or source description, Monte Carlo may require biasing or modification to include a larger volume
wherein the dose will be an average value over a larger volume than desired. The Monte Carlo method can be used to provide a
refinement of the point kernel build-up calculation used with the point kernel method. This can then help point kernel calculations
achieve a required accuracy with optimized efficiency (time, resolution) (27, 37, 38).
7.2.3 Set-up Time—The complexity of three-dimensional problem descriptions in the input files and manipulation of the output
files is where most of the effort is concentrated and can be very time consuming. It may also be necessary to make modifications
to the code to accommodate the specific problem to be solved. If modifications to the code are necessary, revalidation will be
required, particularly if the physics modeled in the code has been changed.
7.3 Selection of Method Type:
7.3.1 The criteria for selection of a method type require input from various sources. Such sources include in-house and outside
modeling expertise, model-based testing history and availability of verified and validated modeling code(s). These criteria should
be documented as per 7.1.
7.3.2 Evaluation of the impact of the code on those items stated in 7.1.1 – 7.1.5 will typically be geared towards minimization
of time for model set-up, execution and evaluation in exchange for exactness of solution set(s).
7.3.3 There are currently no written methods available for determining the optimum code to use. However, some general
guidelines are as follows:
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7.3.3.1 Empirical equations can be developed, evaluated against experimental results and, when found to satisfy written criteria
within the limits established in the documentation, accepted and applied.
7.3.3.2 If empirical equations are unsatisfactory as determined by the user’s criteria, deterministic or stochastic solutions, or both,
may be sought.
NOTE 11—Deterministic or stochastic approaches, or both, may be utilized for the express purpose of supplementing a sparse measurement database so
that empirical relationships can be established and employed.
NOTE 12—Because of the more rigorous physical models used in Monte Carlo codes, these may be considered for the purpose of verifying or validating
performance of a proposed deterministic or empirical solution.
7.3.3.3 Various options are available to the end-user seeking deterministic or stochastic solutions, or both. Software packages
related to these modeling techniques are listed in Annex A1. Refer to Table A3.1 in Annex A3 for guidance.
7.3.3.4 In all cases, validation of model performance should be done using a comprehensive measurement database (dosimetry
results). See Section 8 concerning validation.
8. Verification and Validation of Model Performance
8.1 Model Verification and Validation—Validation compares the code output to results of an appropriate experiment. Verification
confirms that the theory was implemented in a mathematically correct manner. Both verification and validation of a model require
the use of a comprehensive measurement database of dosimetry results and other accepted calculations. In practice, verification
and validation efforts often overlap during model testing.
8.1.1 Model Benchmarking—Model benchmarking is used both to verify a mathematical method and to validate the overall model
construction and underlying physics of the method to produce reliable results. Comparing current model results with previously
well-characterized systems is part of the model testing. Comparing model results with dosimetry for the specific problem being
modeled is strongly recommended whenever possible. Differences between measurement and calculations should be consistent
with uncertainty estimates for both the measurements and the calculations.
8.1.1.1 There are a limited number of referenced benchmark examples in the literature and these may be inadequate in number
to validate a method and inadequate in detail for comparison with the model under consideration. The model of the application
of interest should be as nearly the same as possible to the benchmark example. Benchmark examples may be found in Annex A5.
An example comparing the results of several methods (Monte Carlo, deterministic and semi-empirical) with dosimetry can be
found in Ref (39).
NOTE 13—One or more well-defined problems may be run through the model on the user’s hardware and software platform(s) and compared to accepted
results for execution of the model generated by one or more organizations (typically, this includes, at a minimum, the firm issuing the modeling software).
Input and output are compared, and the modeling package’s performance is deemed verified upon successful completion of the test(s).
NOTE 14—Formal software testing is not addressed in this guide. It is desirable to perform calculations with a modeling code that has undergone a formal
software validation program. The level of validation is commensurate with the application, and must be justified by the user. The intended use of software
may also have GMP or ISO implications. Refer to Annex A2 for references and Guide E482 for further guidance on software validation. Validation of
computer modeling software is a complex issue. In many cases, validation of all aspects of operation of the code under all proposed modeling conditions
is not feasible. The user is advised of the possibility that none of the software packages referenced in Annex A1 may be validated to national or
international standards. The user is also advised to compare the calculation results with the experimental results. If this is not possible it would be
convenient to use, at least, two different computer-modeling codes.
8.2 Particulars of Three-Dimensional Model Construction—Procedures for building and using a three-dimensional model to
integrate code results with dosimetry (verification) are discussed in Annex A5.
8.3 Precautions and Implementation—It is important to test all assumptions for validity and to compare the results against
dosimetry whenever possible.
8.3.1 Dosimetry may be used to “fine tune” the model for the current system. This is an acceptable and recommended practice
when performed by qualified personnel.
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8.4 The verification and validation procedure should be adhered to and docum
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