OFFICIAL ORGAN OF THE RADIATION RESEARCH SOCIETY

RADIATION RESEARCH EDITOR-IN-CHIEF: DANIEL BILLEN

Volume 86, 1981

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M A D E IN T H E U N I T E D STATES OF AMERICA

RADIATION RESEARCH OFFICIAL ORGAN OF THE RADIATION R E S E A R C H S O C I E T Y

Editor-in-Chief: D A N I E L B I L L E N , Univers i ty of T e n n e s s e e - O a k Ridge Graduate School of Biomedical S c i e n c e s , B io logy Div i s ion , Oak Ridge National Laboratory, P .O. Box Y,

Oak Ridge, T e n n e s s e e 37830 Managing Technical Editor: M A R T H A E D I N G T O N , Univers i ty of T e n n e s s e e - O a k Ridge Graduate

Schoo l of Biomedical S c i e n c e s , Bio logy Div i s ion , Oak Ridge National Laboratory, P .O. B o x Y , Oak Ridge, T e n n e s s e e 37830

ASSOCIATE

H. 1. A D L E R , Oak Ridge National Laboratory J. W. B A U M , Brookhaven National Laboratory S. S. B O G G S , Univers i ty of Pittsburgh J. M. B R O W N , Stanford Univers i ty S. S. D O N A L D S O N , Stanford Univers i ty J. D . E A R L E , Mayo Clinic J. J. F I S C H E R , Yale Univers i ty E. W. G E R N E R , Univers i ty of Arizona E. L. G I L L E T T E , Colorado State Univers i ty R. H. H U E B N E R , Argonne National Laboratory J. W. H U N T , Ontario Cancer Institute, Toronto ,

Canada

EDITORS

S. L I P S K Y , University of Minnesota S. O K A D A , Univers i ty of T o k y o , Japan N . L. O L E I N I C K , Case Western Reserve

Univers i ty A. M. R A U T H , Ontario Cancer Institute,

Toronto , Canada M. C. S A U E R , JR. , Argonne National

Laboratory S. P. S T E A R N E R , Argonne National Laboratory R. C. T H O M P S O N , Batte l le , Pacific Nor thwes t

Laboratories J. E . T U R N E R , Oak Ridge National Laboratory S. S. W A L L A C E . N e w York Medical Col lege

OFFICERS OF THE SOCIETY

President: O D D V A R F. N Y G A A R D , National Cancer Institute, National Institutes of Health , Be thesda , Maryland 20205

Vice President and President-Elect: M O R T I M E R M. E L K I N D , Div is ion of Biological and Medical Research , Argonne National Laboratory, Argonne , Ill inois 60439

Secretary-Treasurer: R O B E R T B. P A I N T E R , Laboratory of Radiobio logy , Universi ty of California, San Franc i sco , California 94143

Secretary-Treasurer-Elect: E D W A R D R. EPP, Department of Radiation Medic ine , Massachuset t s General Hospi ta l , B o s t o n , Massachuset t s 02114

Editor-in-Chief: D A N I E L B I L L E N , Univers i ty of T e n n e s s e e - O a k Ridge Graduate School of Biomedical S c i e n c e s , Biology Div i s ion , Oak Ridge National Laboratory ,

P .O. B o x Y, Oak Ridge, T e n n e s s e e 37830 Executive Director: R I C H A R D J. B U R K , JR., 4720 Montgomery Lane ,

Suite 506, B e t h e s d a , Maryland 20014

ANNUAL MEETINGS

1981: May 3 1 - J u n e 4 , Minneapol i s , Minnesota 1982: April 1 8 - 2 2 , Salt Lake City, Utah

Titus C. Evans , Editor- in-Chief V o l u m e s 1 - 5 0 O d d v a r F. Nygaard , Editor- in-Chief V o l u m e s 5 1 - 7 9

VOLUME 86 , 1981

C o u n c i l o r s R a d i a t i o n R e s e a r c h S o c i e t y 1 9 8 0 - 1 9 8 1

PHYSICS M. Inokuti, Argonne National Laboratory H. J. Burki, University of California, Berkeley

BIOLOGY J. S. Rasey, University of Washington A. M. Rauth, Ontario Cancer Institute, Toronto, Canada

MEDICINE H. D. Suit, Massachusetts General Hospital J. A. Belli, Harvard Medical School

CHEMISTRY J. F. Ward, University of California, San Diego M. Z. Hoffman, Boston University

AT-LARGE L. A. Dethlefsen, University of Utah R. M. Sutherland, University of Rochester

CONTENTS OF VOLUME 86

N U M B E R 1, A P R I L 1 9 8 1

W . G . B U R N S , R. M A Y , A N D K. F. B A V E R S T O C K . O x y g e n as a Product of Water Radiolysis in H i g h - L E T Tracks . 1. The Origin of the Hydroperoxy l Radical in Water Radiolys is 1

