Biophysical Theory of Radiation Action: A Treatise on Relative Biological Effectiveness [Reprint 2022 ed.] 9783112618943, 9783112618936

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Klaus Günther / Wolfgang Schulz

Biophysical Theory of Radiation Action

Biophysical Theory of Radiation Action

A Treatise on Relative Biological Effectiveness by Klaus Günther Wolfgang Schulz

with 80 Figures and 32 Tables

AKADEMIE-VERLAG • BERLIN 1983

Dr. sc. n a t . Klaus Günther Dr. rer. nat. Wolfgang Schulz AdW der D D R Zentralinstitut f ü r Elektronenphysik Zentralinstitut f ü r Molekularbiologie

Erschienen im Akademie-Verlag, D D R - 1 0 8 6 Berlin, Leipziger Straße 3—4 © Akademie-Verlag Berlin 1983 Lizenznummer: 202 • 100/518/83 P r i n t e d in t h e German Democratic Republic Gesamtherstellung: V E B Druckhaus „Maxim Gorki", 7400 Altenburg L e k t o r : Christiane Grunow Einband: Annemarie Wagner LSV 1315 Bestellnummer: 762 429 1 (6432) DDR 6 8 , - M

Preface

The science of radiation action on living systems has a rather long history. Practical problems in application of, and protection from, ionizing radiations have always been the main impetus to continue and extend research in this field, so that emphasis has been put on experimentally establishing radiation actions of practical interest rather than systematically developing a general theoretical framework to achieve comprehension of the underlying mechanisms.. It is true, the early period until about the mid-50s was governed, indeed, by such kind of fundamental ideas which were widely accepted (hit principle and target theory), but thenceforth, when the simple models of target theory proved unsuitable in many cases, the development became divergent. At present, there is an obvious gap between the wide variety of facts described in terms of biology, biochemistry, and medicine and, on the other hand, the now well-established physics of radiation interaction with matter. This book has been written with the intention to help fill that gap. Apart from the fundamental scientific aspects of the problem, its relevance to

practice should be stressed. In contrast to the formerly dominant role of gammaand X-rays, it is now a' broad range of ionizing radiations that is increasingly important in the fields of radiotherapy and protection of men from radiation hazards. Due to this development, it gains high practical significance what has always been the central objective of radiation biophysics: to understand and to make predictable the varying relative biological effectiveness (RBE) of radiations. At present, in spite of several efforts made, there is no generally accepted theory of RBE. In our opinion, this shortcoming in regard of RBE and the above-mentioned gap are much the same: to relate the biological effect to the primary physical events requires model building for which the crucial test is just the ability of a model to yield the right RBE for every radiation. By this powerful criterion, inadequate models or ideas about the mechanism of radiation action can largely be ruled out. This was convincingly demonstrated in the past when simple target models were evaluated consistently, in particular by D. E. LEA in his noted work "Actions of

VI

Radiations on Living Cells". The theoretical approaches contained in L E A ' S book may be looked upon as the prototype of what will be found in this monograph. Our primary objective was merely to give a systematic and comprehensive account of our own theoretical work about radiation action at the molecular and cellular levels, ending up in a definite predictive theory of R B E for mammalian cell killing and related effects. Thus, this monograph bears a great resemblance to an outsized original paper, as most of the results presented have not been published before. Nevertheless, the book has become broad in scope, mainly for two reasons. Firstly, the physical and mathematical fundamentals of the theory constitute a profound study in radiation stochastics including new concepts and methods. This part of the monograph could be entitled 'Theoretical microdosimetry'. Secondly, to demonstrate the capability of the theory, it was necessary and possible to discuss a good deal of modern radiation biology, as far as radiation quality is involved. This ranges from DNA breaks to gross tissue and tumour reactions. Numerous data are compendiously reviewed and systematically interpreted in terms of theory. Have a look at the section titles of chapter 7 to realize the topics covered. With a view of enabling research workers of various engagement to make use of this book, we have aimed for making readable each chapter separately, at the expense of some redundancy. For example, the radiobiologist who wants to be informed only on the achievements of the theory in checking with experimental facts can start with chapter 7, eventually omitting subsection 7.2.1. Furthermore, the final procedures of application (mainly survival curve .prediction) are simple and can be practised without having studied the theoretical background. For this particular purpose, chapter 9 has been written as a compendium of the required formulas and instructions with reference to tabulated

Preface

parameters (appendix). On the other hand, potential readers with a general interest just in theory should start with chapter 1 where, first, existing approaches, strictly speaking, their rationale only, are analysed in a critical way and, second, our general model is introduced with regard to its basic principles and expected scope. To body forth, as a whole, the theory based on the latter model, chapter 2 has been included, addressing to a fairly wide circulation. It develops the subject systematically in a simplified version as to the underlying physics, thus keeping it within bounds and presenting, nevertheless, all the essential aspects, theorems, limitations, and perspectives of this theory. From here, one may take a leap direct to chapter 6 or chapter 7. In this way, people with only a limited training in mathematical physics and microdosimetry can easily make themselves familiar with the theory as regards both the essential principles and the applications. The chapters not mentioned above deserve special comments. They would be superfluous in principle unless the strict evaluation of our model required a description of radiation stochastics in more detail than offered by customary microdosimetry. So it was necessary to start with first principles and to give a comprehensively systematic mathematical treatment of radiation fields and absorption events in both micrometre and nanometre regions, introducing new concepts, e.g., to take account of sites with inherent structure in molecular dimensions. The general and profound character of the approach, in spite of its special purpose in the framework of the considered theory of radiation action, will be appreciated, as we hope, by many workers engaged in radiation physics and theoretical fundamentals of microdosimetry. It is important to note that the formalism, which rests upon analytical procedures instead of Monte Carlo techniques, involves the calculation of the usual micro-

Preface

dosimetric parameters, too. In this respect, a great number of numerical results are compiled and compared' with the large body of corresponding experimental data in chapter 5. The intention is twofold: testing our approach and giving an abundant summary of calculated data for photons, fast neutrons, and monoenergetic ions (see appendix). Results for nanometre sites along with modern particle track simulation data are a topic of chapter 6. It is a peculiarity of this monograph that it covers topics which may appear largely unrelated, at first glance. However, as the principal goal is developing and testing a systematic theory which is to build a bridge between physics and biology, the discussion of different areas is quite naturally justified and necessary but should not be confused with profoundly reviewing them like autonomous subjects. Hence, we cannot pretend that we have given an exhaustive and wellweighted account of the present state of affairs in the fields of radiation physics, microdosimetry, action models, and radiobiology. As to the experimental work discussed, emphasis has been placed on results and data rather than fundamentals and methods. In quoting literature we have confined ourselves to what seemed suitable and sufficiant for our purpose, and was easily available to us. We hope that this can be tolerated even though the most recent literature is only scarcely covered. It is another shortcoming that this monograph is not exhaustive with respect to the theory's scope for possible application. In fact, the general idea of taking DNA lesions for the crucial primary radiation damage and the corresponding general formalism are likely to be valid for most of the biological effects observed on various kinds of objects. This may justify the title of the book, but actually the main attention has been focused on the impairment of the proliferative capacity

