a generalized formulation of dual radiation action. · 2012. 5. 22. · a. szechter, w. p....

OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

RADIATION R E S E A R C H

EDITOR-IN-CHIEF: ODDVAR F. NYGAARD

Volume 75, 1978

Academic Press

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RADIATION RESEARCH

OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

Edüor-in-Chief: O D D V A R F . N Y G A A R D , Department of Radiology, Case Western Reserve University, Cleveland, Ohio 44106

ASSOCIATE EDITORS

K . I . A L T M A N , University of Rochester J . H . K I M , Memorial Sloan-Kettering

A . V . C A R R A N O , Lawrence Livermore Laboratory S L I P S K Y , University of Minnesota

R ' i t a n d a i ^ ^ ' B m ™ U ° f P. N E T A , University of Notre Dame

K . H . C L I F T O N , University of Wisconsin J - P. O K U N E W I C K , Allegheny General Hospital

L . A . D E T H L E F S E N , University of Utah T T - ^ T ^ X T . A T ™- • A. M . R A U T H , Ontario Cancer Institute,

J . D . E A R L E , Mayo Clinic Toronto, Canada

C . L . G R E E N S T O C K , Atomic Energy of _ Q X J A 1 ) T I ) A „ . . . , , T , Canada, L t d . , Pinawa, Manitoba B - S H A P I R O , Albert Einstein Medical

' Center, Philadelphia G . M . H A H N , Stanford University

S. P. S T E A R N E R , Argonne National E . J . H A L L , Columbia University Laboratory

J . W. H U N T , Ontario Cancer Institute, ~ ^ ^ „ ^ , ^ 0 T A 1 o • *.a Toronto, Canada R - G . T H O M A S , Los Alamos Scientific

Laboratory M . I N O K U T I , Argonne National

Laboratory P. T O D D , Pennsylvania State University

OFFICERS OF THE SOCIETY

President: W A R R E N K . S I N C L A I R , Office of the Director, Bldg. 202, Argonne National Laboratory, Argonne, Illinois 60439

Vice President and President-Elect: W I L L I A M C . D E W E Y , Department of Radiology & Radiation Biology, Colorado State University, Fort Collins, Colorado 80521

Secretary-Treasurer: R O B E R T B . P A I N T E R , Laboratory of Radiobiology, University of California, San Francisco, California 94143

Editor-in-Chief: O D D V A R F . N Y G A A R D , Department of Radiology, Case Western Reserve University, Cleveland, Ohio 44106

Executive Director: R I C H A R D J . B U R K , J R . , 4720 Montgomery Lane, Suite 506, Bethesda, Maryland 20014

ANNUAL MEETINGS

1979: May 13-19, Sixth International Congress of Radiation Research, Tokyo, Japan 1980: New Orleans, Louisiana 1981: Minneapolis, Minnesota

VOLUME 75, 1978

Councilors Radiation Research Society 1977-1978

P H Y S I C S

W . C. Roesch, B a t t e l l e Pacific N o r t h w e s t Laborator ies M . I n o k u t i , Argonne N a t i o n a l L a b o r a t o r y

B I O L O G Y

E. J . H a l l , College of Physicians & Surgeons J . S. Rasey, U n i v e r s i t y of W a s h i n g t o n

M E D I C I N E

H . R . W i t h e r s , U n i v e r s i t y of Texas, H o u s t o n H . D . Su i t , Massachusetts General H o s p i t a l

C H E M I S T R Y

J . K . Thomas , U n i v e r s i t y of N o t r e D a m e J . F . W a r d , U n i v e r s i t y of Ca l i f o rn ia , Los Angeles

A T - L A R G E

J . E . Cleaver, U n i v e r s i t y of Ca l i f o rn ia , San Francisco P. F a i l l a , Argonne N a t i o n a l L a b o r a t o r y

CONTENTS OF VOLUME 75

N U M B E R 1, J U L Y 1 9 7 8

S U R E NDR A R U S T G I AND P E T E R R I E S Z . E S R of Spin-Trapped Radicals in Aqueous Solutions of Dihydropyrimidine Bases 1

P. C . K E S A V A N , G . J . S H A R M A , AND S. M . J . A F Z A L . Differential Modification of Oxic and Anoxic Radiation Damage by Chemicals. I . Simulation of the Action of Caffeine by Certain Inorganic Radical Scavengers 1 8

S H U J I Y O N E I AND M I K I T A K A T O . X -Ray- Induced Structural Changes in Erythrocyte Membranes Studied by Use of Fluorescent Probes 3 1

N A O N O R I M A T S U S A K A . Whole-Body Retention of 6 5 Z n during Pregnancy and Lactation, and Its Secretion into Milk in Mice 4 6

E . M A R T I N F I E L D E N , O R A Z I O S A P O R A , AND P A M E L A S. L O V E R O C K . The Application of Rapid Lysis Techniques in Radiobiology. I I I . The Effect of Radiosensitizers on the Production of D N A Damage and the Time Course of Its Repair 5 4

R E N £ B E A U P A I N . Effects of X Rays on D N A Synthesis in Developing and Regressing Miillerian Ducts of the Chick Embryo 6 6

G A R Y R. B L A C K B U R N , D O N A L D P. H I G H F I E L D , AND W I L L I A M C . D E W E Y . Studies on the Nuclear Envelope as a Radiation-Sensitive Site in Mammalian Cells : The Absence of an Effect on the Binding of Nuclear Membrane-Associated D N A 7 6

S T E P H E N L . S N Y D E R AND S H A R Y N K . E K L U N D . Radiation-Induced Alterations in the Dis -tribution of Lysosomal Hydrolases in Rat Spleen Homogenates 9 1

S. W A R R E N , O. G A T E S , C . E . B R O W N , R . N . C H U T E , AND M . W. P O R T E R . Spontaneous and Radiation-Induced Benign Tumors in Parabiont Rats 9 8

H O W A R D W. B R A H A M A N D G E O R G E A. S A C H E R . Metabolic and Thermoregulatory Effects of Acute 6 0 C o Radiation in Myomorph Rodents 1 0 8

S T E P H E N A. B E N J A M I N , F L E T C H E R F . H A H N , AND B R U C E B . B O E C K E R . Effects of Chronic Pulmonary Irradiation on Peripheral Lymphocytes and Their Function in the Dog. . . 1 2 1

G I L B E R T W. B E E B E , H I R O O K A T O , A N D C H A R L E S E . L A N D . Studies of the Mortality of A-Bomb Survivors. 6. Mortality and Radiation Dose, 1 9 5 0 - 1 9 7 4 1 3 8

D I A N A M . S C H L E S I N G E R AND R O B E R T L . B R E N T . Effects of X Irradiation during Preimplanta-tion Stages of Gestation on Cell Viability and Embryo Survival in the Mouse 2 0 2

P I E R R E J U L L I E N , D E N I S E B O R N E C Q U E , AND D A V I D S Z A F A R Z . Effects of X Irradiation of One or Both Partners on Mammalian Cell Ability to Form Viable Hybrids 2 1 7

C O R R E S P O N D E N C E

P. S U B R A H M A N Y A M , B . S. R A O , AND U . M A D H V A N A T H . O E R of Fission Neutrons for Induction of Heteroallelic Reversion in Diploid Yeast 2 3 0

T I M L . P A Y , F . A L A N A N D E R S E N , AND G O R D O N L . J E S S U P , J R . Survival of Drosophila melanogasler Pupa Exposed to Ultrasound 2 3 6

N U M B E R 2, A U G U S T 1 9 7 8

Y O S H I N O R I H A Y A K A W A , S U S U M U H A R A S A W A , A T S U S H I N A K A M O T O , K A Z U Y O S H I A M A N O , H I R O S H I H A T A N A K A , AND J U N E G A W A . Simultaneous Monitoring System of Thermal Neutron Flux for Boron-Neutron Capture Therapy 2 4 3

I . E G T V E D T , E . S A G S T U E N , R . B E R G E N E , AND T . H E N R I K S E N . Radiation Damage to Dihydrouracil 2 5 2

