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Dose calculations relating to the use of negative pi-mesons for radiotherapy Henry, Marguerite Irene 1973

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DOSE CALCULATIONS RELATING TO THE USE OF NEGATIVE PI-MESONS FOR RADIOTHERAPY by MARGUERITE IRENE HENRY B . S c , U n i v e r s i t y of Alb e r t a , I9U8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission f o r extensive copying of t h i s thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date d^UJ /J. /?73 - i i -ABSTRACT P h y s i c a l (or absorbed) dose d i s t r i b u t i o n s and b i o l o g i c a l l y e f f e c t -ive d i s t r i b u t i o n s are c a l c u l a t e d i n t h i s thesis f o r (a) monoenergetic beams (b) "shaped" continuous energy spectra of negative pi-mesons. The r e s u l t s of these c a l c u l a t i o n s confirm q u a l i t a -t i v e l y the claims made f o r the advantages of negative pi-mesons f o r r a d i o -therapy and give some quan t i t a t i v e measures of these advantages. The f i r s t and most d e t a i l e d c a l c u l a t i o n s include only dose contributions from primary pions and from the charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n s which occur at the end of the negative pion tracks. The p h y s i c a l dose c a l c u l a t i o n s are based on published data on the •number and energy of the charged p a r t i c l e s from these d i s i n t e g r a t i o n s and on published range-energy-stopping power data f o r the primary pions and for the charged d i s i n t e g r a t i o n products. Two p h y s i c a l dose c a l c u l a t i o n s are made, assuming (a) 2 9 - 0 MeV and (b) 3 5 - 6 MeV t o t a l k i n e t i c energy per pion capture of the charged p a r t i c l e s from the " s t a r s " . These c a l c u l a t i o n s show that , f o r a monoenergetic beam having a 2 0 cm range, the dose at the Bragg peak i s 1 0 to 1 2 times the entrance dose. B i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s are c a l c u l a t e d , both for aerobic and f o r anoxic conditions, using a v a i l a b l e (but uncertain) data f o r the dependence of (a) " r e l a t i v e b i o l o g i c a l e f f e c t i v e n e s s " (RBE) and (b) "oxy-gen enhancement r a t i o " (OER) on the stopping power of the medium. A l l ca l c u l a t i o n s are repeated f o r two d i f f e r e n t assumptions with respect to dependence of "RBE" on stopping power. On the assumptions made, f o r a monoenergetic beam i n the Bragg peak the e f f e c t i v e RBE and the e f f e c t i v e OER - i i i -are approximately 1.9 and 1.65, r e s p e c t i v e l y , f o r the lower.RBE values used and about 2.5 and 1.55> r e s p e c t i v e l y , f o r the higher RBE values. The c a l c u l a t i o n s f o r continuous energy spectra of negative pions demonstrate the p o s s i b i l i t y of s e l e c t i n g a "shaped" spectrum which gives an e s s e n t i a l l y constant dose through a s p e c i f i e d depth with a surface dose which i s only 25 to 30% of t h i s constant dose. For a spectrum chosen to give constant b i o l o g i c a l l y e f f e c t i v e dose from 12 to 20 cm depth, assuming the lower RBE values ( r e f e r r e d to above), the e f f e c t i v e RBE increases from about 1.35 at 12 cm to I.65 at 20 cm and the e f f e c t i v e OER decreases from about 2.00 to 1.75 over the same depth i n t e r v a l . Assuming the higher RBE values, the corresponding range of e f f e c t i v e RBE values i s from 1.6 to 2.1 and the range of e f f e c t i v e OER values I.85 to 1.65-An attempt i s made to estimate corrections f o r the e f f e c t s which were neglected i n the d e t a i l e d c a l c u l a t i o n s , namely, (a) muon and electron contamination o f the incident pion beam, (b) loss of pions from the beam by int e r a c t i o n s with n u c l e i of the medium before coming to re s t and (c) dose contributions from neutrons released i n the " s t a r s " at the end of the pion tracks. When these corrections are made, i t i s shown for a monoenergetic beam of 20 cm range that the r a t i o of the maximum dose i n the Bragg peak to the surface dose i s about 6.5 i n good agreement with published experimental r e s u l t s . A l s o , i t i s shown that , when a l l corrections are taken i n t o account, f o r a "shaped" spectrum which d e l i v e r s a constant p h y s i c a l dose from 12 to 20 cm depth, about 30$ of the t o t a l energy absorbed i n the patient i s absorbed w i t h i n the constant dose region. Calculated values o f RBE and OER are compared with published experimental values but the v a l i d i t y of the comparison i s very questionable. - i v -TABLE OF CONTENTS PAGE TITLE i ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES x ACKNOWLEDGEMENTS x i i 1. INTRODUCTION 1 2. DOSE DISTRIBUTIONS CALCULATED 5 3. CONSIDERATIONS COMMON TO ALL DOSE CALCULATIONS FOR NEGATIVE PI-MESONS 7 3.1. Q u a l i t a t i v e Review of Absorption Processes 7 3.2. D e f i n i t i o n of Absorbed Dose and General Method of C a l c u l a t i o n ' 10 3.3. Energy, Range and Stopping Power Data f o r Charged P a r t i c l e s i n Water 11 3.4. Range Spectra of Charged P a r t i c l e s Released from Di s i n t e g r a t i o n s of Oxygen 22 3.5. Energy Absorption per Centimetre Thickness of S p h e r i c a l S h e l l as a Function of Distance from the Nuclear D i s i n t e g r a t i o n . 27 k. PHYSICAL DEPTH DOSE DISTRIBUTION DUE TO A MONOENERGETIC BEAM OF NEGATIVE PIONS . 33 4.1. Absorbed Dose D i s t r i b u t i o n Neglecting Range S t r a g g l i n g 33 4.2. Absorbed Dose D i s t r i b u t i o n with Range Straggling . . . hk 5. BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTIONS DUE TO MONOENERGETIC BEAMS OF NEGATIVE PI-MESONS . . . . . . . . . 53 5.1. Introduction . . . , . 53 - V -TABLE OF CONTENTS (CONTINUED) PAGE 5.1.1. Relative B i o l o g i c a l Effectiveness and B i o l o g i c a l l y E f f e c t i v e Dose 54 5.2. B i o l o g i c a l l y E f f e c t i v e Dose D i s t r i b u t i o n s under Fully-Oxygenated Conditions 56 5.2.1. R e l a t i v e B i o l o g i c a l Effectiveness and Linear Energy Transfer . . . . 56 5.2.2. S e l e c t i o n of Data f o r "RBE versus Stopping Power" 57 5.2.3. Calculations 58 5.3. B i o l o g i c a l l y E f f e c t i v e Dose D i s t r i b u t i o n under Anoxic Conditions . 65 5.3.1. Oxygen Enhancement Ratio and B i o l o g i c a l l y E f f e c t i v e Dose under Anoxic Conditions . . . 65 5.3.2. Oxygen Enhancement Ratio and Linear Energy Transfer 67 5.3.3. Calculations 67 5.4. E f f e c t i v e RBE and E f f e c t i v e OER at D i f f e r e n t Depths on D i f f e r e n t Assumptions 69 6. DOSE DISTRIBUTIONS FOR CONTINUOUS ENERGY SPECTRA OF NEGATIVE PI-MESONS 73 6.1. S e l e c t i o n of a Continuous Pion Spectrum to Y i e l d a Desired Depth Dose D i s t r i b u t i o n and C a l c u l a t i o n of the Depth Dose D i s t r i b u t i o n Due to the Selected Spectrum 73 6.2. Selected Spectra and Resulting Depth Dose D i s t r i b u t i o n s 79 6.2.1. Continuous Pion Spectrum to Y i e l d Constant Absorbed Dose 80 6.2.2. Continuous Spectra to Y i e l d Constant B i o l o g i c a l l y E f f e c t i v e Doses under Well Oxygenated Conditions . . . . 85 - v i -TABLE OF CONTENTS (CONTINUED) PAGE T. BEAM CONTAMINATION AND DOSE CONTRIBUTIONS NOT INCLUDED IN THE DOSE CALCULATIONS. COMPARISON OF CALCULATIONS WITH EXPERIMENTAL RESULTS AND WITH OTHER CALCULATIONS . . 93 7-1. Beam Contamination and Dose Contributions Not Included i n the Dose Calculations 93 7.1.1. E l e c t r o n and Muon Contamination of the TT" Beam 93 7.1.2. Pions Removed from the Beam before Coming to Rest by Interactions with Nuclei of the Medium 97 7.1.3. Dose Contributions from Neutrons and Photons Released i n the Nuclear Disintegrations . . . 107 7.2. Ratio of the Maximum Dose i n the High Dose Volume to the Dose at the Surface of the Medium. Comparison with Experimental Measurements . 109 7.3. F r a c t i o n of T o t a l Absorbed Energy Which Is Absorbed i n the Treatment Volume 114 7.4. E f f e c t i v e RBE and E f f e c t i v e OER . . . . . . . . . . 119 8. CONCLUSIONS 122 BIBLIOGRAPHY 125 APPENDIX A 129 APPENDIX B 135 - v i i -LIST OF TABLES PAGE I NUMBER AND ENERGY OF PARTICLES RELEASED BY TT ~ CAPTURE IN 160 AND EXCITATION ENERGY OF RESIDUAL NUCLEI 8 II ENERGY SPECTRA OF CHARGED PARTICLES RELEASED BY TT - CAPTURE IN 160 (Number per MeV per TT" capture) . - . 9 III SOURCES OF RANGE AND STOPPING POWER DATA Ik IV RANGE, ENERGY AND STOPPING POWER DATA FOR PROTONS, ALPHA PARTICLES AND 1 *B IONS . . . • l6 V ABRIDGED TABLE OF RANGE, ENERGY AND STOPPING POWER DATA FOR DEUTERONS, TRITONS AND 3HE IONS 20 VI ABRIDGED TABLE OF RANGE, ENERGY AND STOPPING POWER DATA FOR PI-MESONS 21 VII NUMBER OF CHARGED DISINTEGRATION PRODUCTS STOPPING PER CM, dN/dR, AS A FUNCTION OF DISTANCE FROM THE DISINTEGRATION 26 VIII ENERGY ABSORBED PER CENTIMETRE THICKNESS OF SPHERICAL SHELL, dT/dr, AS A FUNCTION OF DISTANCE r p FROM THE DISINTEGRATION 31 IX NUMBER AND ENERGY OF EACH TYPE OF CHARGED PARTICLE RELEASED PER TT ~ CAPTURE: COMPARISON OF TABLE I. WITH RESULTS OF INTEGRATION OF EQUATIONS (10), (11) and (Ik) 32 X KINETIC ENERGY OF CHARGED PARTICLES FROM TT " CAPTURE IN 160: VALUES FROM DIFFERENT SOURCES 38 XI ABSORBED DOSE PER PION PER CM2, D/F, STRAGGLING NEGLECTED, (a) DUE TO PRIMARY PIONS ONLY AND (b) ASSUMING ( i ) 29-0 MeV "STARS", ( i i ) 35-6 MeV "STARS" 0^ XII ABSORBED DOSE PER PION PER CM2 DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER, RANGE STRAGGLING INCLUDED 50 XIII BIOLOGICALLY EFFECTIVE DOSE PER PION PER CM2 DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER UNDER FULLY OXYGENATED CONDITIONS, RANGE STRAGGLING INCLUDED 62 - v i i i -LIST OF TABLES (CONTINUED) PAGE XIV BIOLOGICALLY EFFECTIVE DOSE PER PION PER CM2 DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER UNDER ANOXIC CONDITIONS, RANGE STRAGGLING INCLUDED . . . . . TO XV EFFECTIVE RBE AND EFFECTIVE OER FOR A BEAM OF 82.2 MeV PIONS AT DIFFERENT DEPTHS AND ON DIFFERENT ASSUMPTIONS. . . 72 XVI ABSORBED DOSE DISTRIBUTION DUE TO A "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 29.0 MeV "STARS" 8l XVII BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTION UNDER WELL OXYGENATED CONDITIONS DUE TO "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 29.0 MeV "STARS" AND RBE VALUES FROM FIGURE 9, CURVE A (LOWER VALUES) 86 XVIII BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTION UNDER WELL OXYGENATED CONDITIONS DUE TO "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 29-0 MeV "STARS" AND RBE VALUES FROM FIGURE 9, CURVE B (HIGHER VALUES) . 90 XIX CONVERSION AND SCALING OF CONTINUOUS PION SPECTRA 92 XX MEASUREMENTS AND ESTIMATES OF ELECTRON AND MUON CONTAMINATION OF EXPERIMENTAL PION BEAMS 95 XXI CALCULATION OF THE PERCENTAGE CONTRIBUTIONS OF PRIMARY PIONS, MUON CONTAMINATION AND ELECTRON CONTAMINATION TO THE ABSORBED DOSE AT THE SURFACE OF WATER 98 XXII ABSORBED DOSE PER STOPPING PION PER CM2. DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER, RANGE STRAGGLING INCLUDED AND CORRECTED FOR INTERACTIONS OF PIONS WITH NUCLEI OF THE MEDIUM 100 XXIII ABSORBED DOSE DISTRIBUTION DUE TO A "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 29.0 MeV "STARS" AND CORRECTED . • FOR INTERACTIONS OF PIONS WITH NUCLEI OF THE MEDIUM . . . . 103 XXIV CONVERSION AND SCALING OF CONTINUOUS PION SPECTRUM OF TABLE XXIII • • • 105 XXV ESTIMATED PERCENTAGE CONTRIBUTIONS TO SURFACE AND PEAK ABSORBED DOSES 108 XXVI COMPARISON OF CALCULATED AND EXPERIMENTAL VALUES OF THE RATIO OF PEAK DOSE TO SURFACE DOSE FOR A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER 112 - ix -LIST OF TABLES (CONTINUED) PAGE XXVII TOTAL ENERGY ABSORPTION DUE TO PION SPECTRUM OF TABLE XXIV, COLUMN 7, PLUS ACCOMPANYING ELECTRON AND MUON CONTAMINATION 118 XXVIII PUBLISHED VALUES OF EFFECTIVE RBE AND EFFECTIVE OER IN BRAGG PEAK FOR NEARLY MONOENERGETIC BEAMS OF NEGATIVE PI-MESONS 120 AI FRACTION OF NEGATIVE PIONS OF DIFFERENT ENERGIES SURVIVING AN 8-METRE PATH WITHOUT DECAY . . . . 131 A l l FRACTION OF NEGATIVE PIONS DECAYING IN RANGE R IN WATER. . . 132 AIII RANGES IN WATER AND KINETIC ENERGIES OF PIONS AND MUONS AND KINETIC ENERGIES OF ELECTRONS OF EQUAL MOMENTA 133 - X -LIST OF FIGURES PAGE 1. NUMBER OF PROTONS STOPPING PER UNIT RANGE INTERVAL PER TT - CAPTURE AS A FUNCTION OF DISTANCE FROM THE NUCLEAR DISINTEGRATION 2k . 2. DIAGRAM TO ILLUSTRATE DERIVATION OF EQUATION (13) 29 3. DIAGRAM TO ILLUSTRATE DOSE CONTRIBUTIONS AT DIFFERENT DEPTHS 35 4. CONTRIBUTIONS OF NUCLEAR DISINTEGRATIONS IN PLANE AT DEPTH R^ TO DOSE AT P 35 5. ABSORBED DOSE CURVES DUE TO A MONOENERGETIC BEAM OF NEGATIVE PIONS IN WATER, RANGE STRAGGLING NEGLECTED (Path length i n water = 30 cm) 43 6. DIAGRAM TO ILLUSTRATE DERIVATION OF EQUATION (27) 45 7. DIAGRAM TO ILLUSTRATE CONTRIBUTION OF PIONS STOPPING AT x TO DOSE AT Xp > x 47 8. ABSORBED DOSE CURVES DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER FOR DIFFERENT ASSUMPTIONS RE KINETIC ENERGY OF CHARGED PARTICLES FROM "STARS". (Data from table XII) 51 9. TWO RELATIONSHIPS ASSUMED BETWEEN RELATIVE BIOLOGICAL . EFFECTIVENESS AND STOPPING POWER (Data adapted from references 20 and 21) . . . . . . . . . . 59 10. BIOLOGICALLY EFFECTIVE DOSES DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS FOR DIFFERENT ASSUMPTIONS RE KINETIC ENERGY OF CHARGED PARTICLES FROM "STARS" 63 11. COMPARISON OF ABSORBED DOSE AND BIOLOGICALLY EFFECTIVE DOSES DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS. A l l curves c a l c u l a t e d on assumption of 29 MeV k i n e t i c energy of charged p a r t i c l e s from " s t a r s " 6k 12. ASSUMED DEPENDENCE OF OXYGEN ENHANCEMENT RATIO ON STOPPING POWER (Data from reference 24) 68 13. ADDITION OF NEARLY MONOENERGETIC BEAMS TO OBTAIN A CONSTANT DOSE OVER A SPECIFIED DEPTH 74 14. DIAGRAM TO ILLUSTRATE DERIVATION OF EQUATION (40) 77 - x i -LIST OF FIGURES (CONTINUED) PAGE 15. ABSORBED DOSE AS A FUNCTION OF DEPTH IN WATER DUE TO THE "SHAPED" CONTINUOUS SPECTRUM OF NEGATIVE PIONS SHOWN IN INSET (Assumes 29 MeV k i n e t i c energy of charged disintegration products) 83 16. (a) BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTION UNDER WELL-OXYGENATED CONDITIONS USING RBE VALUES FROM FIGURE 9, CURVE A, AND (b) ABSORBED DOSE DISTRIBUTION, BOTH FOR THE "SHAPED" SPECTRUM OF NEGATIVE PIONS SHOWN IN INSET 88 IT- (a). BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTION UNDER WELL-OXYGENATED CONDITIONS USING RBE VALUES FROM FIGURE 9, CURVE B, AND (b) ABSORBED DOSE DISTRIBUTION, BOTH FOR THE SHAPED SPECTRUM OF NEGATIVE PIONS SHOWN IN INSET 91 18. ABSORBED DOSE CURVE DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER — WITH CORRECTION FOR LOSS OF PIONS BY NUCLEAR INTERACTIONS (For 29 MeV "stars" only) 101 19. ABSORBED DOSE CURVE DUE TO A "SHAPED" SPECTRUM OF NEGATIVE PIONS — WITH CORRECTION FOR LOSS OF PIONS BY NUCLEAR INTERACTIONS (For 29 MeV "stars" only) . . . . . .104 - x i i -ACKNOWLEDGEMENTS This project was c a r r i e d out at The B r i t i s h Columbia Cancer I n s t i t u t e . The author would, l i k e to thank the I n s t i t u t e f o r the many f a c i l i t i e s made ava i l a b l e to her. The wr i t e r wishes to express her appreciation f o r the suggestions and encouragement given her by members of the Physics Depart-ment at the I n s t i t u t e , e s p e c i a l l y Dr. R. 0. Kornelsen and Mrs. M. E. J . Young. The author i s p a r t i c u l a r l y g r a t e f u l to Dr. H. F. Batho f o r h i s continual helpfulness and enthusiasm throughout the time he supervised t h i s p r o j e c t . 1. INTRODUCTION The p o s s i b l e advantages of beams of negative pi-mesons f o r r a d i o -therapy have been recognized f o r several years (1,2,3,4). The f i r s t advan-tage w i l l be an improved "depth dose d i s t r i b u t i o n " i n t r e a t i n g deep malignant tumours, i . e . , i n t r e a t i n g tumours at a depth below the s k i n . In almost a l l r a d i a t i o n treatment of deep tumours, a source i s used which i s external to the patient's body. The energy to t r e a t the malignancy must, therefore, be d e l i v e r e d through the surrounding normal t i s s u e . Since, i n general, there i s not a marked di f f e r e n c e i n s e n s i t i v i t y between normal and malignant c e l l s , a tumour can be t r e a t e d only i f i t can be given an appreciably l a r g e r dose than that to the o v e r l y i n g normal t i s s u e . The b i o l o g i c a l e f f e c t produced by rad-i a t i o n i s r e l a t e d to the i o n i z a t i o n produced i n the i r r a d i a t e d t i s s u e . X-rays and gamma rays, the r a d i a t i o n s conventionally used for radiotherapy, are i n d i r e c t l y i o n i z i n g . The i o n i z a t i o n responsible f o r the b i o l o g i c a l e f f e c t i s produced by the photoelectrons, Compton r e c o i l electrons and/or electron p a i r s produced by absorption of the primary r a d i a t i o n . With an i n d i r e c t l y i o n i z i n g r a d i a t i o n , the dose always f a l l s approximately exponen-t i a l l y with depth i n t i s s u e , except f o r build-up e f f e c t s j u s t below the s k i n . Since the maximum dose i s at or near the s k i n , i t i s always necessary to use " m u l t i p l e - f i e l d " treatment plans to t r e a t deep tumours, i . e . , the tumour i s i r r a d i a t e d from sev e r a l d i f f e r e n t d i r e c t i o n s , a l l f i e l d s overlapping at the s p e c i f i e d treatment volume but each i r r a d i a t i n g a d i f f e r e n t volume of the overlying t i s s u e . With c a r e f u l treatment planning, i t i s p o s s i b l e to get an adequate dose to the tumour without serious damage to surrounding t i s s u e . With an i n d i r e c t l y i o n i z i n g r a d i a t i o n , however, not more than 15 to 20% of the t o t a l energy d e l i v e r e d to the patient i s absorbed i n the s p e c i f i e d t r e a t -ment volume. - 2 -Any charged p a r t i c l e r a d i a t i o n tends to give a b e t t e r depth dose d i s t r i b u t i o n than an i n d i r e c t l y i o n i z i n g r a d i a t i o n since the energy d i s -s i p a t i o n per un i t distance and, therefore, the density of i o n i z a t i o n increase sharply as the p a r t i c l e slows down at the end of i t s track. Unfortunately, electron beams have been the only charged p a r t i c l e r a d i a t i o n r e a d i l y a v a i l -able f o r radiotherapy and, because of the marked s c a t t e r i n g of electrons, the depth dose d i s t r i b u t i o n s obtained are scarcely b e t t e r than those obtained with megavoltage x-rays or cobalt 60 gamma rays. Beams of heavy charged p a r t i c l e s are preferable to ele c t r o n beams since the heavy p a r t i c l e s are sc a t t e r e d much l e s s than electrons and, fur t h e r , the i o n i z a t i o n density at the end of the track of a heavy p a r t i c l e i s much greater than at the end of an el e c t r o n track. I f the energy of the p a r t i c l e s i s chosen so that the p a r t i c l e tracks terminate i n the desired treatment volume, a very favourable r a t i o of energy deposition per un i t mass i n the treatment volume to that i n the o v e r l y i n g t i s s u e can be obtained. The r a t i o of the b i o l o g i c a l e f f e c t i n the treatment volume to that i n the surrounding t i s s u e i s even more favourable than the r a t i o of the p h y s i c a l energy depositions per gram since i t i s known that the denser i o n i z a t i o n near the end of the p a r t i c l e track i s , i n general, b i o l o g i c a l l y more e f f e c t -ive than the sparser i o n i z a t i o n along the track. Negative pi-meson beams have a p a r t i c u l a r advantage f o r r a d i o -therapy. A negative pion, u n l i k e a p o s i t i v e pion or any other p o s i t i v e l y charged p a r t i c l e , a f t e r having d i s s i p a t e d a l l i t s k i n e t i c energy i n t e r a c t s with an atomic nucleus i n the medium causing d i s i n t e g r a t i o n of the nucleus. This d i s i n t e g r a t i o n r e s u l t s i n a la r g e energy release, part of which appears as k i n e t i c energy of short-range heavy charged p a r t i c l e s . This k i n e t i c - 3 -energy i s absorbed i n the immediate v i c i n i t y of the point of d i s i n t e g r a t i o n and increases appreciably the energy absorption per unit mass at the end of the track, i . e . , i t increases the r a t i o of the energy deposition i n the treatment volume to that i n the ove r l y i n g t i s s u e . The process leading to the nuclear d i s i n t e g r a t i o n and the products of the d i s i n t e g r a t i o n are discussed i n a l a t e r s e c t i o n . A second p o s s i b l e advantage of negative pions f o r radiotherapy may be a reduced dependence of the b i o l o g i c a l response on the oxygen supply of the i r r a d i a t e d c e l l s . I t i s known that anoxic or very hypoxic c e l l s are less s e n s i t i v e than well-oxygenated c e l l s to a sparsely i o n i z i n g r a d i a t i o n such as x-rays or gamma rays. Anoxic c e l l s must be exposed to an x-ray-or gamma-ray dose 2.6 to 3.0 times as great as that to well-oxygenated c e l l s to produce the same b i o l o g i c a l e f f e c t . Further, there i s evidence that some of the c e l l s i n a malignant tumour may be very hypoxic or anoxic; these c e l l s w i l l be p a r t i a l l y protected from x-ray or gamma-ray damage. For a densely i o n i z -ing r a d i a t i o n , however, the b i o l o g i c a l e f f e c t produced i s much less dependent on the oxygen supply of the c e l l s . In view of the dense i o n i z a t i o n at the end of the negative pi-meson track, anoxic malignant c e l l s i n a tumour i r r a -diated with negative pions w i l l s u f f e r r e l a t i v e l y greater damage than they would i f the tumour were tr e a t e d with x-rays or gamma rays. I t should be noted, however, that the advantage to be gained from the reduced dependence on the oxygen supply of the c e l l s cannot be estimated with much c e r t a i n t y since the s i g n i f i c a n c e of anoxia or hypoxia i n conventional treatment of tumours with x-rays or gamma rays i s not w e l l established. To date, there have been no accelerators which can produce pi-meson beams of s u f f i c i e n t i n t e n s i t y f o r c l i n i c a l t r i a l s of radiotherapy or f o r the r a d i o b i o l o g i c a l i n v e s t i g a t i o n s which w i l l be e s s e n t i a l before c l i n i c a l t r i a l s - It -can be attempted. However, three accelerators are now under construction — the LAMPF acc e l e r a t o r at Los Alamos, New Mexico, the SIN accelerator near Zurich, Switzerland, and the TRIUMF accelerator at the U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada — which w i l l produce pi-meson beams several hun-dred times the i n t e n s i t y of any now a v a i l a b l e . A l l three accelerators w i l l provide s u f f i c i e n t negative pion i n t e n s i t y f o r the necessary r a d i o b i o l o g i c a l i n v e s t i g a t i o n s and, i f these are encouraging, f o r t r i a l s of radiotherapy. With the prospect o f pion beams sui t a b l e f o r b i o l o g i c a l experiments and f o r therapeutic t r i a l s being a v a i l a b l e i n the near future, p h y s i c a l depth dose d i s t r i b u t i o n s , which can be expected with negative pions under s p e c i f i e d i r r a d i a t i o n conditions, have been c a l c u l a t e d from data now a v a i l a b l e . Also, some b i o l o g i c a l l y e f f e c t i v e doses have been calcu l a t e d , both f o r w e l l -oxygenated and f o r anoxic conditions, though the b i o l o g i c a l data used are much l e s s r e l i a b l e than the data used f o r the p h y s i c a l c a l c u l a t i o n s . 2. DOSE D I S T R I B U T I O N S CALCULATED While the c a l c u l a t i o n s i n t h i s paper are of i n t e r e s t f o r t h e i r relevance to radiotherapy, they have "been made f o r beams of negative pions stopping i n water. This s i m p l i f i e d the calcu l a t i o n s since a l l negative pions stopping i n water produce d i s i n t e g r a t i o n s of oxygen n u c l e i while, i n t i s s u e , d i s i n t e g r a t i o n s of oxygen, carbon, nitrogen and, o c c a s i o n a l l y , heavier e l e -ments a l l occur. The differe n c e between the energy d i s t r i b u t i o n i n water and that i n s o f t t i s s u e w i l l be small. An uncontaminated and p a r a l l e l negative pion beam has been assumed f o r the c a l c u l a t i o n s , with the pions uniformly d i s t r i b u t e d over the f i e l d i r r a d i a t e d . A l s o , the beam has been assumed to be broad compared with the l a t e r a l s c a t t e r i n g caused by multiple small-angle d e f l e c t i o n s . The c a l c u l a -tions are v a l i d i n the centre of the beam but, f o r the examples selected, not within two or three centimetres of the edge. The following depth dose d i s t r i b u t i o n s have been ca l c u l a t e d : (a) p h y s i c a l depth dose d i s t r i b u t i o n s ( i . e . , the energy deposited per gram at d i f f e r e n t depths) due to a monoenergetic beam of negative pions, (b) b i o l o g i c a l l y e f f e c t i v e depth dose d i s t r i b u t i o n s due to the same monoenergetic negative pion beam, (c) the p h y s i c a l depth dose d i s t r i b u t i o n due to a selected energy spec-trum o f negative pions, (d) b i o l o g i c a l l y e f f e c t i v e depth dose d i s t r i b u t i o n s f o r other selected energy spectra of inc i d e n t pions. Of the depth dose d i s t r i b u t i o n s l i s t e d above, the p h y s i c a l d i s t r i -bution f o r a monoenergetic pion beam i s the basic c a l c u l a t i o n . " B i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s f o r a monoenergetic beam have been c a l c u l a t e d - 6 -simply by introducing the appropriate b i o l o g i c a l weighting factors i n t o the ph y s i c a l c a l c u l a t i o n . The c a l c u l a t i o n of ph y s i c a l and b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s f o r continuous energy spectra of incident pions i s straightforward when d i s t r i b u t i o n s are known for monoenergetic beams. The cal c u l a t i o n s of greatest i n t e r e s t f o r radiotherapy are those f o r continuous spectra. The c a l c u l a t i o n s include only dose contributions from charged par-t i c l e s , i . e . , from the primary negative pions and from the charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n s at the ends of the pion tracks. (Over 50% of the k i n e t i c energy of the l a t t e r i s , on the average, absorbed within 0 . 1 cm o f the d i s i n t e g r a t i o n from which the p a r t i c l e s originate.) The dose d i s t r i b u t i o n s as f i r s t c a l c u l ated (sections U , 5 and 6) do NOT include dose contributions from (a) e l e c t r o n and muon contamination of the incident pion beam, (b) pions removed from the beam before coining to r e s t by i n t e r a c t i o n s with n u c l e i of the medium, (c) neutrons and photons released i n the nuclear d i s i n t e g r a t i o n s at the end of the pion tracks (since much of the neutron and photon energy i s absorbed at an appreciable distance from the d i s i n t e g r a t i o n s ) , (d) pi-mesic x-rays accompanying the dis i n t e g r a t i o n s (energy release per pion capture l e s s than 0 . 