VII of mammalian cells and tissues in regard of H B E . This topic being of particular importance, no doubt, other subjects are only touched on or totally omitted in order to keep the book within bounds. For example, this concerns an already accomplished special action model for bacteria as well as model building, in the framework of the general theory, with respect to other objects and endpoints such as viruses, phages, mutations, chromosome aberrations, division delay, cancerogenesis, etc. Furthermore, there are also limitations concerning the radiations considered. This is only a question of data calculation; the corresponding tables are restricted to photons, fast neutrons between 0.25 MeV and 20 MeV, and ions up to 20 MeV/nucl.. Extended calculations are under work in order to make practical application of the R B E theory possible for pions, high-energy neutrons and relativistic ions. We think that this will benefit the modern activities in high-LET radiotherapy as well as fundamentals of radiation protection in manned space research. During the pretty long period of writing this book, S I units were declared to be obligatory. However, a corresponding revision of tables, figures, and the text has been forborne in the hope that this will hardly be realized as a serious defect, particularly as the imperative alterations would almost only refer to the unit of absorbed energy dose. We have used the older unit rad, throughout, whereas the SI unit is Gray = Joule/kg. The difference is by a factor of hundred: 1 Gy = 100 rad. Finally, we should apologize to the reader for our English. Despite having made every effort to eradicate mistakes and uncustomary phrases, we are painfully aware of our very limited success in hushing up the German writer. We believe the book, will nevertheless be readable, at least, and understandable with respect to the language, hoping that the blemishes make the English user smile rather than grieve him.

VIII We are much obliged to our colleagues who have encouraged and supported this work in various ways. I t was only in the kindly and stimulating atmosphere of the Department t of Radiation Biophysics of the Central Institute of Molecular Biology that we were able to accomplish it. Most grateful we are to Prof. H. Abel, director of the department, who has steadily promoted our work. His valuable discussions, helpful criticism and stimulating encouragement merit our inmost gratitude. We are highly indebted to W. Leistner

Preface

for his help in carrying out a great deal of calculations and data selection underlying the figures and tables in chapters 5 and 7. Our obligations to Mrs. Prof. L. Herforth are due to here patiently and thoroughly reading the whole manuscript and giving useful hints. The typing has exclusively been the work of Mrs. E . Homeyer, who has always taken so much care that she has deserved well of this book. Last not least, we thank the Akademie-Verlag, Berlin, for the beneficial co-operation. K . GÙNTHER, W . SCHTTLZ

Contents

Chapter 1

Chapter 2

ReYiew oî theoretical conceptions

1

1.1.

Introduction

1

1.2. 1.2.1. 1.2.2. 1.2.3.

Classical target theory General considerations Single-hit reactions Multi-hit reactions

3 4 5 7,

1.3.

The LET concept

8

1.4. 1.4.1.

Microdosimetry Survey of microdosimetric quantities and relations The problem of constructing theory The theory of dual radiation action

1.4.2. 1.4.3. 1.5. 1.5.1. 1.5.2. 1.5.3. 1.6. 1.6.1. 1.6.2. 1.6.3.

Gross sensitive volume with structure The model of DNA-lesion production The theory of cellular radiation effects Biological models of action and formal analysis Some descriptive approaches survival curves and RBE The two-component theory of diation The track-structure theory survival The molecular theory of survival

to raof cell

10 11 13 15 19 20 22

An approximate approach based on microdosimetry

33

2.1.

Introduction

33

2.2.

Formalism of the microdosimetric version of theory The rationale of the approach Derivation of the fundamental equation The resultant procedure The ^-approximation

2.2.1. 2.2.2. 2.2.3. 2.2.4. 2.3. 2.3.1. 2.3.2. 2.3.3. , 2.3.4. 2.3.5.

24

2.4. 2.4.1.

25

2.4.2.

26

2.4.3.

27

2.5.

30

2.5.1.

Theoretical consequences for the relative biological effectiveness Preliminary notes The limiting cases D = oo and D = 0 Inadequacy of the energy-only concept ^-approximation as a limiting case An approximate general R B E formula Analysis of experimental data Survival curve fitting with unknown r General laws governing RBE-LET relationships Conclusions Primary lesions and the relevant RBE Preliminary reflections

34 34 36 37 39 39 39 40 42 42 45 46 47 50 52 53 53

Contents

X 2.5.2. 2.5.3. 2.6. 2.6.1. 2.6.2. 2.6.3.

Experimental data on DNA-strand breakage The model of DNA-lesion production Appendix More than one effective lesion type Inhomogeneity of the cell population Investigation of the RBE-dose relationship

54 55 56 56 57 58

Chapter 3 The mathematical formalism of the DNAlesion theory of radiation action 3.1. 3.1.1.

Introduction The underlying physical quantities and functions

3.2.

Prerequisites and basic principles of approximation Prerequisites concerning the conditions of irradiation Approximations concerning the description of absorption events

3.2.1. 3.2.2.

59

61 62 63 64

Description of radiation fields

71

3.4.

Description of absorption events

77

3.5.

The model of DNA-lesion production

82

3.6.

The theory of cellular radiation effects

86

3.7.

Formal properties and consistency of the theory Mean DNA-lesion number and statistical independence of lesions as a limiting case The relation of equivalence Generalization of the theory: more than one lesion type involved in the injury process

3.7.2. 3.7.3.

3.8. 3.8.1. 3.8.2.

Energy spectrum of S-rays

4.4.

Primary ionizations and ionization cluster frequencies The W-value Experimental distribution of ions in clusters The algorithm for heavy particles The algorithm for electrons Results

4.4.1. 4.4.2. 4.4.3. 4.4.4. 4.4.5.

93 94 96 99

Appendix: estimation of errors 101 Incorrectly apportioned energy 101 Errors caused by the continuousslowing-down approximation 105

Chapter 4 The underlying physical relationships

Ill

4.1.

Introduction

111

4.2. 4.2.1. 4.2.2.

Linear energy transfer Heavy charged particles (ions) Electrons

112 112 115

116 119 119 121 121 123 124

4.5. 4.5.1. 4.5.2.

Electron range 127 A theory of mean projected range 127 Modified stopping power and modified range 129

4.6.

Secondary electron spectrum of photon radiation 132

4.7.

Secondary charged particle spectra of fast neutrons • 133 ' Principles 133 Elastic scattering 134 Inelastic scattering 135 Nuclear reactions 137

59

3.3.

3.7.1.

4.3.

4.7.1. 4.7.2. 4.7.3. 4.7.4.