B . L . G U P T A , R. M . B H A T , K . R . G O M A T H Y , AND B . S U S H E E L A . Radiation Chemistry of the Ferrous Sulfate-Benzoic Acid-Xylenol Orange System 2 6 9

v

v i C O N T E N T S O F V O L U M E 75

J . F . W A R D AND I . K u o . Radiation Damage to D N A in Aqueous Solution: A Comparison of the Response of the Single-Stranded Form with That of the Double-Stranded Form 278

F R A N C I S J . J O H N S T O N . Radiolytic Reduction of Colloidal Silver Bromide 286 M I C H A E L A. J . R O D G E R S , D A V I D C . F O Y T , AND Z B I G N I E W A. Z I M E K . The Effect of Surfactant

Micelles on the Reaction between Hydrated Electrons and Dimethyl Viologen 296 H A Z E L L . L E W I S , D O N N R. M U H L E M A N , AND J O H N F . W A R D . Serologic Assay of D N A Base

Damage. I . 5-Hydroxymethyldeoxyuridine, A Radiation Product of Thymidine .305 M . S O D I C O F F , A. D . C O N G E R , P. T R E P P E R , AND N . E . P R A T T . Short-Term Radioprotective

Effects of WR-2721 on the Rat Parotid Glands 317 N. E . P R A T T A N D M . S O D I C O F F . Morphological Effects of WR-2721 on the Rat Parotid Acinar

Cell 327 W A N D A R E T E L E W S K A AND W A N D A L E Y K O . Effect of Ionizing Radiation on the Energy

Metabolism of Hog Lymphocytes 336 J . H . D I E L . Local Dose to Lung Tissue from Inhaled 2 3 8 P u 0 2 Particles 348 R A L P H E . D U R A N D . Effects of Hyperthermia on the Cycling, Noncycling, and Ifypoxic Cells

of Irradiated and Unirradiated Multicell Spheroids 373 H . M E T I V I E R , R . M A S S E , N . L E G E N D R E , AND J . L A P U M A . Pulmonary Connective Tissue

Modifications Induced by Internal a Irradiation. I . Effect of Time and Dose on Altera-tions following Inhalation of Plutonium-239 Dioxide Aerosol in Rat 385

W. S T E V E N S , F . W. B R U E N G E R , D . R . A T H E R T O N , D . S. B U S T E R , AND G . H O W E R T O N . The Retention and Distribution of 2 4 1 Am and 6 5 Z n , Given as D T P A Chelates in Rats and of [ , 4 C ] D T P A in Rats and Beagles 397

J . J . VAN H E M M E N , W. J . A. M E U L I N G , AND J . F . B L E I C H R O D T . Effect of Oxygen on Inactiva-tion of Biologically Active D N A by 7 Rays in Vitro: Influence of Metalloporphyrins and Enzymatic D N A Repair 410

H . C H E S S I N , T . M C L A U G H L I N , Z. M R O C Z K O W S K I , W. D . R U P P , AND K . B . L O W . Radio-sensitization, Mutagenicity, and Toxicity of Escherichia coli by Several Nitrofurans and Nitroimidazoles 424

C O R R E S P O N D E N C E

A. S Z E C H T E R , W. P. K O W A L S K Y , AND G . S C H W A R Z . Average Versus Viable Cell Multi -plicity in Survival Determination for Mammalian Cells in Vitro 432

M . R. R A J U , H . I . A M O L S , AND J . B . R O B E R T S O N . Effect of Negative Pions on Cells Plated on Glass and Plastic Surfaces 439

J . L . R E D P A T H AND L . K . P A T T E R S O N . The Effect of Membrane F a t t y Acid Composition on the Radiosensitivity of E. coli K-1060 443

J . H . K I M , S. H . K I M , AND E . W. H A H N . Killing of Glucose-Deprived Hypoxic Cells with Moderate Hyperthermia 448

N U M B E R 3, S E P T E M B E R 1978

F R A N Z S T . G E O R G E , D A V I D W. A N D E R S O N , AND L I N D A M C D O N A L D . The Electron Spectrum at Equilibrium Depth from a Narrow Beam of 6 0 C o Photons 453

G U D R U N A L M C A R L S S O N . Basic Concepts in Dosimetry. A Critical Analysis of the Concepts of Ionizing Radiation and Energy Imparted 462

A L B R E C H T M . K E L L E R E R AND H A R A L D H . R O S S I . A Generalized Formulation of Dual Radia-tion Action 471

A M R A M S A M U N I , M O R D E C H A I C H E V I O N , Y E H E S K E L S. H A L P E R N , Y A E L A. I L A N , AND G I D O N C Z A P S K I . Radiation-Induced Damage in T 4 Bacteriophage: The Effect of Superoxide Radicals and Molecular Oxygen 489

K R I S H A N M . B A N S A L AND R I C H A R D W. F E S S E N D E N . Radiation-Chemical Studies of the Reaction of S 0 4 ~ with Uracil and Its Derivatives 497

Z. D . D R A G A N I C , I . G . D R A G A N I C , AND S. V . J O V V N O V I C . The Radiation Chemistry of Aque-ous Solutions of Cyanamide 508

C O N T E N T S O F V O L U M E 75 v i i

M Y R A N C . S A U E R , J R . , K L A U S H . S C H M I D T , C H A R L E S D . J O N A H , C O N R A D A. N A L E W A Y , AND E D W I N J . H A R T . H i g h - L E T Pulse Radiolysis: 0 2 ~ and Oxygen Production in T r a c k s . . 519

J . H . H E N D R Y AND A. H O W A R D . Neutrons and the Oxygen Effect: Corroboration of the Greater Effectiveness at High L E T for Sensitization by Oxygen at Low Concentrations. 529

T . W. W O N G , G . F . W H I T M O R E , AND S. G U L Y A S . Studies on the Toxicity and Radiosensitizing Ability of Misonidazole under Conditions of Prolonged Incubation 541

D A V I D K . K W A N AND A M O S N O R M A N . Radiosensitivity of Large Human Monocytes 556 J O H N M . Y U H A S AND A L B E R T P. L i . In Vitro Studies on the Radioresistance of Oxic and

Hypoxic Cells in the Presence of Both Radioprotective and Radiosensitizing Drugs. . . 563 J . L A N D R Y AND N . M A R C E A U . Rate-Limiting Events in Hyperthermic Cell Killing 573 C H A N G W. S O N G AND D A N I E L P. G U E R T I N . Combined Effect of X Irradiation and Cell -

Mediated Immune Reaction 586 M . H . B A S S A N T AND L . C O U R T . Effects of Whole-Body y Irradiation on the Activity of

Rabbit Hippocampal Neurons 593 A L M A H O W A R D AND F . G . C O W I E . Induced Resistance in Closlerium: Indirect Evidence for

the Induction of Repair Enzyme 607 N . J . P A R K S , R. R. P O O L , J . R . W I L L I A M S , AND I I . G . W O L F . Age and Dosage-Level De-

pendence of Radium Retention in Beagles 617 R A Y D. L L O Y D , S U S A N S. M C F A R L A N D , D A V I D R. A T H E R T O N , AND C H A R L E S W. M A Y S .

Plutonium Retention, Excretion, and Distribution in Juvenile Beagles Soon After Injection 633

J . L . R E D P A T H , R. M . D A V I D , AND L . C O H E N . Dose-Fractionation Studies on Mouse Gut and Marrow: An Intercomparison of 6-MeV Photons and Fast Neutrons (E = 25 M e V ) . . 642

F I O N A A. S T E W A R T , B A R R Y D. M I C H A E L , A N D J U L I A N A D E N E K A M P . Late Radiation Damage in the Mouse Bladder as Measured by Increased Urination Frequency 649

M . M O H I U D D I N AND S. K R A M E R . Therapeutic Effect of an Elemental Diet on Proline Ab-sorption across the Irradiated Rat Small Intestine 660

A N N O U N C E M E N T S 664

A U T H O R I N D E X F O R V O L U M E 75 666

The Subject Index for Volume 75 will appear in the December 1978 issue as part of a cumu-lative index for the year 1978.