2 2 MeV). An attempt has been made in' s e c t i o n 7 o f the paper to estimate corrections f o r these omissions. A computer was used to carry out most of the c a l c u l a t i o n s but the programs have not been included i n the t h e s i s . 3. CONSIDERATIONS COMMON TO ALL DOSE CALCULATIONS FOR NEGATIVE PI-MESONS This s e c t i o n includes (a) a q u a l i t a t i v e review of the processes by which energy i s absorbed i n water when i t i s exposed to a beam of negative pi-mesons, (b) d e f i n i t i o n of absorbed dose and general method of c a l c u l a t i o n and (c) c e r t a i n data and preliminary c a l c u l a t i o n s from these data required f o r a l l c a l c u l a t i o n s of dose d i s t r i b u t i o n s due to beams of negative pions. 3.1 Q u a l i t a t i v e Review of Absorption Processes A negative pi-meson as i t passes through a medium dis s i p a t e s most of i t s k i n e t i c energy by e x c i t a t i o n and i o n i z a t i o n of the medium as a r e s u l t of Coulomb forces a c t i n g between the pion and the electrons of the medium. Ad d i t i o n a l processes are involved at very low energies. A f t e r a negative pion comes to r e s t i n water, i t i s captured by e i t h e r a proton or an oxygen nucleus to form a pi-mesic atom. I f t h i s atom has been formed with a proton, the pion i s always captured l a t e r by an oxygen nucleus with which i t forms a lower energy system than with the proton. With oxygen, the pion drops through successively lower energy l e v e l s with emission of mesic x-rays. Upon reaching the second lowest or lowest l e v e l , the pion penetrates the nucleus causing i t to d i s i n t e g r a t e . The pion disappears; part of i t s r e s t mass energy overcomes the binding energy of the nucleus and the remainder appears as k i n e t i c energy of the d i s i n t e g r a t i o n products or as e x c i t a t i o n energy of the r e s i d u a l nucleus. This whole process occurs i n a time short compared with the mean l i f e of the pion. Any s i n g l e d i s i n t e g r a t i o n may be any one of many possible reac-t i o n s . From data f o r a large number of d i s i n t e g r a t i o n s , average number and an. average energy spectrum can be determined f o r each type o f p a r t i c l e released. Since the pions have come to r e s t before being absorbed, the - 8 -products are, on the average, i s o t r o p i c a l l y d i s t r i b u t e d about the p o s i t i o n of d i s i n t e g r a t i o n . From a study of p a r t i c l e tracks i n photographic emulsions, Fowler and Mayes (5) obtained some experimental r e s u l t s on the d i s i n t e g r a t i o n of oxygen n u c l e i * . Guthrie, A l s m i l l e r and B e r t i n i ( 6 ) used a Monte Carlo program to estimate average numbers and average energy spectra of the p a r t i c l e s emitted and the average e x c i t a t i o n energy of the r e s i d u a l n u c l e i . Their data have been used i n the c a l c u l a t i o n s of t h i s paper since they are more d e t a i l e d than those of Fowler and Mayes. Table I shows the average number and energy per stopping pion f o r each type of p a r t i c l e produced. Table II gives an energy spectrum f o r each type of charged p a r t i c l e as number of p a r t i c l e s per MeV * The d i s i n t e g r a t i o n s are sometimes r e f e r r e d to as " s t a r s " because of the forked appearance of the tracks i n a photographic emulsion. TABLE I NUMBER AND ENERGY OF PARTICLES RELEASED BY TT AND EXCITATION ENERGY OF RESIDUAL NUCLEI** CAPTURE IN l 6 0 Type of p a r t i c l e Average number per TT~ capture Average k i n e t i c energy per i r ~ capture (MeV) Proton Deuteron T r i t o n 3He Alpha L i Be B C N Neutron •0.310 •0.631 0.101 >2.k2 0.177 60.65 Average e x c i t a t i o n energy of r e s i d u a l n u c l e i 5.85 ** Data from Guthrie, A l s m i l l e r and B e r t i n i (6) - 9 -TABLE I I , ENERGY SPECTRA OF CHARGED PARTICLES RELEASED BY T T " CAPTURE IN 1 6 0 * (Number per MeV per TT - capture) Energy-i n t e r v a l (MeV) Type of p a r t i c l e emitted Protons Deuterons Tritons 3He Alpha p a r t i c l e s Heavy r e c o i l s ( z > 3 ) 0 - 1 1 . 7 8 X IO" 2 5 . 1 6 X 1 0 " 3 3 . 5 6 X IO" 2 1 . 6 l X 1 0 " 1 1-2 8 . 3 6 X IO" 2 1.14 x IO" 2 6 . 1 9 X IO" 2 9 . 8 4 x 10~ 2 2 - 3 1.03 X IO" 1 1.32 x IO" 2 8 . 2 1 X I O - 2 7 . 9 1 x 1 0 " 2 3-4 8 . 8 5 X IO" 2 2 . 0 2 x IO" 2 7 . 6 8 X IO" 2 5.64 x 1 0 _ 2 4-5 8 . 2 9 X I O - 2 1 . 9 8 X IO" 2 8 . 4 3 x IO" 2 4 . 8 5 x 1 0 " 2 5-6 7 . 3 5 X IO" 2 2 . 0 2 x IO" 2 8 . 6 9 X IO" 2 5 . 0 7 x 10~ 2 6 - 7 7 . 0 8 X IO" 2 1 . 8 9 x IO" 2 7 . 2 0 X IO" 2 3 . 2 4 x 1 0 " 2 7 - 8 6 . 5 9 X I O - 2 2 . 2 9 x IO" 2 7 . 2 9 X IO" 2 2 . 5 1 * x 10~ 2 8 - 9 5 . 0 2 X I O - 2 1 . 5 8 X IO" 2 6 . 5 9 X I O - 2 2 . 5 4 x 1 0 " 2 9 - 1 0 4 . 5 5 X IO" 2 1 . T 2 x IO" 2 4 . 9 2 X IO" 2 1 . 3 1 x 1 0 " 2 1 0 - 1 2 4 . 7 0 X IO" 2 1 . 6 5 x 1 0 ~ 2 4 . 3 7 X IO" 2 7 . 8 7 x 1 0 " 3 12-14 3 . 4 6 X I O - 2 1 . 2 1 x IO" 2 3 . 5 8 X IO" 2 4 . 5 9 x 1 0 " 3 14-16 2 . 5 0 X IO" 2 9 . 0 2 X 1 0 " 3 2 . 6 3 X IO" 2 3 . 7 2 x 1 0 " 3 1 6 - 1 8 2 . 2 9 X I O - 2 7.48 x 1 0 " 3 2.04 X l O ^ 2 1 . 5 3 x 1 0 " 3 1 8 - 2 0 1 . 6 6 X IO" 2 6 . 6 0 X 1 0 " 3 1 . 3 6 X IO" 2 8 . 7 5 x 1 0 " 2 0 - 2 5 1 . 2 7 X IO" 2 3 . 8 6 X IO" 3 1 .11 X IO" 2 6 . 1 2 x 1 0 _ 25-30 6 . 3 8 X I O - 3 2 . 1 1 x I O - 3 5 . 3 6 X 1 0 " 3 1 . 7 5 x 1 0 " 30- 40 4 . 8 1 X I O - 3 7 . 9 2 X IO -* 2 . 5 0 X 1 0 " 3 40-50 3 . 8 9 X I O - 3 3 . 9 6 X 1 0 " * 5 . 7 1 X 1 0 " * 5 0 - 6 0 3 . 6 7 X 1 0 " 3 6 0 - 7 0 3.24 X I O - 3 7 0 - 8 0 1 . 2 7 X I O - 3 8 0 - 9 0 7 . 8 7 X 10~* 9 0 - 1 0 0 3 . 9 4 X 1 0 " * * Data from Guthrie, A l s m i l l e r and B e r t i n i (6) - 10 -energy interval per negative pion capture. Guthrie et a l ( 6 ) have given a single energy spectrum for deuterons, tritons and 3He ions and a single spec-trum for a l l the r e c o i l nuclei. Deuterons, tritons and 3He have been treated separately in this paper as described later but, to simplify the calculations, a l l heavy recoil nuclei have been treated as boron 1 1 . The charged particles from the nuclear disintegrations dissipate their kinetic energy i n the medium by essentially the same processes as the negative pions, except for small differences at the ends of the particle tracks due to the difference in the sign of the charge. Since the charged particles travel radially from the disintegrations and are, averaged over a large number of disintegrations, isotropically distributed about them, the average energy absorption is spherically symmetrical about the disintegra-tions. I f the range and stopping power for each type of particle are known as a function of i n i t i a l particle energy, the average energy absorption per centimetre thickness of spherical shell can be calculated from the spectra of table II. 3.2 Definition of Absorbed Dose and General Method of Calculation ABSORBED DOSE is defined (7) as the ENERGY ABSORPTION PER GRAM OF MEDIUM. The special unit of absorbed dose is the RAD where 1 rad = 100 ergs per gram. "Physical dose" and "absorbed dose" are used synonymously i n this paper. The absorbed dose at any point due to primary pions can be calcu-lated i f the number of pions per cm2 and the stopping power of the medium (i.e., energy dissipation per centimetre of particle track) for these pions are known. Also, the absorbed dose at a point due to disintegrations of oxygen nuclei can be determined i f the average distribution of absorbed - 11 -energy around the d i s i n t e g r a t i o n s has been calc u l a t e d . The t o t a l absorbed dose i s the sum of the contributions from primary pions and those from d i s i n t e g r a t i n g oxygen n u c l e i . Absorbed dose c a l c u l a t i o n s i n t h i s paper required, therefore, (a) data f o r the ranges and stopping powers' as a function of p a r t i c l e energy f o r the primary pions and for a l l the charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n s and (b) data f o r the average number of each type of charged d i s i n t e g r a t i o n product stopping per u n i t distance as a function of distance from the d i s i n t e g r a t i o n . These data are given i n the sub-sections which follow. 3.3 Energy, Range and Stopping Power Data for Charged P a r t i c l e s i n Water Ranges and stopping powers as a function of energy were required f o r the following: (a) f o r negative pi-mesons (up to about 100 MeV), (b) f o r the charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n s (see t a b l e I I ) , i . e . , f o r ( i ) protons (up to 100 MeV), ( i i ) deuterons, t r i t o n s , 3He ions and alpha p a r t i c l e s (up to 50 MeV), ( i i i ) boron 11 ions (up to 30 MeV). Due to s t a t i s t i c a l e f f e c t s , p a r t i c l e s of a given type and given i n i t i a l energy do not a l l t r a v e l the same distance i n the medium before com-ing to r e s t . In t h i s paper, RANGE o f a p a r t i c l e of given i n i t i a l energy i s defined as the AVERAGE o f the path lengths of a large number of p a r t i c l e s of the same i n i t i a l energy. As noted p r e v i o u s l y , except at very low energies the energy loss by - 12 -charged particles traversing a medium is almost entirely due to interactions with the electrons of the medium. The electronic stopping power S ^  of the medium is given by the following equation (reference 8): S = dE/dx = (Inrz^NZ/mv2) [ln { 2 m v 2/l(l - 3 2 ) } - 3 2 - Sc] (l) where e = electronic charge m = rest mass of electron E = kinetic energy of the charged particle z = charge number of the particle v = velocity of the particle 8 = v/c For a molecular medium N = number of molecules per unit volume of medium Z = number of electrons per molecule I = mean excitation energy per electron of the molecules of the medium. The term SC in the square bracket is the so-called "shell correction" term, important only at low velocities, and need not be considered in this paper. Equation (l) is sometimes referred to as "Bethe's equation". Equation (l) can be used without modification to calculate the stopping power for a positively charged particle with energy greater than about 1/2 z 2 MeV (9). Below this energy, a positive particle begins to gain and lose orbital electrons and z must be replaced by an effective charge z* which is less than z. As z* decreases with decreasing energy, the electronic stopping power goes through a maximum and then decreases. Data on values of z* are limited and uncertain. For any particle with energy less than about 10 times its rest mass energy (9), elastic collisions between the particle and whole atoms - 13 -( i . e . , the coupled systems of n u c l e i and atomic electrons) become important. The stopping power due to e l a s t i c c o l l i s i o n s with atoms ( c a l l e d "nuclear stopping power" by some authors) f i r s t increases and then decreases with decreasing p a r t i c l e energy. The t o t a l stopping power, e l e c t r o n i c plus atomic, must decrease to zero at zero energy. In the energy i n t e r v a l i n which equation ( l ) i s v a l i d , the range R of a heavy charged p a r t i c l e i s r e l a t e d to i t s energy E by the equation f E R(E) = R 1 ( E 1 ) + / ( 1 / S e l ) dE ( 2 ) * JE where E^ i s equal to or greater than the minimum energy f o r which equation ( l ) i s . v a l i d , R^ i s the corresponding range and S ^ i s given by equation ( l ) . As long as equation ( l ) can be used without modification (above about 1/2 MeV f o r protons), e l e c t r o n i c stopping powers c a l c u l a t e d by d i f f e r -ent authors are i n good agreement — d i f f e r i n g only due to differences i n the value of the mean e x c i t a t i o n energy I and i n the s h e l l corrections used. Ranges may d i f f e r also due to differences i n the value chosen f o r R^ i n equation (2). At energies lower than those f o r which equation ( l ) i s v a l i d with-out m o d i f i c a t i o n , values of stopping powers and ranges as given by d i f f e r e n t sources are not i n good agreement. The choice of low energy data, however, i s not c r i t i c a l since the energy involved i s only a small f r a c t i o n of the t o t a l energy d i s s i p a t i o n . In view o f the u n c e r t a i n t i e s i n range and stopping power data at * The range given by equation (2) i s c a l l e d the c.s.d.a. (continuous slow-ing down approximation) range. The actual slowing down of the p a r t i c l e s i s caused by small (but not i n f i n i t e s i m a l ) d i s c r e t e energy l o s s e s . B i c h s e l estimates that f o r protons the difference between the c.s.d.a. range and the a c t u a l range i s 0.2% or l e s s ( 1 0 ) . - lU -low energies, i t was decided to se l e c t energy-range data f o r t h i s work and to use TOTAL STOPPING POWER as determined from S, . = dE/dR (3) to t For the energy i n t e r v a l i n which equation ( l ) i s v a l i d , S i s e s s e n t i a l l y equal to S , hut, at lower energies, S x . w i l l be the sum of S , and the e l t o t ' e l stopping power due to c o l l i s i o n s with atoms. Table I I I l i s t s the p a r t i c l e s and the energy i n t e r v a l f o r each p a r t i c l e f o r which range and stopping power data were required, the sources of the published range data used and the method of obtaining range data where published data were not a v a i l a b l e . Range data published by N o r t h c l i f f e and S c h i l l i n g ( l l ) f o r protons, alpha p a r t i c l e s and boron 11 ions f o r energies up to 12 MeV per atomic mass uni t were used because these were the most complete low energy data a v a i l a b l e . Range data published by B i c h s e l (12) were used f o r proton energies above I 12 MeV. (The two sources are i n good agreement for energies from 8 to TABLE I I I SOURCES OF RANGE AND STOPPING POWER DATA Type of Maximum Source of data used p a r t i c l e energy (MeV) Range Stopping power Proton 100 N o r t h c l i f f e and S c h i l l i n g ( l l ) and B i c h s e l (12) From equation (3) Alpha 50 N o r t h c l i f f e and S c h i l l i n g ( l l ) From equation (3) 1 I LB ion 30 N o r t h c l i f f e and S c h i l l i n g ( l l ) From equation (3) Deuteron 50 Scaled from proton data T r i t o n 50 Scaled from proton data 3He ion 50 Scaled from alpha data Pi-meson 100 Scaled from proton data - 15 -12 MeV.) Values of energy f o r the s p e c i f i c ranges used i n the c a l c u l a t i o n s of t h i s thesis were then obtained e i t h e r from empirical equations f i t t e d to the selected data or by numerical i n t e r p o l a t i o n . To obtain data f o r energies l e s s than 0 . 0 1 2 5 MeV/amu'(the lowest value given by N o r t h c l i f f e and S c h i l l -ing) , an empirical equation was f i t t e d to the low energy data i n each case and extrapolated to zero range and energy. Stopping powers f o r the required ranges were found by means of equation (3) from the energy-range data e i t h e r by taking the d e r i v a t i v e of the appropriate empirical equation or by numerical methods of d i f f e r e n t i a t i o n . These stopping powers were p l o t t e d against ranges and the curves smoothed s l i g h t l y , where necessary. The consistency of the range, energy and stopping power data was checked f o r each type of p a r t i c l e by numerical i n t e g r a t i o n of the equation E (1/S) dE (4) i where S i s S ^ as found from equation (3). The range, energy and stopping power data used f o r protons, alpha p a r t i c l e s and boron 11 ions are given i n f u l l i n t a b l e IV.* Energy, range and stopping power data were obtained f o r deuterons and t r i t o n s by s c a l i n g proton data and f o r 3He ions by s c a l i n g alpha p a r t i c l e data. The equation used to scale stopping powers i s derived from equa-* Most of the integrations of t h i s t h e s i s were c a r r i e d out numerically using Simpson's r u l e . The range values i n table IV and following tables were chosen to make po s s i b l e integrations of adequate accuracy and, f u r t h e r , to permit several sequential i n t e g r a t i o n s . Table IV as used f o r the c a l -culations i s given i n f u l l as an example but tables V and VI and several of the following tables are abridged versions of the f u l l tables used f o r the c a l c u l a t i o n s — as shown at the bottom of each t a b l e . See appendix B f o r f u r t h e r discussion and explanation. - 16 -TABLE IV RANGE, ENERGY AND STOPPING POWER DATA FOR PROTONS, ALPHA PARTICLES AND 1 : LB IONS Protons Alpha p a r t i c l e s 1 : LB ions Range Energy Stopping Energy Stopping Energy Stopping power power power (cm) (MeV) (MeV/cm) (MeV) (MeV/cm) (MeV) (MeV/cm) 0 0 0 0 0 0 • 0 0.000025 0.0029 174.6 0.0052 3 3 0 . 9 0.0364 1 3 1 9 . 0.000050 0.0082 249.8 0.0157 497-6 0.0722 1 5 4 9 . 0.000075 0 . 0 1 5 2 308.0 0 . 0 2 9 8 631.7 0.1138 1775. 0.000100 0.0236 3 5 7 . 3 0.0471 748.2 0.1609 1 9 9 3 . 0 . 0 0 0 1 2 5 0.0331 4oi.o 0.0671 853.2 0.2134 2206. 0.000150 0.0436 440.6 0 . 0 8 9 7 9 4 9 . 8 0.2712 2415. 0.000175 0.0551 477.2 0.1146 1040. 0 . 3 3 4 1 2621. 0.000200 0 . 0 6 7 4 511.3 0.1417 1125. 0.4022 2826. 0.000225 0.0800 5 3 2 . 0 . 1 7 0 8 1206. 0.4754 3 0 2 8 . 0.000250 0.0960 549. 0.2019 1 2 8 3 . 0 . 5 5 3 6 3 2 3 0 . 0.000275 0.1090 5 6 6 . 0 . 2 3 4 9 1 3 5 7 . 0 . 6 3 6 9 3 4 3 1 . 0.000300 0 . 1 2 3 0 5 8 3 . 0.2697 1428. 0 . 7 2 5 2 3 6 3 1 . 0 . 0 0 0 3 2 5 0 . 1 3 8 0 600. 0 . 3 0 6 3 1 4 9 7 . 0.8184 3 8 3 1 . 0 . 0 0 0 3 5 0 0 . 1 5 2 0 616. 0 . 3 4 4 6 1564. 0.9167 4 0 3 1 . 0 . 0 0 0 3 7 5 0.1680 6 3 3 . 0 . 3 8 4 5 1629. 1.0200 4 2 3 1 . 0 . 0 0 0 4 0 0 0.1840 6 4 5 . 0.4260 1692. 1.1283 4432. 0.00045 0 . 2 1 8 0 666. 0 . 5 1 3 6 1813. 1 . 3 5 9 9 4 8 3 3 . 0.00050 0.2520 670. 0.6072 1929. I.6116 5 2 3 6 . 0.00055 0 . 2 8 5 0 661. 0.7065 2 0 4 0 . 1.8886 5 8 6 3 . 0.00060 0.3180 6 4 5 . 0.8112 2148. 2.191 6236. 0 . 0 0 0 6 5 0 . 3 4 9 0 634. 0.9212 2251. 2 . 5 1 2 6 5 9 9 . 0.00070 0 . 3 8 1 0 625. 1.0380 2 3 0 4 . 2 . 8 5 1 6954. 0.00075 0.4120 612. 1.1550 2325. 3.207 7 3 0 2 . 0.00080 0 . 4 4 3 0 595. 1.2720 2320. 3.581 7643. 0.00100 0.5520 525.0 1 . 7 2 8 0 2 1 9 5 . 5.044 6629. 0.00120 0.6530 4 6 8 . 3 2.150 2010. 6.375 6660. 0 . 0 0 1 4 0 0.7393 4 1 3 . 3 2.532 1 8 0 5 . 7 - 7 0 3 6617. 0.00160 0.8183 379.3 2.872 1615. 9.019 6529. 0.00180 0.8910 348.6 3.178 1480. 10.031 6413. 0.00200 0.9581 323.0 3.464 1385. 11.583 6280. 0.00220 1.0204 301.6 3.732 1300. 12.825 6137. 0 . 0 0 2 4 0 1.0790 285.2 3.984 1245. 14.037 5 9 8 9 . 0.00260 1 . 1 3 4 5 272.3 4.230 1215. 15.220 5 8 3 9 . 0.00280 1 . 1 8 7 9 263.7 4.470 1180. 16.373 5689. 0 . 0 0 3 0 0 1 . 2 4 0 0 251.7 • 4.702 1142. 17.496 5 5 4 1 . 0 . 0 0 3 2 0 1.2886 241.0 4.927 1072. 18.589 5 3 9 5 . 0 . 0 0 3 4 0 1.3364 237.1 5.131 994.8 1 9 . 6 5 4 5252. 0.00360 1.3834 233.0 5 . 3 2 5 935.3 20.69 5 1 1 3 . . 0.00380 1.4296 228.9 5.505 862.2 21.70 4 9 7 8 . 0 . 0 0 4 0 0 1.4750 224.9 5.670 813.7 22.68 4 8 4 7 . 0 . 0 0 4 2 5 1 . 5 3 0 6 220.0 5.872 8 0 4 . 0 23.87 4 6 8 8 . 0.00450 1.5850 215.0 6.072 7 9 4 . 3 25.03 4 5 3 6 . 0 . 0 0 4 7 5 1 . 6 3 8 1 210.0 6.269 7 8 4 . 6 26.14 4 3 9 1 . 0.00500 1.6900 205.0 6.464 774.9 27.22 4 2 5 1 . (Continued on next page) - I T -TABLE IV (CONTINUED) RANGE, ENERGY AND STOPPING POWER DATA FOR PROTONS, ALPHA PARTICLES AND 12-B IONS Protons Alpha p a r t i c l e s 12-B ions Range Energy Stopping Energy Stopping Energy Stopping power power power (cm) (MeV) (MeV/cm) .(MeV) (MeV/cm) (MeV) (MeV/cm) 0.00525 1 . 7 4 0 6 2 0 0 . 0 6 . 6 5 7 7 6 5 . 2 28.27 4 1 1 7 . 0.00550 1 . 7 9 0 0 1 9 5 . 0 6.847 755-5 2 9 . 2 8 3 9 8 8 . 0.00575 1 . 8 3 8 1 190 .0 7 . 0 3 5 745.8 30 .26 3 8 6 5 . 0 . 0 0 6 0 0 1.8850 I85.O 7 . 2 2 0 7 3 6 . 1 3 1 . 2 1 3 7 4 7 . 0 . 0 0 6 2 5 1 . 9 3 0 6 1 8 0 . 0 7.403 726.4 O.OO65O 1 . 9 7 5 0 1 7 5 . 0 7 . 5 8 3 716.7 0 . 0 0 6 7 5 2 . 0 1 8 169.9 7 . 7 6 1 706.9 0.00700 2 . 0 6 0 165.9 7 . 9 3 6 6 9 7 . 2 0 . 0 0 7 2 5 2 . 1 0 1 1 6 1 . 0 8 . 110 687-5 0.00750 2.141 158.6 8 . 2 8 0 677.8 0.00775 2 . 1 8 0 1 5 6 . 3 8 . 4 4 8 668 .1 0.00800 2 . 2 1 9 154 .1 8 .614 6 5 8 . 4 O.OO85 2.295 1 5 0 . 0 8.939 6 3 9 . 0 0.0090 2.369 1 4 6 . 2 9 . 2 5 3 6 1 9 . 6 0.0095 2.441 142.7 9.558 6 0 0 . 2 0 . 0 1 0 0 2.512 139-5 9 . 8 5 3 5 8 0 . 8 0.0105 2 . 5 8 1 136.5 1 0 . 1 3 9 5 6 1 . 4 0 . 0 1 1 0 2 . 6 4 9 133.7 10.414 541.9 0.0115 2.715 131 .1 1 0 . 6 8 1 522.5 0 . 0 1 2 0 2 . 7 8 0 128.7 1 0 . 9 3 6 510.4 0 . 0 1 3 0 2.906 1 2 4 . 2 1 1 . 4 3 8 4 9 2 . 8 0.0140 3.028 1 2 0 . 1 1 1 . 9 2 2 476.9 0.0150 3 . 1 4 6 116.5 1 2 . 3 9 2 462.7 0 . 0 1 6 0 3 . 2 6 1 113 .2 12.848 4 4 9 . 7 0.0170 3 . 3 7 3 1 1 0 . 2 1 3 . 2 9 2 437-9 0 . 0 1 8 0 3.482 1 0 7 . 4 13 .724 4 2 7 . 0 0 . 0 1 9 0 3.588 104.9 1 4 . 1 4 6 4 1 7 . 0 0 . 0 2 0 0 3 . 6 9 2 102.5 14.558 407.7 0.0225 3.941 9 7 . 2 9 15.551 3 8 7 . 1 0.0250 4 . 1 7 9 9 2 . 8 4 1 6 . 4 9 6 3 6 9 . 6 0 . 0 2 7 5 4.406 8 8 . 9 8 17.401 354.4 0.0300 4 . 6 2 4 85 .61 1 8 . 2 7 0 341.1 0 . 0 3 2 5 4 . 8 3 4 8 2 . 6 1 1 9 . 1 0 8 329.3 0.0350 5.037 7 9 . 9 4 1 9 . 9 1 7 318.7 0 . 0 3 7 5 5.234 77.52 20.70 3 0 9 - 2 0.0400 5.425 75 .33 21.46 300.5 0.045 5-792 71 .49 22.93 2 8 5 . 4 0.050 6.141 68 .22 24.32 272.4 0.055 6 .475 65.39 25-66 2 6 1 . 2 0 . 0 6 0 6 . 7 9 5 62.90 2 6 . 9 4 2 5 1 . 4 O.O65 7.104 60.70 2 8 . 1 7 242.7 0.070 7.403 58 .74 29.37 234.9 0.075 7 . 6 9 2 56 .94 30.52 227.9 0 . 0 8 0 7 -973 55.35 31.65 221.5 0.085 8.246 53.88 32 .74 215.7 0.090 8.512 52.53 33 .80 210.4 (Continued on next page) - 18 -TABLE IV (CONTINUED) RANGE, ENERGY AND STOPPING POWER DATA FOR PROTONS, ALPHA PARTICLES AND B IONS Protons Alpha p a r t i c l e s 1 ; LB ions Range Energy Stopping Energy Stopping Energy Stopping power power power (cm) (MeV) (MeV/cm) (MeV) (MeV/cm) (MeV) (MeV/cm) 0.095 8.771 51.28 34.84 205.4 0.100 9.025 50.12 35.86 200.8 0.105 9.273 49.05 36.85 196.6 0.110 9.515 48.04 37.82 192.6 0.115 ' 9-753 47.10 38.78 188.9 0.120 9.986 46.22 39.71 185.4 0.130 io .44o 44.61 41.53 178.9 0.140 10.879 43.19 43.29 173.2 0.150 11.304 41.86 45.00 168.0 0.160 11.717 40.67 46.66 163.3 0.170 12.118 39.59 48.27 159.0 0.180 12.509 38.60 49.84 155.1 0.190 12.890 37.68 51.37 • 151.4 0.200 13.263 36.83 52.87 148.0 0.250 15.013 33.35 0.300 16.612 30.76 0.350 18.097 28.72 0.400 19.491 27.06 0.450 20.81 25.68 0.500 22.06 24.51 0.550 23.26 23.49 0.600 24 . 4 l 22.60 0.700 26.60 21.10 0.800 28.64 19.89 0.900 30.58 18.87 1.000 32.42 18.01 1.100 34.18 17.26 1.200 35.88 l 6 . 6 l 1.300 37.51 16.03 1.400 39.08 15.51 1.650 42.82 14.41 1.900 46.31 13.54 2.150 49.60 12.81 2.400 52.73 12.20 2.650 55.71 11.68 2.900 58.57 11.22 3.150 61.32 10.81 3.400 63.98 10.45 3.925 69.29 9.805 4.450 74.29 9.273 4.975 79-04 8.824 5-500 83.57 8.439 6.025 87.91 8.104 6.550 92.09 7.809 7.075 96.12 7.545-7.600 100.0 7.309 - 19 -t i o n ( l ) . By equation ( l ) , the e l e c t r o n i c stopping powers of two p a r t i c l e s having the same v e l o c i t y are pr o p o r t i o n a l to the squares of t h e i r charges. Further, i f E and E, are the k i n e t i c energies of p a r t i c l e s of re s t mass M a b a and r e s p e c t i v e l y , t h e i r v e l o c i t i e s are equal when E a = ( M a / M b ) E b I t follows that Also, the equation used f o r s c a l i n g ranges can be derived from equation (2) i f we assume that the R-^E^) term i n that equation i s n e g l i g i b l e . With t h i s assumption V W V = ( 2 a / z b ) 2 (V M a } X V V V ^ V V ( 6 ) Equation (5) i s accurate only f o r the energy i n t e r v a l In which equation ( l ) i s v a l i d and equation (6) i s accurate only i f R^ i s n e g l i g i b l e i n comparison with R i n equation (2). In f a c t , the s c a l i n g equations were used f o r a l l energies. The u n c e r t a i n t i e s i n the scaled values of S and R at low energies are probably no greater than the uncertainties i n the data which were scaled Table V i s an abridged t a b l e of the range, energy and stopping power data ( i ) f o r deuterons and t r i t o n s obtained by s c a l i n g the proton data of table IV and ( i i ) f o r 3He ions by s c a l i n g alpha p a r t i c l e data. (Data f o r 3He ions derived by s c a l i n g proton data were i n good agreement with the values obtained by s c a l i n g alpha p a r t i c l e data.) The range and stopping power data f o r pi-mesons were scaled from the proton data of ta b l e IV — again using the s c a l i n g equations (5) and (6) to zero energy. An abridged t a b l e of these data i s given i n table VI. - 20 -TABLE V ABRIDGED TABLE* OF RANGE, ENERGY AND STOPPING POWER DATA FOR DEUTERONS, TRITONS AND 3HE IONS Deuterons Tritons 3He ions Range Energy Stopping power Energy Stopping power Energy Stopping power (cm) (MeV) (MeV/cm) (MeV) (MeV/cm) (MeV) (MeV/cm) 0 0 . 0 0 0 1 0 0 . 0 0 0 2 0 0 0 . 0 1 6 5 0 . 0 4 7 1 0 249 .8 3 5 7 . 4 0 0.0134 0 . 0 3 8 3 0 2 0 2 . 8 2 9 0 . 1 0 0 . 0 5 5 6 0 . 1 6 7 3 0 8 8 3 . 8 1 3 2 9 . . 0 . 0 0 0 3 0 0.00040 0 . 0 8 1 2 0 . 1 3 4 9 4 4 0 . 8 5 0 9 . 3 0 . 0 7 0 8 • 0 . 1 0 9 5 3 5 7 - 7 4 1 5 . 2 0 . 3 1 8 6 0 . 5 0 3 3 1 6 8 7 . 1 9 9 8 . 0 . 0 0 0 6 0 0 . 0 0 0 8 0 0 . 2 4 4 6 0 . 3 6 7 8 5 8 3 . 5 644 .5 0 . 2 0 2 5 0 . 3 1 1 5 5 1 1 . 8 5 6 8 . 0 0 . 9 5 1 0 1 . 4 0 9 0 2 3 6 6 . 2 1 6 7 . 0 . 0 0 1 6 0 0.00240 0.8844 1 . 3 0 5 7 5 9 7 . 5 456.O 0 . 8 1 8 2 1 . 3 2 6 8 665.O 5 9 7 . 0 2 . 7 4 7 3 . 6 9 4 1 3 4 8 . 1 0 3 7 -0 . 0 0 3 2 0 0.00400 1 . 6 3 6 4 1 . 9 1 5 8 3 7 8 . 4 3 2 3 . 1 1 . 7 6 8 1 2 . 1 3 3 503 .7 424 .5 4 . 4 2 3 5 . 0 5 0 8 0 4 . 1 7 6 2 . 9 0 . 0 0 5 0 0 0 . 0 0 6 0 0 2 . 2 1 4 2 . 4 8 0 2 7 7 . 6 2 5 1 . 5 2 . 5 2 8 2 . 8 7 2 3 6 7 . 5 3 2 2 . 5 5 . 7 8 7 6 . 4 7 2 7 1 1 . 4 6 5 9 . 8 0 . 0 0 7 0 0 0 . 0 0 8 0 0 2 . 7 2 0 2 . 9 5 0 234 .6 2 2 5 . 4 3 . 1 7 7 3 . 4 5 5 2 8 9 . 3 2 6 8 . 0 7 . 1 0 6 7 . 6 8 9 6 0 8 . 3 5 5 6 . 0 0 . 0 1 0 0 0 . 0 1 2 0 3 . 3 7 9 3 . 7 6 9 2 0 5 . 1 1 8 5 . 0 3 . 9 5 8 4 . 4 2 1 2 3 8 . 2 2 2 4 . 8 8 . 7 1 8 ' 9 . 6 5 6 488 .3 4 5 0 . 6 0 . 0 1 6 0 0 . 0 2 0 0 4 . 4 3 7 5.023 154 .0 139 .5 5 . 2 6 7 6 . 0 0 6 1 9 8 . 1 1 7 1 . 6 1 1 . 3 4 4 1 2 . 8 5 4 3 9 7 . 1 3 5 9 . 9 0.0300 0.0400 6 . 2 9 2 7 . 3 8 2 1 1 6 . 5 1 0 2 . 5 7 . 5 2 9 8 . 8 3 3 139-4 1 2 2 . 7 1 6 . 1 3 1 1 8 . 9 5 1 301 .1 2 6 5 . 3 0 . 0 6 0 0 0 . 0 8 0 0 9 . 2 4 6 10.848 8 5 . 5 9 7 5 - 3 1 1 1 . 0 6 5 1 2 . 9 8 1 1 0 2 . 4 9 0 . 1 3 23 . 7 8 2 7 . 9 4 2 2 2 . 0 1 9 5 . 6 0 . 1 0 0 0 0 . 1 2 0 0 1 2 . 2 7 9 13 . 5 8 8 6 8 . 2 0 6 2 . 8 9 1 4 . 6 9 4 1 6 . 2 6 0 8 1 . 6 1 7 5 . 2 6 31 .66 3 5 . 0 6 1 7 7 . 3 1 6 3 . 6 0 . 1 6 0 0 . 2 0 0 15.942 18.045 55.34 5 0 . 1 1 1 9 . 0 7 7 2 1 . 5 9 6 6 . 2 2 5 9 . 9 7 4 1 . 1 9 46 . 6 8 1 4 4 . 2 130 .7 0 . 4 0 0 0 . 6 0 0 2 6 . 5 2 3 3 . 2 2 3 6 . 8 2 30.75 31 .74 3 9 - 7 5 4 4 . 0 7 3 6 . 8 0 6 8 . 8 2 8 6 . 3 6 9 6 . 3 5 8 0 . 6 1 1 . 0 0 0 1 . 4 0 0 4 4 . 1 1 5 3 . 1 8 2 4 . 5 0 2 1 . 1 0 5 2 . 7 9 63.64 2 9 . 3 2 2 5 . 