K. F. B A V E R S T O C K A N D W . G . B U R N S . O x y g e n as a Product of Water Radiolys is in H i g h - L E T Tracks . II. Radiobiological Implicat ions 20

B . H . E R I C K S O N . Survival and Renewal of Murine Stem Spermatogonia fo l lowing , ioCo y Radiat ion 34

G . P . R A A P H O R S T A N D E. I. A Z Z A M . Fixation of Potentially Lethal Radiation Damage in Chinese H a m s t e r Cel ls by Anisoton ic So lut ions , Po lyamines , and D M S O 52

P . M . N A N A . Differential Mutagenic R e s p o n s e of G l Phase Variants of Balb/c-3T3 Cells to uv Irradiation 67

G E O R G I L I A K I S . Characterizat ion and Properties of Repair of Potential ly Lethal Damage as Measured with the Help of ß-Arabinofuranosyladenine in Plateau-Phase E A T Cells 77

S I D N E Y M I T T L E R . Effect of Hyperthermia upon Radiation-Induced C h r o m o s o m e L o s s in Mutagen-Sens i t ive Drosophila melanogaster 91

R Y S / A R D O L I N S K I , R O B E R T C . B R I G G S , L U B O M I R S . H N T L I C A , J A N E T S T E I N , A N D G A R Y S T E I N . Gamma-Radia t ion - lnduced Crossl inking of Cell-Specific Chromosomal N o n h i s t o n e Protein-D N A C o m p l e x e s in H e L a Chromatin 102

G L E N N N . T A Y L O R , C R A I G W . J O N E S , P A U L A . G A R D N E R , R A Y D. L L O Y D , C H A R L E S W . M A Y S , A N D K E I T H E. C H A R R I E R . T w o N e w Rodent Models for Actinide Toxic i ty Studies 115

W I R G I L I U S X D U D A . Effect of y Irradiation on the a and ß-Chains of Bov ine Hemoglob in and Globin 123

B R U C E E. M A G U N A N D C H R I S T O P H E R W . F E N N I E . Effects of Hyperthermia on Binding, Internalization, and Degradation of Epidermal Growth Factor 133

Y U - A U N G Y A U , S H U - C H E N H U A N G , P I N - C H I E H H S U , A N D P A O - S H A N W E N G . Gonadal D o s e Obtained from Treatment of Nasal Carc inoma by Ionizing Radiation 147

Y O S H I H I K O Y O S H I I , Y U T A K A M A K I , H I R O S H I T S U N E M O T O , S A C H I K O K O I K E , A N D T S U T O M U K A S U G A . The Effect of Acute Tota l -Head X Irradiation on C 3 H / H e Mice 152

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

J . L. A N T O I N E , G . B . G E R B E R , A . L E O N A R D , F . R I C H A R D , A N D A . W A M B E R S I E . Chromo­s o m e Aberrat ions Induced in Patients Treated with Telecobal t Therapy for Mammary Carcinoma 171

B O O K R E V I E W 178 A C K N O W L E D G M E N T S 180 A N N O U N C E M E N T 183

N U M B E R 2, M A Y 1981

S Y M P O S I U M O N R A D I C A L P R O C E S S E S IN R A D I O B I O L O G Y A N D C A R C I N O G E N E S I S J O H N F . W A R D . S o m e Biochemical C o n s e q u e n c e s of the Spatial Distribution of Ionizing

Radiat ion-Produced Free Radicals 185 C. L. G R E E N S T O C K . Redox P r o c e s s e s in Radiation Bio logy and Cancer 196 J O H N E. B I A G L O W . Cellular Electron Transfer and Radical M e c h a n i s m s for Drug

Metabol ism 212 R O B E R T A. F L O Y D . Free-Radical Events in Chemical and Biochemical React ions Involving

Carcinogenic Arylamines 243 A. M. K E L L E R E R . Proximity Funct ions for General Right Cylinders 264 A. M . K E L L E R E R . Criteria for the Equiva lence of Spherical and Cylindrical Proportional

Counters in Microdos imetry 277

J O O N Y . L E E A N D W I L L I A M A . B E R N H A R D . An ESR Study of H y d r o g e n - B o m b a r d e d 9-Methyladenine 287

K E I S U K E M A K I N G - , N O B U H I R O S U Z U K I , F U M I O M O R I Y A , S O U J I R O K U S H I K A , A N D H I R O Y U K I H A T A N O . A Fundamental Study of A q u e o u s Solut ions of 2-Methyl -2-ni trosopropane as a Spin Trap 294

D. W. W H I L L A N S A N D G. F. W H I T M O R E . The Radiation Reduct ion of Misonidazo le 311 C. C L I F T O N L I N G , H O W A R D B. M I C H A E L S , L E O E. G E R W E C K , E D W A R D R. E P P , A N D E L E A N O R

C. P E T E R S O N . O x y g e n Sensit izat ion of Mammal ian Cells under Different Irradiation Condit ions 325

J O A N B. C H I N A N D A N D R E W M. R A U T H . The Metabol i sm and Pharmacokinet ics of the H y p o x i c Cell Radiosensi t izer and Cyto tox ic Agent , Mison idazo le , in C3H Mice 341