R A D I A T I O N R E S E A R C H 75, 471-488 (1978)

A Generalized Formulation of Dual Radiation Action1

A L B R E C H T M . K E L L E R E R A N D H A R A L D H . R O S S I

Institut für Medizinische Strahlenkunde der Universität Würzburg, Würzburg, Federal Republic of Germany and Radiobiological Research Laboratories, Columbia University,

New York, New York 10027

K E L L E R E R , A. M. , A N D R O S S I , H . H . A Generalized Formulation of Dual Radiation Action. Radial. Res. 75, 471-488 (1978),

Dual radiation action is a process in which cellular lesions are produced as a result of the interaction of pairs of sublesions that are molecular alterations produced by ionizing radiation. Previous formulations of this process have employed a number of simplifying assumptions that limit the accuracy and the range of application of theo-retical analysis. The formulation presented here removes some of these restrictions by introducing three functions that describe the geometry of the sensitive material in the cell, the geometry of the pattern of energy deposition, and the interaction probability of sublesions as a function of their separation. The relation derived is similar to that obtained previously, in that lesion production is found to depend on two terms that are proportional to the first and the second power of the absorbed dose. However, the coefficients of these terms are now derived on the basis of a more realistic treatment.

I N T R O D U C T I O N

T h e t e r m dual radiation action can refer to a v a r i e t y of mechanisms. C o m m o n to these is t h a t sublesions are produced depending on the p a t t e r n of energy transfers t o the cell , a n d t h a t these sublesions can i n t e r a c t i n pairs t o produce lesions w h i c h i n t u r n determine the observed effect. T h e general n o t i o n of d u a l r a d i a t i o n ac t i on thus comprises a broad spec t rum of mechanisms, a n d i t is v e r y d i f f i c u l t t o p rov ide a m a t h e m a t i c a l f o r m u l a t i o n w h i c h covers a l l the possible mechanisms i n such a w a y t h a t the m i c r o d i s t r i b u t i o n of i m p a r t e d energy as w e l l as t h e geometry of the sensitive s tructures i n the cell is accounted for. I t h a d been necessary, therefore, to re ly on var ious s impl i f i cat ions w h i c h res tr i c t the appl i ca -b i l i t y a n d the accuracy of theoret i ca l t reatments . I n the f o l l owing a f o r m u l a t i o n is presented t h a t avoids m a n y — a l t h o u g h n o t a l l — o f the restr ic t ions . T h e neces-sar i l y complex m a t h e m a t i c a l t r e a t m e n t leads t o a r e la t i ve ly s imple general

1 Work supported by E U R A T O M Contract 208-76-7 BIO D, by Contract EY-76-C-02-3243 from the Energy Research and Development Administration, and by Grants C A 12536 and C A 15307, awarded by the National Cancer Institute, D H E W . The U . S. Government's right to retain a nonexclusive royalty-free license in and to the copyright covering this paper, for govern-mental purposes, is acknowledged.

471

0033-7587/78/0753-0471S02.00/0 Copyright © 1978 by Academic Press, Inc.

All rights of reproduction in any form reserved.

472 K E L L E R E R A N D R O S S I

f o r m u l a , a n d i t is fo l lowed b y a n u m b e r of p rac t i ca l appl icat ions . However , before t h i s m a t e r i a l is presented, i t w i l l be useful t o recal l br ie f ly re lated earlier approaches.

L e a (1) analyzed the p r o d u c t i o n of chromosome aberrat ions b y i o n i z i n g r a d i a t i o n s . H e assumed t h a t the sublesions, w h i c h he t e r m e d chromosome breaks, were produced along charged-part ic le t racks . These charged-part ic le t racks were, f o r s i m p l i c i t y and for lack of a more accurate descr ipt ion , t r ea ted as s t r a i g h t l ines w i t h energy transfer rates corresponding to the L E T of the part i c l e . F u r t h e r -more , i t was assumed t h a t a pa ir of chromosome breaks could i n t e r a c t a n d , w i t h a c e r t a i n p r o b a b i l i t y , produce a lesion, i .e. , an observable aber ra t i on such as a d i c e n t r i c chromosome. Lea pos tu la ted t h a t an i n t e r a c t i o n between t w o sublesions c o u l d t a k e place p r o v i d e d t h e y were f o rmed a t a distance less t h a n a specified va lue h.

I n spi te of i t s obvious s impl i f i cat ions , Lea's analysis p roved t o be successful. H i s compar ison of X - r a y - i n d u c e d and neutron - induced chromosome aberrat ions l ed t o t h e conclusion t h a t the c r i t i c a l distance h for the c o m b i n a t i o n of chromo-some breaks is of the order of a micrometer . T h i s resul t has been conf irmed b y var i ous authors {2-4), a n d the works of Lea a n d re lated invest igat ions have exerted a n i m p o r t a n t influence on m a n y subsequent detai led studies of r a d i a t i o n -i n d u c e d chromosome aberrat ions (see, for example, (5-7)).

W i t h the development of mi c rodos imetry a more r igorous t r e a t m e n t of d u a l r a d i a t i o n a c t i o n became possible (2). T h i s approach was based on the p r o b a b i l i t y d i s t r i b u t i o n s of specific energy z i n spherical sites. I t assumed t h a t sublesions are f o r m e d w i t h i n spherical sites i n the cell w i t h a y i e l d p r o p o r t i o n a l t o the energy i m p a r t e d t o the site. F u r t h e r m o r e , i t postulated t h a t sublesions i n a site have a fixed p r o b a b i l i t y of i n t e r a c t i n g pairwise . U n d e r cer ta in s i m p l i f y i n g assumptions t h e expected number of lesions is t h e n p r o p o r t i o n a l to the square of the specific energy i n the site, and the y i e l d S (D) of lesions at a dose D is, therefore, p ropor -t i o n a l t o t h e expectat ion value of the square of specific energy. T h i s led t o t h e l i n e a r - q u a d r a t i c dose-effect r e l a t i o n

S(D) = k? = k{W + Z>2), (1)

where f is a n average of specific energy produced i n i n d i v i d u a l events i n the site. T h i s r e l a t i o n was f o u n d to be applicable t o a wide range of cel lular effects

produced b o t h b y sparsely i on iz ing radiat ions a n d b y neutrons . I n p a r t i c u l a r , i t p r o v e d va luab le i n the pred i c t i on of the very h i g h re la t ive b io logical effectiveness of l o w doses of neutrons .

I n i t s c u r r e n t f o r m , however, the theory of d u a l r a d i a t i o n ac t ion is also based o n var i ous s impl i f i cat ions the most conspicuous of w h i c h is the somewhat a r b i -t r a r y assumpt ion of spherical sensitive regions i n the cell. T h e n u m b e r a n d the s p a t i a l d i s t r i b u t i o n of such sites i n the nucleus of the cell have been le f t unspecif ied. I n f a c t , t h e n o t i o n of spherical sensitive sites w i t h i n the nucleus serves merely as a n a p p r o x i m a t i o n . A more realistic assumpt ion is t h a t sublesions can be produced t h r o u g h o u t the nucleus of the cell b u t can combine w i t h appreciable p r o b a b i l i t y on ly over distances smaller t h a n the d imension of the nucleus.

F O R M U L A T I O N O F D U A L R A D I A T I O N A C T I O N 473

The a p p r o x i m a t i o n i n terms of spherical sensitive sites has been necessitated by the fac t t h a t microdos imetr ic da ta usual ly refer t o spherical sites. M o r e o v e r , the a p p r o x i m a t i o n m a y o f ten be sat is factory a n d suff ic iently accurate t o b r i n g o u t the characterist ic differences i n the actions of var ious types of i o n i z i n g rad iat ions . I t is d o u b t f u l , however, whether cer ta in l i m i t i n g cases are adequate ly t reated b y th i s f o r m a l i s m . A n example is the ac t i on of low-energy X rays . For such rad ia t i ons the par t i c l e ranges are subs tant ia l l y smaller t h a n the site d iameters of one or a few micrometers . E n e r g y transfers produced b y such t r a c k s are, therefore, m u c h more closely spaced t h a n energy transfers r a n d o m l y s i t u a t e d i n the site. T h i s m a y lead t o an increased c o m b i n a t i o n p r o b a b i l i t y of sublesions a n d , accord ingly , t o a n increased biological effectiveness (8-11). As i l l u s t r a t e d i n F ig . 1, the close spacing of energy transfers w i t h i n the site is disregarded i f one refers merely t o the specific energy i n the site. T h e established f o r m a l i s m is, therefore, n o t sufficient, and a more general descr ipt ion is desirable. I t w i l l be presented i n th i s ar t i c le .