2 5 * This abridged table contains every FOURTH value of the f u l l table used f o r the c a l c u l a t i o n s of t h i s paper. - 21 -TABLE VI ABRIDGED TABLE* OR RANGE, ENERGY AND STOPPING POWER DATA FOR PI-MESONS Range Energy Stopping power Range Energy Stopping power (cm) (MeV) (MeV/cm) (cm) (MeV) (MeV/cm) 0 0 0 0.00010 0.0540 629.0 0.0600 2.913 26.96 0.00020 0.1065 422.9 0.0800 3.418 23.73 0.00030 0.1433 321.0 0.1000 3.868 21.49 0.00040 0.1723 267.3 0.1200 4.281 19.81 0.00060 0.2205 224.3 0.160 5.022 17.43 0.00080 0.2627 197.5 0.200 5.685 15.78 0.00160 0.3891 135.1 0.400 8.355 11.60 0.00240 0.4874 112.8 0.600 10.468 9.687 0.00320 0.5718 99-25 1.000 13.898 7.719 o.oo4oo 0.6473 89.88 1.400 16.754 6.647 0.00500 0.7327 81.39 2.400 22.81 5.523 0.00600 0.8108 75.05 3.400 27.92 4.773 0.00700 0.8833 70.08 5-500 36.98 3.947 0.00800 0.9513 66.04 7.600 44.82 3.557 0.0100 1.0768 59.81 10.80 55.57 3.193 0.0120 1.1915 55-15 14.00 65.39 2.958 0.0160 1.3980 48.53 18.00 76.79 2.754 0.0200 1.5824 43.94 22.00 87.50 2.607 0.0300 1.9821 36.70 26.00 97.69 2.494 o.o4oo 2.326 32.29 30.00 107.5 2.4o4 * This abridged table contains every SECOND value of the f u l l t a b l e used f o r the c a l c u l a t i o n s . - 22 -3.4 Range Spectra o f Charged P a r t i c l e s Released from D i s i n t e g r a t i o n s  of Oxygen For the c a l c u l a t i o n s described i n the f o l l o w i n g s e c t i o n , i t was necessary to know, f o r each type of charged p a r t i c l e released from nuclear d i s i n t e g r a t i o n s o f oxygen, the number stopping per u n i t distance per negative pion capture as a function of distance from the d i s i n t e g r a t i o n . These range spectra were obtained from the energy spectra o f t a b l e I I . Table II gives the mean number of p a r t i c l e s released per MeV energy i n t e r v a l per p i minus capture f o r d i f f e r e n t i n t e r v a l s of i n i t i a l energy. Let (dN/dE) be the mean number per MeV f o r one type o f p a r t i c l e f o r i n i t i a l energies i n the i n t e r v a l . say, E to E . Let R and R be the ranges corresponding to these energies, m n m n Then (E - E j (cLN/dR) = ( R n _ R I" ) (dN/dE) (7) n m where (dU/dR) i s the mean number of p a r t i c l e s stopping per u n i t distance per negative pion capture within the range i n t e r v a l R^ to B.^. For protons, values of (dN/dR) were c a l c u l a t e d from equation (7) f o r each value of (dN/dE) given i n t a b l e I I . As an example, i n the energy i n t e r v a l from 10 t o 12 MeV (dN/dE) = 4.70 x 10~"2 protons per MeV per TT" capture E =10 MeV R = 0.1203 cm m m E =12 MeV R = O.167O cm n n Interpolated from t a b l e IV. Therefore, (dN/cLR) = mean number of protons stopping/cm between 0.1203 cm and O.I67O cm from d i s i n t e g r a t i o n = 4.70 x IO" 2 x (12 - 10)/(0.l670 - 0.1203) = 2.01 protons per cm per IT capture Each value o f (dN/dR) was p l o t t e d against the range corresponding to the - 23 -mid-energy of the i n t e r v a l ( i . e . , (dN/dR) of the example above was p l o t t e d at a range of 0.143 cm corresponding to an energy of 11 MeV). A smooth curve was drawn through the points to eliminate the s t a t i s t i c a l f l u c t u a t i o n s of the data generated by the Monte Carlo program used to obtain table II and the curve was extrapolated to zero range on the assumption that dN/dR = 0 f o r R = 0. Figure 1 shows the points c a l c u l a t e d from table II and the smooth curve drawn through the points. The value of dN/dR f o r each required range was then determined from the curve. Table VII i s an abridged table of dN/dR values; r e s u l t s f o r protons are given i n column 2. The procedure f o r f i n d i n g dN/dR f o r alpha p a r t i c l e s f o r the required ranges was i d e n t i c a l with that f o r protons. Abridged r e s u l t s are given i n column 6 of table VTI. Guthrie et a l (6) have given a si n g l e energy spectrum f o r a l l heavy r e c o i l n u c l e i . As already stated, these r e c o i l n u c l e i were a l l treated as ions. In t h i s case, the mean dN/dE values given i n table II were p l o t t e d versus the mid-interval energies and a smooth curve was drawn through the points. The curve was extrapolated to zero energy assuming dN/dE = 0 f o r E = 0. The value of dN/dE was then read from the curve f o r energy E corresponding to each•required range and the value of dN/dR f o r t h i s range was c a l c u l a t e d from dN/dR = (dN/dE) x S (8) where the values of E and S are given i n Table IV. Every fourth value of dN/dR f o r 12-B i s tabulated i n column 7 of the abridged table VII. A smoothed dN/dE versus E curve was obtained f o r the s i n g l e spectrum given i n table I I f o r deuterons, t r i t o n s and 3He n u c l e i by the same procedure as f o r ^ B . The three p a r t i c l e s were then separated by assuming - ns -- 25 -that the proportions of the p a r t i c l e s were constant i n every energy i n t e r v a l and, therefore, equal to the proportions of the t o t a l number of p a r t i c l e s as given i n table I. From table I , Number of deuterons = 0.210 per d i s i n t e g r a t i o n Number of t r i t o n s = O .O65 per d i s i n t e g r a t i o n Number of 3He = 0.035 per d i s i n t e g r a t i o n T o t a l number = 0.310 per d i s i n t e g r a t i o n Therefore, the f r a c t i o n of each p a r t i c l e i s as follows: f ( 2 H ) = 0.210/0.310 = 0.611 f ( 3 H ) = 0.065/0.310 = 0.210 f( 3He) = 0.035/0.310 = 0.113 Then, the number stopping per centimetre f o r any given p a r t i c l e can be found by introducing the appropriate f r a c t i o n into equation (8), i . e . , dN/dR = (dN/dE) x S x f (9) For example, to f i n d the value of dN/dR for deuterons f o r R = 0.040 cm: Deuteron energy f o r range of 0.040 cm = 7-38 MeV (Table V) Stopping power S f o r t h i s range = 102.5 MeV/cm (Table V) dN/dE f o r 7-38 MeV = 1.943 x 10~2/MeV/ir~ capture (from smoothed curve) f ( 2 H ) = 0.677 Therefore, No. of deuterons stopping per cm at 0.040 cm from d i s i n t e g r a t i o n = dN/dR = 1.943 x 10~ 2 x 102.5 x 0.677 = 1.348/cm/Tr" capture The values of dN/dR for. deuterons, t r i t o n s and 3He ions are tabulated i n columns 3, 4 and 5» r e s p e c t i v e l y , of table V I I . A check was made f o r each type of p a r t i c l e to ensure that the t o t a l number and t o t a l energy a f t e r smoothing the dN/dR (or dN/dE) curve d i d not d i f f e r s i g n i f i c a n t l y from the t o t a l number and t o t a l energy as given by - 26 -TABLE VII* NUMBER OF CHARGED DISINTEGRATION PRODUCTS STOPPING PER CM, dN/dR, AS A FUNCTION OF DISTANCE FROM THE DISINTEGRATION Range (cm) dN/dR (number stopping per cm per IT capture) Protons Deuterons Tritons 3He ions Alphas 1 *B ions 0 0 0 0 0 0 0 O.OOOIO 0.500 0.0305 0.0063 0.0551 8.50 311.5 0.00020 1.07 0.1146 0.0238 0.228 18.75 508. 0.00030 1.67 0.249 0.0517 0.522 30.60 627. 0.00040 2.24 0.429 0.0895 0.941 44.75 524. 0.00060 3.67 0.849 0.1942 1.964 81.20 500. 0.00080 5.90 1.364 0.320 2.445 127.3 • 459-0.00160 13.45 2.798 0.904 2.483 136.2 114.7 0.00240 16.50 2.912 1.197 2.244 102.9 22.70 0.00320 18.16 2.848 1.241 1.857 84.30 6.74 o.oo4oo 19.23 2.709 1.206 1.800 72.60 2.42 0.00500 19.81 2.625 1.190 1.673 61.05 0.750 0.00600 19.30 2.589 1.135 1.518 52.40 0 0.00700 17.85 2.571 1.079 1.357 45.05 0.00800 16.48 2.599 1.044 1.197 38.63 0.0100 14.33 2.548 O.986 0.976 29.40 0.0120 12.64 2.420 0.965 0.835 23.15 0.0160 10.42 2.133 0.870 0.631 15.95 0.0200 8.94 1-971 0.746 0.491 11.28 0.0300 6.81 1.617 O.563 O.283 5.70 0.0400 5.48 1.348 0.452 0.177 3.25 0.0600 3.99 0.983 0.310 0.0837 T.440 0.0800 3.15 0.753 0.226 0.0438 0.730 0.1000 2.59 0.592 0.1687 0.0247 0.415 0.1200 2.16 0.475 0.1294 0.0149 0.245 0.160 I.560 0.319 0.0810 0.0054 0.0600 0.200 1.183 0.224 0.0539 0.0018 0 o.4oo 0.452 0.0593 0.0113 0 0.600 0.249 0.0209 0.0031 . 1.000 0.1215 0.0037 0 i.4oo 0.0750 0 2.400 0.0380 3.400 0.0253 5.500 0.0104 7.600 0 * This abridged table contains every FOURTH value of the f u l l t a ble used for the c a l c u l a t i o n s . - 27 -Guthrie et a l i n table I. For any one type of p a r t i c l e T o t a l number per TT - capture = / (dN/dR) dR (10) where the smoothed values of dN/dR (as given i n the f u l l table corresponding to the abridged table VII) are used i n the in t e g r a t i o n and R^ i s the maximum range of the p a r t i c l e released. A l s o , the t o t a l energy imparted to one type of p a r t i c l e i s given by the i n t e g r a l where the dN/dR values are as i n equation (10) and E i s the i n i t i a l k i n e t i c energy of a p a r t i c l e of range R as given i n table IV or V. The r e s u l t s of the integrations of equations (10) and ( l l ) are compared i n table IX with the data of table I. The number of each type of p a r t i c l e released per p i minus capture as determined from equation (10) i s i n s a t i s f a c t o r y agree-ment with table I. The agreement on energy released i s l e s s s a t i s f a c t o r y but i t i s to be noted that t o t a l i n i t i a l energy of one type of p a r t i c l e c a l c u -l a t e d from the spectrum of table I I depends on an a r b i t r a r y assumption regarding the d i s t r i b u t i o n of p a r t i c l e s within each energy i n t e r v a l . For t h i s reason, the t o t a l i n i t i a l energy as calculated from equation ( l l ) does not n e c e s s a r i l y agree with the value of table I. 3.5. Energy Absorption per Centimetre Thickness of Spherical S h e l l as a  Function of Distance from the Nuclear D i s i n t e g r a t i o n charged p a r t i c l e s released i n nuclear d i s i n t e g r a t i o n s , the average energy absorption per centimetre thickness of sp h e r i c a l s h e l l per TT~ capture was calc u l a t e d as a function of distance from the d i s i n t e g r a t i o n . This was determined f o r each type of p a r t i c l e from T o t a l i n i t i a l k i n e t i c energy per TT capture = / E (dN/dR) dR (11) As a step i n c a l c u l a t i n g the absorbed dose at a point due to - 2 8 -(a) the energy loss per centimetre for given range (i . e . , stopping power for given range from table IV or V), (b) the number of particles stopping per centimetre per TT - capture at given distance (dN/dR from table VII). Figure 2(a) shows (qualitatively only) the change in stopping power with dis-tance from the disintegration for a given type of particle of path length R. The stopping power S of this particle at distance r from the disintegration ' z P is a function of the residual path length z where z = R - r ( 1 2 ) P Figure 2(b) shows, for the same type of particle, the number stopping per centimetre per IT - capture as a function of distance from the disintegration. The number of particles of this type stopping between R and (R + dR) i s , therefore, (dN/dR) dR and, since each of these particles dissipates S^ MeV per cm at distance r , their contribution to the energy absorption per centimetre at r^ is S (dN/dR) dR z To find the to t a l energy absorption per centimetre at r due to the one type of particle, we must add the contributions of a l l particles of this type passing r , i.e., Energy absorption per cm per TT~ capture at distance r R m S (dN/dR) dR ( 1 3 ) z Since the particles a l l travel radially from the point of disintegration, (dT/dr) i s the energy absorption per unit thickness of spherical shell per rp - 2 9 -FIGURE 2 DIAGRAM TO ILLUSTRATE DERIVATION OF EQUATION ( 1 3 ) - 30 -TT~ capture at radius rp>* The value of (dT/dr)™ was c a l c u l a t e d f o r each type of p a r t i c l e f o r P selected values of r by i n t e g r a t i o n of equation (13) by means of Simpson's r u l e (see appendix B). The values of r p f o r which the i n t e g r a l was evaluated for each type of p a r t i c l e were the same as the ranges for which data are given i n tables IV or V ( f u l l t a b l e , not the abridged v e r s i o n ) . Also, the r e s i d u a l ranges z f o r which the integrands were evaluated f o r each i n t e g r a l were the same as the ranges i n tables IV or V and, therefore, a l l required values of S z were given i n those t a b l e s . The value of (dN/dR), however, i s a function of R and since, from equation (12), R = r p + z not a l l the required values of (dN/dR) were tabulated i n t a b l e VII ( f u l l v e r s i o n ) . Values not included i n the table were found by i n t e r p o l a t i o n (using an i n t e r p o l a t i o n subroutine a v a i l a b l e on the computer used to carry out the i n t e g r a t i o n s ) . Table VIII gives (dT/dr) f o r each type of p a r t i c l e as a function P of distance from the d i s i n t e g r a t i o n . Column 8 i s the sum of (dT/dr) f o r P a l l types, i . e . , i t i s the sum of columns 2 to 7 i n c l u s i v e . Column 8 i s , therefore, the t o t a l energy absorbed per TT" capture per centimetre t h i c k -ness of s p h e r i c a l s h e l l at distance r p from the d i s i n t e g r a t i o n due to a l l the charged d i s i n t e g r a t i o n products. Since (dT/dr) i s the energy absorption per u - capture per u n i t * In t h i s paper, E i s used to denote the k i n e t i c energy of a p a r t i c l e and T to denote energy absorbed i n the medium. The symbol dE/dR i s the energy d i s s i p a t e d by a p a r t i c l e per centimetre of track ( i . e . , the stop-ping power) where R i s measured along the track. The quantity dT/dL (where L i s length) i s also measured i n MeV per centimetre but the i n t e r -p r e t a t i o n depends on the context. - 31 -TABLE VIII ENERGY ABSORBED PER CENTIMETRE THICKNESS OF SPHERICAL SHELL, dT/dr, AS A FUNCTION OF DISTANCE r FROM THE DISINTEGRATION r P dT/dr (MeV/cm per TT" " capture) (cm) *H 2H 3H 3He '•He n B T o t a l 0 0.00010 0.00020 90.98 91.66 92 .33 23.80 23.93 24.05 10.11 10.16 10.20 21.21 21.53 21.77 956.4 968.5 975-5 3481.0 3204.8 2864.0 4583.5 4320.6 3987.9 0.00030 0.00040 92.98 93.62 24.16 24.25 10.25 10.29 21.91 21.96 9 8 0 . 3 980.4 2510.6 2183.9 3640.1 3314.4 0.00060 0.00080 94.80 95.77 24.40 24.48 10.36 i o . 4 i 21.82 21.50 963.6 934.1 1609.8 1137 .3 2724.8 2223.6 0.00160 0.00240 97.41 97.25 24.35 23.97 10.44 10.29 19.89 18.27 783.7 680.6 258.4 72.15 1194 .2 902.5 0.00320 o.oo4oo 96.24 94.59 23.60 23.28 10.09 9.88 17.11 16.08 6o4.6 541.8 24.26 7-95 775-9 693.6 0.00500 0.00600 91.73 88.31 22.93 22.60 9 . 6 3 9 .38 14.80 13.58 476.5 , 4 2 2 . 0 O.76 0 616.3 555-9 0.00700 0.00800 8 5 . 0 2 82.10 22.27 21.91 9.17 8.96 12.49 11.54 375-7 337.0 504.6 461.5 0.0100 0.0120 77.08 73.03 21.12 20.32 8.59 8.20 10.10 8.93 277.5 233.4 394.4 343.9 0.0160 0.0200 66.71 61.98 18.98 17.81 7.44 6.77 7.12 5.78 172.72 132.41 273.0 224.7 0.0300 o.o4oo 53.33 47.13 15.33 13.36 5.59 4.72 3.65 2.46 76.34 48 .29 154.24 115.96 0.0600 0.0800 38.95 33.27 10.47 8.43 3.49 2.68 1.259 O.692 23.61 12.57 77.78 57.64 0.1000 0.1200 28.99 25.61 6.92 5.76 2.11 1.686 0.402 0.240 7.01 3.76 45.43 37.06 0.160 0.200 20.57 17.14 4.13 3.07 1.132 0.796 0.083 0.019 0.550 0 26.46 21.02 0.400 0.600 9 .21 6.36 0.929 0.344 O.185 0.049 0 10.32 6.75 1.000 i . 4 o o 3-95 2.85 0.048 0 0 4 .00 2.85 2.400 3.400 I . 6 7 3 1.068 1.673 1.068 5.500 7.600 0.319 0 0.319 0 * This abridged table contains every FOURTH value of the f u l l t able used f o r the c a l c u l a t i o n s . - 32 -thickness of s p h e r i c a l s h e l l , i t follows that the t o t a l energy absorption i s given by PR m T o t a l energy absorption per TT capture = / (dT/dr) dr (14) J 0 where R^ i s the maximum range of the d i s i n t e g r a t i o n products. Equation (l4) was evaluated f o r each type of charged d i s i n t e g r a t i o n product using values of (dT/dr) as given i n table V I I I . The r e s u l t s are given i n column 6 of table IX. For each type of p a r t i c l e , the value i n column 6 should be i n close agreement with the t o t a l k i n e t i c energy l i b e r a t e d per stopping pion as given i n column 5 since the two values are c a l c u l a t e d from the same data. As pointed out previously* columns 5 and 6 do not n e c e s s a r i l y agree with column 4. TABLE IX NUMBER AND ENERGY OF EACH TYPE OF CHARGED PARTICLE RELEASED PER TT - CAPTURE: COMPARISON OF TABLE I WITH RESULTS OF INTEGRATION OF EQUATIONS ( 1 0 ) , ( l l ) AND (l4) Number/IT capture Energy/IT - capture (MeV) Type of Equation ( 1 0 ) Equation ( l l ) Equation ( l 4 ) p a r t i c l e Table I PR m / (dN/dR) dR Jo Table I PR m / E (dN/dR) dR Jo / (dT/dr) dr Jo Proton 1 . 2 5 1 . 2 5 0 3 2 0 . 0 3 1 9 . 5 0 4 1 9 . 5 0 9 Deuteron 0 . 2 1 0 0 . 2 0 5 2 2.1+5 2.304 2.304 T r i t o n 0 . 0 6 5 O .O637 0 . 6 8 0.714 0.714 "^ He ions 0 . 0 3 5 .0.0343 0.48 0 . 3 8 5 0 . 3 8 5 Alpha 1 . 0 8 0 1 . 0 6 8 5 1 0 . 6 2 1 0 . 2 7 7 1 0 . 2 7 1 Heavy r e c o i l (taken as ^ B ) 0 . 6 3 1 0 . 6 2 8 8 2 . 4 2 2 . 4 4 4 2 . 4 3 1 Totals 3 6 . 6 8 3 5 . 6 2 8 3 5 . 6 1 4 - 33 -4. PHYSICAL DEPTH DOSE DISTRIBUTION DUE TO  A MONOENERGETIC BEAM OF NEGATIVE PIONS The absorbed dose d i s t r i b u t i o n was calculated f i r s t f o r a beam of negative pions a l l having the same i n i t i a l energy. As already stated, the beam was assumed to be uncontaminated, i n i t i a l l y p a r a l l e l and uniform i n i n t e n s i t y over the f i e l d considered. I t was further assumed that the number of pions per square centimetre was large enough that the dose could be considered constant i n any plane perpendicular to the d i r e c t i o n of the beam. The mean path length of monoenergetic pions has been defined as the "range". The actual path lengths are d i s t r i b u t e d approximately normally about the mean. I t was assumed f i r s t that a l l the pions t r a v e l exactly the range, a f t e r which "range s t r a g g l i n g " was introduced. 4.1. Absorbed Dose D i s t r i b u t i o n Neglecting Range Straggling On the assumption that a l l the pions t r a v e l the range R^, a l l the dis i n t e g r a t i o n s occur at depth R^ i n the medium (see fi g u r e 3) . Then, the dose contributions at d i f f e r e n t depths are as follows. (a) In a plane at depth A, where (R^ - A) i s greater than the maximum range of the charged d i s i n t e g r a t i o n products, the dose i s e n t i r e l y due to primary pions. (b) .In a plane at depth C, where (C - R^) i s l e s s than the maximum range of the charged d i s i n t e g r a t i o n products, the dose i s due to d i s i n t e g r a t i o n products only. (c) In a plane at depth B, where ( R ^ - B ) i s le s s than the maximum range of charged d i s i n t e g r a t i o n products, both primary pions and d i s i n t e g r a -t i o n products contribute to the dose. The absorbed dose at any depth due to primary pions depends on the number of pions per cm2 and on the stopping power of the medium for these pions. Let 2 F = number of pions per cm x = distance from the surface i n the d i r e c t i o n of the beam. Then, 2 Energy absorbed i n an element of thickness dx and 1 cm cross-s e c t i o n perpendicular to the beam = F S„ dx Therefore, 3 Energy ab sorbed per cm — F D i v i d i n g both sides of t h i s equation by the density p of the medium gives Absorbed dose due to primary pions = = F S^/p (15) The average absorbed dose due to the charged d i s i n t e g r a t i o n prod-ucts at any distance r from a nuclear d i s i n t e g r a t i o n follows d i r e c t l y from the energy absorption per unit distance, dT/dr, where dT/dr i s the t o t a l f o r a l l types of charged d i s i n t e g r a t i o n products as given i n column 8 of table VIII. Energy absorbed per TT - capture i n a s p h e r i c a l s h e l l of radius r and thickness dr = (dT/dr) dr Volume of s p h e r i c a l s h e l l = U n r 2 dr Averaging over a large number of d i s i n t e g r a t i o n s , the energy absorption i s uniformly d i s t r i b u t e d over the s h e l l . Therefore, Energy absorption per cm3 per TT - capture at distance r • + +• 1 dT , 1 dT , from d i s i n t e g r a t i o n = W 2 d r x — dr = r ^ r ^  (16) To f i n d the dose at P (see figure U) at a distance X from the plane of the d i s i n t e g r a t i o n s , i t i s necessary to sum the contributions from a l l the d i s i n t e g r a t i o n s which can contribute to the dose at P. Consider an - 35 -D i r e c t i o n of. pion beam - A - H RTT ~ — r — ^ - i -X-- x T Surface of water -XP Plane of nuclear d i s i n t e g r a t i o n s FIGUEE .3 DIAGRAM TO ILLUSTRATE DOSE CONTRIBUTIONS AT DIFFERENT DEPTHS FIGURE k CONTRIBUTIONS OF NUCLEAR DISINTEGRATIONS IN PLANE AT DEPTH R^ TO DOSE AT P - 36 -annular r i n g of radius a and .width da i n the plane of the di s i n t e g r a t i o n s such that a l l parts of the r i n g are at the same distance r from P. Then r 2 = a 2 + X 2 2 Since there are F pions stopping per cm , Number of d i s i n t e g r a t i o n s w i t h i n annulus = F x 2Tra da = F x 2-rrr dr Therefore, from equation ( l 6 ) , 3 Energy absorption per cm at P due to t h i s annulus 1 dT _ • . F dT , *r — x F x 2irr dr = — — dr 4Trr^ dr 2 r dr Div i d i n g by the density of the medium, 1 F dT Absorbed dose at P due to t h i s annulus = — — — dr p 2 r dr Therefore, / 1 F dT Absorbed dose at P due to a l l d i s i n t e g r a t i o n s = D . = / — — — dr ( 1 7 ) b st / p 2 r dr J x where R^ i s the maximum range f o r any charged d i s i n t e g r a t i o n product ( 7 - 6 cm fo r 100 MeV protons). I t i s to be noted that the absorbed dose due to nuclear d i s i n t e g r a t i o n s i s symmetrical about the plane of the d i s i n t e g r a t i o n s . From equations ( 1 5 ) and ( 1 7 )» the absorbed dose at P due to primary pions and charged d i s i n t e g r a t i o n products i s f Rm D = D + D i f dr ( 1 8 ) TT st p p / 2 r dr J x (where one or other of the terms on the r i g h t side may be zero depending on the p o s i t i o n of P). The absorbed dose per pion per cm2 i s , therefore, D DTT D s t V i / I d T , F = -F + T - = T + p / i r - a 7 d r ( l 9 ) J x The energy absorbed per unit volume per pion per unit area due to - 37 -charged d i s i n t e g r a t i o n products w i l l be denoted by dT/dX and, from equa-t i o n ( 1 7 ) 5 i s defined by TR d T D s t / 1 dT , , > dX = P — = / 2 7 d r - d r ( 2 0 ) i x Using t h i s notation, equation ( 1 9 ) becomes ^ = - + ^ ^ ( 2 1 ) F p • p dX V ' The stopping power i s a function of the r e s i d u a l range of the pions and becomes zero f o r any depth greater than R^ (figure 3 ) . The quantity dT/dX i s a function of the distance X from the plane i n which the pions stop, i s symmetrical about t h i s plane and i s equal to zero f o r X greater than the maximum range R^ of the d i s i n t e g r a t i o n products. The average t o t a l k i n e t i c energy of the charged p a r t i c l e s released per negative pion capture i n oxygen as estimated by Guthrie et a l by means o t h e i r Monte Carlo program i s about 3 6 . 7 MeV (see table I) and as c a l c u l a t e d i n t h i s t h e s i s from smoothed spectra from table II i s about 3 5 . 6 MeV. Both values are appreciably greater than estimates based on experimental r e s u l t s -see table X below f o r comparison. I t i s , therefore, u s e f u l to rewrite equation ( 2 l ) as F p p dX With Wt = 1 , the t o t a l k i n e t i c energy assumed to be released i n charged p a r t i c l e s per negative pion capture i s 3 5 . 6 2 MeV (see table IX). I f Wt = 0 . 8 1 4 , t h i s becomes 2 9 . 0 MeV which, on the basis of experimental r e s u l t appears to be a more probable value. I f Wt = 0 , then equation ( 2 2 ) gives the absorbed dose per pion per cni 2 due to the primary pion contribution only. - 38 -TABLE X KINETIC ENERGY OF CHARGED PARTICLES FROM TT CAPTURE IN X 0 : VALUES FROM DIFFERENT SOURCES Source Method K.E. of charged d i s i n t e g r a t i o n pro-ducts per TT - capture Reference Fowler and Perkins Analysis of tracks i n photographic emulsions About 30 MeV (1 ) Fowler Do 29 MeV ( 2 ) Fowler and Mayes Do 2 7 .4 MeV ( 5 ) B a a r l i Adopted to f i t experimental depth dose measurements About 20 MeV (13) S u l l i v a n and B a a r l i Analysis of depth dose measurements 1 9 - 6 MeV (14) Curtis and Raju Calculated to f i t experimental depth dose measurements 29 MeV ( 1 5 ) Holt and Perry (Not stated i n reference) 3 1 . 6 MeV (16) Guthrie et a l Monte Carlo program 3 6 . 6 8 MeV ( 6 ) Present c a l c u l a t i o n s 3 5 . 6 2 MeV Except f o r X = 0 , dT/dX can be calculated by numerical i n t e g r a t i o n of equation ( 2 0 ) using the t o t a l dT/dr values from column 8 , table VIII. The r e s u l t s , divided by the density of water (to change to dose u n i t s ) , are given i n column 6 of "table XI. A value of dT/dX f o r X = 0 was obtained as follows. T o t a l energy absorbed from charged p a r t i c l e s per TT~ capture 'Rm f\ (dT/dr) dr = 2 / (dT/dX) dX ( 2 3 ) - 39 -where the f a c t o r 2 appears i n front of the second i n t e g r a l because the d i s -t r i b u t i o n of energy i s symmetrical on e i t h e r side of the plane at depth R^ (fig u r e 3). The value of the f i r s t i n t e g r a l as given i n column 6 of table IX i s 35.6lU MeV. The value of dT/dX f o r X = 0 was determined so that equa-t i o n (23) was s a t i s f i e d . Table XI includes a l l the data required to p l o t curves of "Absorbed dose per pion per cm2" versus "Depth i n water" for monoenergetic beams of negative pions, range s t r a g g l i n g neglected, (a) due to the primary pion contribution only, (b) assuming ( i ) 29.0 MeV k i n e t i c energy ( i i ) 35-6 MeV k i n e t i c energy of charged p a r t i c l e s released per nuclear d i s i n t e g r a t i o n . Column 3 i s the mass stopping power S^/p of water f o r pions having the r e s i d u a l range given i n column 1. This i s the absorbed dose per pion per cm2 due to the primary pion c o n t r i b u t i o n only. Column h i s the dose con-t r i b u t i o n per pion per cm2 due to charged p a r t i c l e s from the nuclear d i s i n -t egration assuming 29.0 MeV k i n e t i c energy of charged p a r t i c l e s , i . e . , c o l -umn 4 i s ^ w h e r e dT/dX i s c a l c u l a t e d from equation (20). This con-p dX t r i b u t i o n i s , of course, symmetrical about the plane i n which the pions stop. Column 55 which i s the sum of columns 3 and k, gives, therefore, the t o t a l absorbed dose per negative pion per cm assuming 29-0 MeV " s t a r s " . C o l - . umns 6 and 7 correspond to columns h and 5 except Wt = 1.0 was used i n equa-t i o n (22), i . e . , columns 6 and 7 are doses assuming 35-6 MeV " s t a r s " . In table XI, negative r e s i d u a l ranges are shown i n column 1 f o r positions beyond the plane i n which the pions stop. These negative values of z are required i n the next section.--RO-TABLE XI** ABSORBED DOSE PER PION PER CM2, D/F, STRAGGLING NEGLECTED (a) DUE TO PRIMARY PIONS ONLY AND (b) ASSUMING ( i ) 29.0 MeV "STARS", ( i i ) 35-6 MeV "STARS" Residual range z 1 Distance from plane of " s t a r s " X 2 D/F due to primary pions only = S> 3 Wt = 0.814 (29.0 MeV star) Wt = 1.0 (35.6 MeV star) Wt dT* P X dX 4 D/F 5 wt er* P X dX 6 D/F 7 I f»Tf\ 1 (MeV-cm2 (MeV-cm2 (MeV-cm2 (MeV-cm2 (MeV-cm2 V t-iri; \ cmj per g) per g) per g) per g) per g) 30.00 2.404 2.4o4 2.4o4 26.00 2.494 2.494 2.494 22.00 2.607 2.607 2.607 18.00 2.754 2.754 2.754 14.00 2.958 2.958 2.958 10.80 3.193 3.193 3.193 7.600 7.600 3.557 0 3-557 0 3.557 5.500 5-500 3.947 0.017 3.964 0.021 3.968 3.400 3.400 4.773 0.150 4.923 0.184 4.957 2.400 2.400 5.523 0.344 5.867 0.422 5.945 1.400 1.400 6.647 0.830 7.477 1.020 7.667 1.000 1.000 7.719 1.294 9.013 1.590 9.309 0.600 0.600 9.687 2.386 12.073 2.931 12.618 0.400 0.400 11.601 3.770 15•371 4.631 16.232 0.200 0.200 15.788 8.032 23.82 9.867 25.65 0.160 0.160 17.434 10.167 27.60 12.490 29.92 0.1200 0.1200 19.813 13.851 33.66 17.016 36.83 0.1000 0.1000 21.49 16.900 38.39 20.76 42.25 0.0800 0.0800 23.73 21.56 45.29 26.49 50.22 0.0600 0.0600 26.96 29.44 56.40 36.16 63.12 0.0400 o.o4oo 32.29 45.24 77-53 55-57 87.86 0.0300 0.0300 36.70 60.96 97.66 74.89 111.59 0.0200 0.0200 43.94 91.93 135.87 112.93 156.87 0.0160 0.0160 48.53 114.47 163.OO 140.62 189.15 0.0120 0.0120 55.15 150.52 205-7 184.91 240.1 0.0100 0.0100 59.81 177.86 237.7 218.5 278.3 0.00800 0.00800 66.04 216.7 282.7 266.2 332.2 0.00700 0.00700 70.08 242.9 313.0 298.5 368.6 0.00600 0.00600 75.05 276.2 351.3 339.3 4i4.4 0.00500 0.00500 81.39 319.7 401.1 392.8 474.2 o.oo4oo o.oo4oo 89.88 379.1 469.0 465.8 555.7 0.00320 0.00320 99.25 445.8 545.1 547.7 646.9 0.00240 0.00240 112.79 543.7 656.5 668.0 780.8 0.00160 0.00160 135.08 714.0 849.1 877.1 1012.2 0.00080 0.00080 197.45 1183.1 1380.6 . 