M. J . G A L V I N , C. A. H A L L , A N D D . 1. M C R E E . M i c r o w a v e Radiation Effects on Cardiac Muscle Cells in Vitro 358

J . L. G I E S B R E C H T , W. R. W I L S O N , A N D R. P. H I L L . Radiobiological Studies of Cells in Multi­cellular Spheroids U s i n g a Sequential Trypsinizat ion Technique 368

D I E T M A R W. S I E M A N N A N D K A R E N K O C H A N S K I . Combinat ions of Radiation and Mison idazo le in a Murine Lung Tumor Model 387

A N N O U N C E M E N T 398

N U M B E R 3 , J U N E 1981

J O H N C L A R K S U T H E R L A N D A N D K A T H L E E N P I E T R U S Z K A G R I F F I N . Absorpt ion Spectrum of D N A for Wave lengths Greater than 300 nm 399

B R E N T B E N S O N A N D L E S T E R E R I C H . Free Radicals in Pyrimidines: E S R of y-Irradiated 5 - C y c l o h e x e n y l - l , 5-dimethyl Barbituric Acid 411

B. T I L Q U I N , R. V A N E L M B T , C. B O M B A E R T , A N D P. C L A E S . Unsaturated H e a v y Products from y Irradiation of Sol id Forms of 2 ,3 -Dimethylbutane . II. Radical Contribution 419

A. P. H A N D E L A N D W. W. N A W A R . Radiolyt ic C o m p o u n d s from M o n o - , Di- , and Tri-acylg lycerols 428

A . P. H A N D E L A N D W. W. N A W A R . Radiolys is of Saturated Phosphol ip ids 437 S T E V E N A. L E A D O N A N D J O H N F . W A R D . The Effect of y-Irradiated D N A on the Act iv i ty

of D N A Po lymerase 445 J . L E S L I E R E D P A T H , E I L E E N Z A B I L A N S K Y , A N D M A R T I N C O L M A N . Radiat ion, Adr iamyc in ,

and Skin React ions: Effects of Radiation and Drug Fract ionat ion, Hyperthermia , and Tetracycl ine 459

M A R Y A N N S T E V E N S O N , K E N N E T H W. M I N T O N , A N D G E O R G E M. H A H N . Survival and Concanaval in-A-Induced Capping in C H O Fibroblasts after E x p o s u r e to Hyperthermia , Ethanol , and X Irradiation 467

N O R I K O M O T O H A S H I , I T S U H I K O M O R I , Y U K I O S U G I U R A , A N D H I S A S H I T A N A K A . Modif ication of y-Irradiation-Induced Change in Myoglobin by a -Mercaptoprop iony lg lyc ine and Its Related C o m p o u n d s and the Formation of Sul fmyoglobin 479

R A L P H J . S M I A L O W I C Z , J . S. A L I , E Z R A B E R M A N , S T E V E J . B U R S I A N , J A M E S B. K I N N , C H A R L E S G. L I D D L E , L A W R E N C E W. R E I T E R , A N D C L A U D E M. W E I L . Chronic E x p o s u r e of Rats to 100-MHz (CW) Radiofrequency Radiation: A s s e s s m e n t of Biological Effects 488

B A R B A R A C. M I L L A R , O R A Z I O S A P O R A , E. M A R T I N F I E L D E N , A N D P A M E L A S. L O V E R O C K . The Application of Rapid-Lys i s Techn iques in Radiobio logy . IV. T h e Effect of Glycero l and D M S O on Chinese Hamster Cell Survival and D N A Single-Strand Break Pro­duction 506

O T T O G. R A A B E , S T E V E N A. B O O K , N O R R I S J . P A R K S , C L A R E N C E E. C H R I S P , A N D M A R V I N G O L D M A N . Lifet ime Studies of 22<iRa and 90Sr Toxic i ty in B e a g l e s — A Status Report 515

L A W R E N C E S. G O L D S T E I N , T. L. P H I L L I P S , K . K . F U , G. Y . R O S S , A N D L. J . K A N E . Bio logical Effects of Acce lerated H e a v y Ions . I. Single D o s e s in Normal T i s s u e , T u m o r s , and Cells in Vitro 529

L A W R E N C E S. G O L D S T E I N , T. L. P H I L L I P S , A N D G. Y . R O S S . Biological Effects of A c c e l e r a t e d H e a v y Ions. II. Fract ionated Irradiation of Intestinal Crypt Cells 542

J O H N F. T H O M S O N , F R A N K S. W I L L I A M S O N , D O U G L A S G R A H N , A N D E. J O H N A I N S W O R T H . Life Shortening in Mice E x p o s e d to Fiss ion Neutrons and y Rays . I. Single and Short-Term Fractionated Exposures 559