T h e p r i n c i p a l l i m i t a t i o n of the f o r m u l a t i o n g iven here is t h a t i t applies o n l y i f one is deal ing w i t h one t y p e of effect. Since a v a r i e t y of effects, i n c l u d i n g n o t o n l y cell d e a t h b u t also specific in jur ies ( t rans format ions , specified somat ic m u t a t i o n s , etc . ) , appear t o result f r o m d u a l ac t i on (2), i t m u s t be concluded t h a t there are var ious types of lesions (or combinat ions of lesions) responsible for these effects. T h i s , i n t u r n , leads t o the poss ib i l i ty t h a t there can be m u t u a l interference between sublesions w h i c h reduces the frequency of lesions r e s u l t i n g i n any p a r t i c u l a r effect. A l t h o u g h th i s is of substant ia l prac t i ca l i m p o r t a n c e , i t w i l l n o t be considered here.

A n o t h e r , re lated l i m i t a t i o n of the t r e a t m e n t g iven below is t h a t i t disregards compl i cat ions t h a t can arise w h e n the c o m b i n a t i o n p r o b a b i l i t y of a n y p a i r of sublesions is inf luenced b y the presence of other sublesions. I n such c ircumstances t h e t r e a t m e n t is v a l i d on ly w i i e n the specific energy is low.

T h i s presentat ion should therefore be regarded as a ma jor step t o w a r d t h e deve lopment of a s t i l l more general f o rmal i sm.

F I G . 1. Schematic diagram of energy deposition by ( 1 ) a particle of large range and ( 2 ) a particle of short range. The specific energy z is equal in the two sites, but the biological effectiveness is larger for the short-range particle because of the smaller distances between sublesions.


S T A T E M E N T O F T H E P R O B L E M

Sensitive Matrix and Inchoate Distribution of Energy

I o n i z i n g r a d i a t i o n produces a cer ta in microscopic p a t t e r n of energ}^ transfers i n the cell. T o describe th is p a t t e r n one needs the concepts of transfer p o i n t and energy transfer (12). T h e t e r m transfer point designates a p o i n t T where an i on iz ing part ic le undergoes an i n t e r a c t i o n w i t h the i r r a d i a t e d m e d i u m . The t e r m energy transfer designates the energy w h i c h is, a t the p o i n t T, t rans ferred f r o m the r a d i a t i o n field t o the m a t e r i a l .

T h e concept of energy transfers is f u n d a m e n t a l i n the considerations w h i c h fo l low. I t is therefore appropr ia te to state i t s d e f i n i t i o n more f o r m a l l y . I f E is the k inet i c energy of the inc ident i on i z ing part i c le a n d £ E» is the s u m of the kinet ic energies of a l l i on iz ing partic les l eav ing T, t h e n the energy transfer at T is (E — X) Ey). I t w i l l be noted t h a t £ Ev = 0 i f no i on iz ing part ic les are leaving T. Changes of rest mass resu l t ing f r o m the i n t e r a c t i o n (pair p r o d u c t i o n , nuclear t rans format ions ) are disregarded here. A rigorous de f in i t i on w o u l d have to be analogous to t h a t of energy-imparted (18).

T h e t empora l d i s t r i b u t i o n of energy transfers w i l l n o t be considered i n this a r t i c l e ; i.e., instantaneous exposure w i l l be assumed. Formulae der ived earlier (2) for the dose-rate dependence w i l l , however, r emain v a l i d .

T h e microscopic p a t t e r n of energy transfers has been t e r m e d t h e inchoate distribution (12) i n order t o ind icate t h a t i t is the i n i t i a l p a t t e r n w h i c h exists before subsequent processes, such as energy conduct ion or d i f fus ion of free radicals , lead t o a di f ferent spat ia l d i s t r i b u t i o n of i n i t i a l r a d i a t i o n products. F igure 2 represents the s i t u a t i o n schematical ly . E a c h of the heavy dots stands for an energy transfer . T h e l i g h t contour represents the out l ine of the ce l l ; the

F I G . 2. Schematic diagram of a cell and its nucleus taken as the gross sensitive volume. The lightly dotted regions represent the sensitive matrix, i.e., the volume collectively occupied by the loci. The heavy dots represent energy transfers due to a charged particle.


heavy contour indicates the nucleus of the cell. I t is assumed t h a t the nucleus contains a large n u m b e r of smaller regions t e r m e d loci. I f an energy transfer is i n a locus a sublesion m a y result . T h e vo lume col lect ively occupied b y t h e loci w i l l , i n the f o l l owing , be t e r m e d the sensitive matrix. I n F i g . 2 i t is represented b y the l i g h t l y d o t t e d areas. As ind i ca ted i n the f igure, the sensitive m a t r i x w i l l i n general not be u n i f o r m l y dispersed t h r o u g h o u t the nucleus of the cell. Sublesions created anywhere i n the sensitive m a t r i x can i n t e r a c t t o f o r m lesions, b u t the i n t e r a c t i o n p r o b a b i l i t y m a y be smal l i f the i r separation is large.

Formation of Sublesions

The process of r a n d o m f o r m a t i o n of sublesions i n the sensitive m a t r i x m a y be i n t u i t i v e l y c lear; however, i t needs to be specified r igorously . Assume t h a t there are / transfer po ints i n the sensitive m a t r i x a n d t h a t u (i = 1, 2, . . . , / ) are the energy transfers a t these po ints . I t is t h e n postulated t h a t the n u m b e r of sub-lesions produced a t the iih transfer p o i n t fol lows a Poisson d i s t r i b u t i o n w i t h mean ce,. I t m a y be reasonable t o assume t h a t cet- is general ly m u c h smaller t h a n u n i t y . I n th is case ce{ is s i m p l y the p r o b a b i l i t y t h a t a sublesion w i l l be produced b y the energy transfer e„ a n d the p r o b a b i l i t y t h a t m u l t i p l e sublesions are p r o -duced by one transfer can be disregarded. T h e assumpt ion is n o t necessary, however ; one can instead consider the more general Poisson process i n the f o r m a -t i o n of sublesions.

The nature of sublesions is le f t unspecified i n the present context . T h e reason is t h a t there are a t least t w o di f ferent mechanisms i n the cell w h i c h have the characteristics of d u a l r a d i a t i o n act ion . One mechanism is the p r o d u c t i o n of D N A double -s trand breaks as the result of t w o adjacent s ingle-strand breaks. T h e other mechanism is the f o r m a t i o n of chromosome aberrat ions f r o m pairs of chromosome breaks. B o t h mechanisms are re levant t o cel lular r a d i a t i o n bio logy . Ear l i e r analyses ( i , 2) have l i n k e d the l i n e a r - q u a d r a t i c dose-effect relat ions i n cel lular r a d i a t i o n ac t i on to chromosome aberrat ions or another process charac-ter ized by large i n t e r a c t i o n distances of sublesions. T h e f o r m a t i o n of D N A double -s trand breaks, whi le another instance of d u a l r a d i a t i o n ac t ion , takes place at distances w h i c h are so smal l t h a t the i n t e r a c t i o n of energy transfers f r o m separate charged partic les can be disregarded except a t doses m u c h higher t h a n those re levant t o r a d i a t i o n effects on the cells of most higher organisms. O n l y the l inear t e r m i n E q . (1) is i m p o r t a n t , therefore, a n d the f o r m a t i o n of double -s trand breaks f r o m single-strand breaks cannot be the process responsible for the s igmoidal s u r v i v a l curves of m a m m a l i a n cells.