1453.5 1651.O 0.00060 0.00060 224.3 1473.3 1697.6 1810.0 2034. 0.00040 0.00040 267.3 1974.0 224l. 2425. 2692. 0.00030 0.00030 321.0 2382. 2703. 2926. 3247. 0.00020 0.00020 422.9 3012. 3435. 3701. 4124. 0.00010 0.00010 629.0 4191. 4820. 5149. 5778. 0.0 0.0 0 9734. 9734. 11958. 11958. (Continued on next page) - C I -TABLE XI (CONTINUED)** ABSORBED DOSE PER PION PER CM2, D/F, STRAGGLING NEGLECTED (a) DUE TO PRIMARY PIONS ONLY AND (b) ASSUMING ( i ) 2 9 . 0 MeV "STARS", ( i i ) 3 5 - 6 MeV "STARS" Distance D/F due to wt = 0 . 8 1 U Wt = 1 . 0 Residual (29 . 0 MeV star) ( 3 5 . 6 MeV star) -range z from plane primary of " s t a r s " X pions only = Vp Wt dT* P X dX D/F Wt dT* P X dX D/F 1 2 3 1+ 5 6 7 (cm) (cm) (MeV-cm2 (MeV-cm2 (MeV-cm2 (MeV-cm2 (MeV-cm2 per g) per g) per g) per g) per g) - 0 . 0 0 0 1 0 0 . 0 0 0 1 0 1+191. 1+191. 51U9. 51^9. - 0 . 0 0 0 2 0 0 . 0 0 0 2 0 3012. 3012. 3 7 0 1 . 3 7 0 1 . - 0 . 0 0 0 3 0 0.00030 2 3 8 2 . 2 3 8 2 . 2 9 2 6 . 2 9 2 6 . -0.000*10 o.oooi+o 1 9 7 ^ . 0 1971+.0 21+25. 21+25. - 0 . 0 0 0 6 0 0 . 0 0 0 6 0 11+73.3 1 U 7 3 . 3 1 8 1 0 . 0 1 8 1 0 . 0 - 0 . 0 0 0 8 0 0 . 0 0 0 8 0 1 1 8 3 . 1 1 1 8 3 . 1 11+53.5 11+53.5 - 0 . 0 0 1 6 0 0 . 0 0 1 6 0 711+.0 711+.0 8 7 7 - 1 8 7 7 . 1 -0.0021+0 0.0021+0 5 ^ 3 . 7 5 ^ 3 . 7 6 6 8 . 0 6 6 8 . 0 - 0 . 0 0 3 2 0 0 . 0 0 3 2 0 1+1+5.8 1+1+5.8 5 ^ 7 . 7 5^7-7 -o.ooi+oo o.ooi+oo 3 7 9 . 1 3 7 9 . 1 1+65.8 I+65.8 - 0 . 0 0 5 0 0 0 . 0 0 5 0 0 3 1 9 . 7 3 1 9 . 7 3 9 2 . 8 3 9 2 . 8 - 0 . 0 0 6 0 0 0 . 0 0 6 0 0 2 7 6 . 2 2 7 6 . 2 3 3 9 . 3 339 • 3 - 0 . 0 0 7 0 0 0 . 0 0 7 0 0 21+2.9 21+2.9 2 9 8 . 5 2 9 8 . 5 - 0 . 0 0 8 0 0 0 . 0 0 8 0 0 2 1 6 . 7 2 1 6 . 7 2 6 6 . 2 2 6 6 . 2 - 0 . 0 1 0 0 0 . 0 1 0 0 1 7 7 . 8 6 1 7 7 . 8 6 2 1 8 . 5 2 1 8 . 5 - 0 . 0 1 2 0 0 . 0 1 2 0 1 5 0 . 5 2 1 5 0 . 5 2 I 8 U . 9 1 18I+.91 - 0 . 0 1 6 0 0 . 0 1 6 0 111+.1+7 111+.1+7 11+0.62 11+0.62 - 0 . 0 2 0 0 0 . 0 2 0 0 9 1 . 9 3 9 1 . 9 3 1 1 2 . 9 3 1 1 2 . 9 3 - 0 . 0 3 0 0 0 . 0 3 0 0 6 0 . 9 6 6 0 . 9 6 7I+.89 71+.89 -O.Ol+OO o.oi+oo 1+5.21+ 1+5.21+ 5 5 - 5 7 5 5 - 5 7 - 0 . 0 6 0 0 0 . 0 6 0 0 29.1+1+ 29.1+1+ 3 6 . 1 6 3 6 . 1 6 - 0 . 0 8 0 0 0 . 0 8 0 0 2 1 . 5 6 2 1 . 5 6 26.1+9 26.1+9 - 0 . 1 0 0 0 0 . 1 0 0 0 1 6 . 9 0 0 1 6 . 9 0 0 2 0 . 7 6 2 0 . 7 6 - 0 . 1 2 0 0 0 . 1 2 0 0 13 . 8 5 1 . 13 . 8 5 1 1 7 . 0 1 6 1 7 . 0 1 6 - 0 . l 6 0 0 . 1 6 0 IO . 1 6 7 1 0 . 1 6 7 12.1+90 12.1+90 - 0 . 2 0 0 0 . 2 0 0 8.032 8 . 0 3 2 9 . 8 6 7 9 . 8 6 7 -o.i+oo 0.1+00 3 . 7 7 0 3 . 7 7 0 1+.631 1+.631 - 0 . 6 0 0 0 . 6 0 0 2 . 3 8 6 2 . 3 8 6 2 . 9 3 1 2 . 9 3 1 -1 . 0 0 0 1 . 0 0 0 I.29I+ 1.29k 1 . 5 9 0 1 . 5 9 0 -1.1+00 1.1+00 0 . 8 3 0 0 . 8 3 0 1 . 0 2 0 1 . 0 2 0 -2.1+00 2.1+00 0.31+1+ 0.31+1+ 0.1+22 0.1+22 -3.1+00 3.1+00 0 . 1 5 0 0 . 1 5 0 0.181+ 0.181+ - 5 . 5 0 0 5 . 5 0 0 0 . 0 1 7 0 . 0 1 7 0 . 0 2 1 0 . 0 2 1 - 7 . 6 0 0 7 . 6 0 0 0 0 0 0 Energy i n MeV under curve 1 0 7 . ^ 9 136.1+9 11+3.11 * dT/dX ca l c u l a t e d from equation ( 2 0 ) except the value f o r X = 0 was determined to s a t i s f y equation ( 2 3 ) . ** This abridged table contains every SECOND value of the f u l l t a ble used f o r the c a l c u l a t i o n s . - 42 -Dose data of t a b l e XI are p l o t t e d i n fi g u r e 5 against "Depth i n water" i n units of grams per cm2 (a) f o r primary pions only (column 3) and (b) assuming 35-6 MeV " s t a r s " (column 7 ) • The curves are p l o t t e d f o r pions having a path length of 30 cm i n water with the o r i g i n at the surface. The dose curve assuming 29.0 MeV " s t a r s " ( c o l -umn 5) has not been p l o t t e d since, except f o r the height of the peak,' i t can scarcely be d i s t i n g u i s h e d from the curve p l o t t e d f o r the 35-6 MeV s t a r s . where (D/F) i s from table XI, x i s i n centimetres and the upper l i m i t i s R + R = 3 7 - 6 cm. The r e s u l t f o r each dose curve i s shown i n the l a s t l i n e TT m of table XI. I t i s to be noted that the energy under each curve i s the energy contributed by one pion. Considering only the primary pion contribu-t i o n , the energy i s (a) 1 0 7 . 4 9 MeV i n i t i a l k i n e t i c energy of a pion with a range of 30 cm i n water. Including the energy of charged p a r t i c l e s from the nuclear d i s i n t e g r a t i o n , i t i s (b) 1 0 7 . 4 9 + 2 9 . 0 0 = 1 3 6 . 4 9 MeV assuming 2 9 . 0 0 MeV k i n e t i c energy and (c) 1 0 7 . 4 9 + 3 5 . 6 2 = l 4 3 . l l MeV assuming 3 5 - 6 2 MeV k i n e t i c energy of charged d i s i n t e g r a t i o n products. The energy i n MeV under each dose curve can be found by i n t e g -r a t i o n . '37.6 p (D/F) dx ( 2 4 ) 15 20 Depth i n water (g/cm2) FIGURE 5 ABSORBED DOSE CURVES DUE TO A MONOENERGETIC BEAM OF NEGATIVE PIONS IN WATER, RANGE STRAGGLING NEGLECTED (Path length i n water = 30 cm) — Ui+ — U.2. Absorbed Dose D i s t r i b u t i o n with Range Straggling The absorbed dose c a l c u l a t i o n s f o r a monoenergetic beam of pions were made f i r s t on the assumption that a l l the pions t r a v e l l e d the same distance (the range) i n water. I t was then necessary to introduce the e f f e c t of "range s t r a g g l i n g " . I t i s s u f f i c i e n t l y accurate to assume that the actual path lengths of the pions are d i s t r i b u t e d normally about the mean path length or range. The beam can, therefore, be characterized by the range R^ and the root mean square deviation a.* 2 The procedure f o r f i n d i n g the absorbed dose per pion per cm as a function of depth with s t r a g g l i n g was e s s e n t i a l l y the same as the procedure f o r f i n d i n g the energy absorbed per centimetre, dT/dr, around a nuclear d i s -i n t e g r a t i o n . In each case, i t was necessary to use (a) a curve showing the energy d i s s i p a t e d by (or energy absorbed from) a p a r t i c l e per centimetre path as a function of r e s i d u a l path length and (b) a curve showing the number of p a r t i c l e s stopping per unit distance as a function of p o s i t i o n . The curves required f o r the present problem, corresponding to figures 2(a) and 2(b) for the previous problem, are shown i n figures 6(a) and 6(b). Figure 6(a) i s a curve showing ( q u a l i t a t i v e l y only) the "Absorbed dose per pion per cm , D/F" versus "Depth i n water" f o r pions with a path length x. In f a c t , t h i s curve i s the l a s t x cm of any one of the dose d i s t r i b u t i o n s given i n columns 3, 5 and 7 of ta b l e XI, the choice depending on the case being treated. Figure 6(b) i s the normal d i s t r i b u t i o n of stopping pions around the mean path length R^. The equation of the curve i s * Range s t r a g g l i n g was neglected i n a l l cal c u l a t i o n s f o r charged p a r t i c l e s released from nuclear d i s i n t e g r a t i o n s . This omission, however, makes a n e g l i g i b l e d i f f e r e n c e i n the f i n a l d i s t r i b u t i o n s since, even f o r 100 MeV protons, the root mean square deviation i n path i s only about 0.1 cm. - H5 -FIGURE 6 DIAGRAM TO ILLUSTRATE DERIVATION OF EQUATION (27) - h6 -<JN = 1 r-U - R j 2 / ( 2 a 2 ) , , dx a / 2 l f V ° ; where dW/dx i s the number of pions stopping per u n i t distance at distance x and a i s the root mean square deviation i n path length. This equation i s normalized so the t o t a l area under the curve of f i g u r e 6(b) i s unity. The number of pions stopping between x and (x + dx) i s (dK/dx) x dx Each p a r t i c l e stopping i n t h i s i n t e r v a l contributes (D/F) z to the absorbed dose per pion per cm2 at Xp where z = r e s i d u a l path length = x - Xp ( 2 6 ) Therefore, the pions stopping between x and (x + dx) contribute (D/F) z (dW/dx) x dx to the absorbed dose per pion per cm2 at Xp and, considering a l l the pions which can contribute to the dose at Xp, Absorbed dose/pion/cm 2 at Xj. with s t r a g g l i n g f \ D dN dx dx ( 2 7 ) where D denotes the dose when s t r a g g l i n g i s included. The l i m i t s on the S "GI* i n t e g r a l are shown as zero to i n f i n i t y but, i n f a c t , were used as about (R - 5<?) to (R^ + 5o) since dN/dx i s zero beyond these l i m i t s . I t i s to be noted that the r e s i d u a l path length, z, as defined by equation ( 2 6 ) , i s negative f o r x l e s s than Xp and, f o r negative pions, t h i s case must be considered. Figures 7(a) and 7(b) show the dose contribution at Xp due to the d i s i n t e g r a t i o n products from negative pions which have already stopped at some smaller distance x. 0 i i x Xp Depth (a) "Absorbed dose/pion/cm 2" vs "Depth i n water" Depth (b) "Number of pions stopping/cm" vs "Depth i n water" FIGURE 7 DIAGRAM TO ILLUSTRATE CONTRIBUTION OF PIONS STOPPING AT DEPTH x TO DOSE AT x > x - 1+8 -I t can be seen from figures 7(a) and 7(b) that, i n order to deter-mine the complete dose curve with s t r a g g l i n g f o r negative pions, i t was nec-essary to determine the dose f o r values of x^ up to about x = R + 5a + R p Tf m where R^ Is the range of the pions and R^ i s the maximum range of charged d i s i n t e g r a t i o n products ( 7 - 6 cm for 100 MeV protons). For the dose from primary pions only, i t was not necessary to use values of x greater than x = R + 5a P Tf The absorbed dose per pion per cm2 with s t r a g g l i n g was determined at any desired x^ by numerical i n t e g r a t i o n of equation ( 2 7 ) where the required values of D/F without s t r a g g l i n g were taken from table XI and dN/dx was c a l c u l a t e d from equation ( 2 5 ) . In order to c a l c u l a t e dN/dx, however, i t was necessary to choose a p a r t i c u l a r range R^ and a corresponding value of the root mean square deviation a. The c a l c u l a t i o n s i n t h i s paper were made f o r pions having an i n i t i a l energy of 8 2 . 2 MeV f o r which the following values were used: R^ = 2 0 . 0 cm (see table VI) a = 0 . 5 cm* The values of x, f o r which the integrands of equation ( 2 7 ) were calculated, depended on the value of x^ for which the equation was being integrated. The s e l e c t i o n of x was determined by the more r a p i d l y varying of the two functions (D/F) and dN/dx. (a) For small |z|, ( i . e . , x and x nearly equal), (D/F) varies more p z r a p i d l y than dN/dx. In t h i s case, z and (D/F) values as tabulated i n z table XI were used. The value of x for each z was determined from * The values of a as c a l c u l a t e d by two d i f f e r e n t methods ( 1 , 1 7 ) were and 0.1+9 cm. The value 0 . 5 cm was chosen as a convenient number of the r i g h t order. - 49 -equation ( 2 6 ) and dN/dx f o r t h i s x vas c a l c u l a t e d from equation ( 2 5 ) . (b) For large |z|, dN/dx varies more r a p i d l y than (D/F) . For t h i s case, values of x were selected at i n t e r v a l s of about a/4 over the range from (R - 5o) to (R^ + 5cr) and dN/dx was ca l c u l a t e d f o r each x. (D/F) z f o r the z corresponding to each x was found by i n t e r p o l a t i o n between the values tabulated i n table XI. Case (a) above occurs only f o r - x^)| les s than about 5 a . The values of the absorbed dose per pion per cm2 with s t r a g g l i n g , (D /F), determined from equation ( 2 7 ) are given i n table XII. Data are s"Cr included f o r (a) the primary pion c o n t r i b u t i o n only, (b) the t o t a l of the contributions from the primary pion and the nuc-l e a r d i s i n t e g r a t i o n assuming ( i ) a 2 9 - 0 MeV " s t a r " , ( i i ) a 3 5 . 6 MeV " s t a r " . These data are p l o t t e d i n f i g u r e 8 . The t o t a l energy under each curve of figure 8 i s shown i n the l a s t l i n e of table XII and on the f i g u r e . The energies were c a l c u l a t e d from the equivalent of equation ( 2 4 ) , i . e . , 0 2 8 . 8 Energy i n MeV = / p (D t r / F ) <*x ( 2 8 ) Jo where the upper l i m i t i s the maximum depth i n centimetres f o r which dose contributions are shown i n t a b l e XII. Again, i t i s to be noted that the energy under each curve i s the energy contributed by one pion, i . e . , (a) 8 2 . 2 MeV ( i n i t i a l k i n e t i c energy of a pion with a range of 20 cm i n water), (b,i) ( 8 2 . 2 + 2 9 . 0 ) MeV = 1 1 1 . 2 MeV, ( b , i i ) ( 8 2 . 2 + 3 5 - 6 ) MeV = 1 1 7 . 8 MeV. - 50 -TABLE XII ABSORBED DOSE PER PION PER CM2 DUE TO A BEAM OF 8 2 . 2 MeV NEGATIVE PIONS IN WATER, RANGE STRAGGLING INCLUDED Depth i n water Absorbed dose/pion/cm2 (MeV-cm2/g) *P Primary pions P r i . pions plus P r i . pions plus (cm) only 29-0 MeV " s t a r s " 3 5 . 6 MeV " s t a r s " 0.0 2 . 6 7 6 2 .676 2 . 6 7 6 3.0 2 . 7 9 9 2 .799 2 . 7 9 9 6.0 2 . 9 5 9 2 .959 2 . 9 5 9 8.0 3 . 0 9 5 3 . 0 9 5 3 . 0 9 5 10.0 3 . 2 7 0 3 . 2 7 0 3 . 2 7 0 11.2 3.400 3.400 3.400 12.4 3 . 5 6 1 3 . 5 6 1 3 . 5 6 1 13 .45 3 .734 3 . 7 3 8 3 . 7 3 8 14 .50 3 . 9 5 8 3 .978 3 . 9 8 2 15-55 4 . 2 8 7 4 . 3 5 1 4 . 3 6 5 1 6 . 6 0 4 . 8 0 6 4 . 9 6 9 5 . 0 0 6 1 7 . 1 0 5.149 5 . 3 9 7 5 . 4 5 4 1 7 . 6 0 5 . 584 5 . 9 6 4 6 . 0 5 0 17.84 5 . 8 3 3 6 . 3 0 5 6.413 1 8 . 0 8 6 . 1 2 7 6 . 729 6 . 8 6 7 18.32 6 . 5 0 1 7 . 3 2 0 7 . 5 0 7 1 8 . 5 6 7 . 0 1 2 8 .279 8 . 5 6 8 1 8 . 8 0 7 . 7 2 6 10.003 1 0 . 5 2 4 1 9 . 0 4 8 . 6 4 8 1 3 . 0 5 6 1 4 . 0 6 3 1 9 . 2 8 9 . 6 0 2 1 7 . 7 3 5 1 9 . 5 9 4 1 9 . 5 2 1 0 . 1 6 1 2 3 . 2 7 4 2 6 . 2 7 0 1 9 . 7 6 9 . 7 8 8 27.470 3 1 . 5 1 0 20.00 8.249 2 7 . 8 1 6 32 .287 2 0 . 2 4 5 -902 2 3 . 5 8 4 2 7 . 6 2 4 20.48 3.513 1 6 . 6 2 6 1 9 . 6 2 3 2 0 . 7 2 1 . 7 1 6 9.849 1 1 . 7 0 7 2 0 . 9 6 0 . 6 8 2 5 . 0 9 0 6 . 0 9 7 2 1 . 2 0 O .219 2 . 4 9 6 3 . 0 1 6 21.44 O . 0 5 6 6 1 . 3 2 3 1 . 6 1 2 2 1 . 6 8 0 .0117 0 . 8 3 1 1 . 0 1 8 2 1 . 9 2 0 . 0 0 1 9 0 . 6 0 4 0.742 22 .16 0.0003 0 . 4 7 2 O .580 22 .40 0.0000 0 . 3 8 0 0.466 22 .90 0.248 0 . 3 0 5 23.40 0 . 1 6 3 0 . 2 0 0 2 4 . 4 5 0.0634 0 . 0 7 7 9 25 .50 0 . 0 1 9 9 0 . 0 2 4 4 26.55 0 . 0 0 3 8 0.0047 2 7 . 6 0 0.0003 0 . 0 0 0 4 2 8 . 8 0 0 . 0 0 0 0 0 . 0 0 0 0 Energy i n MeV under curve 8 2 . 2 0 1 1 1 . 2 4 1 1 7 . 8 8 ( f i g u r e 8) bO B o i > O) CM o u ft c o • r l ft (U ft <u to o 1=1 ••S O 3 I vn 10 15 Depth i n water (g/cm 2) FIGURE 8 ABSORBED DOSE CURVES DUE TO A BEAM OF 8 2 . 2 MeV NEGATIVE PIONS IN WATER FOR DIFFERENT ASSUMPTIONS RE KINETIC ENERGY OF CHARGED PARTICLES FROM "STARS" (Data from table XII) - 52 -The method of introducing the e f f e c t of s t r a g g l i n g i s not s t r i c t l y -accurate but the erro r r e s u l t i n g i s probably small. A c t u a l l y , the beam was s p e c i f i e d as monoenergetic, i . e . , a l l pions i n i t i a l l y had the same energy ( 8 2 . 2 MeV i n i t i a l energy f o r a range of 20 cm — see table V I ) ; the range s t r a g g l i n g r e s u l t s from the s t a t i s t i c a l nature of the energy l o s s e s . In f a c t , a l l the (D/F) values used i n evaluating the integrands of equation ( 2 7 ) were taken from table XI and the data i n t h i s table are f o r pions which do not s u f f e r range s t r a g g l i n g . In other words, the d i s t r i b u t i o n of act u a l path lengths about the mean or range was assumed i m p l i c i t l y to r e s u l t from a d i s t r i b u t i o n of i n i t i a l energies about the mean of 8 2 . 2 MeV. For example, the pions which stopped at 18 cm were assumed to have an i n i t i a l energy of 7 6 . 8 MeV and the pions which stopped at 22 cm to have an i n i t i a l energy of 8 7 . 5 MeV. From the l a s t l i n e of table XII i t i s seen that t h i s approximation makes no s i g n i f i c a n t change i n the t o t a l energy under the curve but i t may change the shape of the curve s l i g h t l y . E i t h e r curve (b,i) or ( b , i i ) of figu r e 8 i s the desired p h y s i c a l depth dose d i s t r i b u t i o n produced by a beam of 8 2 . 2 MeV negative pions i n water, the choice depending on the assumption regarding the t o t a l k i n e t i c energy of the charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n at the end of the pion track. Curve (a) showing the primary pion contribution only has been included f o r comparison to show the r e l a t i v e magnitude of the contribution of the charged p a r t i c l e s from the nuclear d i s i n t e g r a t i o n . Curves s i m i l a r to those of fi g u r e 8 are often c a l l e d "Bragg curves". - 53 -5. BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTIONS DUE TO MONOENERGETIC BEAMS OF NEGATIVE PI-MESONS 5.1. Introduction B i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s , both f o r f u l l y oxygen-ated and for anoxic conditions, vere c a l c u l a t e d by introducing b i o l o g i c a l weighting factors into the c a l c u l a t i o n of the p h y s i c a l dose d i s t r i b u t i o n as described i n the l a s t s e c t i o n . The cal c u l a t i o n s may be expected to give us e f u l information on the b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s i n soft t i s s u e since the p h y s i c a l dose d i s t r i b u t i o n s i n water and i n soft t i s s u e w i l l be very s i m i l a r . The d i f f e r e n c e s between the b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u -tions and the absorbed dose d i s t r i b u t i o n a r i s e from the f a c t that the b i o l o g -i c a l response of c e l l s or t i s s u e s to a given p h y s i c a l dose depends on various p h y s i c a l and chemical f a c t o r s . The two factors which are considered i n t h i s paper are (a) the l i n e a r energy t r a n s f e r of the r a d i a t i o n (to be defined l a t e r ) and (b) the oxygen supply of the i r r a d i a t e d b i o l o g i c a l m a t e r i a l . The e f f e c t o f l i n e a r energy t r a n s f e r (abbreviated LET) of the r a d i -a t i o n on the b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n was considered f i r s t . In t h i s c a l c u l a t i o n , i t was assumed that differences i n b i o l o g i c a l response resulted only from differences i n the LET of the r a d i a t i o n , i . e . , a l l other factors were assumed to be unchanged. The e f f e c t of anoxia on the b i o l o g i c a l response of the i r r a d i a t e d material was treated as a second problem. Some of the d e f i n i t i o n s required i n discussing the ca l c u l a t i o n s are introduced i n the next s e c t i o n . - 5h -5.1.1. Relative B i o l o g i c a l E f f e c t i v e n e s s and B i o l o g i c a l l y E f f e c t i v e Dose "Relative b i o l o g i c a l e f f e c t i v e n e s s " (abbreviated RBE) i s a measure of the b i o l o g i c a l e f fectiveness of a given r a d i a t i o n as compared with that of some standard r a d i a t i o n . I t i s defined ( l 8 ) by pgg _ Absorbed dose of standard r a d i a t i o n Absorbed dose of given r a d i a t i o n to produce the SAME BIOLOGICAL EFFECT under the SAME CONDITIONS OF IRRADIATION. In measurements of RBE, i t i s customary to use orthovoltage x-rays (220 or 250 kV) as the standard. The RBE i s very dependent on the b i o l o g i c a l e f f e c t or end-point chosen. The "same conditions of i r r a d i a t i o n " implies that a l l conditions, such as dose ra t e , oxygen supply, etc., must be e s s e n t i a l l y the same f o r the exposures to the given r a d i a t i o n and to the standard r a d i a t i o n . The b i o l o g i c a l l y e f f e c t i v e equivalent of the p h y s i c a l dose i n rads i s c a l l e d the " b i o l o g i c a l l y e f f e c t i v e dose" i n t h i s paper and i s defined as follows: BIOLOGICALLY EFFECTIVE DOSE = ABSORBED DOSE x RBE ( 2 9 ) Where the absorbed dose i s contributed by a mixture of r a d i a t i o n s , each with a d i f f e r e n t RBE, then the b i o l o g i c a l l y e f f e c t i v e dose i s c a l c u l a t e d as B i o l o g i c a l l y e f f e c t i v e dose = 2~.(RBE). X D. (30) i 1 1 where D^ i s the absorbed dose cont r i b u t i o n from r a d i a t i o n of r e l a t i v e b i o l o g -i c a l e f f ectiveness (RBE)^. For a continuous d i s t r i b u t i o n of RBEs (as i n the present case), t h i s becomes P B i o l o g i c a l l y e f f e c t i v e dose = / (RBE) x dD (31) JO where D i s the t o t a l absorbed dose. Although equations (30) and (3l) are used to c a l c u l a t e the b i o l o g -i c a l l y e f f e c t i v e dose due to a mixture of r a d i a t i o n s , the v a l i d i t y of adding e f f e c t i v e doses due to rad i a t i o n s of d i f f e r e n t RBE i s open to question. The RBE determined f o r a given r a d i a t i o n may no longer be v a l i d i f there i s a d d i t i o n a l exposure to r a d i a t i o n of a d i f f e r e n t RBE. I t i s u s e f u l to derive an expression f o r the e f f e c t i v e or mean RBE f o r a mixed r a d i a t i o n . I f there i s a continuous d i s t r i b u t i o n of RBE, from equation (31) B i o l o g i c a l l y e f f e c t i v e dose due to absorbed P dose D of given r a d i a t i o n = / (RBE) dD Jo ; Also, B i o l o g i c a l l y e f f e c t i v e dose due to absorbed TD dose D of standard r a d i a t i o n = J dD = D Jo These equations compare the b i o l o g i c a l l y e f f e c t i v e doses f o r EQUAL ABSORBED DOSES. To determine RBE, i t i s necessary to compare absorbed doses of the two ra d i a t i o n s f o r EQUAL BIOLOGICALLY EFFECTIVE DOSES. Now, the absorbed doses f o r equal b i o l o g i c a l l y e f f e c t i v e doses must be i n inverse proportion to the b i o l o g i c a l l y e f f e c t i v e doses f o r equal absorbed doses, i . e . , /(RBE) dD _ S t d f o r equal B.E.D. = .E .D.) ^ y e n f Q r D = C _ u g i v e n (B.E.D.; s t d I " Jo where D i s absorbed dose and " b i o l o g i c a l l y e f f e c t i v e dose" i s abbreviated as B.E.D. The r a t i o (D .,/D . ) for equal b i o l o g i c a l e f f e c t s i s , by d e f i n i -s t d given t i o n , the RBE and, f o r a mixed r a d i a t i o n as i n the present case, w i l l be c a l l e d the e f f e c t i v e or mean RBE where RBE = J(RBE) dD j J dD (32) - 56 -5 . 2 . B i o l o g i c a l l y E f f e c t i v e Dose D i s t r i b u t i o n s under F u l l y Oxygenated  Conditions The dose d i s t r i b u t i o n s c a l c u l a t e d i n t h i s s ection are v a l i d f o r conditions of adequate oxygen supply since the b i o l o g i c a l data used were measured on well-oxygenated b i o l o g i c a l material. (The requirement f o r " w e l l -oxygenated" i s discussed i n s e c t i o n 5 - 3 . 1 . ) 5 - 2 . 1 . Relative B i o l o g i c a l E f f e c t i v e n e s s and Linear Energy Transfer The " l i n e a r energy t r a n s f e r " (abbreviated LET) of the r a d i a t i o n i s believed to be the most important p h y s i c a l factor determining the RBE of the r a d i a t i o n f o r a given b i o l o g i c a l e f f e c t . LINEAR ENERGY TRANSFER of a charged p a r t i c l e t r a v e r s i n g a medium i s defined ( 19) as "the energy l o c a l l y imparted to the medium per unit distance t r a v e l l e d by the charged p a r t i c l e " . Relating the RBE to the LET implies that the b i o l o g i c a l e f f e c t of a r a d i a t i o n i s determined only by the " l o c a l l y absorbed" energy and, therefore, by the i o n i z a t i o n density i n the immediate v i c i n i t y of the p a r t i c l e track. The d e f i n i t i o n of LET i s not u s e f u l , however, unless the phrase " l o c a l l y imparted" i s defined. The appropriate d e f i n i t i o n depends on the problem considered. The stopping power S of the medium f o r a given charged p a r t i c l e i s greater than the LET i f part of the energy d i s s i p a t e d by the p a r t i c l e i s absorbed at some distance from the p a r t i c l e track. This i s the case when energetic d e l t a rays are produced by heavy charged p a r t i c l e s . Lacking d e t a i l e d information on the distance from the p a r t i c l e track at which the energy i s absorbed and l a c k i n g information on the appropriate s p e c i f i c a t i o n of " l o c a l l y absorbed", i t i s usual to assume that the LET equals the stop-ping power; t h i s i s the assumption i n t h i s t h e s i s . On t h i s assumption, the RBE i s taken to be a function of stopping power. - 57 -5.2.2. S e l e c t i o n of Data f o r "RBE versus Stopping Power" In order to c a l c u l a t e a b i o l o g i c a l l y e f f e c t i v e dose, i t was necess-ary to assume a r e l a t i o n s h i p between stopping power and RBE and to weight the stopping power f o r each p a r t i c l e by the appropriate RBE*. Measurements of RBE as a function of LET published by Barendsen, Walter, Fowler and Bewley (20) and as a function of stopping power published by Todd (2l) were selected as most relevant to the present c a l c u l a t i o n s . Since the LET i s assumed i n t h i s paper to be equal to the stopping power, the r e s u l t s of Bar-endsen et a l and those of Todd were taken to be comparable. Their r e s u l t s were based on s u r v i v a l (maintenance of reproductive capacity) of human k i d -ney c e l l s i n well-oxygenated t i s s u e culture exposed to radiations of d i f f e r -ent LET. Both i n v e s t i g a t o r s determined RBE vs LET (or S) curves f or d i f f e r -ent l e v e l s of c e l l s u r v i v a l from 1% to 80%. Their r e s u l t s are i n good qual-i t a t i v e agreement, each f i n d i n g that with increasing LET the RBE f i r s t increased from unity to a maximum and then decreased sharply. Each obtained approximately the same maximum RBE f o r a given l e v e l of c e l l s u r v i v a l and each found that the RBE determined f o r a given LET was greater f o r high per-centage s u r v i v a l than f or low s u r v i v a l , i . e . , greater f o r small doses than for large doses. Barendsen et a l , however, observed the maximum RBE for an LET of approximately 1200 MeV per cm whereas Todd obtained maximum RBE f o r a stopping power of about 2300 MeV per cm. In view of the uncertainty i n the RBE f o r any given stopping power, two d i f f e r e n t curves were assumed. These curves are shown i n fig u r e 9-Curve A i s an approximate average of the r e s u l t s of Barendsen et a l and those * Equation (31) requires that, to f i n d the b i o l o g i c a l l y e f f e c t i v e dose, each element of absorbed dose be weighted by the appropriate RBE. Each element of absorbed dose i s , however, proportional to the stopping power of the medium f o r the p a r t i c l e c o n t r i b u t i n g the dose element and i t has already been assumed that t h i s i s the only factor which a f f e c t s the RBE. Hence, weighting each stopping power by an RBE i s , i n f a c t , weighting each e l e -ment of absorbed dose by an RBE as required by equation ( 3 l ) . - 58 -of Todd f o r 20% c e l l s u r v i v a l and curve B i s an approximate average of t h e i r r e s u l t s f o r 80% s u r v i v a l . One b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n was ca l c u l a t e d weighting each stopping power by the RBE read from curve A; a sec-ond d i s t r i b u t i o n was c a l c u l a t e d weighting each stopping power by the RBE read from curve B. 5.2.3. Calculations The c a l c u l a t i o n of the b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n was the same as the c a l c u l a t i o n of the p h y s i c a l dose d i s t r i b u t i o n except that wherever the stopping power S appeared i n an equation i t was replaced by (RBE x S) where the RBE was the value read from fig u r e 9 (curve A or B) f o r the given stopping power. For