NCRP 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 LEGAL NOTICE This report was prepared by the National Council on Radiation Protection and Mea- surements (NCRP). The Council strives to provide accurate, complete and useful information in it.r eports. 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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 utilized by any information storage and retrieval system without written permission from the copyright owner, except for brief quotation in critical articles or reviews. 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.