J O H N F. T H O M S O N , F R A N K S. W I L L I A M S O N , D O U G L A S G R A H N , A N D E. J O H N A I N S W O R T H . Life Shortening in Mice E x p o s e d to Fiss ion N e u t r o n s and y Rays . II. Duration-of-Life and Long-Term Fractionated Exposures 573

M A R Y J. O R T N E R , M I C H A E L J . G A L V I N , A N D D O N A L D I. M C R E E . S tudies on Acute in Vivo Exposure of Rats to 2450-MHz M i c r o w a v e Radiation. 1. Mast Cells and Basophi l s 580

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

P . V . H A R I H A R A N , S. E L E C Z K O , B. P . S M I T H , A N D M. C. P A T E R S O N . Normal Rejoining of D N A Strand Breaks in Ataxia Telangiectas ia Fibroblast L ines after L o w X-Ray Exposure 589

A U T H O R I N D E X FOR V O L U M E 86 598 The Subject Index for V o l u m e 86 will appear in the D e c e m b e r 1981 issue as part of a

cumulat ive index for the year 1981.

R A D I A T I O N RESEARCH 86 , 2 6 4 - 2 7 6 (1981)

Proximity Functions for General Right Cylinders 1

A. M. KELLERER Institut für Medizinische Strahlenkunde der Universität Würzburg,

Versbacher Str.5. 8700 Würzburg, Federal Republic of Germany

K E L L E R E R , A. M . Proximity Funct ions for General Right Cyl inders . Radiat. Res. 86 , 2 6 4 - 2 7 6 (1981).

Distributions of d is tances b e t w e e n pairs of points within geometrical objec t s , or the c lose ly related proximity functions and geometr ic reduction factors , have applicat ions to dosimetric and microdosimetr ic calculat ions . For c o n v e x bodies these funct ions are linked to the chord-length distributions that result from random intersect ions by straight l ines. A synops i s of the most important relations is g iven. The proximity funct ions and related functions are derived for right cyl inders with arbitrary cross sec t ions . The solut ion util izes the fact that the squares of the d i s tances b e t w e e n two random points are s u m s of independ­ently distributed squares of d is tances parallel and perpendicular to the axis of the cyl inder. Ana logous formulas are derived for the proximity functions or geometr ic reduction factors for a cyl inder relative to a point. This requires only a minor modification of the so lut ion.

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

The distributions of distance between pairs of points within geometrical objects were first utilized by Berger ( / ) in dosimetric computations. These point-pair distributions have broad applicability in calculations of absorbed dose from radio­nuclides [see (2-11)]; they are also relevant to microdosimetry (12-14). Analytical expressions can be given for configurations such as spheres, slabs, or spherical shells. A solution for cylinders that contains one quadrature without singularities is derived in the present article. As in an earlier article dealing with chord-length distributions (15), the solution will be obtained for cylinders with arbitrary cross section; the formula for circular cylinders results as a special case.

The result is applicable to calculations of absorbed doses with cylindrical sources or receptors. Because of the utilization of cylindrical detectors the solution is also relevant to microdosimetry. The accompanying article (16) uses the results of the present study for an assessment of the degree of equivalence achievable between spherical and cylindrical microdosimetric detectors.

The distance distribution of a geometrical object is essentially equivalent to two other concepts, the proximity function and the geometric reduction factor. The interrelations between the three concepts are given in Section 2.1; Sections 2.2 and 2.3 deal with the connection to the chord-length distributions that result when the geometrical body is randomly intercepted by straight lines. Readers interested only in the solution for cylinders may first ignore Sections 2.2 and 2.3, but may consult them for equations required in practical applications.

1 Work supported by Euratom Contract 208-76-7 BIO D.

264 0033-7587/81/050264-13$02.00/0 Copyright C> 1981 by Academic Press. Inc. All rights of reproduction in any form reserved.

P R O X I M I T Y F U N C T I O N S FOR C Y L I N D E R S 265

F i c . 1. Diagram illustrating the definition of the proximity function of a s i te , 5 . P is a random point in S. The integral proximity funct ion, S(.v), equals the e x p e c t e d vo lume represented by the shaded reg ion; the differential proximity funct ion, A ( A ) , equals the expec ted surface indicated by the circular line s e g m e n t B \ and the geometr ic reduction factor, U(x), equals the ratio of B to the total surface of the s p h e r e .

For brevity, various considerations in this article will refer to only one of the related concepts, for example, the proximity function. It should be realized that reference could equally be made to the other concepts.

2 . P R O X I M I T Y F U N C T I O N S A N D S I M I L A R C O N C E P T S A N D T H E I R R E L A T I O N S TO T H E C H O R D - L E N G T H D I S T R I B U T I O N S

2.1 Proximity Functions, Distance Distribution, and Geometric Reduction Factor The integral proximity function, 5 ( A ) , of a region S is equal to the expected

volume of the region that is contained in a sphere of radius x centered at a random point of S. The differential proximity function s(x) is the derivative of 5(x) , i.e., s(x)dx is the expected volume of 5 contained in a spherical shell of radius x and thickness dx that is centered at a random point of S. These notions are indicated schematically in Fig. 1.