example, the r i g h t side of equation (13) became rR m (RBE) S (dN/dR) dR 2J Since t h i s i s no longer the energy absorption per centimetre at distance r , the notation on the l e f t side of the equation has been changed a l s o . The b i o l o g i c a l equivalent of equation (13) i s now written TR m (dB/dr) = /• (RBE) S (dN/dR) dR ( 3 3 ) * » i r P where B = b i o l o g i c a l l y e f f e c t i v e equivalent of p h y s i c a l energy absorption T. With the i n t r o d u c t i o n of RBE i n each equation and the replacement of T by B, equations ( 2 0 ) , ( 2 2 ) and ( 2 7 ) become equations ( 3 4 ) , ( 3 5 ) and ( 3 6 ) , respec-t i v e l y , as follows: Stopping power (MeV/cm) FIGURE 9 TWO RELATIONSHIPS ASSUMED BETWEEN RELATIVE BIOLOGICAL EFFECTIVENESS AND STOPPING POWER (Data adapted from references 20 and 21) - 60 -B i o l o g i c a l l y e f f e c t i v e dose/pion/cm 2 ( s t r a g g l i n g neglected) _ (B.E.D.) _ ( R E E ) STT Wt dB , . F p p dX B i o l o g i c a l l y e f f e c t i v e dose/pion/cm 2 ( s t r a g g l i n g included) POO P Jo z In order to c a l c u l a t e the b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s for a monoenergetic pion beam corresponding to the Bragg curves p l o t t e d i n figure 8, i t was necessary to repeat a l l the steps of c a l c u l a t i n g the absorbed dose d i s t r i b u t i o n s . S p e c i f i c a l l y , i t was necessary to c a l c u l a t e (a) the b i o l o g i c a l equivalent of table VIII using equation (33) instead of equation (13), i . e . , a t a b l e of values of (dB/dr) as a function of distance r from the nuclear d i s i n t e g r a t i o n , (b) the b i o l o g i c a l equivalent of table XI using equations (34) and (35) instead of equations ( 2 0 ) and ( 2 2 ) , i . e . , a table of b i o l o g i c a l l y e f f e c t i v e doses per pion per cm2, range s t r a g g l i n g neglected, and (c) the b i o l o g i c a l equivalent of table XII using equation (36) instead of equation ( 2 7 ) , i . e . , a t a b l e of b i o l o g i c a l l y e f f e c t i v e doses per pion per cm2, range s t r a g g l i n g included. The r e s u l t of the above procedure depends, of course, on the r e l a -t i o n s h i p assumed between RBE and stopping power. Since two d i f f e r e n t curves were assumed i n t h i s work (curves A and B of figure 9), the whole procedure was repeated twice — f i r s t , with RBE values read from curve A and, second, with values from curve B. While i t was necessary to calculate the equivalents of tables VIII and XI for each set of RBE values, only the f i n a l r e s u l t s corresponding to - 61 -table XII are given here. The b i o l o g i c a l l y e f f e c t i v e doses per pion per cm , range s t r a g g l i n g included, are given i n table XIII for both sets of RBE values. The doses using the lower RBE values (fi g u r e 9> curve A) are tabu-l a t e d i n columns 2 , 3 and k and those for the higher RBE data (figure 9» curve B) i n columns 5 S 6 and 7- The unit i n which these doses are given i s the b i o l o g i c a l l y e f f e c t i v e equivalent of the p h y s i c a l dose u n i t , MeV-cm2/g. The dose data of table XIII for the higher RBE values (columns 5> 6 and 7) have been p l o t t e d i n f i g u r e 1 0 . These curves are very s i m i l a r i n shape to those of fi g u r e 8 but d i f f e r i n the values of the ordinates. Both figures i l l u s t r a t e the e f f e c t of the k i n e t i c energy of charged p a r t i c l e s released i n nuclear d i s i n t e g r a t i o n s on the dose d i s t r i b u t i o n s . The lower RBE data of table XIII (columns 2 , 3 and k) have not been p l o t t e d since the curves d i f f e r only q u a n t i t a t i v e l y from those of fi g u r e 1 0 . Figure 11 has been p l o t t e d to i l l u s t r a t e the e f f e c t of d i f f e r e n t values of RBE. The three curves are (a) absorbed dose (data from table XII, column 3 ) , (b) b i o l o g i c a l l y e f f e c t i v e dose using the lower RBE values (data from table XIII, column 3 ) , (c) b i o l o g i c a l l y e f f e c t i v e dose using the higher RBE values (data from table XIII, column 6 ) . A l l curves are p l o t t e d f o r 29 MeV " s t a r s " . The lower and upper curves are, of course, the curves for 29 MeV stars as already p l o t t e d i n figures 8 and 10 (but note the change i n scale of ordinates from figure 8 to fi g u r e l l ) . Values of RBE and e f f e c t i v e "oxygen enhancement r a t i o " , OER, are shown at several depths on the b i o l o g i c a l l y e f f e c t i v e dose curves of f i g -ures 10 and 1 1 . OER i s defined and the method of c a l c u l a t i o n i s discussed i n - 62 -TABLE XIII BIOLOGICALLY EFFECTIVE DOSE PER PION PER CM2 DUE TO A BEAM OF 8 2 . 2 MeV NEGATIVE PIONS IN WATER UNDER FULLY OXYGENATED CONDITIONS, RANGE STRAGGLING INCLUDED B i o l o g i c a l l y e f f e c t i v e dose/pion/cm 2 Depth i n RBE data from f i g . 9, curve A RBE data from f i g . 9, curve B water (lower RBE values) (higher RBE values) *P Primary Plus Plus Primary Plus Plus pions 29.0 MeV 35-6 MeV pions 29.0 MeV 35.6 MeV (cm) only " s t a r s " " s t a r s " only " s t a r s " " s t a r s " 1 2 3 4 5 6 7 0.0 2.676 2.676 2.676 2.676 2.676 2.676 3.0 2 . 7 9 9 2 . 799 2 . 799 2 . 799 2 . 799 2 .799 6.0 2.959 2.959 2.959 2.959 2.959 2.959 8.0 3 . 0 9 5 3 . 0 9 5 3 . 0 9 5 3 . 0 9 5 3 . 0 9 5 3 . 0 9 5 10.0 3 . 2 7 0 3 . 2 7 0 3 . 2 7 0 3 .270 3 . 2 7 0 3 . 2 7 0 11.2 3.400 3.400 3.400 3.400 3.400 3.400 12.4 3 . 5 6 1 3 . 5 6 1 3 . 5 6 1 3 . 5 6 1 3 . 5 6 1 3 . 5 6 1 13.45 3 . 734 3 . 7 3 8 3 . 7 3 9 3 .734 3-739 3.740 14.50 3 . 9 5 8 3 . 9 8 0 3 . 9 8 6 3 . 9 5 8 3 . 9 8 2 3 . 9 8 8 15.55 4 . 2 8 7 4 . 3 5 9 4 .375 4 . 2 8 7 4.363 4 . 3 8 0 1 6 . 6 0 4 . 8 0 6 4 . 9 8 8 5 . 0 2 9 4 . 8 0 6 4 . 9 9 7 5 . 0 4 1 17.10 5.149 5-425 5.488 5.149 5 . 4 3 8 5 .504 1 7 . 6 0 5 . 5 8 4 6.005 6.101 5 .584 6 . 0 2 4 6 . 125 17.84 5.833 6 . 3 5 8 6 . 4 7 8 5.833 ' 6 . 3 8 3 6 . 5 0 8 1 8 . 0 8 6 . 1 2 7 6 . 8 0 8 6 . 9 6 3 6 . 128 6.848 7 . 0 1 2 1 8 . 3 2 6 . 5 0 3 7 . 4 8 6 7.711 6 . 5 0 3 7 -582 7 . 8 2 8 1 8 . 5 6 7 . 0 2 1 8 . 7 6 1 9 . 158 7 . 0 2 5 9 . 0 6 3 9 .529 1 8 . 8 0 7 . 7 5 6 11.464 12.311 7 . 7 6 9 12 .417 13 .479 1 9 . 0 4 8 . 7 3 1 1 6 . 9 3 2 1 8 . 8 0 6 8 . 7 6 8 19 . 5 0 1 21 .953 1 9 . 2 8 9.787 2 6 . 1 8 9 2 9 . 9 3 7 9 . 8 7 1 31 .829 36.846 19.52 10 .487 3 8 . 1 1 9 44.433 10 .635 48.054 56 .604 19 .76 10 . 2 4 6 48.314 57.013 10.455 6 2 . 2 8 5 74 . 128 20.00 8 . 7 5 9 5 1 . 1 6 0 60.848 8.993 66 .814 8 0 . 0 2 6 20 .24 6 . 3 5 4 44 .421 53 .120 6 . 5 6 2 5 8 . 3 9 2 70 .235 20.48 3 . 8 3 1 31 .462 37.776 3 .978 41 .397 49 .947 20 .72 1 . 8 9 3 1 8 . 2 9 5 22.043 1 . 9 7 6 23 .934 28 .952 20 .96 O .760 8 . 9 6 2 10 .836 0 . 7 9 7 11.530 13 .982 21.20 0 . 2 4 7 3.955 4 . 8 0 2 0 . 2 6 0 4 . 9 0 8 5-970 21.44 0 . 0 6 4 3 1 . 8 0 4 2.202 0 . 0 6 8 0 2 . 1 0 7 2 . 5 7 3 21.68 0.0135 0 . 9 9 7 1.222 0.0143 1 . 0 9 3 1.339 21.92 0.0023 0 . 6 8 2 0.838 0 . 0 0 2 4 0.723 0.887 . 22 .16 0.0003 0 . 5 2 5 0.644 0.0003 0.550 0 .675 22.40 0.0000 0 . 4 2 1 0 . 5 1 7 0.0000 0.440 0.541 22.90 O .278 0.339 O .289 0.355 23.40 0 . 1 8 2 0.224 0 . 191 0 . 2 3 5 2 4 . 4 5 0 . 0 7 1 3 0 . 0 8 7 6 0 . 0 7 5 8 0 . 0 9 3 1 25.50 0 . 0 2 2 7 0 . 0 2 7 9 0.0244 0 . 0 2 9 9 26.55 0.0047 0 . 0 0 5 8 0.0049 0 . 0 0 6 0 2 7 . 6 0 0.0005 0 . 0 0 0 6 0.0005 0 . 0 0 0 6 2 8 . 8 0 0.0000 ' 0.0000 0.0000 0.0000 These data p l o t t e d i n f i g . 10 A l l closes calculated using higher RBE values from curve B, figure 9 (Data p l o t t e d from table XIII, columns 5 , 6 and 7) (a) P r i . pion contribution only, i . e . , no "s t a r " energy (b) Assuming 2 9 . 0 MeV k i n e t i c energy of charged p a r t i c l e s from " s t a r s " (c) Assuming 3 5 . 6 MeV k i n e t i c energy of charged p a r t i c l e s from " s t a r s " RBE OER 1, 2, 00 70 RBE = 2.1+8 OER = 1 . 5 5 RBE = 2.1+0 OER = 1 . 5 7 RBE = 2 . 5 5 OER = 1.5l+ RBE = 2.1+9 OER = 1 . 5 6 RBE = 1 . 2 0 OER = 2.1+9 10 15 Depth i n water (g/cm2) 20 25 FIGURE 10 BIOLOGICALLY EFFECTIVE DOSES DUE TO A BEAM OF 8 2 . 2 MeV NEGATIVE PIONS FOR DIFFERENT ASSUMPTIONS RE KINETIC ENERGY OF CHARGED PARTICLES FROM "STARS" CM —. B M o CM >H s <D o • ft • > a CO o S3 •H ft rH CM 0) S ft o <D u to CD o ft T) u a o CU o !> •H •H ft -P O u CD 0) «H ft <rH (1) CD to >> o H t3 H cd -tf a CD •H rQ bo rH o O H to O rO •H < m 80 6o 40 20 (a) Absorbed dose (Data from table XII, column 3) (b) B i o l o g i c a l l y e f f e c t i v e dose using lower RBE values from curve A, fi g u r e 9 (Data from table XIII, column 3) (c) B i o l o g i c a l l y e f f e c t i v e dose using higher RBE values from curve B, figu r e 9 (Data from table XIII, column 6) 0 RBE = 2.40 OER =1.57 RBE = 1.84 OER = 1.65 I RBE = 2.49 OER =1.56 RBE = 1.89 OER = 1.63 RBE = 1.20 OER = 2.49 RBE = 1.13 OER = 2.53 10 20 25 ON 15 Depth i n water (g/cm2) FIGURE 11 COMPARISON OF ABSORBED DOSE AND BIOLOGICALLY EFFECTIVE DOSES DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS A l l curves calculated on assumption of 29 MeV k i n e t i c energy of charged p a r t i c l e s from " s t a r s " - 65 -a l a t e r s e c t i o n . Values of RBE were ca l c u l a t e d from equation ( 3 2 ) . The b i o l o g i c a l l y e f f e c t i v e dose which i s the numerator of equation (32) can be read from table XIII f o r s p e c i f i e d conditions. The denominator of the equa-t i o n , the absorbed dose f o r the same conditions, i s given i n table XII. For example, B i o l o g i c a l l y e f f e c t i v e dose at 20 "cm assuming 29 MeV s t a r and RBE values from figure 9, curve B = 66.8l (table XIII) Absorbed dose under same conditions = 27.82 (table XII) Therefore, from equation ( 3 2 ) , RBE = 66.81/27.82 = 2.1+0 The c a l c u l a t i o n s of a l l the RBEs and OERs shown on figures 10 and 11 are given i n table XV which follows i n section 5.k. 5.3. B i o l o g i c a l l y E f f e c t i v e Dose D i s t r i b u t i o n Under Anoxic Conditions The d i s t r i b u t i o n under anoxic conditions d i f f e r s from that under conditions of adequate oxygenation because (a) the b i o l o g i c a l e f f e c t of a given r a d i a t i o n dose i s , i n general, smaller under anoxic conditions, (b) the dependence of the e f f e c t on the oxygen supply of the i r r a d i a t e d b i o l o g i c a l m a terial decreases with increasing density of i o n i z a t i o n , i . e . , with in c r e a s i n g LET of the r a d i a t i o n , and (c) f o r a beam of negative pions, LET i s a function of depth i n the medium. 5.3.1. Oxygen Enhancement Ratio and B i o l o g i c a l l y E f f e c t i v e Dose under Anoxic Conditions The dependence of b i o l o g i c a l e f f e c t on the oxygen supply of the i r r a d i a t e d m a terial i s measured, f o r any given r a d i a t i o n , by the "oxygen enhancement r a t i o " (abbreviated OER) which i s defined (22) as follows: - 66 -THE OXYGEN ENHANCEMENT RATIO f o r a GIVEN RADIATION i s the r a t i o of the r a d i a t i o n dose under anoxic conditions to the dose under f u l l y oxy-The measured OER i s r e l a t i v e l y independent of the b i o l o g i c a l e f f e c t observed For example, the OER f o r x-rays or gamma rays as measured f o r a v a r i e t y of b i o l o g i c a l e f f e c t s , both i n c e l l s and i n t i s s u e s , l i e s between 2.5 and 3.0. The OER i s , however, quite dependent on the r a d i a t i o n . A value of, say, 2.7 i s t y p i c a l f o r a low LET r a d i a t i o n such as x-rays or gamma rays but the OER f a l l s to unity f o r high LET r a d i a t i o n . and (31) i s to be considered as the e f f e c t i v e dose under f u l l y oxygenated conditions since the RBE data used were determined under normal atmospheric conditions*. As already noted, the b i o l o g i c a l effectiveness of a given absorbed dose i s , i n general, reduced by anoxia, i . e . , the oxygen enhance-ment r a t i o i s , i n general, greater than unity. The d e f i n i t i o n of " b i o l o g i -c a l l y e f f e c t i v e dose under anoxic conditions" used i n t h i s paper i s the following: BIOLOGICALLY EFFECTIVE DOSE UNDER ANOXIC CONDITIONS where RBE values measured under f u l l y oxygenated conditions are used i n both * While the x-ray or gamma-ray s e n s i t i v i t y of b i o l o g i c a l material i n the absence of oxygen i s t y p i c a l l y about 35 to h0% of the s e n s i t i v i t y under f u l l y oxygenated conditions, the s e n s i t i v i t y increases very r a p i d l y with very small concentrations of oxygen (23). I t i s very d i f f i c u l t to meas-ure oxygen concentrations i n t i s s u e but, for d i l u t e c e l l suspensions i n water where oxygen concentrations can be measured, the x-ray s e n s i t i v i t y r i s e s to nearly 70% of maximum f o r about 5 micromoles of dissolved oxygen per l i t r e and to nearly 90% of maximum for about 25 micromoles disso l v e d per l i t r e . The l a t t e r concentration i s approximately the eq u i l i b r i u m genated conditions to produce equal (or equivalent) e f f e c t s . The b i o l o g i c a l l y e f f e c t i v e dose as defined i n equations ( 2 9 ) , (30) (37) (Continued on next page) - 67 -equations ( 3 1 ) and ( 3 7 ) . The e f f e c t of anoxia i s taken i n t o account by the introduction of OER i n equation ( 3 7 ) . the e f f e c t i v e RBE, i t may be shown that the mean or e f f e c t i v e OER i s given 5 . 3 . 2 . Oxygen Enhancement Ratio and Linear Energy Transfer I t has already been stated that the OER varies with the LET of the r a d i a t i o n . The dependence of OER on LET has been measured by Barendsen and co-workers ( 2 ^ ) . Their r e s u l t s are p l o t t e d i n f i g u r e 1 2 . In t h i s f i g u r e , OER has been p l o t t e d against stopping power since, as i n the consideration of RBE, i t i s assumed that the l i n e a r energy t r a n s f e r i s equal to the stopping power. The curve of f i g u r e 12 was used i n t h i s thesis to determine the OER f o r any given stopping power. 5 . 3 . 3 . Calculations The c a l c u l a t i o n of the b i o l o g i c a l l y e f f e c t i v e dose under anoxic conditions i s i n every respect s i m i l a r to the c a l c u l a t i o n of e f f e c t i v e dose under f u l l y oxygenated conditions except for the b i o l o g i c a l weighting f a c -tors introduced into the p h y s i c a l c a l c u l a t i o n . To c a l c u l a t e the e f f e c t i v e dose under oxygenated conditions, every stopping power S appearing i n the p h y s i c a l c a l c u l a t i o n was replaced by (RBE x S); to c a l c u l a t e the e f f e c t i v e dose under anoxic conditions, every S i n the p h y s i c a l c a l c u l a t i o n i s replaced by [(RBE/OER) X S] where the same RBE i s used i n the two cases and the OER * (Continued from previous page) concentration of oxygen dissol v e d i n water under a p a r t i a l oxygen pressure of 15 mm Hg. Any experimental measurements under normal atmospheric conditions are, therefore, to be considered as measurements under f u l l y oxygenated conditions. By an argument s i m i l a r to that used i n d e r i v i n g an expression f o r by ( 3 8 ) 2.75 1.25 1.00 10 10 2 10 3 10 Stopping power (MeV/cm) FIGURE 12 ASSUMED DEPENDENCE OF OXYGEN ENHANCEMENT RATIO ON STOPPING POWER (Data from reference 2k) - 69 -corresponding to the given S i s read from figure 12. The equations corres-ponding to equations (33) to (36) are not given here as the required m o d i f i -cations are ohvious. To c a l c u l a t e the b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s under anoxic conditions, i t was necessary to repeat a l l the steps of the c a l c u l a -t i o n of absorbed dose d i s t r i b u t i o n or of b i o l o g i c a l l y e f f e c t i v e dose d i s t r i -butions under f u l l y oxygenated conditions. The f i n a l r e s u l t only ( i . e . , the b i o l o g i c a l l y e f f e c t i v e doses per pion per cm2 under anoxic conditions, s t r a g -g l i n g included) are given i n table XIV. As i n tables XII and XIII, these are doses due to a beam of 82.2 MeV pions. While a l l OER values used i n these c a l c u l a t i o n s were read from the curve of f i g u r e 12, two d i f f e r e n t sets of r e s u l t s are given i n table XIV, one for the lower RBE values read from figure 9 5 curve A, and the other f o r the higher RBE values from curve B (each f o r primary pions only and f o r two d i f f e r e n t "wei ghts" of " s t a r s " ) . The dose unit i n table XIV i s the same as i n table XIII, i . e . , i t i s the b i o l o g i c a l l y e f f e c t i v e equivalent of the ph y s i c a l dose u n i t , MeV-cm2/g. The e f f e c t i v e doses given i n table XIV have not been p l o t t e d but are required f o r the c a l c u l a t i o n s of e f f e c t i v e OER i n the next s e c t i o n . 5.h. E f f e c t i v e RBE and E f f e c t i v e OER at D i f f e r e n t Depths on D i f f e r e n t  Assumptions The e f f e c t i v e RBE and e f f e c t i v e OER f o r an 82.2 MeV beam of nega-t i v e pions were c a l c u l a t e d by means of equations (32) and (38) at several depths i n water using the data of tables XII, XIII and XIV. These values were c a l c u l a t e d using (a) the lower RBE values from figure 9* curve A, (b) the higher RBE values from fig u r e 9 9 curve B, - 70 -TABLE XIV BIOLOGICALLY EFFECTIVE DOSE PER PION PER CM2 DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER UNDER ANOXIC CONDITIONS, RANGE STRAGGLING INCLUDED B i o l o g i c a l l y e f f e c t i v e dose/pion/cm 2 Depth i n RBE data from f i g . 9, curve A RBE data from f i g . 9 , curve B water (lower RBE values) (higher RBE values) *P Primary Plus Plus Primary Plus Plus pions 2 9 . 0 MeV 3 5 - 6 MeV pions 2 9 . 0 MeV 3 5 . 6 MeV (cm) only " s t a r s " " s t a r s " only " s t a r s " " s t a r s " 1 2 3 1+ 5 6 7 0 . 0 0 . 9 9 1 0 . 9 9 1 0 . 9 9 1 0 . 9 9 1 0 . 9 9 1 0 . 9 9 1 3 . 0 1 . 0 3 6 1 . 0 3 6 1 . 0 3 6 1 . 0 3 6 1 . 0 3 6 1 . 0 3 6 6 . 0 1 . 0 9 5 1 . 0 9 5 1 . 0 9 5 1 . 0 9 5 1 . 0 9 5 1 . 0 9 5 8 . 0 1 . 1 1 + 7 1 . 1 1 + 7 1 . 1 1 + 7 I . 1 I + 7 1 . 1 1 + 7 1 . 1 1 + 7 1 0 . 0 1 . 2 1 1 1 . 2 1 1 1 . 2 1 1 1 . 2 1 1 1 . 2 1 1 1 . 2 1 1 1 1 . 2 1 . 2 5 9 1 . 2 5 9 1 . 2 5 9 1 . 2 5 9 1 . 2 5 9 1 . 2 5 9 12.4 1 . 3 1 8 1 . 3 1 8 1 . 3 1 8 1 . 3 1 8 1 . 3 1 8 1 . 3 1 8 13.1+5 1 . 3 8 3 1 . 3 8 1 + 1 . 3 8 5 1 . 3 8 3 1 . 3 8 1 + 1 . 3 8 5 1 1 + . 5 0 1.1+66 1.1+75 1.1+77 1.1+66 1.1+75 1.1+77 1 5 . 5 5 1 . 5 8 8 I . 6 1 6 1 . 6 2 2 1 . 5 8 8 1 . 6 1 8 I . 6 2 I + 1 6 . 6 0 1 . 7 8 0 1 . 8 5 1 I . 8 6 7 1 . 7 8 0 I . 8 5 I + 1 . 8 7 1 1 7 . 1 0 1 . 9 0 8 2 . 0 1 1 + 2 . 0 3 8 1 . 9 0 8 2 . 0 1 9 2 . 0 1 + 5 1 7 . 6 0 2 . 0 6 8 2 . 2 3 1 2 . 2 6 8 2 . 0 6 8 2 . 2 3 9 2 . 2 7 8 17.81+ 2 . 1 6 1 2 . 3 6 1 + 2.1+10 2 . 1 6 1 2 . 3 7 5 2.1+21+ 1 8 . 0 8 2 . 2 7 0 2 . 5 3 9 2 . 6 0 0 2 . 2 7 0 2 . 5 5 9 2 . 6 2 5 1 8 . 3 2 2.1+09 2 . 8 2 8 2 . 9 2 1 + 2.1+10 2 . 8 8 7 2 . 9 9 7 1 8 . 5 6 2 . 6 0 2 3.1+52 3 . 6 1 + 7 2 . 6 0 6 3 . 6 6 3 3 .901+ 1 8 . 8 0 2 . 8 7 7 l+.9 1 +5 5.1+18 2 . 8 9 1 5 . 6 3 7 6 . 2 6 5 1 9 . 0 1 + 3 . 2 1 + 6 8 . 2 0 6 9.31+0 3 . 2 8 1 1 0 . 1 0 0 1 1 . 6 5 9 1 9 . 2 8 3 . 6 5 3 1 3 . 9 9 8 1 6 . 3 6 2 3 - 7 2 2 1 8 . 1 8 2 21.1+86 1 9 . 5 2 3 . 9 3 3 2 1 . 7 3 1 + 2 5 . 8 0 2 1+.01+5 2 9 . 1 2 0 31+.850 1 9 . 7 6 3 . 8 6 3 2 8 . 6 3 3 31+.293 1+.007 3 9 . 0 2 5 1+ 7 . 0 2 6 2 0 . 0 0 3 . 3 2 0 3 0 . 9 9 1 37.311+ 3.1+70 1+2.629 5 1 . 5 7 7 2 0 . 2 1 + 2.1+20 2 7 . 1 9 0 3 2 . 8 5 0 2 . 5 1 + 5 3 7 . 5 6 2 1+5.561+ 20.1+8 1.1+65 1 9 . 2 6 7 2 3 . 3 3 1 + 1 . 5 ^ 9 2 6 . 6 2 5 32.351+ 2 0 . 7 2 0 . 7 2 7 1 1 . 0 7 3 13.1+37 0 . 7 7 2 1 5 . 2 3 2 1 8 . 5 3 6 2 0 . 9 6 0 . 2 9 3 5 . 2 5 3 6 . 3 8 9 0 . 3 1 2 7 . 1 3 2 8 . 6 9 0 2 1 . 2 0 0 . 0 9 5 ! + 2 . 1 6 3 2 . 6 3 6 0 . 1 0 2 2.81+8 3.1+75 21.1+1+ 0 . 0 2 5 0 0 . 8 7 5 1 . 0 7 0 0 . 0 2 6 8 1 . 0 8 3 1 . 3 2 5 2 1 . 6 8 0 . 0 0 5 2 0.1+21+ 0 . 5 2 0 O . O O 5 6 0.1+83 0 . 5 9 2 2 1 . 9 2 0 . 0 0 0 9 0 . 2 7 0 0.331 0 . 0 0 0 9 0 . 2 9 0 0 . 3 5 6 2 2 . 1 6 0 . 0 0 0 1 0 . 2 0 3 0 . 2 1 + 9 0 . 0 0 0 1 0 . 2 1 1 + 0 . 2 6 3 22.1+0 0 . 0 0 0 0 0 . 1 6 2 0 . 1 9 9 0 . 0 0 0 0 0 . 1 7 0 0 . 2 0 9 2 2 . 9 0 0 . 1 0 6 0 . 1 3 1 0 . 1 1 2 0 . 1 3 7 23.1+0 0 . 0 7 0 5 0 . 0 8 6 6 0 . 0 7 1 + 0 0 . 0 9 1 0 21+.1+5 0 . 0 2 7 8 0.031+2 0 . 0 2 9 1 + 0 . 0 3 6 1 2 5 . 5 0 0 . 0 0 9 1 0 . 0 1 1 2 0 . 0 0 9 3 0 . 0 1 1 5 2 6 . 5 5 0 . 0 0 1 5 0 . 0 0 1 9 0 . 0 0 1 5 0 . 0 0 1 9 2 7 . 6 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0.0000 - 71 -assuming k i n e t i c energy of charged d i s i n t e g r a t i o n products from nuclear d i s -integrations equal to (a) 29.0 MeV, (b) 35 - 6 MeV. The method of c a l c u l a t i n g RBE from equation ( 3 2 ) has already been discussed. The c a l c u l a t i o n of OER from equation (38) i s s i m i l a r . For exam-p l e , assuming a 29 MeV " s t a r " and RBE values from fig u r e 9» curve B, at 20 cm depth B i o l o g i c a l l y e f f e c t i v e dose under f u l l y oxygenated conditions = 66.8l B i o l o g i c a l l y e f f e c t i v e dose under anoxic conditions = 42.63 Therefore, from equation (38), OER = 66.81/42.63 = 1 . 5 7 The values of RBE and OER for a l l conditions f o r which they were calculated are given i n table XV. The table includes a l l the values shown on figures 10 and 11. TABLE XV EFFECTIVE RBE AND EFFECTIVE OER FOR A BEAM OF 82.2 MeV PIONS AT DIFFERENT DEPTHS AND ON DIFFERENT ASSUMPTIONS RBE data from f i g . 9» curve A RBE data from. f i g . 9> curve B (lower RBE values) (higher RBE values) Absorbed B i o l o g i c a l l y B i o l o g i c a l l y Depth dose e f f e c t i v e dose* e f f e c t i v e dose* OPT? KB.fi. l U J i l j UHJJA (cm) (MeV-cm2 F u l l y Anoxic F u l l y Anoxi c per g) oxygenated oxygenated Assuming 29.0 MeV "s t a r " 0 2.676 2.676 0.991 1.00 2.70 2.676 0.991 1.00 2.70 15-55 4.351 4.359 1.616 1.00 2.70 4.363 1.618 1.00 2.70 18.08 6.729 6.808 2.539 1.01 2.68 6.848 2.559 1.02 2.68 19-04 13.056 16.932 8.206 1.30 2.06 19.501 10.100 1.49 1.93 19-52 23.274 38.119 21.734 1.64 1.75 4 8 . 0 5 4 29.120 2.07 1.65 20.00 27.816 51.160 30.991 1.84 1.65 66.814 42.629 2.40 1.57 20.48 16.626 31.462 19.267 1.89 1.63 41.397 26.625 2.49 1.56 20.96 5.090 8.962 5.253 1.76 1.71 11.530 7.132 2.27 1.62 21.92 0.6o4 0.682 0.270 1.13 2.53 0.723 0.290 1.20 2.49 22.90 0.248 0.278 0.106 1.12 2.62 0.289 0.112 1.17 2.58 Assuming 35.6 MeV "s tar" 0 2.676 2.676 0.991 1.00 2.70 2.676 0.991 1.00 2.70 15.55 4.365 4.375 1.622 1.00 2.70 4.380 1.624 1.00 2.70 18.08 6.867 6.963 2.600 1.01 2.68 7.012 2.625 1.02 2.67 19.04 14.063 18.806 9-340 1.34 2.01 21.953 11.659 1.56 1.88 19.52 26.270 44.433 25.802 1.69 1.72 56.604 34.850 2.16 1.62 20.00 32.287 60.848 37.314 1.89 • 1.63 80.026 51.577 2.48 1.55 20.48 19.623 37.776 23.334 1.93 1.62 49.947 32.354 2.55 1.54 20.96 6.097 10.836 6.389 1.78 1.70 13.982 8.690 2.29 1.61 21.92 0.742 0.838 0.331 1.13 2.53 0.887 0.356 1.20 2.49 22.90 0.305 0.339 0.131 1.11 2.59 0.355 0.137 1.16 2.59 * The uni t i s the b i o l o g i c a l l y e f f e c t i v e equivalent of "MeV-cm2/g". - 73 -6. DOSE DISTRIBUTIONS FOR CONTINUOUS ENERGY SPECTRA  OF NEGATIVE PI-MESONS The high dose regions of the dose d i s t r i b u t i o n curves f o r mono-energetic beams of negative pions as p l o t t e d i n figures 8 and 10 are too narrow f o r radiotherapy since required treatment volumes are usu a l l y several centimetres i n thickness (say, 4 to 10 or 12 cm). I t w i l l , therefore, be necessary to s e l e c t a continuous energy spectrum of negative pions f o r r a d i o -therapy, the energy i n t e r v a l included i n the spectrum depending on the thickness to be treated. 6.1. S e l e c t i o n of a Continuous Pion Spectrum to Y i e l d a Desired Depth Dose  D i s t r i b u t i o n and C a l c u l a t i o n of the Depth Dose D i s t r i b u t i o n Due to  the Selected Spectrum The depth dose d i s t r i b u t i o n desired i s , i n general, a d i s t r i b u t i o n of constant dose through the thickness of the s p e c i f i e d treatment volume with the lowest p o s s i b l e dose outside t h i s volume. The method of approaching such a d i s t r i b u t i o n by ad d i t i o n of a f i n i t e number of beams of nearly monoenergeti pions i s shown q u a l i t a t i v e l y i n figure 13. In t h i s f i g u r e , f i v e beams having ranges of 12, 14, l 6 , 18 and 20 cm have been added (using hypothetical Bragg curves f o r i l l u s t r a t i v e purposes o n l y ) . The f i v e beams used are shown as dotted or dashed l i n e s and the t o t a l dose d i s t r i b u t i o n obtained by ad d i t i o n as a s o l i d l i n e . I t i s evident from the diagram that, to obtain a dose that i s approximately constant over the depth to be tr e a t e d (12 to 20 cm i n the example), the beam of greatest range must have the greatest i n t e n s i t y and the i n t e n s i t y of each of the other beams must be decreased with decreasing range. The dose at 20 cm depth i s contributed mainly by the beam of 20 cm range while the dose at 12 cm depth i s due to the beam of 12 cm range plus contribu tions from a l l beams of greater range. The f i v e beams selected do not produce a constant dose over the treatment thickness but, by increasing the 32 2k 16 8 0 / / \\ \\ / / / / V i i I / A /' V Y \/ h \! 1 \ t 1 \ i"\ 1 \ 4 i / \ / v ^ : ' " ~ " \ \ ' y > • \ \ \ \ \ \ v v. V 0 1+ 8 12 16 20 2k Depth (cm) FIGURE 13 ADDITION OF NEARLY MONOENERGETIC BEAMS TO OBTAIN A CONSTANT DOSE OVER A SPECIFIED DEPTH - 75 -number of monoenergetic beams, the dose may be made as uniform as desired provided the i n t e n s i t y of each beam i s adjusted appropriately. I t w i l l not be po s s i b l e to add monoenergetic beams, as i n the example above, to obtain a beam suitable f o r radiotherapy since a l l the a v a i l -able beams w i l l consist of continuous energy spectra of pions.* I t i s pos s i b l e , however, by s e l e c t i n g a "shaped" continuous spectrum to obtain a dose d i s t r i b u t i o n which i s e s s e n t i a l l y constant over a s p e c i f i e d depth. The procedure f o r s e l e c t i n g the desired spectrum i s described i n general terms i n t h i s s e c t i o n and applied to s p e c i f i c problems i n the follow-ing s e c t i o n . The f i r s t spectrum selected i s chosen to include pions with ranges covering the required thickness of treatment and "shaped" a r b i t r a r -i l y — but of the general shape suggested by figu r e 13. The dose d i s t r i b u -t i o n which would be produced by t h i s spectrum i s ca l c u l a t e d ; the shape of the spectrum i s then adjusted by repeated t r i a l and error u n t i l an acceptable dose d i s t r i b u t i o n i s obtained. The method of f i n d i n g the dose d i s t r i b u t i o n produced by a given pion spectrum i s s i m i l a r to that of f i n d i n g (dT/dr) , the energy absorbed P per centimetre per 7r~ capture at distance r p from a nuclear d i s i n t e g r a t i o n [see equation (13)], or to that of f i n d i n g (D /F) , the dose per pion per s t r X p cm2, s t r a g g l i n g included, at depth Xp [see equation (27)]. Figure l U ( a ) shows the dose J as a function of depth where J denotes the dose per pion per cm 2, absorbed or b i o l o g i c a l l y e f f e c t i v e , for a * The pion beams a v a i l a b l e from the SIN accelerator (Zurich), from LAMPF (Los Alamos) and from TRIUMF (University of B r i t i s h Columbia) w i l l a l l be produced by bombarding a s o l i d target with a nearly monoenergetic beam of medium-energy protons (500 to 800 MeV, depending on a c c e l e r a t o r ) . This r e s u l t s i n the production of a continuous energy spectrum of pions from a maximum energy o f about 180 MeV less than the proton energy down to a few MeV (25). - 76 -monoenergetic beam (range s t r a g g l i n g included) of range R. In f a c t , the curve of " j " versus "Depth" might be p l o t t e d from any of the dose data given i n table XII, XIII or XIV. For example, to cal c u l a t e the absorbed dose at depth Xp due to a given spectrum of negative pions, assuming 29 MeV k i n e t i c energy of charged p a r t i c l e s released per stopping pion, the dose data f o r J i n f i g u r e lU(a) would be those from table XII, column 3. These data are used on the assumption that the shape of the dose curve on e i t h e r side of the peak i s independent of the i n i t i a l pion energy and that only the length of the plateau preceding the peak depends on i n i t i a l energy. On t h i s assump-t i o n , f o r a beam of, say, 12 cm range, the absorbed dose at any depth i n figure lh(a) would be the absorbed dose at 8 cm greater depth i n table XII since the data i n t h i s table are f o r a beam having a range of 20 cm.