Chapter 5 Implications concerning microdosimetry

140

5.1.

Introduction

140

5.2. 5.2.1.

The mathematical formalism 142 Mean values z F , z D . and related quantities 142 Event size spectrum ft(z) by inverse Laplace transformation 148

5.2.2. 5.3. 5.3.1. 5.3.2. 5.3.3.

Comparison with experimental data Gamma- and X-rays Fast neutron radiations Ion radiations

151 153 156 162

Chapter 6 Evaluation of the model of DNA-lesion production 176 6.1.

Introduction

176

6.2.

Methodic particulars

177

6.3. 6.3.1.

General results 182 Spherical target and threshold number of ionizations 182 Lesion frequency versus radiation quality 192

6.3.2. 6.4.

DNA-strand breaks

193

XI

Contents 6.4.1. 6.4.2. 6.4.3.

Experimental data and theoretical results 193 Non-rejoining single-strand breaks 198 A fit for a special experimental data set 199

8.2.3. 8.2.4. 8.2.5.

Virtual microdosimetric parameters 268 Results for X-rays and f a s t neutrons 269 Results for monoenergetic ion radiations 271

Chapter 7 Mammalian cell killing

202

7.1.

Introduction

202

7.2.

Survival curve analysis for monoenergetic ion radiations Description of the method Results for oxic conditions Results for anoxic conditions The minimum DNA content as a result of formal analysis

7.2.1. 7.2.2. 7.2.3. 7.2.4. 7.3. 7.3.1. 7.3.2. 7.3.3. 7.3.4. 7.3.5.

Ion radiations R B E and O E R as a function of LET Further examples of survival curve prediction Response-modifying agents and radiation quality Mixed radiations Non-mammalian eukaryotic cells

7.4. 7.4.1. 7.4.2. 7.4.3.

Past neutron radiations Preliminary notes Survival curve prediction R B E and O E R in dependence on neutron energy 7.4.4. R B E and O E R for mixed fields of neutrons and photons 7.4.5. The relationship between X-ray sensitivity and neutron R B E 7.4.6. R B E and cell cycle 7.4.7. R B E for normal and malignant tissue reactions relevant to neutron therapy 7.4.7.1. Fundamentals 7.4.7.2. Gross reactions of dose-limiting normal tissues 7.4.7.3. Tumour reactions

206 206 208 211

8.3. 8.3.1. 8.3.2. 8.3.3. 8.3.4. 8.3.5. 8.4.

217

The ultimate general formalism of the theory of cellular radiation effects Fundamentals Photon radiations Fast neutrons Ion radiations Numerical examples

274 274 277 277 278 280

Evaluation of R B E in LETapproximation for monoenergetic ions 280

217 218

Chapter 9

221

Concise instructions for practical calculations 285

223 225 227 232 232 234 242

9.1.

Introduction

285

9.2. 9.2.1. 9.2.2. 9.2.3.

Calculation of dose-effect relations The general formalism Mixed radiations The 'zero-LET' dose-effect relation Photon radiations Fast neutrons Ion radiations

286 286 287

244

9.2.4. 9.2.5. 9.2.6.

248 250

9.3. 9.3.1.

252 252 254 258

Chapter 8

9.3.2. 9.3.3. 9.3.4. 9.3.5.

Expressing a dose-effect relation by a sum of exponential functions Geometrical characterization of a given survival curve The coefficients B' and (basic method) The coefficients Ji/A and (extended method) Special mathematical models of cell survival Appendix : derivation of the mathematical formalism

287 288 288 289 290 290 291 292 294 295

Re-formulation of the theory

262

8.1.

Introduction

262

Appendix

297

8.2. 8.2.1. 8.2.2.

Virtual microdosimetry Formal considerations The general concept of microdosimetry

263 263

References

340

265

Subject index

351

virtual

Chapter 1

Review of theoretical conceptions 1.1. Introduction The injurious effect of ionizing radiations on living matter has been known since the beginning of this century, and attempts to understand these actions have constituted a good deal of early biophysics. Any theoretical considerations and model building require the phenomena to be described in quantitative terms with respect to both the acting physical agent (radiation) and the biological effect. For the former the radiation dose D now defined as the energy absorbed per unit mass1) has been chosen. In principle, this is an arbitrary choice,2) but energy dose is the only quantity that (a) can be readily defined (and measured) for every ionizing radiation and (b) is the main determinant of the biological effect. Different quantitative measures of biological effects are in use,3) but the most

) Rigorously, the dose at a given point has to be defined as the expected value of specific energy in a site round the point as the site diameter tends to zero (see section 1.4.1). 2 ) An imaginable alternative would be, e.g., the fluence of charged particles. 3 ) There are principally three types: (i) quantifiable degree of damage to a single object (e.g., seedling root growth inhibition, cell

important one in regard to theoretical considerations is the rate of reacting (or non-reacting) biological units such as viruses, bacteria, and single eukaryotic cells. The proportion of units which do not show a certain well-defined reaction is genetally denoted by the symbol S indicating 'survival' since the effect considered in most cases is loss of the capacity for unlimited proliferation. Based on these definitions, an experimental dose-effect relation S(D) is an inputoutput relation for the 'black box': biological object. Any theory aiming at a satisfactory understanding and prediction of S(D) has to be based on both a proper description of the primary physical events (the input) and hypotheses regarding the chain of biological events that follow (the interior of the black box). It is the latter that raises principal problems since the

a

division delay, scaled skin reaction, biochemical changes); (ii) mean number of countable lesions (e.g., DNA breaks, chromosome aberrations); (iii) probability of an object to react in a specified way (scored as a reaction frequency on a large population of objects).

2 hypotheses expressed as models cannot be verified by means of S(D) alone. This signifies a lack of information: different models can equally well account for a measured dose-effect relation. Therefore, with very few exceptions, numerous efforts of this kind, starting with classical target theory, have finally yielded nothing but mathematical descriptions, rather than genuine explanations and predictions of S{D). in terms of largely abstract conceptions and parameters such as number of hits, number of targets, repair rates, biological states with transition coefficients, etc. A radical alternative consists in renouncing this kind of theorizing about dose-effect curves themselves, envisaging radiobiophysical theories t h a t refer only to the question of how the relation S(D) changes with changing sort, composition, and energy of radiation. In other words, the subject of such kind of theory is the relative biological effectiveness, RBE, 1 ) of different ionizing radiations. The advantage is t h a t the 'black box' may almost remain black while the variation of input (radiation quality) is a matter of physics only. I n reality the things are not so simple, but conceptually it is very important to distinguish between the two aspects of theoretical approaches: bare dose-effect curves as such on the one hand, and R B E on the other hand. This has often not been done clearly enough, and a number of models suggested contain more assumptions restrictive to ^(ZJythan necessary if the question of R B E only is treated. This first chapter, corresponding to the 1

) The relative biological effectiveness is defined as the ratio of doses, RBE = D^/D, that produce equal effects under equal conditions, where D refers to the radiation in question and Dx is the reference-radiation dose. The common usage is taking X-rays (mostly 250 kVp) for the latter, but no standard choice has been adopted in practice. Therefore, the reference radiation will be always indicated in cases where confusion may be possible.