On the one hand , the p r o d u c t i o n of chromosome aberrat ions is more closely l i n k e d to cell death or specific cel lular i n j u r i e s ; on the other hand , i t is compl icated b y the fact t h a t the sublesions, i.e., chromosome breaks, are possibly themselves represented by double -s trand breaks. T h e descr ipt ion of the f o r m a t i o n of chromo-some breaks b y i n d i v i d u a l energy transfers serves, therefore, merely as an a p -p r o x i m a t i o n . I n a c t u a l i t y , the efficiency of the i r p r o d u c t i o n m a y depend on r a d i a t i o n q u a l i t y , and , as ind i ca ted earlier (14), modi f icat ions of the f o r m a l t r e a t m e n t m a y be requ i red to account for t h i s effect. T h i s , however, w i l l n o t be a


subject of the present art ic le . The t r e a t m e n t g iven here is therefore more read i ly app l i ed t o t h e process whereby D N A s ingle -strand breaks combine t o f o r m d o u b l e - s t r a n d breaks. However , whi le a p p l i c a t i o n t o th i s case is more s t r a i g h t -f o r w a r d a n d m a y help i n v i sua l i z ing the process, i t is—as stated above—of somewhat less pragmat i c impor tance t h a n the a p p l i c a t i o n t o the f o r m a t i o n of chromosome aberrat ions.

D i f f u s i o n of p r i m a r y r a d i a t i o n products such as free radicals is disregarded i n the present context i n order t o reduce the c o m p l e x i t y of the m a t h e m a t i c a l t r e a t m e n t . I n s t e a d i t is s i m p l y assumed t h a t each sublesion is f o r m e d at a n energy transfer p o i n t , i.e., t h a t the spat ia l d i s t r i b u t i o n of sublesions is a subset of the inchoate d i s t r i b u t i o n i n the sensitive m a t r i x . T h e assumpt ion t h a t sublesions are produced precisely a t the energy transfer po ints is n o t essential, a n d a subse-q u e n t ar t i c l e w i l l generalize the result of the present w o r k so t h a t i t includes di f fus ion. N o substant ia l changes i n the m a t h e m a t i c a l f o r m a l i s m result f r o m t h i s f u r t h e r general izat ion.

Formation of Lesions

Pairs of sublesions can combine t o f o r m a lesion. T h e average c o m b i n a t i o n p r o b a b i l i t y of t w o sublesions w i t h m u t u a l distance x is designated b y g(x)} a n d th i s is also t e r m e d the interaction probability as a f u n c t i o n of distance x between t w o sublesions.

I f o n l y t w o sublesions are present, the p r o b a b i l i t y of the f o r m a t i o n of a lesion is g(x). T h e s i t u a t i o n is more compl icated i f there are three sublesions. T h e expected n u m b e r n of lesions w i l l t h e n depend on a l l three distances Xu, xu, x 23, where the indices refer t o the three sublesions. T h e f u n c t i o n a l r e l a t i o n n(xu, xu, %2d) need n o t be simple because the p r o b a b i l i t y t h a t sublesion 1 w i l l combine w i t h sublesion 2 w i l l be d imin ished because of the poss ib i l i ty t h a t 1 w i l l combine w i t h 3, a n d b o t h c o m b i n a t i o n probabi l i t i es for 1 are d imin i shed because of the poss ib i l i ty t h a t 2 w i l l combine w i t h 3. T h e t e r m saturation of sublesions m a y refer t o t h i s i n t e r r e l a t i o n .

T h e s a t u r a t i o n of sublesions, however, can be disregarded whenever a l l sub-lesions have a smal l c o m b i n a t i o n p r o b a b i l i t y , so t h a t sublesions are m u c h more numerous t h a n lesions. One can t h e n s i m p l y set

n = g(xl2) + g(xu) + g(xn) (2)

and t h e analogous f o r m u l a

n = L g(xik) (i, k = 1, . . J ) (3) i,k

i


breaks w o u l d be i n v o l v e d i n a double -s trand break. S imi lar considerations a p p l y to the f o r m a t i o n of chromosome aberrat ions, since the i r number is m u c h smal ler t h a n t h a t of chromosome breaks ( 1 ) . I t is less cer ta in whether s a t u r a t i o n of sublesions can be disregarded for densely i on i z ing rad iat ions , b u t th i s w i l l n o t be examined i n the present art i c le .

Mean Number of Lesions

E q u a t i o n (3) specifies the expected number n of lesions for a g iven constellation (spat ia l arrangement ) of sublesions i n a cell. T h e ac tua l number v of lesions w i l l f o l low a Poisson d i s t r i b u t i o n w i t h the mean value n .

However , t h e s i t u a t i o n is even more i n v o l v e d . T h e q u a n t i t y n is t h e expected number of lesions resu l t ing f r o m a specified constel lat ion of sublesions. A t a cer ta in dose, d i f ferent specific energies m a y be i m p a r t e d to the sensitive m a t r i x ; b u t even w h e n the same specific energy is i m p a r t e d a di f ferent inchoate p a t t e r n of energy transfers m a y result . T h e conste l lat ion of sublesions w i l l v a r y accord-i n g l y , a n d one w T i l l deal w i t h a cer ta in d i s t r i b u t i o n of n . T h e d i s t r i b u t i o n dD(v) of the ac tua l number v of lesions is t h e n a superposit ion of Poisson d i s t r i b u t i o n s of di f ferent mean values n. I n cer ta in cases i t m a y be feasible to ca lculate t h i s d i s t r i b u t i o n c f o M . However , t h i s is no t a t t e m p t e d i n the present t r e a t m e n t ; instead, o n l y the expected n u m b e r v of lesions per cell a t a specified dose D is ca lculated. T h i s expected n u m b e r of lesions is designated b y the s y m b o l S (D):

8(D) = v{D) = £ vdD(v). (4) o

One m u s t note t h a t t h i s q u a n t i t y does n o t necessarily determine t h e observed effect, even i f the effect is a f u n c t i o n S(v) of the n u m b e r of lesions. As has been p o i n t e d o u t also b y others (16) one cannot generally equate the observed q u a n t i t y S(D) = S(v) w i t h S ( v ) :

S ( P ) = S(t vdnW) * = £ S(v)dD(v). (5) 0 0

I f , for example, s u r v i v a l is a n exponent ia l ly decreasing f u n c t i o n of the n u m b e r of lesions,

S(v) = € T * , (6)

t h e n s u b s t i t u t i o n of t h e mean va lue of v for the ac tua l averaging procedure leads t o a n overest imate of the effect :

S(p) = e-k*


permi t s one to calculate 8 ( D ) i f the geometry of the sensitive m a t r i x and the m i c r o d i s t r i b u t i o n of energy transfers arc k n o w n . I t is f o u n d t h a t th i s r e la t i on depends on ly on g(x) and on t w o funct ions w h i c h m a y be called the p r o x i m i t y f u n c t i o n of the energy transfers a n d the p r o x i m i t y f u n c t i o n of the sensitive m a t r i x . These funct ions are in t roduced before the general f o r m u l a for 8 ( D ) is der ived .

D E F I N I T I O N O F T H E P R O X I M I T Y F U N C T I O N S

I t is f ound t h a t the q u a n t i t y 8 ( D ) , the expected number of lesions, can be expressed b y a simple i n t e g r a l over t w o funct ions s(x) and t(x). These funct ions are discussed i n the present section. T h e y are closely re lated . s(x) characterizes the geometry of the sensitive m a t r i x ; t(x) characterizes the spat ia l d i s t r i b u t i o n of energy transfers. s(x) and t(x) are s imi lar to a f u n c t i o n w h i c h i n m a t h e m a t i c a l morpho logy (17, 18) is called the covariance of a geometr ical ob j e c t ; however, the t e r m proximity function is used here.

The Function s(x)

T h e p r o x i m i t y f u n c t i o n , s(x), is the produc t of the vo lume of the sensitive m a t r i x and the p r o b a b i l i t y density of distances between t w o po ints r a n d o m l y chosen i n the m a t r i x (19).