* Figure l ^ ( b ) shows the assumed range spectrum. The number of pions per cm 2 with range between R and (R + dR) i s (dF/dR) dR and each of these pions contributes a dose J z [see figu r e l ^ ( a ) ] at X p where z i s the r e s i d u a l range of the monoenergetic beam and z = R - x (39) Therefore, T o t a l dose at x^ = / J z (dF/dR) dR (ko) J R X where R-j_ and Rg are the minimum and maximum ranges included i n the spectrum * The assumption that the shape of the peak of the dose curve f o r a mono-energetic beam i s independent of the i n i t i a l energy of the beam i s obviously an approximation since range s t r a g g l i n g increases with range. For example, f o r a beam having a range of 20 cm, the root mean square deviation i n path length, a, i s approximately 0.5 cm while f o r a beam of range 12 cm, a i s approximately 0.3 cm. In f a c t , i n these c a l c u l a t i o n s , a = 0.5 cm was used f o r pions of a l l ranges from 10.88 cm to 21.Uk cm (see page 8 l ) . As a r e s u l t of t h i s approximation, the dose curves p l o t t e d i n figures 15> 16 and 17 are not quite as steep as they should be f o r depths of 10 to 12 cm but the erro r i s not s i g n i f i c a n t . - 77 -0 Depth h 2 H (a) Dose J as a function of depth FIGURE 14 DIAGRAM TO ILLUSTRATE DERIVATION OF EQUATION ( 4 0 ) - 78 -as shown i n f i g u r e 14(h). The t o t a l dose ca l c u l a t e d from t h i s equation w i l l he determined by the quantity used f o r J . I t should be noted that Units of t o t a l dose = Units of J x cm"2 since F has been defined as number of pions per cm2. As i n evaluating equation ( 2 7 ) , z must be allowed to take negative values since a monoenerge-t i c beam of range R may contribute to the dose at Xp greater than R. Equa-t i o n ( 4 0 ) was integrated numerically but, for t h i s i n t e g r a t i o n , the "tr a p e z o i d a l " r u l e rather than Simpson's rule was used.* In a l l examples c a l c u l a t e d i n the following sec t i o n , spectra were chosen to y i e l d approximately constant dose from 12 to 20 cm depth. In order to obtain constant dose between these l i m i t s , i t was found necessary to use range spectra extending from a lower l i m i t of 10.88 cm to an upper l i m i t of 21.44 cm [ i . e . , Rj = 10.88 cm and R 2 = 21.44 cm i n fi g u r e l 4 ( b ) ] . Over t h i s range, the spectrum was s p e c i f i e d by ordinates at i n t e r v a l s of 0.48 cm i n e i t h e r d i r e c t i o n from 20.0 cm, i . e . , for R = (20.0 ± 0.48 m) cm (Ul) from 10.88 cm to 21.44 cm where m was an integer. Also, the values of Xp f o r which equation ( 4 0 ) was evaluated were chosen so that a l l values s a t i s f i e d the equation Xp = (20.0 ± 0.48 n) cm (42) where n was an integer (but not a l l values of n were used). At depths at which the t o t a l dose as c a l c u l a t e d from equation (Uo) was changing r a p i d l y , i t was evaluated f o r Xp at i n t e r v a l s of 0.48 cm but, i n the low dose region, * Integration using the t r a p e z o i d a l r u l e was considered adequate f o r the evaluation of equation (40) since the value of the integrand does not vary r a p i d l y with change of abscissa. Use of the t r a p e z o i d a l r u l e sim-p l i f i e d the r e c a l c u l a t i o n of the dose d i s t r i b u t i o n a f t e r each t r i a l - a n d -error adjustment of the spectrum. l a r g e r i n t e r v a l s were used. Since a l l values of R which were used s a t i s f i e d equation (hi) and a l l values of x^ s a t i s f i e d equation (1+2), i t follows from equation ( 3 9 ) that a l l values of z s a t i s f i e d the equation z = ± 0.1+8 (m ± n) cm (1+3) i . e . , values of the r e s i d u a l range z were at i n t e r v a l s of 0.1+8 cm on e i t h e r side of zero. In f a c t , i t w i l l he seen i n the next s e c t i o n that values of J z were used i n equation (1+0) f o r z at i n t e r v a l s of 0.1+8 cm from -6.21+ cm to 20.61+ cm. The required values of J 2 were read from or found "by i n t e r p o l a t i o n or extrapolation from table XII, XIII or XIV, depending on the dose required. The value of J z f o r any s p e c i f i e d z i s read i n these tables at Depth = ( 2 0 . 0 - z) cm since a l l the tables are f o r 8 2 . 2 MeV pions which have a range of 2 0 . 0 cm i n water. The general method of determining the dose d i s t r i b u t i o n due to a s p e c i f i e d continuous pion spectrum, as described above, i s applied to selected problems i n the following se c t i o n . 6 . 2 . Selected Spectra and Resulting Depth Dose D i s t r i b u t i o n s Spectra were determined by t r i a l and err o r to y i e l d (a) constant absorbed dose, (b) constant b i o l o g i c a l l y e f f e c t i v e dose under well-oxygenated conditions using RBE values from ( i ) curve A, f i g u r e 9» ( i i ) curve B, f i g u r e 9» for depths from 12 to 20 cm. The dose data used i n a l l cases were those - 80 -c a l c u l a t e d on the assumption that K i n e t i c energy of charged p a r t i c l e s released i n TT capture = 2 9 - 0 MeV since t h i s appears to be a more probable value than 3 5 • 6 MeV. No spectra were chosen to y i e l d constant b i o l o g i c a l l y e f f e c t i v e doses under anoxic conditions but, f o r the spectra determined to y i e l d con-stant b i o l o g i c a l l y e f f e c t i v e doses under well-oxygenated conditions (cases (b, i ) and ( b , i i ) of the previous paragraph), b i o l o g i c a l l y e f f e c t i v e doses under anoxic conditions were c a l c u l a t e d at several depths i n order to determine the e f f e c t i v e OER at these depths. 6 . 2 . 1 . Continuous Pion Spectrum to Y i e l d Constant Absorbed Dose The "shaped" range spectrum of pions selected to y i e l d constant absorbed dose and the r e s u l t i n g absorbed dose d i s t r i b u t i o n are given i n columns 1 to 3 of table XVI. The range spectrum, i . e . , the number of pions per cm2 per u n i t range i n t e r v a l , dF/dR (where F i s the number per cm 2), at d i f f e r e n t ranges, as used i n equation (40), i s given i n the second column of the t a b l e . The dose data used f o r J z i n equation (4o) were from column 3 of table XII with a d d i t i o n a l values obtained by i n t e r p o l a t i o n and extrapolation as already described. The absorbed doses at d i f f e r e n t depths, as determined from equation (40), are given i n the t h i r d column of table XVI. The t o t a l number of pions per cm2 i n the selected spectrum, as shown i n the l a s t l i n e of column 2 , table XVI, was c a l c u l a t e d from the equation T o t a l number where R^ and R2 have the same meanings as i n equation (40). The values given - 81 -TABLE XVI ABSORBED DOSE DISTRIBUTION DUE TO^  "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 2 9 . 0 MeV "STARS" Range and depth 1 dF/dR 2 ABSORBED DOSE From spectrum of column 2 3 Scaled to average of 100 rads* 4 (cm) (cm - 3) (MeV/g) (rads) 0.32 15.31 41 .89 4 . 16 1 6 . 6 0 45 .42 6.08 17.52 47.93 8.00 18.88 51.66 8 . 9 6 1 9 . 9 2 54 .50 9 . 9 2 21.46 58.71 10.88 0.000 25.30 69.22 11.36 0.100 29.52 80.77 11.84 0.400 33 .92 9 2 . 8 1 12.32 0 . 4 0 5 36.22 99.10 12 .80 0 . 4 1 0 36.73 100.49 13 . 2 8 0 . 4 1 8 36.73 100.49 13.76 0 . 4 2 7 36.66 100.30 1 4 . 2 4 0.437 36.59 100.11 14.72 0 . 4 5 0 36.55 100.00 15.20 0.466 36.56 100.03 15.68 0.486 36.58 100.08 1 6 . 1 6 0.505 36.57 100 .06 16.64 0.525 36.53 99.95 1 7 . 1 2 0.547 36.48 99 .81 1 7 . 6 0 0.575 36.44 99-70 1 8 . 0 8 0.607 36.30 99-32 1 8 . 5 6 0.643 36.37 99.51 19 .04 0.688 36.57 100 .06 1 9 . 5 2 O . 7 6 3 36.87 100.88 20.00 1.000 35 .18 96.25 20.48 0 . 9 0 0 28.37 77 .62 2 0 . 9 6 0 . 5 0 0 17 • 30 47.33 21.44 0.000 7.37 20 .16 2 1 . 9 2 2.44 6.68 22.88 O .58 1.59 23.84 0.25 0 . 6 8 T o t a l no./cm2 : 5.401 * Numerical values of column 3 m u l t i p l i e d by 2.736 to give average dose of 100 rads from 12.32 to 19-52 cm. - 8 2 -in column 3 are the absorbed doses in MeV per gram at different depths due to this total number of pions per cm2 where the pions are distributed in range in the proportions given in column 2. The absorbed dose data of column 3, in MeV per gram, have been scaled upwards so the average dose over the constant dose region is 100 rads. The scaled data are given in rads in column h of the table and are plotted in figure 15. The absolute energy spectrum required to yield these absorbed doses is shown as an inset in the figure. This spectrum is derived in follow-ing paragraphs. A spectrum of particles is more commonly and more usefully speci-fied as an energy spectrum rather than a range spectrum. The range spectrum of table XVI was f i r s t converted to an energy spectrum and then scaled upwards by the factor required to produce the absorbed doses given in column 4 of table XVI. The range spectrum was converted to an energy spectrum by means of the following equation. dF/dE = (dF/dR) (dR/dE) = (dF/dR)/S (1+5) where dF/dR = number of pions/cm2 per unit range interval at range R dF/dE = number of pions/cm2 per MeV energy interval at energy E corresponding to range R S = stopping power of water for pions of energy E The resulting dF/dE values are given in column 5 of table XIX (p.92). (The dF/dR values in column k of the table are from table XVI.) The total number of pions per cm2 in the converted spectrum, as given in the last line of column 5» was calculated from r E2 Total number of pions/cm2 = / (dF/dE) dE (46) JEX oi 1 1 1 J 1 1 1 0 . 4 . 8 12 16 20 24 Depth in water (g/cm2) FIGURE 15 ABSORBED DOSE AS A FUNCTION OF DEPTH IN WATER DUE TO THE "SHAPED" CONTINUOUS SPECTRUM OF NEGATIVE PIONS SHOWN IN INSET (Assumes 29 MeV kinetic energy of charged disintegration products) - 84 -where E^ and E 2 are the minimum and maximum energies, r e s p e c t i v e l y , i n the energy spectrum. This t o t a l number must agree, within the accuracy of the numerical i n t e g r a t i o n s , with the t o t a l number determined from the range spectrum. In column 6 of ta b l e XIX, the dF/dE values of column 5 have been m u l t i p l i e d by a f a c t o r equal to the r a t i o of corresponding absorbed doses i n columns 4 and 3 of table XVI. These are the absolute dF/dE values required to obtain the absorbed dose d i s t r i b u t i o n of figure 15 and are the values p l o t t e d i n the in s e t of that f i g u r e . The required m u l t i p l y i n g f a c t o r was calc u l a t e d as follows: Average dose (from 12.32 cm to 1 9 - 5 2 cm) from column 3 , table XVI = 3 6 . 5 5 MeV/g 1 MeV/g = 1 . 6 0 2 x 1 0 ~ 6 ergs/g = 1 . 6 0 2 x IO" 8 rads Therefore, f o r an average dose of 100 rads, 100 Q Required m u l t i p l y i n g f a c t o r = 1 . 7 0 8 x 10 3 6 . 5 5 x 1 . 6 0 2 x 1 0 - 8 (The absorbed doses i n column 4 , table XVI, are 1 . 7 0 8 x 1 0 8 times the corres-ponding doses i n column 3 and, therefore, the pion spectrum must be scaled by t h i s f a c t o r . However, the r a t i o of the numerical values of absorbed doses i n columns 4 and 3 of table XVI i s only 1 0 0 / 3 6 . 5 5 = 2 . 7 3 6 since the f a c t o r 1 0 8 / 1 . 6 0 2 i s absorbed i n the change of unit.) No b i o l o g i c a l l y e f f e c t i v e doses have been c a l c u l a t e d from the spectrum of table XVI since t h i s spectrum would be of l i t t l e i n t e r e s t f o r b i o l o g i c a l experiments. - 85 -6.2.2. Continuous Spectra to Y i e l d Constant B i o l o g i c a l l y E f f e c t i v e Doses under Well Oxygenated Conditions. Column 2 of table XVII gives a range spectrum of pions selected to y i e l d approximately constant b i o l o g i c a l l y e f f e c t i v e doses under v e i l oxygen-ated conditions from a depth of 12 cm to 20 cm using the lower RBE values from curve A of f i g u r e 9. The r e s u l t i n g b i o l o g i c a l l y e f f e c t i v e doses at d i f f e r e n t depths are given i n column 3 of the table (where the u n i t i s the b i o l o g i c a l l y e f f e c t i v e equivalent of the absorbed dose u n i t , MeV/g). These doses were c a l c u l a t e d from equation (40) using, f o r J , dose values from column 3 of table XIII. The t o t a l number of pions per cm2, ca l c u l a t e d from equation (44), to y i e l d the b i o l o g i c a l l y e f f e c t i v e doses of column 3 i s given i n the l a s t l i n e of column 2. In a d d i t i o n to the doses under well oxygenated conditions, b i o l o g i c a l l y e f f e c t i v e doses under anoxic conditions ( j values from column 3, table XIV) at a few depths and absorbed doses ( j values from column 3, table XII) at several depths were c a l c u l a t e d f o r the range spectrum of column 2. These values were used to c a l c u l a t e the e f f e c t i v e RBE and. the e f f e c t i v e OER at d i f f e r e n t depths as shown i n columns 6 and 7 of table XVII. From equation (32), _ B i o l , e f f . dose under well oxygenated conditions Absorbed dose and, from equation (38), OER - B i o l , e f f . dose under w e l l oxygenated conditions B i o l . e f f . dose under anoxic conditions where, i n each case, the two doses of the r a t i o are calculated at the same depth. - 86 -TABLE XVII BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTION UNDER WELL OXYGENATED CONDITIONS DUE TO "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 29.0 MeV "STAR" AND RBE VALUES FROM FIG. 9, CURVE A (LOWER VALUES) Range and depth 1 dF/dR 2 B i o l o g i c a l l y e f f e c t i v e dose Absorbed dose 5 RBE 6 OER 7 Scaled doses B i o l . e f f e c t i v e (l.6l6 X c o l . 3) 8 Absorbed (l.6l6 X c o l . 5) 9 Oxygen-ated 3 Anoxic 4 (cm) (cm - 3) * * (MeV/g) # (rads) 0.32 18.23 6.75 18.23 1.00 2.70 29.46 29.46 it.16 19.80 19.80 32.00 32.00 6.08 20.93 20.93 33.82 33.82 8.00 22.67 8.39 22.66 1.00 2.70 36.63 36.62 8.96 24.02 38.82 9.92 26.13 26.03 42.23 42.06 10.88 0.000 34.14 13.94 31.53 1.08 2.45 55.17 50.95 11.36 0.200 44.56 72.01 11.84 0.540 55.26 42.72 89.30 69.04 12.32 0.542 60.62 97.96 12.80 0.546 61.81 31.15 . 45.62 1.36 1.98 99.88 73.72 13.28 0.551 61.78 99.84 13.76 0.558 61.67 45.00 ' 99-66 72.72 14.24 0.569 6l.6l 99.56 14.72 0.583 61.73 44.33 99.76 71.64 15.20 0.598 61.84 99-93 15.68 o.6i4 61.95 100.11 16.16 0.631 62.04 32.57 43.19 1.44 1.91 100.26 69.80 16.64 0.649 62.07 100.31 17.12 0.668 62.07 100.31 17.60 0.688 62.00 41.45 100.19 66.98 18.08 0.710 61.83 99.92 18.56 0.732 61.60 39.72 99.55 64.19 19.04 0.756 61.75 99.79 19.52 O.783 62.75 35.84 38.07 1.65 1.75 ioi.4o 61.52 20.00 1.000 61.44 35.66 99.29 57.63 20.48 0.900 51.03 28.55 82.46 46.14 20.96 0.500 31.52 18.88 17.39 l.8l 1.67 50.94 28.10 21.44 0.000 12.97 20.96 21.92 3.70 1.94 2.47 1.50 1.91 5.98 3.99 22.88 O.67 1.08 23.84 0.29 0.26 0.47 0.42 T o t a l /cm2 = = 6.393 Unit i s the b i o l o g i c a l l y e f f e c t i v e equivalent of "MeV/g" § Unit i s the b i o l o g i c a l l y e f f e c t i v e equivalent of the "rad" - 87 -The doses i n column 3 were scaled up to obtain an average b i o l o g i c a l l y e f f e c t i v e dose under w e l l oxygenated conditions of 100 units from 1 2 . 8 0 cm to 2 0 . 0 0 cm where the u n i t i s the b i o l o g i c a l l y e f f e c t i v e equivalent o f the absorbed dose u n i t , the rad. (There i s , at the present time, no generally accepted name f o r t h i s unit.) Also, the absorbed doses of column 5 were scaled up by the same f a c t o r . The scaled values are given i n columns 8 and 9 of the table and have been p l o t t e d i n f i g u r e l 6 . The range spectrum of column 2 , t a b l e XVII, was converted by means of equation ( 4 5 ) to an energy spectrum. The o r i g i n a l range spectrum and the corresponding energy spectrum are given i n columns 7 and 8 , r e s p e c t i v e l y , of table XIX (p. 9 2 ) . The t o t a l number of pions per cm2 i n t h i s spectrum, as determined from equation ( 4 4 ) or equation ( 4 6 ) , i s given i n the l a s t l i n e of each column. The energy spectrum of column 8 has been scaled up i n column 9 to the absolute energy spectrum required to produce the doses given i n columns 8 and 9 of table XVII. The s c a l i n g f a c t o r i s 1 . 0 0 9 x 1 0 8 c a l c u l a t e d as follows: Average of the numerical values of dose from 1 2 . 8 0 cm to 2 0 . 0 0 cm i n column 3 , table XVII = 6 1 . 8 7 M u l t i p l y i n g f a c t o r to get a numerical average of 100 over same range of depths i n column 8 = 1 0 0 / 6 1 . 8 7 = 1 . 6 l 6 This i s the f a c t o r by which the numerical values of doses i n columns 3 and 5> table XVTI, were m u l t i p l i e d to get the numerical values i n columns 8 and 9 -The spectrum must be scaled up by the a d d i t i o n a l f a c t o r 1 0 8 / 1 . 6 0 2 to allow f o r the change of dose units from columns 3 and 5 to columns 8 and 9-Therefore, M u l t i p l y i n g f a c t o r f o r spectrum = = 1 . 0 0 9 x 1 0 8 T 1 r 0 4 8 12 . 1 6 20 24 Depth i n water (g/cm 2) F I G U R E 16 (a) B I O L O G I C A L L Y E F F E C T I V E DOSE D I S T R I B U T I O N UNDER WELL-OXYGENATED CONDITIONS U S I N G RBE VALUES FROM FIGURE 9 , CURVE A , AND (ID) ABSORBED DOSE D I S T R I B U T I O N , BOTH FOR THE "SHAPED" SPECTRUM OF N E G A T I V E P I O N S SHOWN I N I N S E T - 89 -The scaled energy spectrum of column 9> table XIX, i s p l o t t e d i n the i n s e t of figure l6. Table XVIII, columns 10 to 12 of table XIX and fig u r e IT are the exact equivalents of table XVII, columns 7 to 9 of table XIX and fig u r e 16, r e s p e c t i v e l y , except a l l RBE values used were from curve B of fig u r e 9. The J values used i n equation (40) f o r c a l c u l a t i o n of b i o l o g i c a l l y e f f e c t i v e z doses under f u l l y oxygenated conditions were, therefore, taken from table XIII, column 6, and those for c a l c u l a t i o n of doses under anoxic conditions from column 6 of table XIV. While the two pion spectra used i n t h i s s e c t i o n were selected to y i e l d approximately constant b i o l o g i c a l l y e f f e c t i v e doses under oxygenated conditions, the absorbed dose d i s t r i b u t i o n as w e l l as the b i o l o g i c a l l y e f f e c t -ive dose d i s t r i b u t i o n has been c a l c u l a t e d and p l o t t e d f o r each spectrum since the absorbed dose d i s t r i b u t i o n can be checked by d i r e c t p h y s i c a l measurement whereas the b i o l o g i c a l l y e f f e c t i v e d i s t r i b u t i o n can only be checked by much more d i f f i c u l t b i o l o g i c a l measurements. - 90 -TABLE XVIII BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTION UNDER WELL OXYGENATED CONDITIONS DUE TO "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 2 9 - 0 MeV "STAR" AND RBE VALUES FROM FIG. 9 , CURVE B (HIGHER VALUES) Range and depth 1 dF/dR 2 B i o l o g i c a l l y e f f e c t i v e dose Absorbed dose 5 RBE 6 OER 7 Scaled doses B i o l . e f f e c t i v e ( 1 . 2 0 8 X c o l . 3) 8 Absorbed ( 1 . 2 0 8 X c o l . 5) 9 Oxygen-ated 3 Anoxic 4 (cm) (cm - 3) # (MeV/g) # (rads) 0 . 3 2 2 0 . 2 7 7 . 5 1 2 0 . 2 7 1 . 0 0 2 . 7 0 24 .49 2 4 . 4 9 4 . 1 6 2 2 . 0 4 2 2 . 0 4 2 6 . 6 2 2 6 . 6 2 6 . 0 8 23 .33 2 3 . 3 3 2 8 . 1 8 2 8 . 1 8 8 . 0 0 25.32 9 . 3 8 25.30 1 . 0 0 2 . 7 0 3 0 . 5 9 30 .56 8 . 9 6 2 6 . 8 8 3 2 . 4 7 9 - 9 2 2 9 . 4 3 2 9 . 2 3 3 5 . 5 5 3 5 - 3 1 1 0 . 8 8 0 . 0 0 0 41 .73 1 8 . 0 2 3 5 . 9 0 1 . 1 6 2.32 5 0 . 4 l 4 3 . 3 7 1 1 . 3 6 0 . 2 9 2 5 7 . 9 6 7 0 . 0 2 11.84 0 . 6 2 7 7 3 . 7 0 48.71 8 9 . 0 3 58.84 12.32 0 . 6 2 9 8 1 . 2 7 9 8 . 1 7 1 2 . 8 0 0 . 6 3 2 8 2 . 8 2 4 5 . 0 6 5 1 . 5 1 1 . 6 1 1.84 1 0 0 . 0 5 6 2 . 2 2 1 3 . 2 8 0 . 6 3 8 8 2 . 8 0 . 1 0 0 . 0 2 13 .76 0.645 82.64 5 0 . 5 5 9 9 . 8 3 6 1 . 0 6 1 4 . 2 4 0 . 6 5 3 8 2 . 5 5 9 9 . 7 2 1 4 . 7 2 O .665 8 2 . 5 4 4 9 . 4 3 9 9 . 7 1 5 9 . 7 1 1 5 . 2 0 0 . 6 7 9 8 2 . 6 1 9 9 . 7 9 1 5 . 6 8 0 . 6 9 2 8 2 . 7 4 9 9 - 9 5 1 6 . 1 6 0 . 7 0 7 8 2 . 8 0 46 .62 4 7 . 5 7 1 . 7 4 1 . 7 8 1 0 0 . 0 2 5 7 . 4 6 16.64 0 . 7 2 4 8 2 . 8 9 100.13 1 7 . 1 2 0.742 8 2 . 9 6 1 0 0 . 2 2 1 7 . 6 0 0 . 7 6 1 8 2 . 9 8 4 5 . 1 2 1 0 0 . 2 4 5 4 . 5 0 1 8 . 0 8 0 . 7 8 2 8 2 . 9 0 1 0 0 . 1 4 1 8 . 5 6 0 . 8 0 2 8 2 . 7 1 42 .80 9 9 - 9 1 5 1 . 7 0 1 9 - 0 4 0 . 8 2 4 8 2 . 7 9 1 0 0 . 0 1 1 9 . 5 2 0.849 8 3 . 4 8 5 0 . 7 0 3 9 . 9 6 2 . 0 9 1 . 6 5 100.84 48 .27 2 0 . 0 0 1 . 0 0 0 8 1 . 1 5 3 6 . 7 9 98.03 4 4 . 4 4 20.48 O .926 6 7 . 5 4 29.23 8 1 . 5 9 3 5 . 3 1 2 0 . 9 6 O .506 41 .79 2 6 . 3 5 1 7 - 7 7 2 . 3 5 1 . 5 9 50.48 2 1 . 4 7 2 1 . 4 4 0 . 0 0 0 1 6 . 9 7 2 0 . 5 0 2 1 . 9 2 4 . 5 8 2 . 5 5 2 . 5 4 1 . 8 0 1 . 8 0 5 . 5 3 3 . 0 7 2 2 . 8 8 0 . 7 2 0 . 8 7 23.84 0 . 3 1 0 . 2 6 0 . 3 7 0 . 3 1 T o t a l /cm2 = = 7 . 0 9 2 * Unit i s the b i o l o g i c a l l y e f f e c t i v e equivalent of "MeV/g" if Unit i s the b i o l o g i c a l l y e f f e c t i v e equivalent of the "rad" 100 cu CO o •tf o > •H •P O <D « H <H CU o •rH bO o rH O •H pq 3 > cu S I CM s a o Tot? i l pior 5 . 3 5 x is / cm2 1 0 8 5 0 6 0 TO 8 0 Energy (MeV) 9 0 (a) B i o l o g i c a l l y e f f e c t i v e dose (b) Absorbed dose RBE = 1.61 OER = 1.84' RBE = 1 . 7 4 OER = 1 . 7 8 RBE = 2 . 0 9 j OER = 1 . 6 5 1 vo 1 2 1 6 Depth i n water (g/cm2) FIGURE 1 7 (a) BIOLOGICALLY EFFECTIVE DOSE DISTRIBUTION UNDER WELL-OXYGENATED CONDITIONS USING RBE VALUES FROM FIGURE 9 , CURVE B, AND (b) ABSORBED DOSE DISTRIBUTION, BOTH FOR THE SHAPED SPECTRUM OF NEGATIVE PIONS SHOWN IN INSET TABLE XIX CONVERSION AND SCALING OF CONTINUOUS PION SPECTRA Range, energy and stopping power data f o r pions (from table VI) Spectra f o r constant b i o l o g i c a l l y e f f e c t i v e doses* spectrum i o r cons uanu absorbed dose RBE data from curve A, f i g . 9 (lower RBE values) RBE data from curve B, f i g . 9 (higher RBE values) Range R Energy E Stopping power, S dF/dR dF/dE 1 . 7 0 8 x 1 0 8 x c o l . 5 dF/dR dF/dE 1 . 0 0 9 x 1 0 8 x c o l . 8 dF/dR dF/dE 0 . 7 5 4 x l 0 8 x c o l . 11 1 2 3 4 5 6 7 8 9 10 11 12 (cm) (MeV) (MeV/cm) (cm - 3) (cm 2 MeV - 1) ( 1 0 7 per cm2-MeV) (cm- 3) ( c n f2 -MeV - 1) ( 1 0 7 per cm2-MeV) (cm- 3) (cm 2 -MeV - 1) ( 1 0 7 per cm2-MeV) 10.88 1 1 . 3 6 11.84 12.32 1 2 . 8 0 1 3 . 2 8 13.76 1 4 . 2 4 14.72 1 5 . 2 0 15.68 1 6 . 1 6 16.64 17.12 1 7 . 6 0 1 8 . 0 8 1 8 . 5 6 19.04 19.52 2 0 . 0 0 20.48 2 0 . 9 6 21.44 5 5 . 8 5 5 7 . 3 5 5 8 . 8 5 60.33 61.79 63-24 64.68 66.10 6 7 . 5 0 6 8 . 9 0 70.28 71.65 73.00 74.35 75-68 77.01 78.33 79.63 80.93 8 2 . 2 2 83.50 84.77 86.04 3.186 3.145 3.106 3.070 3.036 3.003 2 . 9 7 3 2 . 9 4 3 2 .915 2 . 8 8 9 2.863 2 . 8 3 9 2 . 8 1 6 2 . 7 9 3 2.772 2 . 7 5 1 2.731 2 . 7 1 2 2.693 2 . 6 7 6 2 . 6 5 8 2 . 6 4 2 2 . 6 2 6 0 . 0 0 0 0.100 0 . 4 0 0 0 . 4 0 5 o.4io 0.418 0 . 4 2 7 0.437 0 . 4 5 0 0.466 0.486 0.505 0.525 0 .547 0.575 O .607 0.643 0.688 0.763 1 . 0 0 0 0 . 9 0 0 0.500 0 . 0 0 0 0 . 0 0 0 0 . 0 3 2 0 . 1 2 9 0 . 1 3 2 0.135 0.139 0 . 1 4 4 0 . 1 4 8 0 . 1 5 4 0 . 1 6 1 0 . 1 7 0 0 . 1 7 8 0 . 1 8 6 0 . 1 9 6 0 . 2 0 7 0 . 2 2 1 0.235 0.254 0 . 2 8 3 0.374 0.339 0 . 1 8 9 0 . 0 0 0 0.00 0.54 2 . 2 0 2.25 2.31 2.38 2.45 2.54 2.64 2 . 7 6 2.90 3.04 3.18 3.35 3.54 3.77 4.02 4.33 4.84 6.38 5-78 3.23 0 . 0 0 0.000 0 . 2 0 0 0.540 0.542 0 . 5 4 6 0 . 5 5 1 0 . 5 5 8 0 . 5 6 9 0 . 5 8 3 0 . 5 9 8 0 . 6 l 4 0.631 0.649 0.668 0.688 0 . 7 1 0 0.732 • 0 . 7 5 6 0.783 1 . 0 0 0 0 . 9 0 0 - 0 . 5 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 6 4 0 . 1 7 4 0.177 0 . 1 8 0 0 . 1 8 3 0 . 1 8 8 0 . 1 9 3 0 . 2 0 0 0 . 2 0 7 0 . 2 1 4 0 . 2 2 2 0 . 2 3 0 0.239 0.248 0 . 2 5 8 0 . 2 6 8 0 . 2 7 9 0 . 2 9 1 0.374 0.339 0 . 1 8 9 0 . 0 0 0 0.00 0.64 1.75 1.78 1 . 8 1 1.85 1 . 8 9 1.95 2 . 0 2 2 . 0 9 2 . 1 6 2 . 2 4 2.33 2 .4 l 2.50 2 . 6 0 2 . 7 0 2 . 8 1 2.93 3.77 3.42 1 . 9 1 0 . 0 0 0.000 0.292 0 . 6 2 7 0.629 0 . 6 3 2 0.638 0.645 0.653 0.665 0 . 6 7 9 0 . 6 9 2 0.707 0 . 7 2 4 0.742 0 . 7 6 1 0 . 7 8 2 0 . 8 0 2 0 . 8 2 4 0.849 1 . 0 0 0 0 . 9 2 6 0.506 0 . 0 0 0 0 .000 0 . 0 9 3 0 . 2 0 2 0 . 2 0 5 0 . 2 0 8 0 . 2 1 2 0 . 2 1 7 0 . 2 2 2 0 . 2 2 8 0.235 0 . 2 4 2 0.249 0 . 2 5 7 0 . 2 6 6 0 . 2 7 5 ' 0 . 2 8 4 0 . 2 9 4 0.304 0 . 3 1 5 0.374 0 . 3 4 8 0 . 1 9 2 0 . 0 0 0 0.00 0.70 1.52 1.54 1.57 1 . 6 0 1.64 1 . 6 7 1 . 7 2 1.77 1.82 1.88 1.94 2 . 0 0 2.07 2 . 1 4 2 . 2 1 2.29 2.38 2 . 8 2 2.63 1 .44 0 . 0 0 T o t a l number per cm2 5.401 5.401 9 2 . 2 5 6.393 6.393 64.51 7 .092 7-093 53.48 * Under f u l l y oxygenated conditions - 93 -7. BEAM CONTAMINATION AND DOSE CONTRIBUTIONS NOT INCLUDED  IN THE DOSE CALCULATIONS. COMPARISON OF CALCULATIONS  WITH EXPERIMENTAL RESULTS AND WITH OTHER CALCULATIONS In t h i s s e c t i o n , the dose d i s t r i b u t i o n s c a l c u l a t e d i n the previous sections are compared with experimental r e s u l t s and with other c a l c u l a t i o n s . Before u s e f u l comparisons can be made, however, i t i s necessary to make some estimates of the e f f e c t s of dose contributions, i n c l u d i n g those due to beam contamination, which were neglected i n the c a l c u l a t i o n s . 7 . 1 . Beam Contamination and Dose Contributions Not Included i n  the Dose C a l c u l a t i o n s . 7 . 1 . 1 . E l e c t r o n and Muon Contamination of the TT- Beam A l l negative pion beams which have been investigated experimentally have been contaminated with electrons and muons and, under p r a c t i c a l condi-tions , i t does not appear that i t w i l l be possible to avoid these contamin-ants . t I t has been suggested ( 1 5 , 26) that most of the electron contamin-ati o n a r i s e s from decay of ne u t r a l pions produced i n the target i n which the negative pions are produced. Neutral pions have a mean l i f e of about 1 0 - 1 6 seconds and, therefore, almost a l l decay within the production target. About 99% decay to two high-energy gamma rays and these, i n turn, are absorbed mainly by ele c t r o n - p o s i t r o n p a i r production. In a l l experiments with charged pions, the pions are "transported" from the production target to the experi-mental p o s i t i o n by a system of focussing and bending magnets. This system determines the momenta of the p a r t i c l e s transported since the motion of a charged p a r t i c l e i n a magnetic f i e l d i s determined by i t s momentum. I t follows, therefore, that a l l electrons o r i g i n a t i n g i n the pion production target and having momenta within the momentum range transmitted by the beam - 9 4 -transport system w i l l pass through the system. Negative pions having an energy of 35 MeV w i l l be contaminated with 104 MeV electrons and 108 MeV pions w i l l be contaminated with 203 MeV electrons (see appendix A). The best estimates of the ele c t r o n contamination to be expected are based on experimental measurements which are considered i n a l a t e r paragraph. The muon contamination arises from decay of pions i n f l i g h t between the production target and the p o s i t i o n at which they come to r e s t . The decay process i s TT~—> vr + v y ( 4 7 ) where i s the muon antineutrino. The mean-life of the pions i s 2 . 