Theoretical conceptions

topic of the entire monograph, deals with approaches (or aspects of approaches) concerning the R B E problem. Even with this restriction to observe, completeness cannot be contemplated and would not be suitable in view of the numerous attempts published over a period of several decades. Instead, we will follow only the principal lines of development of this field of research for which also the common term microdosimetry has come in use, in a general sense. However, throughout the present monograph this vague denotation is not adopted, because a branche of science should be defined by fundamental principles, not by envisaged aims with the principles left open. Accordingly, in this book 'microdosimetry' signifies a special way of physical description of the random spatial patterns of energy deposition in matter without the implication of a definite R B E theory. I n this restricted sense the concept of microdosimetry is reviewed in section 1.4.1, and it underlies a great part of what follows. I n discussing theoretical conceptions, it is presumed that the reader is familiar with the essential experimental facts, since comprehensively reporting the latter would have made this chapter too lengthy, inasmuch as a good deal of the material can be found in chapter 7 (for a survey s e e A l p e r , 1979).

As S(D) is a probability, namely the probability for each single object of a homogeneous population to survive a given dose of a given radiation (i.e., on well-defined conditions of experiment), the underlying processes of action must be of stochastic nature. I t is a matter of fact t h a t two kinds of stochastics are involved: physical stochastics and biological stochastics

(Hug and K e l l e k e r ,

1966).

To

avoid any models or approximation concepts in advance, the following general characterizations may be given. Physical stochastics means t h a t despite specified dose and radiation a large variety of spatial patterns of radiation

Classical t a r g e t t h e o r y

energy deposition is engendered by quantum-mechanical chance. Varying elements of these patterns are: tracks of primary and secondary charged particles, alongtrack excitations and .ionizations, and clusters of the latter. Physical stochastics obeys known physical laws, but a complete description in terms of random variables (position and intensity of all energy transfer points) is practically impossible (GÜNTHER, 1 9 6 8 b ) .

To explain what biological stochastics means, suppose that identical patterns of energy deposition have occurred in, from outward appearances, identical biological objects. In general, only part of the objects will show the reaction under consideration as a matter of chance. Molecular and cell biology can tell us a lot of reasons for this behaviour, but on the other hand the assumption of 'identity' of biological objects is rather fictive. Never will two objects be identical in all respects, so that no clear distinction can be made between sheer biological stochastics and influence of inhomogeneity of a given population (biological variability). Nevertheless, a conceptual distinction should be made, and, in a more pragmatic way, let us speak of nothing but biological stochastics unless some kind of biological variability can be clearly specified (e.g., asynchrony of a cell population). To separate physical stochastics from biological stochastics is not trivial. The prerequisite for this to be meaningful is that the physical stochastics does not essentially depend on the particular structure of the biological object. To give a counter-example, the distribution of energy deposition by U.V. radiation is governed by the sort and structure of the absorbing molecules (their electronic energy levels). By contrast, ionizing radiations consist of charged particles energetic enough for the binding energies of electrons in biological molecules to be of minor importance. As it were, the fast charged particles almost 'feel' nothing but

3

a homogeneous 'sea' of free electrons with only the number per unit volume (electron density) being relevant. Therefore, physical stochastics can be treated as independent, approximately. On the other hand, it is only the physical stochastics that changes when varying radiation quality. Just for these reasons, theoretical approaches to RBE can in some way be separated from complete theories of dose-effect relations. In the sequel we will repeatedly refer to the conception of physical and biological stochastics, since this will prove to be a good guide line for a unifying discussion of different theories or models, in each case asking about the special way in which both kinds of stochastics are taken into account.

1.2. Classical target theory To begin with a restricting comment, the following critical discussion will only deal with the rationale of target theory in a rather abstract manner omitting details, variants, and interpretations as well as the historical development and the numerous applications and successes of target theory; inasmuch as there exist a number of excellent monographs. D E S S A U E R ' S book "Quantenbiologie" re-edited and supplemented by SOMMERMEYER ( 1 9 6 4 ) is a comprehensive survey of the theory's development including the entire range of applications. The work of TIMOFEEFFRESSOVSKY and ZIMMER ( 1 9 4 7 ) places emphasis on the analysis of genetic effects. The monograph by LEA (1946) is of particular value for the present purposes because of the efforts made to use a correct description of physical stochastics in the framework of simple models of action, which is most closely related to modern approaches. An instructive critical analysis of the fundamentals of target theory was given by H U G and K E L L E R E R (1966).

Classical t a r g e t t h e o r y

energy deposition is engendered by quantum-mechanical chance. Varying elements of these patterns are: tracks of primary and secondary charged particles, alongtrack excitations and .ionizations, and clusters of the latter. Physical stochastics obeys known physical laws, but a complete description in terms of random variables (position and intensity of all energy transfer points) is practically impossible (GÜNTHER, 1 9 6 8 b ) .

To explain what biological stochastics means, suppose that identical patterns of energy deposition have occurred in, from outward appearances, identical biological objects. In general, only part of the objects will show the reaction under consideration as a matter of chance. Molecular and cell biology can tell us a lot of reasons for this behaviour, but on the other hand the assumption of 'identity' of biological objects is rather fictive. Never will two objects be identical in all respects, so that no clear distinction can be made between sheer biological stochastics and influence of inhomogeneity of a given population (biological variability). Nevertheless, a conceptual distinction should be made, and, in a more pragmatic way, let us speak of nothing but biological stochastics unless some kind of biological variability can be clearly specified (e.g., asynchrony of a cell population). To separate physical stochastics from biological stochastics is not trivial. The prerequisite for this to be meaningful is that the physical stochastics does not essentially depend on the particular structure of the biological object. To give a counter-example, the distribution of energy deposition by U.V. radiation is governed by the sort and structure of the absorbing molecules (their electronic energy levels). By contrast, ionizing radiations consist of charged particles energetic enough for the binding energies of electrons in biological molecules to be of minor importance. As it were, the fast charged particles almost 'feel' nothing but

3

a homogeneous 'sea' of free electrons with only the number per unit volume (electron density) being relevant. Therefore, physical stochastics can be treated as independent, approximately. On the other hand, it is only the physical stochastics that changes when varying radiation quality. Just for these reasons, theoretical approaches to RBE can in some way be separated from complete theories of dose-effect relations. In the sequel we will repeatedly refer to the conception of physical and biological stochastics, since this will prove to be a good guide line for a unifying discussion of different theories or models, in each case asking about the special way in which both kinds of stochastics are taken into account.

