Table Of ContentNCRP REPORT No. 108
CONCEPTUAL BASIS FOR
CALCULATIONS OF
ABSORBED-DOSE
DISTRIBUTIONS
Recommendations of the
NATIONAL COUNCIL ON RADIATION
PROTECTION AND MEASUREMENTS
Issued March 31, 1991
Sexond Reprinting February 1, 1995
National Council on Radiation Protection and Measurements
7910 WOODMONT AVENUE / Bethesda, MD 20814
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Library of Congress Cataloging-in-Publication Data
Conceptual basis for calculations of absorbed-dose distributions.
p. cm.-(NCRP report; no. 108)
Includes bibliographical references and index.
ISBN 0-929600-16-9
1. Radiation dosimetry. 2. Ionizing radiation-Measurement.
I. National Council on Radiation Protection and Measurements.
11. Series.
QC795.32.RX66 1991 91-9135
539.7'22'0287-dc20 CIF'
Copyright 8 National Council on Radiation
Protection and Measurements 1991
All rights reserved. This publication is protected by copyright. No part of this publica-
tion may be reproduced in any form or by any means, including photocopying, or
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Preface
The idea for this report emerged, in the early 1970's, from the need
of an NCRP Scientific Committee to characterize the beta-ray depth-
dose distribution in connection with immersion doses. It was realized,
however, that the calculation of such a distribution was only a small
part of the very much larger task concerned with the theoretical,
mathematical and computational concepts involved in the develop-
ment of absorbed-dose distributions in general. To address this issue
in an allencompassing manner, the NCRP formed Scientific Commit-
tee 52 on the Conceptual Basis of Calculations of Dose Distribution.
In either external or internal irradiation, the absorbed dose is
usually non-uniform in any structure and, in particular, in the
human body. This non-uniformity is to be distinguished from the
stochastic variations that exist even in regions where the dose is
uniform and that are the subject of microdosimetry and not this
report. Many illustrations of absorbed dose non-uniformity come to
mind: for example, the absorbed-dose distributions from hot parti-
cles, from internal emitters, from radiation therapy, from radiation
accidents and from environmental radiation. There can even be addi-
tional non-uniformity with respect to time of the non-uniform distri-
bution, for example, in the redistributions of administered radioac-
tivity in the body.
For all absorbed-dose calculations, there is a source (or sources) of
radiation and a receptor (or receptors) of some of the energy of this
radiation, with or without intervening material between the source
and receptor. The calculation of absorbed-dose distributions requires
specification of the sources and receptors, characterization of their
geometrical relationships and consideration of the physical interac-
tions of the radiations involving attenuation, scattering and the
production of secondary radiations. All these processes are consid-
ered in the basic transport equation, the general theorems and prop-
erties and the methods of solution of which are described in the
transport theory.
The report is a systematic presentation, discussion and compila-
tion of all the concepts involved. It contains some complicated mathe-
matics that will be of interest to the mathematically knowledgeable,
but that should not discourage those not mathematically inclined.
iv / PREFACE
The text of the report contains detailed explanations of all the con-
cepts and of the consequences of the equations so that, even omitting
the mathematics, a broad and comprehensive understanding can be
obtained of what is entailed in the calculation of an absorbed-dose
distribution.
The cutoff date for the report is about two years ago and, hence,
the report is lacking in the most current references. However, this
field does not evolve at a rapid pace and the current literature is,
therefore, not abundant and can be reviewed easily.
In accord with the recommendations of NCRP Report No. 82, SI
Units in Radiation Protection and Measurements, as of January 1990,
only SI units are used in the text. Readers needing factors for conver-
sion of SI to conventional units are encouraged to consult Report No.
82.
This report was prepared by NCRP Scientific Committee 52 on
Conceptual Basis of Calculations of Dose Distributions. Serving on
the Committee during the preparation of this report were:
Harald H. Rossi, Chairman
105 Larchdale Avenue
Upper Nyack, New York
R. G. Alsmiller, Jr. William C. Roesch
Engineering Physics and 1646 Butternut
Mathematics Division Richland, Washingtan
Oak Ridge National Laboratory
Oak Ridge, Bnnessee
Martin J. Berger Lewis V. Spencer
5011 Elm Street Post Office Box 87
Bethesda, Maryland Hopkinsville, Kentucky
Albrecht M. Kellerer Marco A. Zaider
GSF Institut fiir Strahlungbiologie Radiological Research
Neuherberg, Germany Laboratory
Columbia University
New York, New York
NCRP Secretariat: Thomas Fearon (1976-80)
James A. Spahn, Jr. (1981)
James T. Walker (1982-84)
Constantine J. Maletskos (1985-91)
The Council wishes to express its appreciation to the Committee
members for the time and effort devoted to the preparation of this
report.