6 x 1 0 ~ 8 seconds. The path length of the pions i n a p r a c t i c a l "beam transport system" i s 7 to 10 metres. Assuming an 8-metre path, about 75% of a 35-MeV pion beam and about 50% of a 108-MeV pion beam decay i n f l i g h t (see appendix A). The k i n e t i c energies of the muons r e s u l t i n g from pion decay are l i m i t e d only by the requirements of conserving energy and momentum i n the decay process. However, only a f r a c t i o n of the muons produced w i l l s a t i s f y the momentum requirements f o r transmission through the beam transport system. As a f i r s t approximation, i t i s assumed that a contaminating muon (wherever i t o r i g i n a t e s i n the beam transport system) must have the same momentum as the decaying pion from which i t originates i n order to be transmitted. On t h i s assumption, 35 MeV pions w i l l be contaminated with 43 MeV muons and 108 MeV pions with 124 MeV muons. Attempts have been made to measure or estimate the ele c t r o n and muon contamination of two experimental beams — one produced i n the 1 8 4-inch synchrocyclotron at the Lawrence Radiation Laboratory, U n i v e r s i t y of C a l i f o r n i a , and the other i n the CERN synchrocyclotron. The r e s u l t s are summarized i n table XX. Estimates by Perry and Hynes ( 2 6 ) of the electron TABLE XX MEASUREMENTS AND ESTIMATES OF ELECTRON AND MUON CONTAMINATION OF EXPERIMENTAL PION BEAMS Source of pion beam Pion energy (MeV) Method of estimating contamination Percentage contamination at surface Reference e y 1 8 4-inch synchrocyclotron, Lawrence Radiation Laboratory 9 6 . 2 ± 4 . 0 Time-of-flight measurements 25 10 Curtis and Raju ( 3 , 1 5 ) CERN 600 MeV synchro-cyclotron 8 4 . 3 Contamination estimated to give best f i t of experimental depth dose curves 23 14 Turner et a l ( 2 7 ) * 16 13 Armstrong and Chandler ( 2 8 ) * * Independent estimates of the contamination of the same experimental beam - 96 -and muon contributions to the peak absorbed dose (see table XXV — p . 1 0 8 ) due to a negative pion beam produced by the Nimrod synchrotron, Rutherford High Energy Laboratory, appear consistent with the LRL and CERN contamination estimates. The s i g n i f i c a n c e of the beam contamination i s the e f f e c t i t has on the depth dose d i s t r i b u t i o n . This depends on the energies and ranges of the contaminating p a r t i c l e s and on the processes by which the energy i s absorbed. Assuming the contaminating electrons have energies of about 100 to 200 MeV, t h e i r path lengths i n water are i n excess of 30 cm. Because of the marked s c a t t e r i n g of ele c t r o n s , however, the depths (measured i n the o r i g i n a l d i r e c t i o n of the beam) at which they stop vary from zero up to the maximum path length. Further, more than h a l f the energy w i l l be d i s s i p a t e d by production of bremsstrahlung. As a r e s u l t of the v a r i a t i o n i n the depth at which the electrons stop and the penetrating power of the bremsstrahlung, the electrons contribute a nearly uniform background dose, probably increasing s l i g h t l y with depth ( 1 5 ) . On the assumption that the contaminating muon has the same momentum as the decaying pion from which i t a r i s e s , i t s range i s approxi-mately 50% greater than that of the pion (see appendix A). A l l experimental absorbed dose curves f o r nearly momoenergetic pion beams show a peak due to muons beyond the pion peak ( 1 5 , 2 7 , 29) though t h i s peak i s r e l a t i v e l y much l e s s pronounced than the peak f o r negative pions since the muons do not cause nuclear d i s i n t e g r a t i o n s . Estimates of the percentage contributions of the contaminants to the absorbed dose at the surface of the medium and at the peak of the pion curve are given i n ta b l e XXV ( p . 1 0 8 ) . The contributions at the surface were ca l c u l a t e d on the assumption that the contaminating p a r t i c l e s had momenta - 97 -equal to the momenta of the incident pions and that the r e l a t i v e absorbed doses were p r o p o r t i o n a l to the product of the number of incident p a r t i c l e s of each type and the stopping power of the medium f o r these p a r t i c l e s . The c a l c u l a t i o n s are shown i n table XXI. The percentage contributions at the peak of the absorbed dose curve as. shown i n table XXV are taken d i r e c t l y from the references. 7 . 1 . 2 . Pions Removed from the Beam before Coming to Rest by Interactions with Nuclei of the Medium In the c a l c u l a t i o n s of sections 4, 5 and 6 i t was assumed that a l l pions incident on the medium come to r e s t at the depth i n the medium determined by t h e i r i n i t i a l energy. In f a c t , an appreciable percentage of the pions i n t e r a c t with n u c l e i of the medium and are "removed" from the beam before coming to r e s t . These pions contribute to the dose at the surface of the medium but may make no contribution or reduced contribution to the dose i n the peak or high dose region. The data a v a i l a b l e on nuclear i n t e r a c t i o n s of pions i n water are very l i m i t e d . C u r t i s and Raju ( 1 5 ) have used the equation N = N e ~ ° - 0 1 1 t 7 x (48) o where N i s the number of pions incident on the surface and N i s the number o * at depth x where x i s i n centimetres. This equation has been adopted i n the present work. On the basis of equation (48), i n order to have one negative pion stopping and producing a nuclear d i s i n t e g r a t i o n at range R i t i s necessary to have N x pions at depth x where 0.0147(R-x) N x = e (49) TABLE XXI CALCULATION OF THE PERCENTAGE CONTRIBUTIONS OF PRIMARY PIONS AND MUON AND ELECTRON CONTAMINATION TO THE ABSORBED DOSE AT THE SURFACE OF WATER P a r t i c l e Fraction of incident p a r t i c l e s Momentum of incident p a r t i c l e s (MeV/c) Mean energy of incident p a r t i c l e s (MeV) Stopping power of water (MeV/cm) Relative dose con t r i b u t i o n Percentage of t o t a l absorbed dose Curtis and .Raju ( 15) 0 . 6 5 190 ± 5 9 6 . 2 2 . 5 1 0 . 6 5 x 2 . 5 1 = 1 . 6 3 2 6 7 . 9 0 . 1 0 i t 1 1 1 . 7 2 . 2 5 0 . 1 0 x 2 . 2 5 = 0 . 2 2 5 9 . 3 e~ 0 . 2 5 11 1 8 9 . 5 2 . 2 * 0 . 2 5 x 2 . 2 = 0 . 5 5 0 2 2 . 8 Turner et a l (27) TT-0 . 6 3 175 84 . 3 2 . 6 5 0 . 6 3 x 2 . 6 5 = 1 . 6 7 0 6 6 . 9 0.14 1! 9 8 . 8 2 . 3 1 0.14 x 2.31 = 0.323 1 2 . 9 e~ 0 . 2 3 I t 174 .5 2 . 2 * 0 . 2 3 x 2 . 2 = 0 . 5 0 6 2 0 . 2 * Approximate values f o r c o l l i s i o n losses only-- 99 -For example, to obtain one " s t a r " at 20 ( ± 0 . 5 ) cm, i t i s necessary to have 1 . 3 4 pions of 8 2 . 2 MeV energy incident on the surface. I t follows that the energy deposition and, t h e r e f o r e , the doses (absorbed and b i o l o g i c a l l y e f f e c t i v e ) due to primary pions at any depth x as already c a l c u l a t e d must 0.0147(R-x) be m u l t i p l i e d by the f a c t o r e to correct for the a d d i t i o n a l primary i o n i z a t i o n due to pions which are l a t e r removed from the beam before producing a nuclear d i s i n t e g r a t i o n . Any of tables XII, XIII or XIV can be corrected by introducing t h i s f a c t o r . I t i s to be noted that the correction i s applicable only to the contribution from primary pions and not to the " s t a r " c o n t r i b u t i o n s . Table XXII corresponds to table XII but includes the c o r r e c t i o n for the increased primary i o n i z a t i o n near the surface. The data for 3 5 . 6 MeV " s t a r s " have been omitted from table XXII. Each value i n column 2 of table XXII i s the corresponding value of column 2 , table XII, m u l t i p l i e d by 0.0147(R-x) e ; the d i f f e r e n c e between corresponding values i n columns 2 and 3 , table XXII, i s the same as the d i f f e r e n c e between corresponding values i n columns 2 and 3 , table XII, since the " s t a r " contributions are not corrected. The data from table XXII for 29 MeV " s t a r s " has been p l o t t e d i n f i g u r e 1 8 . The uncorrected curve from f i g u r e 8 has been shown as a dashed l i n e f o r comparison. I t i s to be noted that table XXII and f i g u r e 18 give the absorbed dose per pion STOPPING per cm2, NOT the absorbed dose per pion INCIDENT per cm2. The corrections of t a b l e XIII and XIV and figures 10 and 11 have not been included i n t h i s paper; the method of c o r r e c t i o n would be i d e n t i c a l with that f o r table XII and f i g u r e 8 . The r a t i o of peak to entrance dose would be decreased i n a l l cases as i l l u s t r a t e d by f i g u r e 18 but, f o r the b i o l o g i c a l l y e f f e c t i v e dose c a l c u l a t i o n s , the e f f e c t i v e RBEs and OERs, as - 100 -TABLE XXII ABSORBED DOSE PER PION STOPPING PER CM2 DUE TO A BEAM OF 82.2 MeV NEGATIVE PIONS IN WATER, RANGE STRAGGLING INCLUDED AND CORRECTED FOR INTERACTIONS OF PIONS WITH NUCLEI OF THE MEDIUM Depth i n water X P (cm) Absorbed dose/pion stopping/cm 2 (MeV-cm2/g) Primary pions only P r i . pions plus 29.0 MeV " s t a r s " 0.0 3.0 6.0 8.0 10.0 11.2 12.4 13.45 14.50 15.55 16.60 17.10 17.60 17.84 18.08 18.32 18.56 18.80 19-04 19.28 19.52 19.76 20.00 20.24 20.48 20.72 20.96 21.20 21.44 21.68 21.92 22.16 22.40 22.90 23.40 24.45 25.50 26.55 27.60 28.80 3.591 3.594 3.635 3.692 3.788 3.870 3.982 4.111 4.291 • 577 .052 .373 .785 .021 .302 .664 7.162 7.863 8.771 9.704 10.233 9.823 8.249 4.5 5. 5 6, 6  6, 881 488 698 672 0.215 0.0554 0.0114 0.0018 0.0003 0.0000 3. 3. 3. 3. 3. 3. 591 • 594 635 692 788 870 3.982 4.115 4.311 4.641 5.215 5.621 6.165 6.493 6.904 7.483 8.429 l0.i4o 13.179 17.837 23.346 27.505 27.816 23.563 16.601 9.831 5.080 2.492 1.322 0.831 0.6o4 0.472 0.380 0.248 O.163 0.0634 0.0199 0.0038 0.0003 0.0000 Energy absorbed per pion stopping (MeV) 92.39 121.43 FIGURE 18 ABSORBED DOSE CURVE DUE TO A BEAM OF 8 2 . 2 MeV NEGATIVE PIONS IN WATER — WITH CORRECTION FOR LOSS OF PIONS BY NUCLEAR INTERACTIONS (For 29 MeV " s t a r s " only) - 102 -given i n table XV, would not be changed s i g n i f i c a n t l y . When i n t e r a c t i o n s of pions with n u c l e i of the medium are taken in t o account, the continuous spectra to y i e l d constant doses over a s p e c i f i e d treatment thickness must be changed. The surface dose i s , of course, increased r e l a t i v e to the constant dose over the s p e c i f i e d depth. When the e f f e c t of nuclear i n t e r a c t i o n s i s included, i t i s necessary to d i s t i n g u i s h between the spectrum of stopping pions and the spectrum of incident pions. Column 2 of table XXIII gives the range spectrum of stopping pions used to obtain the absorbed doses of column 3 . These doses were c a l c u l a t e d from equation (40) but the values of J„ for a monoenergetic beam were taken from column 3 of table XXII, r e p l a c i n g those of column 3 , table XII, as used i n the previous c a l c u l a t i o n of absorbed dose due to a continuous spectrum (table XV and fi g u r e 1 5 ) . The absorbed doses at each depth, scaled to an average dose of 100 rads from 1 2 . 3 2 to 1 9 - 5 2 cm, have been p l o t t e d i n figu r e 19 with the corresponding curve from f i g u r e 15 included f o r comparison. The con-version from the range spectrum of stopping pions of column 2 , table XXIII, to the corresponding energy spectrum of incident pions i s shown i n table XXIV. In t h i s t a b l e , each value f o r the range spectrum of incident pions i n column 5 0 . 0 1 H 7 R i s the corresponding value of column k m u l t i p l i e d by e . The range spectrum of column 5 has been converted by means of equation (4-5) to the energy spectrum of column 6 and t h i s energy spectrum has been scaled i n column 7 to the absolute energy spectrum required to produce the dose data of column 4 , table XXIII and fi g u r e 1 9 . This scaled spectrum i s shown as the ins e t i n figu r e 1 9 . Tables XVII and XVIII and figures l 6 and 17 have not been corrected f o r l o s s of pions by i n t e r a c t i o n s with n u c l e i of the medium but figu r e 19 i l l u s t r a t e s the q u a l i t a t i v e e f f e c t to be expected i n a l l cases. The only - 103 -TABLE XXIII ABSORBED DOSE DISTRIBUTION DUE TO "SHAPED" CONTINUOUS PION SPECTRUM ASSUMING 2 9 . 0 MeV "STARS" AND CORRECTED FOR INTERACTIONS OF PIONS WITH NUCLEI OF THE MEDIUM Range and depth dF/dR (stopping) ABSORBED DOSE From spectrum of column 2 Scaled to average of 100 rads* (cm) 0.32 4 . 16 6 . 0 8 8 . 0 0 8 . 9 6 9 - 9 2 1 0 . 8 8 1 1 . 3 6 1 1 . 8 4 12.32 1 2 . 8 0 13 . 2 8 13 . 7 6 1 4 . 2 4 1 4 . 7 2 1 5 . 2 0 1 5 . 6 8 1 6 . 1 6 16.64 1 7 . 1 2 1 7 . 6 0 1 8 . 0 8 1 8 . 5 6 19.04 1 9 . 5 2 2 0 . 0 0 20.48 2 0 . 9 6 2 1 . 4 4 2 1 . 9 2 2 2 . 8 8 23.84 T o t a l no. stop-ping/ cm2 (cm - 3) 0 . 0 0 0 0 . 2 0 0 0 . 3 7 2 0 . 3 7 7 0 . 3 8 3 0 . 3 9 5 0 . 4 0 8 0 . 4 2 3 0 . 4 4 0 0 . 4 5 8 0 . 4 7 7 0 . 4 9 8 0 . 5 2 2 0 . 5 4 9 0 . 5 7 9 O .616 O .656 0 . 6 9 7 0 . 7 6 2 1 . 0 0 0 0 . 9 0 0 0 . 5 0 0 0 . 0 0 0 5 . 3 8 2 (MeV/g) 1 9 . 4 6 1 9 . 9 6 2 0 . 4 3 2 1 . 3 8 2 2 . 2 1 23 . 5 8 27 .76 31.84 3 5 . 2 8 3 6 . 7 7 3 7 . 0 8 3 7 - 0 8 3 7 - 0 7 3 7 . 0 7 3 7 . 0 7 3 7 . 0 8 3 7 . 0 9 3 7 . 0 9 3 7 - 1 0 3 7 . 1 1 3 7 . 1 1 3 7 . 0 6 3 6 . 9 6 3 6 . 9 8 3 7 . 0 8 3 5 . 2 5 2 8 . 3 8 1 7 . 2 9 7-2. 37 44 O .58 0 . 2 5 (rads) 5 2 . 5 2 5 3 . 8 7 55.14 5 7 . 7 0 5 9 . 9 4 63.64 7 4 . 9 2 8 5 . 9 4 9 5 . 2 2 9 9 . 2 4 1 0 0 . 0 8 1 0 0 . 0 8 1 0 0 . 0 5 1 0 0 . 0 5 1 0 0 . 0 5 1 0 0 . 0 8 1 0 0 . 1 1 . 1 0 0 . 1 1 1 0 0 . 1 3 1 0 0 . 1 6 1 0 0 . 1 6 1 0 0 . 0 2 9 9 . 7 6 9 9 - 8 1 1 0 0 . 0 8 95-14 7 6 . 6 0 46.67 1 9 . 8 9 6 . 5 9 1 0, 57 67 * Numerical values of column 3 m u l t i p l i e d by 2 . 6 9 9 to give average dose of 100 rads from 12.32 to 19-52 cm. FIGURE 19 ABSORBED DOSE CURVE DUE TO A "SHAPED" SPECTRUM OF NEGATIVE PIONS — WITH CORRECTIONS FOR LOSS OF PIONS BY NUCLEAR INTERACTIONS (For 29 MeV " s t a r s " only) TABLE XXIV CONVERSION AND SCALING OF CONTINUOUS PION SPECTRUM OF TABLE XXIII Range, energy and stopping power data Spectrum f o r constant for pions (from table VI) absorbed dose Range Energy Stopping dF/dR dF/dR dF/dE 1 . 6 8 5 x 1 0 8 R E power, S stopping incident incident x column 6 1 2 3 4 5 6 7 (cm) (MeV) (MeV/cm) (cm - 3) (cm - 3) (cm - 2-MeV _ 1) (l0 7/cm 2-MeV) 1 0 . 8 8 5 5 . 8 5 3 . 1 8 6 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 0 . 0 0 1 1 . 3 6 5 7 - 3 5 3.145 0 . 2 0 0 0 . 2 3 6 0 . 0 7 5 2 1 . 2 6 1 1 . 8 4 5 8 . 8 5 3 . 1 0 6 0 . 3 7 2 0.443 0 . 1 4 2 5 2 . 4 0 1 2 . 3 2 6 0 . 3 3 3 . 0 7 0 0 . 3 7 7 0 . 4 5 2 0 . 1 4 7 2 2.48 1 2 . 8 0 6 1 . 7 9 3 . 0 3 6 0 . 3 8 3 0 . 4 6 2 0 . 1 5 2 3 2 . 5 7 13 . 2 8 6 3 . 2 4 . 3 . 0 0 3 0 . 3 9 5 0 . 4 8 0 0 . 1 5 9 9 2 . 6 9 1 3 - 7 6 64 .68 2 . 9 7 3 0 . 4 0 8 0 . 4 9 9 0 . 1 6 8 0 2 . 8 3 1 4 . 2 4 6 6 . 1 0 2 . 9 4 3 0.423 0 . 5 2 1 0 . 1 7 7 2 2 . 9 9 14 . 72 6 7 . 5 0 2 . 9 1 5 0 . 4 4 0 0 . 5 4 6 0 . 1 8 7 4 3 . 1 6 1 5 . 2 0 6 8 . 9 0 2 . 8 8 9 0 . 4 5 8 0 . 5 7 3 0 . 1 9 8 2 3 . 3 4 1 5 . 6 8 7 0 . 2 8 2 . 8 6 3 0 . 4 7 7 0 . 6 0 1 0 . 2 1 0 3 . 5 4 1 6 . 1 6 7 1 . 6 5 2 . 8 3 9 0 . 4 9 8 0 . 6 3 2 0 . 2 2 2 3 . 7 5 16.64 7 3 . 0 0 2 . 8 1 6 0 . 5 2 2 0 . 6 6 7 0.237 3 . 9 9 17 . 12 7 4 . 3 5 2.793 0 . 5 4 9 0 . 7 0 6 0 . 2 5 3 4 . 2 6 1 7 . 6 0 7 5 . 6 8 2 . 7 7 2 0 . 5 7 9 0 . 7 5 0 0 . 2 7 1 4 . 5 6 1 8 . 0 8 7 7 . 0 1 2 . 7 5 1 O .616 0 . 8 0 4 0 . 2 9 2 4 . 9 2 1 8 . 5 6 7 8 . 3 3 2.731 O .656 0 . 8 6 2 0 . 3 1 6 5.32 1 9 . 0 4 7 9 . 6 3 2 . 7 1 2 0 . 6 9 7 0 . 9 2 2 0.340 5 - 7 3 1 9 . 5 2 8 0 . 9 3 2 . 6 9 3 0 . 7 6 2 1 . 0 1 5 0 . 3 7 7 6 .35 2 0 . 0 0 8 2 . 2 2 2 . 6 7 6 1 . 0 0 0 1.342 0 . 5 0 1 8 . 4 5 20.48 8 3 . 5 0 2 . 6 5 8 0 . 9 0 0 1 . 2 1 6 0 . 4 5 8 7-71 2 0 . 9 6 84 .77 2 . 6 4 2 0 . 5 0 0 0 . 6 8 0 0 . 2 5 8 4 . 3 4 2 1 . 4 4 86.04 2 . 6 2 6 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0.00 • To t a l number per cm2 5 . 3 8 2 6 . 9 1 6 1 1 6 . 5 - io6 -s i g n i f i c a n t change i n the dose curve i s the increase i n dose near the surface. There i s , however, an appreciable increase i n the number of incident pions required to produce a given constant dose at a s p e c i f i e d depth. Figures 18 and 19 show the changes i n the dose d i s t r i b u t i o n s (as compared with figures 8 and 1 5 ) due to increased primary i o n i z a t i o n when the inc i d e n t beam i s increased to allow f o r loss of pions by nuclear i n t e r a c t i o n s i n the medium. These curves do not, however, include any dose contributions due to pions a f t e r they have i n t e r a c t e d with n u c l e i or due to any of the secondary products r e s u l t i n g from the i n t e r a c t i o n s . The p a r t i c l e s r e s u l t i n g from the i n t e r a c t i o n s include secondary charged pions, neutral pions, neutrons and a l l the charged p a r t i c l e s which are released i n nuclear d i s i n t e -grations. Part of the energy of these p a r t i c l e s w i l l be absorbed outside the primary beam but most of the k i n e t i c energy of the protons and a l l that of the heavier charged p a r t i c l e s w i l l be absorbed i n the immediate v i c i n i t y of . the nuclear i n t e r a c t i o n i n which they are produced. The most u s e f u l c a l c u l a -tions of the dose contributions due to nuclear i n t e r a c t i o n s have been made by Monte Carlo methods since the energy absorption i n the medium often involves several sequential processes which can be best followed by the Monte Carlo method. There are, however, no c a l c u l a t i o n s reported which are applicable to the centre of a broad beam which i s the case treated i n t h i s paper. In a d d i t i o n to removal of incident pions from the beam by i n t e r -actions with n u c l e i of the medium, some pions are l o s t by decay i n the medium before coming to r e s t . This decay r e s u l t s i n increased muon contamin-ation but the l o s s of pions i s l e s s than k% for a 100 MeV pion beam and s t i l l smaller f o r lo v e r energy beams (see appendix A). I t has been neglected i n view of the uncertainty i n the muon contamination of the incident beam and - 107 -the uncertainty i n the pion l o s s "by nuclear i n t e r a c t i o n s . 7.1.3 Dose Contributions from Neutrons and Photons Released i n the Nuclear D i s i n t e g r a t i o n s The dose contributions of the charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n s which r e s u l t from the capture of stopped negative pions by oxygen n u c l e i have been included i n the dose c a l c u l a t i o n s of the -previous sections since more than 50% of the k i n e t i c energy of these charged p a r t i c l e s i s absorbed within 0 . 1 cm and nearly 90% within 2 cm of the nuclear d i s i n t e g r a t i o n i n which they are released (see table VIII and reference 30), i . e . , almost a l l the k i n e t i c energy of the charged p a r t i c l e s i s absorbed i n the high dose region. On the other hand, dose contributions due to neutrons released i n the nuclear d i s i n t e g r a t i o n s and to photons emitted by excited r e s i d u a l n u c l e i have not been included i n the c a l c u l a t i o n s . Even though the energy of the neutrons and photons released per d i s i n t e g r a t i o n i s approxi-mately twice the k i n e t i c energy of the charged p a r t i c l e s (see table I ) , much of t h i s energy i s absorbed outside the high dose region. Curtis and Raju ( 1 5 ) and Turner et a l ( 2 7 ) have c a l c u l a t e d the neutron co n t r i b u t i o n to the peak dose f o r nearly monoenergetic pion beams. These estimates are included i n table XXV. The c o n t r i b u t i o n of photons emitted by excited r e s i d u a l n u c l e i has been neglected since i t i s small i n comparison with the other contributions to the peak dose. Monte Carlo c a l c u l a t i o n s by S c h i l l a c i and Roeder (3l) ind i c a t e that the maximum dose due to these photons i s l e s s than 3% of the dose due to neutrons i n the peak dose region. TABLE XXV ESTIMATED PERCENTAGE CONTRIBUTIONS TO SURFACE AND PEAK ABSORBED DOSES Curtis and Raju ( 15) Turner et a l (27) Averages# Cu r t i s and Raju ( 1 5 ) Turner et a l (27) Perry and Hynes ( 2 6 ) Averages## Pion energy (MeV) 96.2 84.3 Mean pion range (cm) 25 .4 20 .8 Percentage contributions Surface dose Peak dose Primary TT-Secondary charged p a r t i c l e s Neutrons \i~ contamination e~ contamination 68 (TO)** 9 ( 9 ) 23 (21) * 67 13 20 67.5 11.0 21.5 30 50 6 4 10 30 .9 50.2 6 . 8 4.3 7 -8 3 10 30.5 50.0 6.5 4.0 9-0 100 100 100.0 100 100.0 100.0 * Calculated assuming contamination as shown i n table XX. See table XXI f o r c a l c u l a t i o n s . ** Bracketed values scaled from figure 3 of reference 15-# Averages consistent with an incident beam composed of 64% pions, 12% muons and 24% electrons. Cf. table XX. ## Perry and Hynes' values not considered i n the averages. - 109 -7 . 2 . Ratio of the Maximum Dose i n the High Dose Volume to the Dose at the  Surface of the Medium. Comparison with Experimental Measurements. The r a t i o of the absorbed dose i n the peak of the pion curve to that at the surface of the medium i s considered f i r s t f o r a monoenergetic beam since t h i s i s the only case which can be compared with presently a v a i l -able experimental measurements. The dose r a t i o of i n t e r e s t f o r radiotherapy, namely, the r a t i o of the b i o l o g i c a l l y e f f e c t i v e dose i n the constant high-dose region due to a "shaped" continuous pion spectrum to the dose at the surface of the medium, i s considered b r i e f l y l a t e r . The c a l c u l a t e d r a t i o of peak to surface absorbed dose f o r a mono-energetic beam of 20 cm range as shown i n table XXII and fi g u r e 18 i s 7 - 7 5 . Measured r a t i o s as reported by Cur t i s and Raju ( 1 5 ) 5 Turner et a l ( 2 7 ) and Perry and Hynes ( 2 6 ) are 2 . 8 7 , 2.13 and 1 . 6 7 , r e s p e c t i v e l y . The measured r a t i o s , however, include the e f f e c t s of (a) contamination of the incident beam, (b) i n t e r a c t i o n of pions with n u c l e i of the medium before coming to r e s t , (c) energy released i n nuclear d i s i n t e g r a t i o n s when the pions come to r e s t , (d) momentum spread of the experimental beam, (e) l a t e r a l small-angle s c a t t e r ( i . e . , multiple scatter) on the dose measured at the peak of a small experimental beam. These factors must be taken i n t o account i n making any comparisons between calculated and measured r a t i o s . Table XXVT shows the comparison a f t e r appropriate corrections have been made, both to the c a l c u l a t e d and to the experimental r e s u l t s , to make them comparable. The c a l c u l a t e d r a t i o was c a l c u l a t e d on the assumption of a 29 MeV . " s t a r " . To be comparable with measured r a t i o s , the c a l c u l a t e d value must - 110 -be corrected f o r (a) lo s s of pions before coming to re s t by in t e r a c t i o n s with n u c l e i of the medium, (b) contributions of neutrons from nuclear d i s i n t e g r a t i o n s to the peak dose, (c) contributions of muon and electron contamination, both at the surface and at the peak. The c o r r e c t i o n f o r loss of pions by nuclear i n t e r a c t i o n s i n the medium.has already been made (table XXII and figu r e 1 8 ) . The corrections f o r the neutron c o n t r i b u t i o n and f o r the contributions of the contaminants, as shown i n table XXVI, are based on the average values given i n table XXV. The f i n a l value of 6 . 5 0 i s the c a l c u l a t e d r a t i o f o r a contaminated monoenergetic beam of 20 cm range. The experimental values have a l l been corrected to a range of 20 cm for purposes of comparison. This involves two corrections which can be i l l u s t r a t e d f o r the beam measured at the Lawrence Radiation Laboratory ( 1 5 ) -The mean i n i t i a l pion energy was 9 6 . 2 MeV and the corresponding range 25.4 cm. To correct to 20 cm range, the surface dose has to be m u l t i p l i e d by a f a c t o r 2 . 6 8 / 2 . 5 1 since the stopping power of 9 6 . 2 MeV pions i s 2 . 5 1 MeV/cm as compared with 2 . 6 8 MeV/cm for 8 2 . 2 MeV pions. The surface dose 0.147 x 5.4 must, however, be decreased by a f a c t o r e to correct f o r pions l o s t by nuclear i n t e r a c t i o n s i n the greater range of the experimental beam. For the LRL beam, the net c o r r e c t i o n f a c t o r to be applied to the surface dose i s 0 . 9 8 5 ; the c o r r e c t i o n to the peak to surface dose r a t i o i s the r e c i p r o c a l of t h i s f a c t o r . The net c o r r e c t i o n i s small i n a l l the experimental cases considered. The c o r r e c t i o n of the experimental values f o r l a t e r a l spread of the - I l l -beam due to multiple Coulomb s c a t t e r i n g i s more important. This c o r r e c t i o n i s necessary to permit comparison with the calculated value since the l a t t e r i s v a l i d f o r the centre of a large f i e l d ( i . e . , the e f f e c t of multiple sca t t e r n e g l i g i b l e ) whereas the experimental values were measured f or small f i e l d s . The c o r r e c t i o n depends on the s i z e of the f i e l d and, a l s o , on the s i z e of the dosimeter used. In making the c o r r e c t i o n , i t has been assumed (15) that multiple s c a t t e r i n g r e s u l t s i n a Gaussian l a t e r a l d i s t r i b u t i o n of a p e n c i l beam and that the root mean square deviation (as projected on any axis perpendicular to the i n i t i a l beam) i s given by 0.95 a (perp.) = 0.076 x R cm (50) at the range of the beam. Knowing the r.m.s. deviation, the dose at any distance from the nominal edge of the beam r e l a t i v e to the dose at the centre of a large beam can be c a l c u l a t e d . For a large dosimeter, t h i s dose must be integrated over the cross-section of the dosimeter to f i n d the dose which would be measured at the centre of a large f i e l d . The data on the si z e of the experimental beams, the cross-sections of the dosimeters used and the resultant corrections are shown i n table XXVI. In the case of the CERN beam, i t was necessary to correct the dose reading at the surface, as wel l as at the peak, since the incident beam did not cover the f u l l cross-section of the dosimeter. In a l l cases, i t was assumed that the f i e l d was uniformly i r r a d i a t e d at the surface. The l a r g e s t corrections to the experimental values are those required to correct them to the corresponding values f o r a monoenergetic beam. In f a c t , t h i s c o r r e c t i o n was determined f o r each beam by making the inverse c o r r e c t i o n to the c a l c u l a t e d value. For example, f or the beam measured by Curti s and Raju (15) at the Lawrence Radiation Laboratory, the standard deviation i n range r e s u l t i n g from the standard deviation i n - 112 -TABLE XXVI COMPARISON OF CALCULATED AND EXPERIMENTAL VALUES OF THE RATIO OF PEAK DOSE TO SURFACE DOSE FOR A BEAM OF 8 2 . 2 MeV NEGATIVE PIONS IN WATER CALCULATED ( A l l c a l c u l a t i o n s assume 29 MeV "stars") Ratio without any corrections (table XII and figure 8) 1 0 . 