1.2. Classical target theory To begin with a restricting comment, the following critical discussion will only deal with the rationale of target theory in a rather abstract manner omitting details, variants, and interpretations as well as the historical development and the numerous applications and successes of target theory; inasmuch as there exist a number of excellent monographs. D E S S A U E R ' S book "Quantenbiologie" re-edited and supplemented by SOMMERMEYER ( 1 9 6 4 ) is a comprehensive survey of the theory's development including the entire range of applications. The work of TIMOFEEFFRESSOVSKY and ZIMMER ( 1 9 4 7 ) places emphasis on the analysis of genetic effects. The monograph by LEA (1946) is of particular value for the present purposes because of the efforts made to use a correct description of physical stochastics in the framework of simple models of action, which is most closely related to modern approaches. An instructive critical analysis of the fundamentals of target theory was given by H U G and K E L L E R E R (1966).

4

Theoretical conceptions

1.2.1. General considerations The leading idea of target theory was to directly connect two facts: the shape of dose-effect curves 8(D) and the discontinuous quantum-like absorption of radiation energy by matter. 8(D) is found to be always a gradually decreasing function, often of purely exponential character: S(D) = exp (— xD) (inactivation of enzyme molecules, viruses, bacteria). Also if it has a shoulder, there does not exist a threshold dose below which S(D) = 1 would hold. This behaviour was attributed to the random occurrence of so-called hits (with a mean number proportional to dose) in one or several targets. With nothing more said this conception (hit principle) may be general enough; the crucial point, however, is the introduction of two specifying assumptions : (a) the number of hits in a target, and the number of targets damaged by hits, is assumed to determine unambiguously whether the object does or does not react (neglect of biological stochastics), and (b) hits are assumed, beforehand, to be statistically independent (to obey Poisson statistics) in any case, prior to deciding what kind of physical events are hits. Numerous variants of such deterministic models can be, and have been, considered. The most common ones are the singletarget-multi-hit model S(D)

=

E

k=o

(*D)k ifc!

(1.1)

which assumes that m hits are necessary and sufficient as a trigger for the effect, and the multi-target-single-hit model 8(D) = 1 — (1 - e-w'>,

(2.18)

(0) we may roughly identify with for using it in Eq. (2.17) in order to calculate S(D) for other radiations. However, it is not difficult in practice to avoid this approximation. One has only to reverse the relation (2.17) such t h a t each of the 'measured' o t ^ ^ appears as the result of the integral calculated with the corresponding right «/;2 up to at least D = zD. If this is fulfilled, the RBE-dose relation does not appreciably depart from (2.36), in case that |/80'(0)| 1 for, e.g., fast neutrons and X values which are not correlated with zD in dependence on neutron energy (as a consequence of admitting kjkx to vary with radiation quality; cf. section 1.4.3). While such a correlation is

Approach by microdosimetry

just the central postulate of DRA theory, even if its most general version is adopted, the present theory implies more latitude in this respect. It is true that, according to Eq. (2.38), k[kx is largely, apart from the influence of £>0, a matter of the molecular level of lesion formation, as in the case of the DRA-limit of the theory where k[kx = (r[rx)2 holds strictly (cf. Eq. (2.36)), but A is a complex dependency on both fx{z) and r with S0(D) interfering, and reduces to zD only in the limiting case dealt with in subsection 2.3.4 (DRAlimit). Hence, the present theory has good prospects to account better for the experimental findings. To prove this definitively, one would have to implement analyses like those mentioned above, however with only r being treated as a freely adjustable parameter for each radiation quality. In principle, this is the topic of the following section as well as the entire chapter 7, but treated explicitly in terms of dose-effect curves, which are, on the one hand, measured direct (in contrast to RBE) and, on the other hand, easier to derive from the theory than the quantities (2.38) to (2.40), moreover avoiding the approximation (2.37).

2.4. Anal/sis of experimental data In this section the ability of the DNAesion theory to predict the shape of survival curves in dependence on radiation quality will be demonstrated. This means that the primary lesion RBE, r, is treated as a freely adjustable parameter for each radiation, which evidently allows to achieve a fit of each survival curve in one point at least, or in the final slope. Only the shape of the entire curve can thus be considered as a genuine consequence of theory. Apart from this test, the obtained estimates of r themselves and their dependence on LET are of great interest. The second parameter involved is the size of the gross sensitive volume, which is

46 (section 2.6.3). Of course, the adequacy of the expression (2.37) will depend on the particular S0(D), and its general applicability is, therefore, hard to prove strictly, but there is considerable evidence for the errors to rarely exceed 10 per cent. The thus-established approximate algorithm of direct calculating the R B E of radiations is interesting, primarily, from the theoretical point of view. The formulas exhibit obviously how the R B E is influenced by radiobiological features of the particular object in terms of S0(D) and the pertinent characteristic parameters S0'(0),

=

/ ) „ « !

2[2 - j> - (1 - p) q] (1 + D/X) X

P + Vi>2 + 4[2 - p - (1 - p) q? (1 + D/A) D/X ' (2.53)

The corresponding strict solution to the equation 80{D x RBE^) = S(D) is obtained straightforwardly using the expressions (2.49) and (2.50) and eliminating a 1 and a 2 by introducing q and A:

+ 1 / / 1 - pi2Y _ I^JL y \ i —p )

JL _ l - p

l — p

e

e-zBli +

e-gxm

(2.54)

-(ï-i

where B y setting »2 = ? 6 are retaken into account by completely disregarded for ions. considering them to be generated at (d) For incident electrons, only the random in the medium independently of fluctuations of the path segment length inside the SV which are due to random curvature of the track are neglected, but Other elementary processes by which phothe average effect of track curvature upon tons or neutrons create charged particles can the path segment (lengthening) is taken be ignored for the energy range considered into consideration. throughout this book.

66

General mathematical formalism

(e) In the case of primary incident ions the decrease of energy upon traversing through the SV is neglected. These approximation principles on which the mathematics of the subsequent sections is based may be explained and justified by the following arguments. Principle 1 is warrantable by the large mean free path of photons and neutrons in matter (see footnote 3 at page 63). Secondary particles produced simultaneously by successive Compton effects and nuclear collisions, respectively, are very unlikely together to affect the same SV, which has microscopic dimensions. Principles 2 and 3 must be considered in close connection since the errors introduced by them compensate for each other to a certain degree depending on the value of 8. %-ray cut-off is a well-known method in radiation biophysics, first introduced by LEA (1946). But there are important differences between our procedure and most preceding work (BARENDSEN, 1 9 6 6 ; KELLEREE, 1 9 6 8 a, b ; CORMACK, 1 9 6 8 ; BEWLEY, 1 9 6 8 ; ODA et al., 1971).