A n equivalent de f in i t i on w h i c h conveys the meaning of the f u n c t i o n s(x) somewhat more d i r e c t l y is the f o l l o w i n g :

Select points, P, randomly in the matrix. Then s(x) dx is the average partial volume of the matrix which is contained in a shell of radius x and thickness dx centered at P.

Some examples m a y serve t o i l l u s t r a t e the f u n c t i o n s(x). T h e simplest case is t h a t of a u n i f o r m extended m e d i u m . I n th i s case one obtains

s(x) = 4TTX 2 . (8)

A somewhat more compl icated case is t h a t of a spherical d o m a i n of d iameter d. T h e f u n c t i o n s(x) for th i s case can be readi ly obta ined f r o m a f o r m u l a g iven b y K e n d a l l and M o r a n (20, E q . (2 .125) ) ; i t is also l i n k e d t o the chord - l ength d i s t r i -b u t i o n (21) resul t ing f r o m the r a n d o m traversa l of a sphere:

/ 3x x^ \ s(x) = 4:TX2(l I ) , x < d. (9)

\ 2d 2d*/

T h i s f u n c t i o n was u t i l i z e d earlier i n the context of microdos imetr ic computat ions (19, 22)}

Other geometries m i g h t also be considered. T h u s one obtains the f o l l owing f u n c t i o n for a spherical shell of d iameter d and ( in f in i tes imal ) thickness A r :

s(x) = SrAr, x < d. (10)

T h i s result is c i ted here w i t h o u t d e r i v a t i o n . 2 I n the earlier article (19) the term distance distribution was used and the notation tf(x) was

applied instead of s(x).


One may note t h a t these formulae also app ly i f loci are r a n d o m l y d i s t r i b u t e d w i t h o u t the specified configurations. T h e reason is t h a t s(x) w i l l t h e n s t i l l account for the collective response of the cells. I f clusters of loci are r a n d o m l y d i s t r i b u t e d w i t h i n a cer ta in conf igurat ion , the formulae w i l l have to be modi f ied . T h i s is considered i n the examples at the end of th i s art ic le .

The Function t(x)

W h e n bio logical specimens are i r r a d i a t e d , the frequency of pairs of sublesions t h a t are i n sufficient p r o x i m i t y to in terac t depends on t w o factors. T h e first factor is the a v a i l a b i l i t y of ne ighbor ing l o c i ; th is is described b y the f u n c t i o n s(x). The second factor is the frequency of energy transfers spaced closely enough to affect ne ighbor ing loc i . T o account for th i s second factor , one uses a f u n c t i o n tüix), w h i c h is analogous to the f u n c t i o n s(x) b u t relates t o the p a t t e r n of energy transfers and not to the geometry of the sensitive m a t r i x . T h e index D is added to indicate t h a t the f u n c t i o n to(x) depends on absorbed dose. I t w i l l be seen, however, t h a t tD(x) separates convenient ly i n t o a t e r m t(x) w h i c h is independent of absorbed dose a n d a second t e r m w h i c h is p r o p o r t i o n a l to absorbed dose and independent of r a d i a t i o n q u a l i t y .

The f u n c t i o n ID(X) has been i n t r o d u c e d and u t i l i z e d i n the context of m i c r o -dosimetric c omputat i ons (19, 22); i t characterizes the inchoate d i s t r i b u t i o n i n the same w a y t h a t the f u n c t i o n s(x) characterizes the sensitive m a t r i x . T h e de f in i t i on of tn(x), therefore, is largely analogous to the d e f i n i t i o n of s(x):

Select energy transfers, e, randomly in the irradiated medium. Then t(x)dx is the expected sum of energy transfers contained in a shell of radius x and thickness dx centered at the transfer point ivhere the energy transfer € occurred.

One must note t h a t the r a n d o m selection is between energy transfers ra ther t h a n between transfer po ints . Accord ing ly , each transfer p o i n t T has a selection p r o b a b i l i t y p r o p o r t i o n a l t o the associated energy transfer (19).

A l t h o u g h the f u n c t i o n tD(x) is analogous to the f u n c t i o n s(x), i t differs i n t h a t i t has the d imens ion of energy d i v i d e d b y l eng th , w rhile s(x) has the d imension of vo lume d i v i d e d b y l ength .

T h e p r o x i m i t y f u n c t i o n tD (x) alw rays separates i n t o t w o terms. T h e first t e r m , w h i c h i n the f o l l o w i n g is designated t(x), is the znJratrack c o n t r i b u t i o n ; i.e., i t represents energy transfers produced b y the same event 3 as the energy transfer a t p o i n t T. T h i s t e r m depends on r a d i a t i o n q u a l i t y b u t n o t on absorbed dose. T h e second t e r m represents energy transfers b y other , uncorre lated charged part ic les . T h i s m t e r t r a c k c o n t r i b u t i o n is p r o p o r t i o n a l t o absorbed dose and independent of r a d i a t i o n q u a l i t y . One obtains

tD(x) = t(x) + 4TX2PD, (11)

where p is the dens i ty of the i r r a d i a t e d m e d i u m . T h e simplest f o r m u l a for t(x) results i f one regards the charged part i c le as

s l o w i n g d o w n cont inuous ly w i t h s topp ing power L w h i l e m o v i n g along an i n f i n i t e

3 An event (18) designates the imparting of energy by an ionizing particle and/or its secondaries.


s t r a i g h t l ine. T h e n one has t(x) = 2 L . (12)

As one can readi ly show, the dose average, L D , replaces the va lue L i f a d i s t r i b u -t i o n of L E T values is assumed.

I f , instead of an i n f i n i t e par t i c l e t r a c k , one of range R is considered wh i l e the l inear energy transfer is t a k e n to be constant , the f o l l o w i n g r e l a t i o n resu l t s :

t(x) = 2 L ( 1 - z/R), x < R. (13)

T h i s is a drastic s impl i f i ca t i on w h i c h does n o t account for the v a r y i n g s topp ing power along the t r a c k . F u r t h e r m o r e , the s ta t i s t i ca l fluctuations of energy loss along the t r a c k and the r a d i a l extension of the t r a c k are neglected. I n spite of i t s inadequacy E q . (13) is u t i l i z e d i n n u m e r i c a l examples at the end of th i s ar t i c le . T h e reason is t h a t th i s a p p r o x i m a t i o n has been used i n an analysis of the effect of low-energy X rays on cell s u r v i v a l a n d on m u t a t i o n s t h a t appeared to be inconsistent w i t h the convent ional t r e a t m e n t of d u a l r a d i a t i o n ac t ion (28). N u m e r i c a l examples, even i f based on E q . (13), show t h a t no inconsistencies arise w i t h the present more r igorous t r e a t m e n t .

G E N E R A L F O R M U L A F O R D U A L R A D I A T I O N A C T I O N

T h e concepts and quant i t i es defined i n the preceding sections can be u t i l i z e d t o o b t a i n a f o r m u l a for the mean n u m b e r of lesions. A l t h o u g h the discussion refers t o the mean y i e l d of lesions per cell, i t is read i ly apparent how the resu l tant f o r m u l a can be modi f ied to a p p l y to other cases, for example, to an i r r a d i a t e d m e d i u m such as D N A i n aqueous so lut ion .

T h e mean number 8 of lesions per cell is equal t o the mean n u m b e r m of sub-lesions per cell m u l t i p l i e d b y the mean c o m b i n a t i o n p r o b a b i l i t y p of a sublesion

8 = i(vm). (14) T h e factor i enters the r e la t i on because t w o combined sublesions result i n on ly one lesion.

T h e mean number of sublesions per cell is

m = cpVD, (15)

where D is the absorbed dose, V is the (average) v o l u m e of the sensitive m a t r i x , p is the density , and c is a constant i n t r o d u c e d earlier, i n the section deal ing w i t h the f o r m a t i o n of sublesions.