Dividing s(x) by the volume, V, of S one obtains, as can be shown (12), the density of distances between pairs of random points in S (see Fig. 1). Berger ( / ) had earlier termed this the "pair distance distribution,'' p(x):

p(x) = s(x)/V. (1) The proximity functions or distance distributions can also be defined for surfaces

or linear structures in three-dimensional space, R3. Volume is then replaced by surface or length. Since such structures may be contained in one- or two-dimen­sional linear subspaces the case of general dimensionality is of interest. The sub­sequent formulas in this section will therefore apply to arbitrary dimensions; where this is not the case separate relations will be quoted for three-dimensional space, /?3, and two-dimensional space, R2. V and S designate volume and surface in /?3 , and A and C designate area and circumference in /?2-

The function p(x) has the advantage that it is a properly normalized probability distribution; the nonnormalized function, s(x), on the other hand, is more generally

266 A. M. K E L L E R E R

applicable, because it exists also for unbounded structures, such as infinite lines or areas or infinite cylinders.

At small values of x the proximity function of a volume goes toward ATTX2 and that of an area toward ITTX. A related quantity that converges toward 1 at x = 0 can be more practical in numerical applications; it will be used interchangeably with s(x) or p(x):

U(x) = S{X)/4TTX2 = P(X)V/4TTX2 (in Ä3)

= S(X)/2TTX = p(x)Al2irx (in 7?2). This quantity has been termed the geometric reduction factor by Berger (7) , and it is frequently used in dose calculations for internal emitters.2 If a spherical shell of radius x is centered at a random point of S, then U(x) is equal to the average fraction of this shell that lies within S (see Fig. 1).

2.2 Chord-Length Distributions Chord-length distributions result when geometric configurations are randomly

intercepted by straight lines. There are different modes of randomness that lead to different distributions of chord length (75 ,77 ,18) . Three important types that are related to each other and are also linked to the proximity functions are indicated in Fig. 2.

The condition where a site S is exposed to a uniform, isotropic fluence of straight infinite random lines has been termed /x-randomness (77). A second condition, I-randomness (interior radiator randomness), results if random points are chosen within 5 and straight lines are laid through these points with random orientation (77). i-randomness results from the same condition if rays originate from the ran­dom points (15). The distribution p(x) of distance between two random points in S is indicated in the last panel of Fig. 2.

The probability densities of the intercepts, x, for the different types of random­ness are designated by / m ( J C ) , / I ( J C ) , and fi(x). The sum distributions—for con­venience summed from the right—are designated by Fß(x), F{(x), and F{(x). The mean values are designated by JcM, xu and xx. For example,

2 The quantity is c o m m o n l y called average geometr ic reduction factor Va(x), and a related concept ( s e e Sec t ion 4) is called geometr ic reduct ion factor ^(x). A different s y m b o l , U(x) is c h o s e n here to avoid confus ion with energy fluence (19).

P R O X I M I T Y F U N C T I O N S FOR C Y L I N D E R S 267

Fß(x) = fß(s)ds and xß = xfß(x)dx = Fß(x)dx. (3) The next section is a condensed summary of essential interrelations between

the different functions. 2.3 Relations between the Proximity Function and the Chord-Length Distributions

Kingman (18) has given the important relation between the chord-length dis­tributions for I- and /x-randomnesses:

/,(*) = xfß(x)/xß. (4) A somewhat more complicated relation holds for i-randomness (15):

f{(x) = Fß(x)/xß. (5) Finally one obtains for convex sites:

U(x) = Fx(x) = Fß(s)dslxß. (6) The relation holds because a random shift x of a random point in a convex body S will lead with probability F-^x) to a point still in 5 . In R3 this probability is equal to the fraction, U(x), of a spherical surface of radius x that is contained in S, if the shell is centered at P. In R2 an analogous argument applies.

The separate concepts Fx(x) and U(x) are required, because F{(x) and U(x) differ for nonconvex structures.

By using Eqs. (4-6) one can also relate the chord-length densities to the deriva­tives of the geometric reduction factor of convex sites:

-U'(x) =Mx) [-U'(0) = / , (0 ) = l/JcM (see Eq. (5))]; (7) U"(x) =Mx)/xß = / , ( * ) / * , ( 8 )

where the mean chord length, xß, is given by the Cauchy theorem that applies to convex sites [see (18)]:

xß = 4V/S (in Ä3) ( 9 ) = TTAIC (in R2).

From Eqs. (4-6) one obtains by partial integration the relations between the moments (n = 0, 1, 2 . . .):

x'ßl+2l(n + \)(n + 2)xß =x\l+l/(n + \)(n + 2) = xf+1/(n + 1) =

(in R3)

xtlU(x)dx (10)

(11) = X»-1AI2TT (in R2)

The indices /x, I, i, and p refer to the densities fß(x),fi(x),fi(x), and p(x). For R3 and n = 0 to 4 these important relations are listed explicitly in Section 3 [see Eqs. (17-21)].