As shown in Figure 3.2, S-tracks with E' > 8 neither are neglected nor are those with E' <

Fig. 3.2. Decomposition of a pattern of joint primary and secondary particle tracks (a) into single independent incident particle tracks (b) according to the approximation principle 2. Dashed tracks indicate S-electrons with an initial energy E' < d.

be chosen in dependence on the dimensions of the SV (see below). As it were, by virtue of principle 2, the radiation field itself is replaced by a spectrally equivalent artificial one without explicit reference to the SV, whereas principle 3 forbids crossing the SV boundary by dependent particles (E' < 8). Again, exact energy and particle balance evidently is maintained between 3 a and 3 b, i.e. only assignments of track sections and the corresponding energy depositions from one event to another take place in such a way that the same absorbed energy appears more concentrated in a smaller number of events. Principle 2, on the contrary, causes splitting of events into several, i.e. the number of events is increased and their energies are decreased. In this way, principle 3 partly compensates for the errors due to principle 2 and vice versa. Figure 3.1 illustrates the consequences of the approximation principles 2 and 3. Provided that 8 has been chosen appropriately (see below), most of events fall into the types I and II, i.e. nothing is altered and they are treated correctly. I l l and IV show the splitting of events caused by principle 2. Events of the type V are omitted according to 3 a, while 3 a and 3 b together are responsible for the type VI modification. The joint action of both the approximation principles is indicated by the more complex examples V I I and VIII. An open question as yet is the choice of the cut-off energy ] + W

0

] + IT.].

(3.30)

The case of primary incident ions is most simple. These particles always traverse the SV completely along a straight path without being stopped (approximation principles 4c and 4e), which implies that ¥Y 0 ) must directly depend on E and V . This yields

dEViM(E) J dVw(V)

OO

[!?,(•)] = or0 2 7 /

XÏY">(JS?, I').

00

(3.31)

In the case of ions generated by fast neutrons, stopping in the SV as well as the occurrence of starters (3.29 b) must be taken into account. Therefore, let us consider the quantity as a function of and the initial and the final energies of the track segment inside the SV. EW, on its part, depends on the initial energy E and the residual path

E

E,

At given dose, the particle fluence „) and secondary ions (Vi) as well. It should be emphasized that the formalism as such is not restricted to the model of DNA-lesion production. It can also be dtp pW(E) used to gain information on the single SE LX(E) event size spectrum f ^ j ) or /j(z). Neither the cylindrical shape nor the smallness of X Z being the probability that an ion at given energy and path length in the target brings about a DNA-lesion. Therefore, Gi(E) is the probability of the production of a DNA-lesion by an ion that crosses a target with energy E. The function r(E) referring to electrons as well as ions has a similar meaning except that particles starting inside the target are additionally included in this quantity implicitly (i.e. r(E) > 1 is possible).

+ / dET}'*W», E) re(E) (3.67)

o

7T 4

Z; g/cm 3 °°

z J dEEqM{E) 2; / i (E, E 00 Gi(E) = J dI'wdl') E kVi (E, V, L) + L^(E)~Qlll J 0

0

with the abbreviations 0

W

V

0

00

V

d l'util')

00

= / (H'u^nrfcexpt-fcZ'xrffi.i,)], 0 V (3.70)

81 = 4 V, x

)

This

8

follows from the where

( K e l l e r e r , 1971a).

7 Giinthei'/Schulz

0

X

l

L °}(E) re(E).

(3.72)

2 ) For electrons, the fluence correction L£>(E)j general relation is the surface of the body due to track curvature is involved in addition, in

86

General mathematical formalism

The first term of the right-hand side is equal to the expression (3.70) (save for the difference between and >/>; while the second term describes the starters inside the target. Thus, the representations (3.72), (3.70) for r ^ E ) and Gi(E), respectively, clearly exhibit the difference: neither slowing-down nor start of particles is comprehended in Gi(E). However, a representation like (3.71) is also possible for Gi(E) (integration by parts of the integral (3.70)): OO

Q{(E) = -Qth J ai'ut(l') 27 b,Ki(E,£,) 0 v X exp l-eil'Xi(E,Zy)].

(3.73)

In this representation, the difference between Gi(E) and F^E) according to Eq. (3.71) seems to consist only in the neglect of slowing-down in Gi(E) (see the exponents). In other words, the starters are automatically taken into account by changing nothing but the exponent in a way to describe internal slowing-down. This is an important facilitating property of the theory. For the numerical evaluation of r(E), it is advantageous to introduce the new integration variable

x—

i.)

to replace J':

r(E)

=

ln

-liEb.j

*

o

H

r.

X exp

-I

AE'

Loo(E')

Neglect of internal slowing-down would mean exp [ . . . ] = x. It is then easy to show the equivalence r^E) = Gj(E) (integration by parts and return to the variable V). Hence, the starters are taken into account only by the departure of exp [ . . . ] from x.

3.6. The theory of cellular radiation effects Biological radiation effects that are due to DNA-lesions do not only depend on the mean rate of lesion production of the particular radiation. One must also know the statistics of the discrete number of lesions per reacting unit (cell). Generally speaking, the lesions are not distributed

'at random' among the objects of a great population. This distribution, U^D), being the probability of incidence of -k DNA-lesions of a certain type in a reacting unit upon irradiation with the. dose D of a particular radiation, is the topic of this section. It depends on geometric parameters (mean thickness I, mean projected area ((5,) T

M ( E , E

e

)

-

[ | ( 6 , Z )

s M ( E , S)

Z ) ]

+

f

d E

i ' ( E

e

, Z )

e

, „ „ lim I I = 0 ,

6,

6 X

/

d E

' T M ( E , E

e

' ) M

e

e

( E

e

' , E

X ^ o

)

e

o =

M ( E , 6,)

1 ( 5 ; , Z )

J

-

x

M ( E , 6 ) £ ( d , Z )

, it d77 lim x - A , 4 Q V da;

•

=

(1

„

e

h

Q A E

t

' T M ( E , E

e

)

' ) M

e

e

( E

' , d , )

e

4 +

f

d E

M ( E , E

e

Î ' ( E

6

/

d E

0

M

e

( E

' T M ( E , E

e

e

\ E

e

e

' ) f

d E

)

(3.114)

x Ç ' ( E ' , Z ) .

71 I, d/7

lim

s d E

II = 0 at x = 0 is obvious by inspection. The validity of the second relation (3.115) will be demonstrated only for ion radiations, which will make analogous straightforward calculations for the other radiations evident. For the limiting process x 0, in Eq. (3.104) the terms linear in x need to be retained only. Consequently, regarding the function f we can use Eq. (3.109) with higher order terms omitted. This makes K¡(0) and Ke proportional to x, and now the evaluation of d77/dz according to Eq. (3.104) is straightforward:

%

. OO +

/

d

E

J

f

d E T ^ H E , « » , E )

f

d E

e

T

i

^

E

M

^ > ( E , E

e

. E

e

)

e

) r

r

r

e

e

( E

e

f

)

( E )

( E

e

(3.116)

]

where the relations (3.26 to 3.28) have been used. Look at Eq. (3.67) to see that this expression equals s , Q . E . D . , provided that the sum T

i

^ ( E

i

' - ° > , E )

+

E )

OO e


~.