PREFACE, 1 V
Especial thanks are due to Marco Zaider for his contribution to
the editing of scientific aspects of this report.
Warren K. Sinclair
President
Bethesda, Maryland
5 March 1991
Contents
.
....................................
1 Introduction 1
1.1 The Concept of Absorbed Dose .................. 1
.........
1.2 Dose Measurement and Dose Calculation 3
1.3 Elements of Dose Calculations .................. 4
.
2 Transport Formalisms .......................... 6
..................
2.1 Concepts in Dose Calculations 6
...........................
2.2 Transport Equation 8
.
3 Sources ....................................... 14
3.1 Specification of Sources ......................... 14
3.2 Simplified Representations of Sources ............ 15
.
.....................................
4 Receptors 17
.
.................................
5 Cross Sections 20
..............................
5.1 Schematization 20
5.2 General Aspects of Required Cross Sections ....... 22
.
6 Transport Theory-General Theorems and
.....................................
Properties 26
6.1 Integral Form of the Transport Equation .......... 26
.........
6.2 Iterative Solutions (Orders of Scattering) 27
6.3 Density Scaling Theorem ...................... 28
6.4 Fano's Theorem .............................. 29
.........................
6.5 Energy Conservation 29
................................
6.6 Superposition 30
....................
6.7 Adjoint Transport Equation 31
6.8 Reciprocity .................................. 33
6.9 Transport Equations in Commonly Used Coordinate
....................................
. Systems 34
..........
7 Transport Theory-Methods of Solution 36
.................................
7.1 Introduction 36
7.2 Radiation Equilibrium and Space-Integrated
.............................
Radiation Fields 38
. .
7.3 Continuous Slowing-Down Approximation (CSDA) 40
.............
7.4 Numerical Integration Over Energy 43
. .
7.5 Elementary Problems Involving Particle Direction 45
'
7.5.1 Thin-Foil Charged Particle Problems ........ 45
7.6 Penetration Studies ........................... 47
.....................
7.6.1 The Moment Method 47
.........
7 .6.2 Discrete-Ordinates Transport Codes 49
CONTENTS 1 vii
7.6.2.1 Neutron-Photon Transport .......... 49
7.6.2.2 Dosimetry Calculations By the Method
of Discrete Ordinates ..............
7.7 Spectral Equilibrium and Related Concepts .......
7.7.1 Aspects Applicable to All Radiations ........
7.7.2 Electrons ..............................
7.7.3 Photons and Neutrons ....................
7.8 Radiation Quasi-equilibrium ...................
7.8.1 Transient Equilibrium ...................
7.8.2 Non-uniform Sources .....................
7.8.3 Non-uniformity in the Internal Dosimetry of
Radionuclides ...........................
7.8.4 Non-uniform Media ......................
.
8 Monte-Carlo Methods ...........................
8.1 Principles ..................................
8.2 Analog Monte-Carlo and Variance-Reduction
Bchniques ..................................
8.3 Transport Codes .............................
8.3.1 Neutron-Photon Transport at Energies 520
MeV ..................................
8.3.2 Electron-Photon Cascades .................
8.3.3 Nucleon-Meson Transport at Energies >20
MeV ..................................
8.3.4 Dosimetric Calculations ..................
9. Geometric Considerations .......................
9.1 Absorbed Dose in Receptor Regions ..............
9.2 Reciprocity Theorem ..........................
9.3 Isotropic Point-Source Kernels ..................
9.4 Point-Pair Distance Distributions and Geometric
Reduction Factors ............................
.
10 Calculation of the Dose Equivalent ...............
List of Symbols ....................................
.
Appendix A Information about Cross Sections for
Transport Calculations .............................
A.l Photon Cross Sections ........................
A.l.l Photoelectric Effect ......................
A.1.2 Fluorescence Radiation and Auger Electrons .
.
A 1.3 Incoherent (Compton) Scattering ...........
A.1.4 Pair Production ........................
A.1.5 Coherent (Rayleigh) Scattering ............
A.1.6 Photonuclear Effect .....................
A.1.7 Attenuation Coefficient ..................
A.1.8 Energy-Absorption Coefficient .............
A.1.9 Photon Cross-Section Compilations .........
viii / CONTENTS
A.2 Cross Sections for Charged Particles ............. 113
A.2.1 Elastic Scattering of Electrons by Atoms .... 113
A.2.2 Elastic Scattering of Protons by Atoms ...... 117
A.2.3 Scattering of Electrons by Atomic Electrons . . 119
A.2.4 Scattering of Protons by Atomic Electrons ... 120
.................