T h e combinat i on p r o b a b i l i t y for a sublesion is determined b y the expected n u m b e r of neighbor ing sublesions m u l t i p l i e d b y the appropr iate , distance-dependent i n t e r a c t i o n p r o b a b i l i t y g(x).

s(x)/4:wx2 is the expected f r a c t i o n of the spherical shell of radius x centered a t an energy transfer (or sublesion) w h i c h belongs to the sensitive m a t r i x . T h e q u a n t i t y tD(x)dx is the expected energy i m p a r t e d t o the spherical shell. I f either the r a d i a t i o n field is isotropic a n d / o r the cells are r a n d o m l y or iented , the f r a c t i o n s(x)/4;irx2 of th is energy w i l l on the average be i m p a r t e d t o the sensitive m a t r i x . 4

4 A somewhat more complicated relation applies if neither of the two conditions is fulfilled. The directional dependence of the functions s(x) and t(x) must then be taken into account.


The expected n u m b e r of sublesions i n the spherical shell is equal, therefore, t o otD(x)s(x)/'4:Trx2dz. I n t e g r a t i n g th i s expression over a l l distances and i n c l u d i n g the i n t e r a c t i o n p r o b a b i l i t y g(x) one obtains the f o l l o w i n g expression for the average c o m b i n a t i o n p r o b a b i l i t y of a subles ion:

V = c / Jo

00 g(x)tD(x)s(x) dx. (16)

47TZ 2

This i m p o r t a n t a n d s t r i k i n g l y s imple re la t i on l inks the three aspects w h i c h determine the i n t e r a c t i o n of sublesions:

(1) distance-dependent i n t e r a c t i o n p r o b a b i l i t y , (2) microscopic d i s t r i b u t i o n of energy transfers, (3) geometry of the sensitive m a t r i x .

I n s e r t i n g t h i s expression a n d E q . (15) i n t o E q . (14), one obtains the equat ion for the average n u m b e r of lesions per ce l l :

c2pV g(x)tD(x)s(x) 8(D) = D / dx. (17)

2 J 0 4wx2

One derives t h e exp l i c i t f o r m of t h i s equat ion b y inser t ing the expression for the p r o x i m i t y f u n c t i o n tü(x) w h i c h has been g iven i n E q . ( 11 ) :

c2PV / rg(x)t(x)s(x) \ 8(D) = D I / dx + PD / g(x)s(x)dx J

2 \Jo 4wx2 JO / or

&(D) = k(HD + D2). (18)

T h e coefficient k is g iven b y t h e r e l a t i o n

c2p2V r™ k = / s(x)g(x)dxy (19)

2 Jo

b u t i t need n o t be considered i n the f o l l owing . However , the q u a n t i t y £ plays a centra l r o l e : 5

r™ s(x)g(x)t(x) I r 0 0 / dx / s(x)g(x)dx. (20)

Jo 4:irpx2 I Jo

One concludes f r o m E q . (18) t h a t d u a l r a d i a t i o n ac t i on leads to a l i n e a r - q u a d r a t i c dose dependence regardless of the geometry of the sensitive m a t r i x and of the t y p e of r a d i a t i o n . T h e q u a n t i t y £ depends on the funct ions s(x) a n d g(x) w h i c h charac-terize t h e i r r a d i a t e d specimens and the f u n c t i o n t(x) w h i c h characterizes the r a d i a t i o n .

6 T h e symbol £ is used here in order to indicate the close relation to the microdosimetric quantity f (see discussion of the site model).


I t is convenient t o ut i l i ze a combined f u n c t i o n :

00 4>(x) = g(x)s(x) g(x)s(x)dx. (21)

T h i s normal i zed f u n c t i o n can be considered as the d i s t r i b u t i o n of " a v a i l a b l e " loc i i n distance x f r o m a sublesion. T h e w o r d avai lable is used to indicate t h a t t h e i n t e r a c t i o n p r o b a b i l i t y g(x) enters th i s f u n c t i o n i n a d d i t i o n to the distance d i s t r i b u t i o n s(x).

W i t h t h i s de f in i t i on one obtains the final f o r m of the basic r e l a t i o n :

A t a dose equal to £ the l inear component i n the dose-effect r e la t i on is j u s t equal t o the q u a d r a t i c component . T h e l inear component (intratr&ck act ion) dominates for l ower doses whi le the quadrat i c component ( in ter t rack act ion) dominates for larger doses.

E q u a t i o n s (22) and (23) are the essential result of the present s t u d y , and i t is desirable, therefore, to give a v i sua l i za t i on of the q u a n t i t y £. T h i s can be achieved b y observ ing t h a t £ is p r o p o r t i o n a l t o the expected number of ne ighbor ing sub-lesions f r o m the same part i c le t r a c k a r o u n d a sublesion r a n d o m l y selected. £ is o b t a i n e d b y i n t e g r a t i n g the product of the "ava i lab le l o c i " (x) a n d the " c o n d i -t i o n a l dose" t(x)/4iwpx2 a t the loc i .

T h e q u a n t i t y t(x)/4tirpx2 is equal to the expected energy i m p a r t e d to the shell of rad ius x d i v i d e d b y the mass i n th is shell . Therefore, the q u a n t i t y can be con-sidered as the condi t i ona l dose due to the same part i c le t r a c k a t a distance x f r o m a r a n d o m l y selected energy transfer . Absorbed dose is a lways an expectat ion va lue , a n d i t is i n l ine w i t h the general principles of p r o b a b i l i t y t h a t a c ond i t i ona l expec tat ion value m a y differ f r o m the overal l expectat ion value. I t is correct, therefore , to speak about a conditional absorbed dose i n the v i c i n i t y of energy t r a n s f e r s :

T h i s , however , is merely a m a t t e r of i n t e r p r e t a t i o n . I t does n o t affect the equa-t ions , a n d one can use these equations w i t h o u t i n t r o d u c i n g the concept of a c o n d i t i o n a l absorbed dose.

T h e f u n c t i o n t(x) can be computed for any t y p e of r a d i a t i o n , and one m i g h t also consider possible methods of exper imenta l d e t e r m i n a t i o n . O n the other hand , i t m a y appear t h a t Eqs. (22) a n d (23) are of l i t t l e a p p l i c a b i l i t y , i f there is no detai led knowledge of s(x) and g(x). V e r y general conclusions fo l low, however, w i t h o u t such deta i l ed knowledge. For example, one can read i ly show- t h a t £ must be larger t h a n A ( d ) , i f d is e ither the m a x i m u m linear d imension of the sensitive m a t r i x or t h e m a x i m u m c o m b i n a t i o n distance of sublesions. E v e n for sparsely

S ( Z > ) = k(£D + D2) (22) w i t h

(23)

A O ) = t(x)/4:irpx\ (24)


ioniz ing rad iat ions th i s excludes values of £ below a few thousand r a d , whenever d is less t h a n r o u g h l y 20 n m .

As stated earlier, the n u m e r i c a l examples u t i l i ze E q . (13), w h i c h is o n l y a rough a p p r o x i m a t i o n of the ac tua l d i s t r i b u t i o n s t(x). For purposes of i l l u s t r a t i o n th is is sat is factory . F u t u r e q u a n t i t a t i v e studies, however, w r i l l r equ i re more realistic data , and therefore i t w i l l be necessary t o expand previous c o m p u t a t i o n s (19, 24) w h i c h have der ived the funct ions t(x) for cer ta in radiat ions .

The Site Model

T h e t e r m site model refers t o the case w r hich w7as t reated earlier i n t h e t h e o r y of dua l r a d i a t i o n act ion . I t is assumed t h a t loci are r a n d o m l y dispersed over a spherical site of d iameter d. F u r t h e r m o r e , the c o m b i n a t i o n p r o b a b i l i t y g(x) is t a k e n to be constant . One t h e n obtains f r o m E q . (9) , a n d f r o m E q . (21) where t h e combinat i on p r o b a b i l i t y cancels b y d i v i s i o n :

I t w'as shown earlier (19) t h a t t h i s is equal t o t h e microdos imetr ic q u a n t i t y f . E q u a t i o n (26) has i n fact been used (22) as an efficient t o o l t o calculate f f r o m s imula ted part i c l e t racks (25-27).