(4.63), and so to arrive at 1 L£(E,

1)

LM(E)

1 din fe(E) di? 2

(4.66)

The approximation used becomes poor for small E and, on the other hand, we know from the outset that Be(E, I) must change into Be(E) at small E. This is the reason for the settlement (4.61) with the boundary between the two regions chosen to be 1/2 for the sake of continuity in compliance with Eq. (4.57). x ) As the x)

By contrast, Eq. (4.59) requires that boundary to be I rather than J/2 because of Z = Be(E}

132

Physical relationships

spectrum qJ-^(E) in a definite way (see the final equations in Tab. I, appendix), the following expressions represent the differential interaction cross-section per absorber electron, which is least cumbersome. The incident photon energy is denoted by Ey. I t is expressed also by the dimensionless parameter

approach described is no exact solution, its reliability has been examined b y numerical tests concerning the required equivalence of the equations (4.59) and (4.60). Quite satisfactory results have been obtained.

4.6. Secondary electron spectrum of photon radiation

«

The production of energetic electrons by photons (y-rays, X-rays) is governed by three distinct interaction processes : photoelectric effect, Compton effect, and pair

m0c2 =

= Eyl(m0c2),

0.511 M e V . (4.68)

T h e Convpton effect (scattering of photons

by electrons) can be described very well by the fundamental Klein-Nishina formula (EVANS, 1958):

/

«V fium

=

E -

oîE.

\2 /J_ E)

E ^

y

- E

¥y

_

_2 Ey «

E\

E

I

0

production. The latter process can be neglected for the present purposes, since it requires a photon energy larger than twice the electron rest mass energy, m0c2 = 0.511 MeV, and its contribution remains smaller than one per cent for photon energies up to 2 MeV in water (EVANS, 1958). Hence, the initial spectrum,

(see

qeiy)(E)

section

energy 3.3),

of

secondary electrons consists of two terms, qeM(E)

=

q(â(E)

+

q%h(E),

(4.67)

where the subscripts G and ph refer to the Compton and photoelectric effects, respectively. As there is no need for normalizing the

for Ef = 0, whereby a discontinuity in L ^ ( E , l ) would occur, which must be attributed to the approximation involved in the expression (4.66). However, the present approach is preferred primarily because it would be unreasonable to assume that the total energy E of an electron will be always lost within the distance I when its residual mean projected range is just equal to I. Thus, this principal shortcoming of the initial ansatz I = Re(E, Ej) happens to be compensated for by the present approach, though in an arbitrary manner.

for

E


E„

The so-called Compton spectrum is given by 2a

EmiX = j - ^ - E

r

.

edge

:

of

(4.69)

the

(4.70)

The photoelectric effect, which becomes predominating for small photon energies (below 0.06 MeV in the case of water), produces almost monoenergetic electrons, because the incident photon is completely annihilated and its energy becomes the energy of the recoil electron, save for the ionization potential to be overcome. In principle there is no unique value for the latter, but 0.5 keV is a good general estimate for low-atomic-number media (water), where the K-shell electrons practically alone act as the target. Therefore, and because of Ey 0.5 keV, it is felt to be sufficient taking the S-function S(E —Ey+ 0.5 keV) for the description of the photoelectron spectrum. The total collision cross-section per electron, to be added as a independent factor, cannot be obtained accurately enough by theory (EVANS, 1968). Therefore, an empirical expression given by VICTOREEN (1943) has been used. With the parameters given for

132

Physical relationships

spectrum qJ-^(E) in a definite way (see the final equations in Tab. I, appendix), the following expressions represent the differential interaction cross-section per absorber electron, which is least cumbersome. The incident photon energy is denoted by Ey. I t is expressed also by the dimensionless parameter

approach described is no exact solution, its reliability has been examined b y numerical tests concerning the required equivalence of the equations (4.59) and (4.60). Quite satisfactory results have been obtained.

4.6. Secondary electron spectrum of photon radiation

«

The production of energetic electrons by photons (y-rays, X-rays) is governed by three distinct interaction processes : photoelectric effect, Compton effect, and pair

m0c2 =

= Eyl(m0c2),

0.511 M e V . (4.68)

T h e Convpton effect (scattering of photons

by electrons) can be described very well by the fundamental Klein-Nishina formula (EVANS, 1958):

/

«V fium

=

E -

oîE.

\2 /J_ E)

E ^

y

- E

¥y

_

_2 Ey «

E\

E

I

0

production. The latter process can be neglected for the present purposes, since it requires a photon energy larger than twice the electron rest mass energy, m0c2 = 0.511 MeV, and its contribution remains smaller than one per cent for photon energies up to 2 MeV in water (EVANS, 1958). Hence, the initial spectrum,

(see

qeiy)(E)

section

energy 3.3),

of

secondary electrons consists of two terms, qeM(E)

=

q(â(E)

+

q%h(E),

(4.67)

where the subscripts G and ph refer to the Compton and photoelectric effects, respectively. As there is no need for normalizing the

for Ef = 0, whereby a discontinuity in L ^ ( E , l ) would occur, which must be attributed to the approximation involved in the expression (4.66). However, the present approach is preferred primarily because it would be unreasonable to assume that the total energy E of an electron will be always lost within the distance I when its residual mean projected range is just equal to I. Thus, this principal shortcoming of the initial ansatz I = Re(E, Ej) happens to be compensated for by the present approach, though in an arbitrary manner.

for

E


E„

The so-called Compton spectrum is given by 2a

EmiX = j - ^ - E

r

.

edge

:

of

(4.69)

the

(4.70)

The photoelectric effect, which becomes predominating for small photon energies (below 0.06 MeV in the case of water), produces almost monoenergetic electrons, because the incident photon is completely annihilated and its energy becomes the energy of the recoil electron, save for the ionization potential to be overcome. In principle there is no unique value for the latter, but 0.5 keV is a good general estimate for low-atomic-number media (water), where the K-shell electrons practically alone act as the target. Therefore, and because of Ey 0.5 keV, it is felt to be sufficient taking the S-function S(E —Ey+ 0.5 keV) for the description of the photoelectron spectrum. The total collision cross-section per electron, to be added as a independent factor, cannot be obtained accurately enough by theory (EVANS, 1968). Therefore, an empirical expression given by VICTOREEN (1943) has been used. With the parameters given for

133

Fast neutrons

water it can be written in the form

Tab. 4.7. Composition of wet tissue by weight fractions Hydrogen Carbon Nitrogen Oxygen

X B(E - E

y

+ 0.5 keV).