A.2.5 Electron Bremsstrahlung 122
A.2.6 Continuous Slowing-Down Approximation ... 126
A.2.7 Stopping Power ......................... 128
.......................
A.3 Neutron Cross Sections 139
..............
A.3.1 Classification of Interactions 139
......................
A.3.2 Data Compilations 143
A.3.3 Kerma Factors ......................... 145
~ . N4uc lear Cross Sections for Charged Particles at
..............................
High Energies 147
A.4.1 Interactions of Pions below 100 MeV ....... 147
A.4.2 Nuclear Interactions of Hadrons above 100
MeV ................................. 153
.
Appendix B Examples of Absorbed-Dose and Dose-
Equivalent Calculations ............................ 167
B.1 Absorbed Dose from Neutrons in Tissue-Equivalent
...................................
Material 167
B.2 Shielding of Manned Space Vehicles Against
Galactic Cosmic-Ray Protons and Alpha Particles . . 172
B.3 Skyshine for Neutron Energies 5400 MeV ........ 178
.
Appendix C A Compilation of Geometric Reduction
....................
Factors for Standard Geometries 185
C.l The Autologous Case (A = B) .................. 185
C.2 The Heterologous Case (A # B) ................ 188
References ....................................... 197
1. Introduction
1.1 The Concept of Absorbed Dose
The effects of radiation on matter are initiated by processes in
which atoms and molecules of the medium are ionized or excited.
Over a wide range of conditions, it is an excellent approximation to
assume that the average number of ionizations and excitations is
proportional to the amount of energy imparted to the medium by
ionizing radiation1 in the volume of interest. The absorbed dose, that
is, the average amount of energy imparted to the medium per unit
mass, is therefore of central importance for the production of radia-
tion effects, and the calculation of absorbed-dose distributions in
irradiated media is the focus of interest of the present report. It
should be pointed out, however, that even though absorbed dose is
useful as an index relating absorbed energy to radiation effects, it is
almost never sufficient; it may have to be supplemented by other
information, such as the distributions of the amounts of energy
imparted to small sites, the correlation of the amounts of energy
imparted to adjacent sites, and so on. Such quantities are termed
stochastic quantities. Unless otherwise stated, all quantities consid-
ered in this report are non-stochastic. A discussion concerning sto-
chastic quantities is given in ICRU Report 33 (ICRU, 1980).
The absorbed dose, D, is defined (ICRU, Report 33) as the quotient
of d by dm:
where d2 is the mean energy imparted by ionizing radiation to matter
of mass dm.
The energy, E,i mparted to the volume containing dm is defined as
'Ionizing radiation consists of directly ionizing and indirectly ionizing radiation.
Directly ionizing radiations are charged particles (electrons,p ositrons, protons, alpha
particles, heavy ions) with sufficient kinetic energy to ionize or excite atoms or
molecules. Indirectly ionizing radiations are uncharged particles (photons, neutrons)
that set in motion directly ionizing radiation (charged particles) or that can initiate
nuclear transformations.
2 / 1. INTRODUCTION
where Ei, (E,,,) is the sum of energies of all the charged and
uncharged ionizing particles that enter (leave) the volume, excluding
rest mass energies, and XQ is the algebraic sum of all changes
(decreases: positive sign; increases: negative sign) of rest-mass
energy in mass-energy transformations occumng in the volume.
A few clarifications of Equation (1.1) are necessary at this point.
The energy imparted results from random discrete energy deposition
events by individual ionizing particles andlor their secondaries. The
quantity is therefore stochastic in nature and governed by a (nor-
E
malized) probability distribution function fv(d where V is the volume
containing m. The mean value of e,
is the quantity referred to in the definition, Equation (1.1).I n addi-
tion, this equation implies a limiting process2 such that V +. 0. The
absorbed dose, Dfi), is thus defined at a given position ? in the
irradiated object (see footnote 2), and is a non-stochastic quantity.
Furthermore, in general Dc) changes with and this variation is
termed the "dose distribution."
A second important aspect refers to the temporal pattern of dose
accumulation. Let
be the dose increment at ? during the time interval CtJt+dtl. This
equation defines the absorbed-dose rate, D@,t). This aspect is impor-
tant in understanding the relation between dose and biological effect.
A final remark concerns the relation between.dose and its stochas-
tic counterpart, the specific energy, z, defined as
z = dm. (1.5)
The specific energy is always measured in a non-zero volume, V, and
its mean value, 2, has the value of the average absorbed dose, Dv, in
that volume of
2Because of the discrete manner in which energy is imparted, the limiting process
V- 0 is obviouely an idealization. At all times the volume V should contain a large
number of atoms and molecules.