T h e f u n c t i o n (x) is g iven i n F i g . 3 (solid l ine) for a site d iameter 0.6 /xm. As has been stated before, t h i s is the d i s t r i b u t i o n of loc i i n distance f r o m a r a n d o m l y selected sublesion (or locus).

T h e q u a n t i t y £ depends on the f u n c t i o n t(x), i .e., on r a d i a t i o n q u a l i t y . As a n example, electrons of di f ferent i n i t i a l energies are considered, and l(x) is t a k e n f r o m E q . (13). W i t h electron ranges for water (28) one t h e n obtains results w h i c h are p l o t t e d i n F ig . 4 as sol id l ines ; the site diameters have been chosen as 0.6 a n d 0.4 /xm.

The Distance Model

T h e site model was u t i l i z e d i n earlier w o r k t o p e r m i t the app l i ca t i on of m i c r o -dosimetr ic data as t h e y are obta ined exper imenta l ly . A more realistic a s s u m p t i o n , however, is t h a t loci are contained i n a sensitive m a t r i x w h i c h m a y be dispersed t h r o u g h o u t t h e nucleus of the cell , and t h a t sublesions w i t h i n th i s m a t r i x can combine w i t h a p r o b a b i l i t y g(x) t h a t decreases w i t h distance x.

F o r i n t e r a c t i o n distances smal l compared to t h e d iameter of the m a t r i x , one can disregard t h e l a t t e r and use t h e f u n c t i o n s(x) f r o m E q . (8) w h i c h refers t o a n i n f i n i t e site.

N U M E R I C A L E X A M P L E S

x < d, (25)

a n d according t o E q . (23)

(26)


* i T

F I G . 3. T h e functions (x) for the site and the distance model. Ful l line: site model; E q . (25) with d = 0.6 pm. Broken line: distance model; E q . (29) with a = 0.3/um. Dotted line: distance model; E q . (30) with a = 0.15 urn.

I f a s l owly v a r y i n g i n t e r a c t i o n p r o b a b i l i t y is considered one m i g h t set

g(x) = Ce~^^\ (27) or a l t e r n a t i v e l y ,

g(x) = Ce~xla. One t h e n obtains f r o m E q . (27)

(x) = (ir*/6)a*x2e-


1 0 0 0

5 / R A D

500

0 10 100

E L E C T R O N ENERGY / KeV

F I G . 4. T h e quantity £ for electrons as a function of their initial energy, with t(x) approximated by E q . (13). Ful l lines: site model; E q . (25) with d = 0.4 pm (upper curve) and d — 0.6 pm (lower curve). Broken lines: distance model; E q . (29) with a — 0.2 pm (upper curve) and a = 0.3 fxm (lower curve). Dotted lines: distance model; E q . (30) with a = 0.1 fim (upper curve) and a = 0.15 Aim (lower curve).

simple distance model . Therefore , the f in i te d iameter of the cell nucleus need n o t be considered i n ca l cu la t ing £.

F r o m a comparison of the curves i n F ig . 4 one m i g h t conclude t h a t t h e v a r i o u s models of d u a l r a d i a t i o n ac t i on a lways lead t o r o u g h l y t h e same results . T h i s w o u l d t h e n i m p l y t h a t exper imenta l d a t a w h i c h do n o t fit the s imple site m o d e l w i l l be e q u a l l y inconsistent w i t h other models of d u a l r a d i a t i o n ac t ion , as r e c e n t l y concluded w i t h some qual i f i cat ions {23). T h e a r g u m e n t is erroneous however , as is shown i n the next section.

Enhanced Short Range Interaction

T h e three models represented b y Eqs. (25), (29), and (30) are, as seen i n F i g . 4, s imi lar i n t h a t the m a j o r i t y of the p o t e n t i a l react ion partners l ie a t a n i n t e r m e d i a t e distance f r o m a sublesion. I n r e a l i t y the s i t u a t i o n m a y be di f ferent , inso far as short - range i n t e r a c t i o n can p l a y a larger role.

One p o s s i b i l i t y is t h a t t h e i n t e r a c t i o n p r o b a b i l i t y declines steeply a t s m a l l distances b u t s t i l l has a considerable t a i l t o larger distances. T h u s one m i g h t deal w i t h t h e f u n c t i o n

T h e effectiveness of short -range part ic les can be great ly enhanced b y t h e first

g(x) = Ci«r(*/«i) 8 w i t h Ci ^> C 2 a n d ax « a 2 . (31)


E L E C T R O N E N E R G Y / K e V

F I G . 5. The quantity £ for electrons as a function of their initial energy, with t{x) approximated by E q . (13). Dotted lines: site model; E q . (25) with d = 0.4 /um (upper curve) and d = 0.6 pm (lower curve). Ful l line: distance model; E q . (31) with Ct/Ci = 0.004, a t = 0.05 pm and a 2 = 1 pm. Broken line: distance model ( E q . (27) with a = 1 pm) for a random arrangement of clusters in a large site ( E q . (32) J with d = 0.1 pm and r = 0.004).

t e r m , whi le t h a t of long-range part ic les m a y be l i t t l e affected. One must therefore expect a shi f t of the m a x i m u m values of £ t o w a r d lower part i c le energies.

T h i s is indeed t h e case, as can be seen f r o m F ig . 5. T h e sol id curve i n th i s figure results for the distance model w i t h E q . (31) and ai = 0.05 /zm, a 2 = 1 Mm, and C 2 / C i = 0.004.

A comparison w i t h the site model (do t ted curves) shows t h a t at higher energies the values £ correspond to larger diameters (d = 0.6 / t m ) , a t lower energies to smaller diameters (d = 0.35 /*m).

A recent p u b l i c a t i o n (11) deals w i t h a comparat ive s t u d y of cell k i l l i n g and m u t a t i o n i n d u c t i o n b y 1.5-keV photons and higher-energy radiat ions . The s t a t i s t i -cal accuracy of the da ta is n o t sufficient to quote £ values w i t h m u c h accuracy. I t has been stated, however, t h a t the results for the high-energy radiat ions require larger site diameters t h a n those consistent w i t h the da ta for 1.5-keV photons. T h e poss ib i l i ty of d u a l ac t ion i n i t s usual i n t e r p r e t a t i o n has therefore been excluded, and i t has also been stated t h a t the a l t e rnat ive t r e a t m e n t i n terms of the distance model w i l l n o t overcome a l l of the dif f icult ies (28). As seen f r om the example i n F ig . 5 t h i s does n o t necessarily fo l low.

S imi lar results are obta ined i f , instead of a short-range component i n g(x), one considers a sensitive m a t r i x w h i c h consists of clusters of loci i n a larger site. I f spherical clusters of d iameter d are r a n d o m l y d i s t r i b u t e d i n a large ( inf inite)

F O R M U L A T I O N O F D U A L R A D I A T I O N A C T I O N 4 8 7

site one has

/ 3*c x^ \ s(x) = 47TX 2 ( 1 h — J + 4:TX2T, ( 3 2 )

\ 2d 2 d V

where r is the f r a c t i o n of the site filled w i t h the clusters w h i c h belong to the sensitive m a t r i x .

T h e b roken l ine i n F ig . 5 is a so lut ion for th i s case. T h e d iameter d of the clusters is t a k e n t o be 0 . 1 pm, r is 0 . 0 0 4 , and g{x) is set p r o p o r t i o n a l t o e x p [ — (x/aY~] w i t h a = 1 /xm. T h e overa l l result is s imi lar to t h a t for the distance model w i t h E q . ( 3 1 ) . I t is ev ident t h a t the finite d iameter of the nucleus of the cell can be read i l y t a k e n i n t o account. However , t h i s has l i t t l e influence on the results.

I t should be noted t h a t t h e evidence p r o v i d e d b y electron microscopy (29) does i n fact ind i cate t h a t D N A is arranged i n clusters d u r i n g interphase.

R E C E I V E D : N o v e m b e r 1 8 , 1 9 7 7 ; R E V I S E D : A p r i l 1 9 , 1 9 7 8

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a generalized formulation of dual radiation action. · 2012. 5. 22. · a. szechter, w. p....

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