(4.71)

4.7. Secondary charged particle spectra of fast neutrons 4.7.1. Principles Neutrons produce energetic charged particles by interaction with atomic nuclei. In biological tissue, the following types of interaction account for more than 98 per cent of absorbed dose for neutron energies

E„ < 20 MeV

(AUXIER

et

al.,

1968): (i) elastic scattering by hydrogen, carbon, and oxygen; (ii) inelastic scattering by carbon and oxygen; (iii) nuclear reactions that produce a-particles from struck carbon, nitrogen, and Oxygen nuclei. Only these processes are considered in the following. qi'^ )(E),

initial

the

energy

spectra

l

^ ,

0.102

0.123

0.035

0.740

0.103

0.761

0.035

0.101

With few exceptions, the present calculations have been performed for the socalled ICRU-tissue (ICRU Report 10 b, 1964) with the simplified composition shown in Table 4.7 (oxygen stands for higher elements too). The exceptions are only some results in chapter 5 labelled to be for Shonka plastic with a composition also given in Table 4.7 (according to CASWELL a n d COYNE, 1974).

The definition (4.72) provides a simple relation between q^>(E) and the energy transfer

in

question (see section 3.3), refer to the recoil nuclei as well as the a-particles from (n, a) reactions. For the ease of notation, let us stipulate that the subscript i be an index of both the kind of the charged particle and the particular target and reaction type it stems from. q ^ ^ E ) is related to the differential cross-section, doijdE, expressed in terms of E, the energy of the secondary considered. Since the overall normalization of the spectra is unimportant (see the final equations in Tab. I, appendix), one can write q i M(E) = N

ICRUtissue SHONKA plastic

referred to as kerma: locally absorbed dose per unit neutron fluence (neutrons per centimetre squared). If K t denotes the kerma for the sheer element i, the following relations are.obvious to hold: •"max

Ki =

tf,

Ki=

Wi

f

E) with the general W.

MeV1/2 - kinetic energy of n

qV J d EqW(E) f d Vw{V) I X j J

dl"Va,r(E,

Ejf)(E, I")-,

o E0 - E, ErV\E0 - E,V -

I")), (4.94)

where reference is made to the considerations underlying Eq. (3.28). Substitution according to (4.93) and evaluation along the lines of section 3.4, in particular using the relations (3.35 a) and (3.32), reveals that [ ï ^ , , ] becomes the sum of the usual expressions, [V^] + [tf^], as given by Eq. (3.34) and, in addition, a negative term ['i / a _ r ] corJ . which quantifies the effect of a-r-correlation. This term contains the ¿"-integral in the form r fdl"(E, I"))

X^- Wr{E0 - E, ErW(E0 — E,V — I")), which can be transformed by d/dZ' = —d/dI", partial integration, and introducing the integration variable E' = E^(E, I"). This ends up in oo oo E i^.rlcorr = QV J dEqW(E) J dVu(V) J dE' 0 0 E^(E,l') x

dVJE, E>)

—

ErU) (E0 -E,V-

Ra(E) +

K(E'))). (4.95)

139

Fast neutrons Now it is easy to give this correction term the explicit form in which it has to be included in the final expressions (3.105) and (3.71), respectively. Because (4.95) involves one integration more than usual (cf. Eq. (3.39)), it has been decided, for the sake of simplicity, to reduce the initial energy spectrum to the three S-functions (4.91).

Another reaction of carbon nuclei is the break-up into three a-particles. 1 ) The energy transferred to the a-particles reads + Q ~~ En'y where the residual energy, En', of the secondary neutron can be estimated as an average (see Tab. 4.10). The distribution of the alpha energy, however, is very difficult to obtain. There are various reaction channels (modes of interaction) which would have to be analysed and weighted by the corresponding partial cross-sections. An example was given by C A S W E L L and C O Y N E (1972), and the shape of the distribution they found for 14-MeV neutrons suggests assuming tentatively, as a rough approximation, ~

for

a-particle, and according to the abovementioned estimation of E ( B A C H and CASWELL, 1968) we t h u s arrive a t

£ m a x = 1.186(tf, + Q - 0.267 Y^n MeV1'2). (4.97) For 14 MeV, this edge of the spectrum is in accord with the 'experimental' one, as well as the slope (In 2 0 ) / E m i x . The crucial test, however, is the comparison with the theoretical upper limit of alpha energy, which is known for any incident neutron energy and can be obtained, evidently, by Eq. (4.90) with E* = 0 and cos 0 = 1 . As easily verified, the correspondence of both expressions for 2?raax is indeed close enough to take the exponential distribution (4.96) with the constant parameter In 20 for a good description of the alpha energy spectrum at any incident neutron energy above the threshold -Q(M + ,1 )jM = 7.88 MeV.

E < Emax. (4.96)

This ansatz implies the relation E = 0.281i7 max for the mean energy of an

r)

The correlation of the three particles will be neglected because of mathematical complication, and can be neglected because of the relatively long average particle range.

Chapter 5

Implications concerning microdosimetry 5.1. Introduction The mathematical formalism developed in chapter 3 is a rather general tool for the calculation of mean values as well as distributions of various random quantities associated with the stochastics of radiation absorption. This general approach is there evaluated in detail only with respect to the model of DNA-lesion production and the theory of cellular radiation effects (statistics of DNA-lesions). I n this chapter, the same formalism will be used to obtain the customary microdosimetric parameters, i.e. the single event size spectrum /j(z) and its mean values zF, zD, and others surveyed in section 1.4.1. T h e data thus obtained, which are extensively tabulated in the appendix, are important for three reasons: to test the approximations involved in the general formalism (see section 3.2) by comparison with other computational and direct experimental results, to extend the availability of numerical data in microdosimetry, and to evaluate the approximate (microdosimetric) version of t h e radiation action theory as established in chapter 2. The general basis for the following can be given a t once without a lengthy derivation, since t h e model of DNA-lesion

production dealt with in section 3.5 rests solely on fi(j), the single event size spectrum in terms of the number, j, of ionizations in the sensitive volume. Look only a t the initial equation (3.1) from a formal point of view to realize t h a t , taking 1 ) =

h = -1,

= 0 , £2 = c ,

(5.1)

one can define a function

m =

E (1 — e"W)/i(7)

(5-2)

Zf j=o

which is identical with the quantity s for the parameters (5.1). Hence, t h e

funda-

mental function (£) can be calculated b y means

of

the

explicit

final

formulas

(3.67 to 71) for s, substituting the parameters (5.1) and re-replacing — /, by the 4 cross-section,

=

J {&) r /

v

dz

1

i I1 -

/.«»(z),

\

m

/i (A, (2). (5.18)

0

The explicit formula (5.3) shows the same splitting of 0(t) into an ion track core term and a 8-electron term. Hence, the quantities in question are readily obtained in the same way as where Eqs. (5.12), (5.13):

,(