2 6 8 A. M. K E L L E R E R

F I G . 3. Diagram illustrating the computat ion of the distribution of d i s tances , x, b e t w e e n t w o random points in a cyl inder in terms of the independent random variables y and z.

Although it is of no direct importance in the present context, one may note the striking fact that the third moment for I- or i-randomness is independent of the shape of a convex body in R3:

~xj = 44 - 3V/TT, ( 1 2 ) while an analogous relation holds in R2:

I f = 3 4 = 3A/77. ( 1 3 )

This concludes the general considerations. The subsequent section gives solu­tions for cylinders.

3. P R O X I M I T Y F U N C T I O N S FOR C Y L I N D E R S

A formula requiring a numerical integration was derived previously (15,20) for the chord-length distributions FM(x) of general cylinders. By a further integration one could, according to Eq. ( 6 ) , obtain the geometric reduction factor or the proximity function. A disadvantage of this procedure is that the integrals contain various singularities. The functions s(x) or U(x) are, however, considerably simpler than the complicated chord-length distributions for ^t-randomness. In fact, there is as indicated in Fig. 3 a direct solution that constructs the distribution of point-pair distances, x, for the cylinder from the distribution of distances, y, per-

P R O X I M I T Y F U N C T I O N S FOR C Y L I N D E R S 269 pendicular to the axis of the cylinder (horizontal distances) and the distribution of distances, z, in the direction of the axis (vertical distances). The method utilizes the fact that x2 is the sum of/ 2 and z2, and the horizontal and vertical distances y and z are independent random variables.

The formal derivation is given separately in the Appendix. One obtains the following equation for the proximity function of a right cylinder of height h and with arbitrary cross section:

with

s(x) = 2x

_i - (Max (0, x2 - d2))in

1 - z \sc((x2 - z2)1/2) h J (x2 - z2)1

z2 = Min U , h), x < (h2 + d2

(14)

(15) where sc(y) is the proximity function of the cross section of the cylinder, and d is the diameter (i.e., maximum width) of the cross section.

The equation in this general form is the essential result of this article. To use the result for complicated cross sections one needs to derive sc(y) numerically; this may require separate integrations or Monte Carlo methods.

For a circle and a rectangle analytical expressions of sc(y) are listed in the Appendix. For a circular cylinder of diameter </ one obtains with Eq. (A.6)

1 _ ^ ( [ c o s " 1 ((JC2 - z2)mld) h {{x2 - z2)(d2 - (x2 - z2)))ll2/d2]dz. (16)

Corresponding equations hold for the pair-distance density, p(x) = 4s{x)/(hd27r), or the quantity U(x) = S(X)/(4TTX2).

The integrals in Eq. (14) or (16) are readily evaluated since they contain no singularities. Figure 4 represents solutions (/(JC) for various values of the elonga­tion, hid, of the cylinder. The function U(x) is plotted instead of s(x) because this permits higher accuracy at small values of x.

Equations (10) and (11) for the moments can be used to check the numerical accuracy of the results [see Eqs. (19) and (22)] and to derive the mean values xn .v,, and.v,, [see Eqs. (17) and (20)]. These mean values are plotted in Fig. 5 together with the mean chord length for /n-randomness, iM = dh/(d/2 + h) = - l / ( / ' ( 0 ) .

Separately listed for/? = 0 to 4, and with c = V/4TT, the relations from Eqs. (10) and (11) have the form:

U(x)dx = ex. x, = x{!2 = xl^Xfj (17)

xU(x)dx = ex'1 = xV2 = xV6 = x'i/6x^

x2U(x)dx = c = x\ß = x'i/\2 = x4J\2xfX,

(18)

(19)

270 A. M. K E L L E R E R

0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 3 . 2 1 . 4

1

. 6 0 . 8

F I G . 4 . The geometrical reduction factors , U(x), for right circular cyl inders of e longat ions 1 / 1 6 , 1 / 8 , 1 /4 , 1/2, l / (2) , / 2 , 1 , 21 / 2 , 2 , 4 , 8 , and 1 6 . The larger axis is taken to be of unit length, and the ratio o f the larger to the smaller axis is g iven as parameter with s o m e of the funct ions . The differential proximity function s(x) is equal to 4TTX2U(X).

x3U(x)dx = cxp = jcl/4 = xV20 = x%/20xß,

x4U(x)dx = cxl = xV5 = xf/30 = x%/30x, c(d2/4 + h2/6).

(20)

(21) (22)

The integrals run from 0 to the maximum value of*. Equations (17) to (21) hold generally for convex bodiesAn R3. Equation (22) is restricted to circular cylinders; it is based on the fact that x% is the sum of the mean squared distance, d2/4, for point pairs inside a circle and the mean squared distance, /z2/6, for point pairs on a line segment.

P R O X I M I T Y F U N C T I O N S F O R C Y L I N D E R S 271

.2 I 1 1 1 1 1 1 1 1.2 .05 .1 .2 .5 1 2 5 10 20

ELONGATION F I G . 5. M e a n chord length, i M , for ^ - r a n d o m n e s s , mean chord length, xh for i -randomness , and

mean d i s t a n c e , xf), b e t w e e n t w o random points for circular cyl inders of various e longat ions . The v a l u e s are g iven relative to the length of the smaller ax is . The mean chord length, xlt for I -randomness is equal to 2xh and is therefore not plotted.

In certain applications it is practical to utilize the limiting form of the solution for very long and for very flat cylinders. For long cylinders (h > d) and for moderate values of x one can use the limiting form of Eq. (16) for infinite height; for large values of x one can disregard the radial extension of the cylinder. With these two approximations Eq. (16) reduces to

s(x) = Sx T [cos"1 ((x2 - z2)ll2/d)

- ((x2 - z2)(d2 - (x2 - z2)))ll2/d2]dz, for x < h = d27r(\ - x/h)/2 for x>d, (23)

with the limit values X(x = d\ x{ « 0.662c/; xp = A/3. (24)

For flat cylinders (h < d) and for moderate values of x one can use formulas for the infinite slab; for large x one can disregard the vertical extension and use, with inclusion of the factor /?, the formula for the disk [see Eq. (A.6)]. This leads to

s(x) = 4TTX2(\ - xllh) for x <h = lirhx forh<x<d (25)

= 4hx[ cos"1 I i-J - ^ (d2 - x2)112} for h < x < d, with the limit values

xß = 2h; Jci « (In (dlh) + 0.3069)-A/2 (see (20)); Xtj = 64<//45TT - 0.4527c/. (26)

The solution for a right cylinder with square cross section, i.e., a rectangular parallelepiped, is obtained by inserting Eq. (A.7) into Eq. (14). This leads to a

272 A. M. K E L L E R E R

. 4

0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4

X F I G . 6. The geometrical reduct ion factors , U(x), for regular paral lelepipeds with t w o s ides equal t o d

and o n e side equal to h. Curves are g iven for e longat ions hid = 1/16, 1/8, 1/4, 1/2, l / (2 ) , / 2 , 1, 21 / 2 , 2 , 4 , 8 , and 16. T h e larger side is taken to be of unit length. The ratio of the larger to the smaller side is g iven as parameter with s o m e of the funct ions . The differential proximity funct ion, s(x), is equal to 4-rrx2 U(x). more complicated expression than Eq. (16), but the integration is also straight­forward without singularities. Figure 6 gives the resulting functions U(x).

4. P R O X I M I T Y F U N C T I O N S O F A C Y L I N D E R R E L A T I V E T O A P O I N T

4.1 Generalized Definition of the Proximity Function and Related Concepts The pair distance distribution, p(x), and the proximity function, s(x), relate to

the distances between pairs of points picked at random in the specified region. Berger previously pointed out ( / ) that an entirely similar procedure, using a pair distribution function, can also be applied in considerations of the transfer of energy from a source region to any other region in the medium. This notion is closely related to concepts such as the absorbed energy fraction that is used in dose cal-

P R O X I M I T Y F U N C T I O N S FOR C Y L I N D E R S 273 cuiations for internal emitters (1-11). One can define a proximity function of a region S (the source region) relative to a region R (the receptor region). The integral proximity function SRS(x) of S relative to R is the expected volume of S that is contained within a distance up to x from a random point in R. The differential proximity function sRS(x) is the derivative of SRS(x).

The proximity function sRS(x) is equal to the pair-distance distribution pRS(x) for random points picked in R and S, multiplied by the volume Vs of the source region:

SRS(X) = Vs-pRS(x) = VS/VR'Ssr(X). (27) The nonnormalized function is required because it is applicable also to unbounded source regions.

It is again practical to introduce the geometrical reduction factor URSW = SRS(X)/4TTX2 = VS/VR'USR(x). (28)

URS(x) is the probability that a random displacement of magnitude* from a random point in R leads to a point in S. This is equal to the expected fraction of a spherical surface of radius x that is contained in S, if the sphere is centered at a random point of R. 4.2 Solutions for the Cylinder

Of special importance for the calculation of absorbed dose in and around ex­tended sources is the simple case where the receptor region is a single point R. Various solutions for this case have been obtained by Berger ( / ) , among them the one for infinite cylinders. In the following, the solution for finite cylinders will be given. The derivation requires only slight modifications of the solution utilized in Section 3; it is also relegated to the Appendix.

For simplicity, the indices of the functions sRS(x) and URS(x) will be omitted in the remainder of this section; it will be understood that the functions refer to the cylindrical source region and to a point of specified location. For easier reference s(x) can be called a point proximity function of 5 .

The position of the point R will (in addition to suitable horizontal coordinates) be specified by its vertical distance b from the face of the cylinder and away from the cylinder. To simplify the formulas only nonnegative values of b will be con­sidered. For points between the two planes through the faces of the cylinder (b < 0) the solution can evidently be expressed as the sum of two solutions with b = 0.

As shown in the Appendix, one obtains the following point proximity function for the right cylinder with arbitrary cross section and with height h:

sc((x2 - z2)1/2) , dz, (29) s(x) = X

with (x2 - z2)1/2

z, = Max (/?, (Max (0, x2 - y22))m) and z2 = Min (b + h, (x2 - yl)112) (30) and

(y2 + b2)m < x < ((b + h)2 + y22)m. (31)

274 A. M. K E L L E R E R

sc(y) is the point proximity function of the cross section of the cylinder relativ e to the reference point, /?; yt and y2 are the minimum and the maximum of y.

In the special case of a circular cylinder of radius r one obtains with Eq. (A.9) from the Appendix and with the distance a from the axis of the cylinder

s(x) = 2x C O S " ' l ^ ' - 1 ' " 2aix'-z*y* ^ 02)

Equations (30) and (31) hold with yt = Max (0, a - r) and y2 - a + r. The solution does not apply for a = 0; in this case one obtains from Eq. (29),

with sQ(y) - liry for v < r: s(x) = 2rrx(z2 - z,), a < x < ((a + hf + r2)m. (33)

The geometric reduction factor, U(x) = S(X)/4TTX2, commonly designated by ty(x), has previously been given for the special case of infinite circular cylinders ( / ) .

A P P E N D I X : D E R I V A T I O N O F T H E S O L U T I O N FOR R I G H T C Y L I N D E R S /. Density of x - (y2 + z2)m from Independent Densities of y and z

Let* be the distance between two random points in the cylinder, and>> and z the horizontal and vertical distances. Then y and z are independently distributed, and JC2 = y2 + z2.

As a first step a general expression for the density p(x) as a function of the densities p^z) and p2(y) of y and z will be derived. Insertion of actual expressions for py(z) and p2(y) will be a second step.

To make the derivation more transparent, it is helpful to introduce separate symbols, X = JC2 , Y = v2, and Z = z2, for the squares of the random variables and also separate symbols, n(X), ^ ( Z ) , and TT2(Y), for the densities of these squares. Because of the additivity, X = Y + Z, and the independence of K and Z one has the familiar convolution relation

7T(X) X

7T2(X ~ Z)7Tl(Z)dZ. ( A . l )

The relation between the density of the random variable x and the density of its square X is

dx TT(X) = p(x) — = p{x)l2x\ (A.2) dX

analogous relations hold for TTX(Z) and TT2(Y).

By inserting these relations and dZ = 2zdz into Eq. (A.l) one obtains p(x)/2x =

and therefore p(x) = x

w - O T g ^ (A3) , 2(x2 - z2)"2 2z 'PMS - zV«)pM

(.v2 - z2)"2

P R O X I M I T Y F U N C T I O N S FOR C Y L I N D E R S 2 7 5

2 . Proximity Function for the Cylinder One readily obtains the distance distribution for a line segment of length h:

Pl(z) = 2(1 - zlh)lh. (A .5 ) By inserting this into Eq. (A.4) and switching fromp(x) andp2(y) to the proximity functions one obtains the general solution, Eq. ( 1 4 ) .

The proximity function for a circular surface of diameter d is (77)

s(x) = 4x\ cos"1 (i-j - (d2 - x2)112 x < d. (A .6 )

This together with Eq. ( 1 4 ) leads to the solution, Eq. ( 1 6 ) , for circular cylinders. The proximity function for a square of side length d is somewhat more compli­

cated [see also (18) for the general case of a rectangle]:

s(x) = 2x < d2

4x + TT, (A .7 )

7T - 2 - 4 cos 1 — + 4 1 — - 1 A1

x d2 — d < x < 2md.

The formula for a regular parallelepiped is therefore not given in explicit form. However, the numerical integration of Eq. ( 1 4 ) with Eq. (A.7) is readily per­formed and contains no singularities; the solutions are given in Fig. 6 .

3. Solution for a Cylinder Relative to a Point With the coordinate b ( ^ 0 ) , as defined in Section 4 , one obtains the distribution

of vertical distances from the point R to the cylinder: px(z)=\lh for b < z < b + h. (A .8 )

By inserting this into Eq. (A.4) and switching to the point proximity functions, one obtains the solution, Eq. ( 2 9 ) , for the general cylinder.

For a circular cross section with radius r and for the distance a of the point from the center one obtains

sc(y) = 2y cos"1 j^Max j ^ - 1 , jj ; Max (a - r, 0 ) < y < a + r. (A .9 )

By inserting this into Eq. ( 2 9 ) one obtains the solution, Eq. ( 3 2 ) , for the circular cylinder. RECEIVED: April 2 9 , 1 9 8 0 ; REVISED: November 5 , 1 9 8 0

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