4 Corrected only f o r l o s s of pions by nuclear i n t e r a c t i o n s (table XXII and f i g u r e 1 8 ) 7 . 7 5 Corrected f o r neutron c o n t r i b u t i o n at the peak (co r r e c t i o n f a c t o r = 0 . 8 7 0 / 0 . 8 0 5 — see table XXV) 8 . 3 7 Corrected for muon and e l e c t r o n contributions at the surface ( c o r r e c t i o n f a c t o r = 0 . 6 7 5 — see table XXV) 5 - 6 5 Corrected f o r muon and e l e c t r o n contributions at the peak (c o r r e c t i o n f a c t o r = 1 . 0 0 / 0 . 8 7 — see table XXV) 6.50* EXPERIMENTAL LRL CERN . NIMROD Reference ( 1 5 ) ( 2 7 ) ( 2 6 ) Beam momentum (MeV/c) 190 ± 5 175 ± 4 . 7 156 ± 11 Energy (MeV) 9 6 . 2 ± 4 . 0 8 4 . 3 ± 3 . 7 6 9 . 8 ± 8 . 7 Range (cm) 2 5 . 4 ± 1 . 6 2 0 . 8 ± 1 . 4 1 5 . 5 ± 3 . 0 Beam s i z e Width = 7-6 cm E l l i p t i c a l (Semi-axes = 2 . 0 and 3 - 5 cm at surface) Diameter of 80% dose at range = 2 . 0 cm Detector Small 6 . 0 cm diam. 1 . 0 cm di am. Ratio of peak dose to surface dose (a) Uncorrected** 2 . 8 7 2.1-3 1 . 6 7 (b) Cprrected to 20 cm range 2 . 8 7 x 1 . 0 8 3 x ( 2 . 5 1 / 2 . 6 8 ) = 2 . 9 1 2 . 1 3 1 . 6 7 x 0 . 9 3 6 x ( 2 . 8 7 / 2 . 6 8 ) = 1 . 6 8 (c) Corrected f o r mul-t i p l e s c a t t e r and detector s i z e 2 . 9 1 / 0 . 9 9 = 2 . 9 4 2 . 1 3 x ( 0 . 7 3 2 / 0 . 6 2 3 ) = 2.46 1 . 6 8 / 0 . 9 6 2 = 1 . 7 5 (d) Corrected to mono-energetic ( 8 2 . 2 MeV) beam 7.13 5.48 6 . 3 8 * Assuming an r.m.s. deviation i n momentum of 0.5%, the c a l c u l a t e d r a t i o i s reduced to 6 . 0 ** Scaled from published graphs: figure 3 , reference 1 5 ; f i g u r e 2 , reference 2 7 ; f i g u r e 3, reference 2 6 . - .113 -momentum of 5 MeV/c was 1.6 cm. When t h i s range spread (assumed Gaussian) was included i n the c a l c u l a t i o n s , the peak dose of 27.8 MeV-cm2/g as shown i n table XXII and fi g u r e 18 was reduced to 11.47 MeV-cm2/g, i . e . , the peak dose was reduced by a f a c t o r of 0.413. I t was assumed that the measured value would have been increased by the r e c i p r o c a l of t h i s f a c t o r , had i t been measured f o r a s t r i c t l y monoenergetic beam. The corrections of a l l the experimental values to the corresponding values f o r monoenergetic beams are shown i n the l a s t l i n e of table XXVT. The corrected experimental values of 7-13, 5-48 and 6.38 should be comparable with each other and with the corrected c a l c u l a t e d value of 6.50 since a l l values are presumably f o r a contaminated monoenergetic beam of 20 cm range. Differences may r e s u l t from differences i n contamination but t h i s does not appear to be a source of large discrepancy since the estimates of the contamination of the beams were very s i m i l a r . Differences may also r e s u l t from error i n the estimate of the momentum spread of each beam; i n two cases, at l e a s t , t h i s estimate was admittedly uncertain. Since the c o r r e c t i o n f o r momentum spread was the la r g e s t c o r r e c t i o n introduced, i t appears to be the most probable cause of discrepancy. The good agreement between the experimental values and the c a l c u -l a t e d value, a f t e r a l l corrections have been made, suggests that the c a l c u -l a t i o n s and the measurements can be accepted with some confidence. I t suggests, f u r t h e r , that f a i l u r e to include the dose contributions of secon-dary pions and of secondary products r e s u l t i n g from pion i n t e r a c t i o n s with n u c l e i has not introduced a large error i n the cal c u l a t e d value, i . e . , these contributions may increase the peak dose as much or nearly as much as the surface dose and, the r e f o r e , may not change the peak to surface absorbed dose r a t i o s i g n i f i c a n t l y . - I l k -In p r a c t i c e , i t w i l l not be possible to obtain a peak to surface dose r a t i o as high as the c a l c u l a t e d value of 6 . 5 0 since i t w i l l not be possible to obtain a s t r i c t l y monoenergetic beam. Assuming a standard deviation i n momentum of 0 . 5 % with a well-designed beam transport system, the corresponding standard deviation i n range f o r 20 cm range i s 0 . 2 5 cm ( i n a d d i t i o n to the 0 . 5 cm standard deviation due to range s t r a g g l i n g which has already been included). When t h i s a d d i t i o n a l range spread i s introduced, the c a l c u l a t e d peak to surface dose r a t i o of 6 . 5 0 i s reduced to about 6 . 0 . For the continuous spectrum used f o r fi g u r e 1 9 , the r a t i o of the constant absorbed dose i n the high dose region to the surface dose i s 1 0 0 / 5 2 . 3 = 1 . 9 1 - This i s f o r an uncontaminated beam and does not make any allowance f o r the neutron con t r i b u t i o n to the high dose region. I f the average percentage dose contributions as shown i n table XXV are assumed to be applicable f o r pions of a l l ranges, the contributions of contaminants and neutrons reduce t h i s r a t i o to 1 . 9 1 x ( 0 . 6 7 5 / 0 . 8 0 5 ) = 1 . 6 0 . For radiotherapy, the quantity of p r a c t i c a l i n t e r e s t i s the r a t i o of the b i o l o g i c a l l y e f f e c t i v e dose i n the constant dose region to that at the surface. The most important but most uncertain f a c t o r i n determining t h i s r a t i o i s the e f f e c t i v e RBE. A very approximate c a l c u l a t i o n f o r the case p l o t t e d i n figu r e 17 suggests a value of about 2 . 5 or 2 . 6 f o r the r a t i o when a l l corrections have been made. 7 . 3 . F r a c t i o n of T o t a l Absorbed Energy which i s Absorbed i n the  Treatment Volume In radiotherapy, the energy absorbed i n the prescribed treatment volume i s u s e f u l ; energy absorption i n the patient outside t h i s volume i s , i n general, undesirable. The f r a c t i o n of the t o t a l absorbed energy which i s absorbed i n the treatment volume depends on the s i z e and depth of t h i s - 115 -volume. The.fraction has been estimated f o r the example p l o t t e d i n figure 1 9 -In f i g u r e 1 9 , the high dose volume was taken as the depth from 12 to 20 cm and the average dose over t h i s depth, due to primary pions and charged d i s i n t e g r a t i o n products only, as 100 rads. The average dose would, i n f a c t , be increased by the dose contributions from " s t a r " neutrons and from muon and ele c t r o n contamination. Using the average values shown i n table XXV, the average dose over the 8 cm depth was estimated as 1 0 0 / 0 . 8 0 5 = 1 2 4 rads. Therefore, i n the treatment depth Useful energy absorption/cm 2 = 124 x 8 = 992 g-rads. This i s the energy absorption from an incident beam of l . l 6 x 1 0 9 pions/cm 2 (see table XXIV). In estimating the t o t a l energy absorption i n the p a t i e n t , the rest mass energy released by disappearance of pions and muons must be taken into account i n ad d i t i o n to the k i n e t i c energy of the incident pions and of the contaminating p a r t i c l e s . I t was assumed that a l l incident pions caused di s i n t e g r a t i o n s of oxygen n u c l e i , i . e . , as previously stated, decay of pions to muons i n the medium was neglected. For each incident pion i t was assumed that T o t a l energy absorption/incident pion = [KE(TT) + 29 + 1 3 ] MeV (51) where the 29 MeV i s the k i n e t i c energy of the charged d i s i n t e g r a t i o n products and 1 3 MeV i s a d d i t i o n a l energy absorbed from neutrons and photons released by nuclear d i s i n t e g r a t i o n s . * * By i n t e g r a t i o n under a curve c a l c u l a t e d by means of a Monte Carlo program and published by Armstrong and Chandler (figure 3 , reference 2 8 ) , i t can be c a l c u l a t e d that the t o t a l energy absorbed per incident pion i n a cross-section of 50 cm radius to a 3 0 cm depth i s 1 3 3 . 2 MeV. The k i n e t i c energy of the inc i d e n t pion was 84.6 MeV and the program used the same data as that of Guthrie et a l ( 6 ) f o r the nuclear d i s i n t e g r a t i o n s , i . e . , 3 6 . 7 MeV k i n e t i c energy of charged p a r t i c l e s per " s t a r " . Assuming that a l l t h i s k i n e t i c energy i s absorbed wi t h i n the 50 cm radius c y l i n d e r , then 1 1 . 9 MeV (Continued on next page) - 116 -In estimating the energy absorbed from the incident muon contam-i n a t i o n , the k i n e t i c energy of the muons was cal c u l a t e d on the assumption that they had momentum equal to that of the incident pions. I t was assumed a l l t h i s k i n e t i c energy was absorbed i n the patien t . Further, i t was assumed that when a muon decayed i t s r e s t mass energy was, on the average, shared approximately equally by the electron and the two neutrinos r e s u l t i n g from the decay (see appendix A), i . e . , an average of about 35 MeV k i n e t i c energy per e l e c t r o n , and that a l l the k i n e t i c energy of the electron was absorbed i n the pa t i e n t . Therefore, T o t a l energy absorption/incident muon = [KE(y) + 35] MeV (52) For the contaminating electrons, a l s o , the k i n e t i c energy was cal c u l a t e d on the assumption that they had momentum equal to that of the incident pions. As already stated, the k i n e t i c energy i s i n the range from 100 to 200 MeV and more than h a l f t h i s energy would be d i s s i p a t e d by produc-t i o n of bremsstrahlung. Some of the electrons and some o f the bremsstrahlung energy would escape from the pa t i e n t . I t was assumed, however, that two-thi r d s of the k i n e t i c energy of the electrons would be absorbed i n the patient, i . e . , T o t a l energy absorption/incident electron = 2/3 x KE(e) (53) Combining equations ( 5 l ) 5 ( 5 2 ) and (53) on the basis of the assumed contamination (6U% pions, 12% muons and 2h% e l e c t r o n s ) , the energy absorption * (Continued from previous page) of neutron and photon energy from the di s i n t e g r a t i o n " must be absorbed within the c y l i n d e r to account for the t o t a l of 133.2 MeV. On the assumption made i n the present c a l c u l a t i o n , that the k i n e t i c energy of the charged d i s i n t e g r a t i o n products i s only 29 MeV, the t o t a l of the k i n e t i c energy of the neutrons and the e x c i t a t i o n energy of the r e s i d u a l nucleus may be expected to be a few MeV greater than shown i n table I. For t h i s reason, the 11.9 MeV con t r i b u t i o n from neutrons and photons has been increased to 13 MeV i n equation ( 5 l ) . - 117 -from pions and associated contaminants, i n MeV per incident pion, i s as follows: Energy absorption i n MeV from pions and contaminants/incident pion 0.12 r, m/ \ 1 0.24 (2 = T t Q t = [KE(ir) + 29 + 1 3 ] + ~ ^ [KE(y) + 35] + ^ - x KE(e) ( 5 4 ) Equation ( 5 4 ) has been used to calculate the t o t a l energy absorption i n the patient due to the energy spectrum of column 7 , table XXIV, i . e . , due to the spectrum required to obtain the s o l i d dose curve of figure 19• The data f o r the c a l c u l a t i o n are given i n table XXVII. The t o t a l energy absorp-t i o n of 20921 MeV/cm2 as shown i n the l a s t l i n e of column 7 of t h i s table was c a l c u l a t e d from T o t a l energy absorption from pions and contaminants TE 2 per cm2 of incident beam = / T^ q^ ^  dE ( 5 5 ) The t o t a l energy absorption of 20921 MeV/cm2 converts to 3352 g-rads/cm 2. This energy absorption r e s u l t e d i n a u s e f u l energy absorption of 992 g-rads/cm 2. Therefore, F r a c t i o n of t o t a l energy absorbed i n treatment volume = 9 9 2 / 3 3 5 2 = 0 . 2 9 6 This f r a c t i o n holds only f o r the p a r t i c u l a r example for which i t was c a l c u -l a t e d but may be taken as a t y p i c a l value. Since the e f f e c t i v e RBE i n the treatment volume i s greater than unity, the t o t a l energy absorption i n the patient f o r the b i o l o g i c a l l y e f f e c t -ive equivalent of an absorbed dose of 100 rads from 12 to 20 cm depth would be appreciably l e s s than the 3352 g-rads/cm 2 c a l c u l a t e d above — perhaps, 2500 g-rads/cm 2 or s l i g h t l y l e s s . - 118 -TABLE XXVII TOTAL ENERGY ABSORPTION DUE TO PION SPECTRUM OF TABLE XXIV, COLUMN 7 , PLUS ACCOMPANYING ELECTRON AND MUON CONTAMINATION Pion range 1 E(TT) 2 dF/dE* (incident) 3 E(y) 4 E(e) 5 T t o t * * 6 T t o t x dF/dE 7 (cm) (MeV) ( 1 07 per cm2-MeV) (MeV) (MeV) (MeV) ( l 07/cm 2) 1 0 . 8 8 1 1 . 3 6 1 1 . 8 4 12.32 5 5 . 8 5 5 7 - 3 5 5 8 . 8 5 6 0 . 3 3 0 . 0 0 1 . 2 6 2.40 2.48 6 7 . 1 9 6 8 . 8 8 7 0 . 5 7 7 2.23 1 3 6 . 2 8 1 3 8 . 4 1 1 4 0 . 5 3 1 4 2 . 6 0 1 5 1 . 1 1 5 3 . 4 1 5 5 . 8 1 5 8 . 1 0 . 0 1 9 3 . 3 3 7 3 . 9 3 9 2 . 1 1 2 . 8 0 1 3 . 2 8 1 3 . 7 6 1 4 . 2 4 6 1 . 7 9 6 3 . 2 4 6 4 . 6 8 6 6 . 1 0 2 . 5 7 2 . 6 9 2 . 8 3 2 . 9 9 7 3 . 8 7 7 5 . 5 0 7 7 . 1 1 7 8 . 6 9 144.64 146.64 1 4 8 . 6 2 1 5 0 . 5 6 1 6 0 . 4 1 6 2 . 6 1 6 4 . 9 1 6 7 . 1 4 1 2 . 2 4 3 7 . 4 4 6 6 . 7 4 9 9 . 6 1 4 . 7 2 1 5 . 2 0 1 5 . 6 8 1 6 . 1 6 6 7 . 5 0 6 8 . 9 0 7 0 . 2 8 7 1 . 6 5 3 . 1 6 3 . 3 4 3 . 5 4 3 . 7 5 8 0 . 2 5 8 1 . 8 1 8 3 . 3 4 8 4 . 8 6 152.46 1 5 4 . 3 5 1 5 6 . 2 0 1 5 8 . 0 3 1 6 9 . 2 1 7 1 . 4 1 7 3 . 5 1 7 5 . 6 5 3 4 . 7 5 7 2 . 5 6 1 4 . 2 6 5 8 . 5 16.64 1 7 . 1 2 1 7 . 6 0 1 8 . 0 8 7 3 . 0 0 7 4 . 3 5 7 5 - 6 8 7 7 - 0 1 3 . 9 9 4 . 2 6 4 . 5 6 4 . 9 2 8 6 . 3 6 8 7 . 8 5 8 9.32 9 0 . 7 9 1 5 9 . 8 3 1 6 1 . 6 1 1 6 3 . 3 6 1 6 5 . 1 1 1 7 7 - 7 1 7 9 . 8 1 8 1 . 8 1 8 3 . 9 7 0 9 . 0 • 7 6 5 . 9 8 2 9 . 0 9 0 4 . 8 1 8 . 5 6 1 9 . 0 4 1 9 . 5 2 2 0 . 0 0 7 8 . 3 3 7 9 . 6 3 8 0 . 9 3 8 2 . 2 2 5 . 3 2 5 . 7 3 6 . 3 5 8 . 4 5 9 2 . 2 5 9 3 . 6 8 9 5 . 1 0 9 6 . 5 2 1 6 6 . 8 3 1 6 8 . 5 2 1 7 0 . 2 0 1 7 1 . 8 6 1 8 5 - 9 1 8 7 . 9 1 8 9 . 9 1 9 1 . 9 989.O 1 0 7 6 . 7 1 2 0 5 . 9 1 6 2 1 . 6 2 0.48 2 0 . 9 6 2 1 . 4 4 8 3 . 5 0 8 4 . 7 7 8 6.04 7-71 4 . 3 4 0 . 0 0 9 7 - 9 3 9 9-31 1 0 0 . 7 0 1 7 3 . 5 1 1 7 5.13 1 7 6 . 7 5 1 9 3 . 8 1 9 5 . 7 1 9 7 . 7 1 4 9 4 . 2 8 4 9 . 3 0 . 0 T o t a l energy (MeV/cm2) = = 2 0 9 2 1 . * Energy spectrum for dose curve ( s o l i d l i n e ) of fig u r e 19 — from table XXIV, column 7 . data ** Calculated from equation ( 5 4 ) . - 119 -l.h. E f f e c t i v e REE and E f f e c t i v e OER The e f f e c t i v e RBEs and OERs cal c u l a t e d on d i f f e r e n t assumptions f o r an uncontaminated beam of monoenergetic negative pions have been given i n table XV. While the " s t a r " energy assumed had a s i g n i f i c a n t e f f e c t on the peak dose, i t had r e l a t i v e l y l i t t l e e f f e c t on the e f f e c t i v e RBE and OER cal c u l a t e d at the peak-, the RBE data used f o r the c a l c u l a t i o n s had a much greater e f f e c t . As would be expected, the calculated RBE i s higher and the OER lower on the shallow side of the Bragg peak than on the deep side. With respect to RBE, t h i s conclusion has been confirmed q u a l i t a t i v e l y by experi-mental measurements made by B a a r l i and Bianchi (32) using a nearly mono-energetic pion beam produced i n the CERN accelerator.. I n c l u s i o n of the contr i b u t i o n to the peak dose of neutrons released i n the nuclear d i s i n t e g r a t i o n s should have l i t t l e e f f e c t on the c a l c u l a t e d values of RBE and OER since the neutrons contribute r e l a t i v e l y high LET r a d i a t i o n . For a contaminated beam, however, the muon and el e c t r o n c o n t r i -butions would decrease the e f f e c t i v e RBE and increase the e f f e c t i v e OER i n the Bragg peak. For the contributions of neutrons and contaminants assumed i n t h i s paper, a rough c a l c u l a t i o n indicates that these contributions would decrease the e f f e c t i v e RBE i n the Bragg peak by l e s s than 10% and would increase the e f f e c t i v e OER by l e s s than 3%. Table XXVIII i s a summary of measured and ca l c u l a t e d values of e f f e c t i v e RBE and e f f e c t i v e OER f o r negative pions which have been published by various i n v e s t i g a t o r s . The experimental values were a l l measured with pion beams with a r e l a t i v e l y small energy spread. The spread of the measured RBE values emphasizes the marked dependence of the RBE f o r a given r a d i a t i o n on the b i o l o g i c a l e f f e c t considered and on the conditions of - 120 -TABLE XXVIII PUBLISHED VALUES OF EFFECTIVE RBE AND EFFECTIVE OER IN BRAGG PEAK FOR NEARLY MONOENERGETIC BEAMS OF NEGATIVE PI-MESONS B i o l o g i c a l i n d i c a t o r Pion energy (MeV) Dose rate (rads/hr) RBE OER Reference Calculated f o r a contaminated beam s i m i l a r to that from 1 8 4-inch synchrocyclotron at LRL 1 . 8 - 2 . 0 ( 3 3 ) Estimated from c a l c u l a t e d t i s s u e f o r uncontaminated range LET spectrum i n beam of 15 ± 2 cm 1 . 8 -2 . 3 * I . 9 6 -1 . 7 9 * ( 3 4 ) Polyploidy induction i n asc i t e s tumour c e l l s 90 ± 4 5-9 2 . 1 5 ( 3 5 ) P r o l i f e r a t i v e capacity of a s c i t e s tumour c e l l s (a) 5-day o l d tumours 90 ± k 5-9 5-4 ( 3 6 ) (b) 2-day o l d tumours 90 ± 4 25-30 2.9-5.4 ( 3 7 ) Root t i p growth i n V i c i a faba 90 ± 4 30 3 . 0 1 . 3 - 1 . 5 ( 3 8 ) Do TO ± 8 . 7 25 2 . 0 1 . 7 ( 3 9 ) Chromosome aberrations i n V i c i a faba root t i p s 90 ± it 30 2 . 4 - 3 . 8 1 . 7 - 1 . 9 (40 , 4 1 ) Mutations i n yeast 30 1.4 1-9 (42) P r o l i f e r a t i v e capacity of human kidney ( T - l ) c e l l s 30-60 2 . 2 - 2.4 1 . 5 - 1 . 9 (43) S u r v i v a l of spermatagonia i n mice 84 ± it It 3 . 7 (44) * RBE increased from 1 . 8 going through the Bragj to 2 . 3 and OER decreased from 1 . 9 6 to 1 . 7 9 i n 2; peak (from 13 to 17 cm). - 121 -i r r a d i a t i o n . The values of RBE i n table XXVIII tend to be higher than the values c a l c u l a t e d i n t h i s paper for monoenergetic pion beams. In view of the low i n t e n s i t y of a v a i l a b l e beams, i t was necessary to use r e l a t i v e l y s e n s i t i v e b i o l o g i c a l systems f o r the measurements of RBE which have been reported. I t i s possible that the b i o l o g i c a l effectiveness of pion r a d i a t i o n i s greater fo r these systems than for l e s s s e n s i t i v e systems. A l t e r n a t i v e l y , the RBE for given LET (stopping power) may be higher than assumed i n fi g u r e 9- This question can only be answered by measurements with high i n t e n s i t y beams under conditions which simulate as c l o s e l y as possible those which w i l l be relevant to radiotherapy. The c a l c u l a t e d values of OER are somewhat lower than most of the experimental measurements. Again, b i o l o g i c a l experiments with higher i n t e n s i t y beams w i l l be required to determine the appropriate values. The e f f e c t i v e values of RBE and OER shown on figures 16 and IT fo r broad energy spectra follow d i r e c t l y from the values c a l c u l a t e d f o r monoenergetic beams as given i n table XV. These values cannot be compared d i r e c t l y with experimental r e s u l t s since no measurements have been made with broad energy spectra but the q u a l i t a t i v e comparisons f o r monoenergetic or nearly monoenergetic beams would be expected to hold f o r continuous spectra. - 122 -8. CONCLUSIONS The c a l c u l a t i o n s of t h i s paper give q u a l i t a t i v e support to the claims made with respect to the advantages of beams of negative pi-mesons f o r radiotherapy, namely, (a) b e t t e r depth dose d i s t r i b u t i o n s , p a r t i c u l a r l y better b i o l o g i c a l l y e f f e c t i v e dose d i s t r i b u t i o n s , (b) a greater percentage of the t o t a l energy absorbed within the s p e c i f i e d treatment volume, (c) lower oxygen enhancement r a t i o . The c a l c u l a t i o n s also give some quantitative measures of the advantages to be gained. The r e l i a b i l i t y of these quantitive measures, however, i s l i m i t e d by (a) the accuracy of the input data, (b) the neglect of some dose contributions i n the d e t a i l e d dose c a l c u l a t i o n s . The accuracy of the c a l c u l a t i o n s i n section -3 of the dose c o n t r i -butions from charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n s at the end of the pion tracks depends only on the accuracy of the input data since the methods used were e s s e n t i a l l y exact. While there were uncertain-t i e s at low energies i n the range-energy-stopping power data used, t h i s probably had l i t t l e e f f e c t on the f i n a l r e s u l t s since, as already pointed out, only a small f r a c t i o n of the t o t a l energy absorption was within the range of the uncertain data. The other source of uncertainty i n the c a l c u l a t i o n s of section 3 i s the choice of data f o r the spectra of the charged p a r t i c l e s released i n the nuclear d i s i n t e g r a t i o n s . So f a r as the absorbed dose c a l c u -l a t i o n s of section 4 are concerned, the t o t a l energy released per TT- capture i s the most important f a c t o r ; the uncertainty i n t h i s f a c t o r i s shown i n - 123 -table X (p.38). However, the effect of t h i s uncertainty on the absorbed dose calculations i s l i m i t e d by the fact that only a f r a c t i o n of the absorbed energy i s contributed by the charged p a r t i c l e s from the nuclear disintegra-tions . The p r i n c i p a l l i m i t a t i o n on the accuracy of the absorbed dose •calculations of section 4 i s almost certainly due to neglect of (a) loss of pions by interactions with nuclei of the medium, (b) dose contributions from muon and electron contamination and (c) dose contributions from "star" neutrons. Apart from these omissions, inaccuracy arises only from errors i n the data of section 3 and these probably have a r e l a t i v e l y small e f f e c t . In the calculations of b i o l o g i c a l l y e f f e c t i v e doses i n section 55 the results are subject to a l l the uncertainties i n the absorbed dose calculations of section k plus a major uncertainty i n the selection of the RBE data used. There does not, however, at the present time appear to be any method of selecting the appropriate RBE data with any greater certainty. The dose di s t r i b u t i o n s for "shaped" continuous energy spectra of pions, as calculated i n section 6, are as r e l i a b l e as the input data from the previous sections and subject to the same l i m i t a t i o n s i n accuracy. This section emphasizes the necessity for radiotherapy of selecting an energy spectrum of the desired shape. An attempt has been made i n section 7 to estimate corrections due to the factors which were neglected i n the detailed calculations of the previous sections. The corrections have been based mainly on published experimental r e s u l t s . With reasonable estimates of the corrections, the agreement between the calculations and experimental measurements appears to be good. - 124 -Since computer programs were written to carry out the c a l c u l a t i o n s of sections 3 to 6 , these r e s u l t s can he r e v i s e d by re-running the program as b e t t e r input data become a v a i l a b l e . This does not apply, however, to the considerations of s e c t i o n 7- I t appears that the methods used i n t h i s s e c t i o n , i . e . , comparison with experimental r e s u l t s and with the r e s u l t s of Monte Carlo programs, f u r n i s h the most r e l i a b l e corrections for the factors which were neglected i n the d e t a i l e d c a l c u l a t i o n s . I t should be emphasized that , f o r radiotherapy, the most important requirement appears to be more r e l i a b l e b i o l o g i c a l data i n c l u d i n g , but not l i m i t e d t o , b e t t e r RBE data. - 125 -BIBLIOGRAPHY 1 . P. H. FOWLER and D. H. 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Proceedings o f the International Congress on Protection Against Accelerator and Space Radiation, A p r i l 2 6 - 3 0 , 1 9 7 1 , CERN, Geneva, Switzerland, CERN 7 1 - l 6 , volume 1 , 2 2 0 - 2 3 0 , 1 9 7 1 . 2 7 . J . E. TURNER, H. A. WRIGHT and R. N. HAMM and J . BAARLI, J. DUTRANNOIS and A. H. SULLIVAN, "Some Studies i n Pion Beam Dosimetry". Pro-ceedings of the I n t e r n a t i o n a l Congress on Protection Against Accelerator and Space Radiation, A p r i l 2 6 - 3 0 , 1 9 7 1 , CERN, Geneva, Switzerland, CERN 7 1 - l 6 , volume 1 , 2 3 1 - 2 4 5 , 1 9 7 1 . 2 8 . T. W. ARMSTRONG and K. C. CHANDLER, "Monte Carlo Calculations of the Dose Induced by Charged Pions and Comparison with Experiment". Radiation Research, 5 2 , 2 4 7 - 2 6 2 , 1 9 7 2 . - 127 -2 9 . P. N. DEAN and D. M. HOLM, "Pion Stopping Region V i s u a l i z a t i o n E x p e r i -ments". Radiation Research, 48, 2 0 1 - 2 0 5 , 1 9 7 1 . 30. J. R. DUTRANNOIS and R. N. HAMM, J . E. TURNER and H. A. WRIGHT, "Analysis of Energy Deposition i n Water around the S i t e of Capture of a Negative Pion by an Oxygen or Carbon Nucleus". Physics i n Medicine and Biology, 1 7 , 7 6 5 - 7 7 0 , 1 9 7 2 . 31. M. E. SCHILLACI and D. L. ROEDER, "Dose D i s t r i b u t i o n i n Tissue from Star Neutrons and Photons". P u b l i c a t i o n MP-3-154, Los Alamos S c i e n t i f i c Laboratory, U n i v e r s i t y of C a l i f o r n i a , Los Alamos, N. M., 1 9 7 2 . 32. J . BAARLI and M. BIANCHI, "Observed Variations of R.B.E. Values i n the Stopping Region of a 95 MeV Negative Pion Beam". International Journal of Radiation Biology, 2 2 , 1 8 3 - 1 8 6 , 1 9 7 2 . 3 3 . S. B. CURTIS, "Human Kidney C e l l Oxygen-Enhancement Ratios for Fast-Neutron Beams and a P r e d i c t i o n for Negative Pion Beams". P u b l i c -at i o n UCRL-19456, Lawrence Radiation Laboratory, U n i v e r s i t y of C a l i f o r n i a , Berkeley, C a l i f . , 1 9 7 0 . 34. P. D. HOLT, "A C a l c u l a t i o n of the L.E.T. D i s t r i b u t i o n from Negative P i Mesons i n Tissue". Proceedings of the International Congress on P r o t e c t i o n Against Accelerator and Space Radiation, A p r i l 2 6 - 3 0 , 1 9 7 1 , CERN, Geneva, Switzerland, CERN 7 1 - 1 6 , volume 1, 2 4 6 - 2 5 9 , 1 9 7 1 . 3 5 . W. D. L0UGHMAN, J . M. FE0LA, M. R. RAJU and H. S. WINCHELL, "RBE of TI- Beams i n the Bragg Peak Region Determined with Polyploidy Induction i n Mammalian C e l l s I r r a d i a t e d i n Vivo". Radiation Research, 3 4 , 5 6 - 6 9 , 1 9 6 8 . 3 6 . J . M. FE0LA, C. RICHMAN, M. R. RAJU, S. B. CURTIS and J . H. LAWRENCE, " E f f e c t of Negative Pions on the P r o l i f e r a t i v e Capacity of A s c i t e s Tumor C e l l s (Lymphoma) i n Vivo". Radiation Research, 3 4 , 70 - 7 8 , 1 9 6 8 . 37. J- M. FEOLA, M. R. RAJU, C. RICHMAN and J . H. LAWRENCE, "The RBE of Negative Pions i n 2-Day-01d Ascites Tumors". Radiation Research, 44., 637-648, 1 9 7 0 . 38. M. R. RAJU, N. M. AMER, M. GNANAPURANI and C. RICHMAN, "The Oxygen E f f e c t of TI ~ Mesons i n V i c i a faba". Radiation Research, 4 l , 135-144, 1 9 7 0 . 39- B. M. WINSTON and R. J . BERRY and D. R. PERRY, "Response of V i c i a faba to I r r a d i a t i o n under Aerobic and Hypoxic Conditions with the Neg-at i v e Pion Beam from Nimrod i n the "Plateau" and "Peak" I o n i z a t i o n Regions". B r i t i s h Journal of Radiology, 45_, 5 5 0 , 1972. 40. S. P. RICHMAN, C. RICHMAN, M. R. RAJU and B. SCHWARTZ, "Studies of . V i c i a faba Root Meristems I r r a d i a t e d with a T I ~ Beam". Radiation Research, Supplement 7 , 1 8 2 - 1 8 9 , 1967-- 128 -41. M. GNANAPURANI, M. R. RAJU and C. RICHMAN, "Chromatid Aberrations Induced by TI ~ Mesons in Vicia faba root meristem c e l l s " . Inter-national Journal of Radiation Biology, 21, 49-56, 1972. 42. M. R. RAJU, M. GNANAPURANI, B. STACKLER, B. I. MARTINS, U. MADHVANATH, J. HOWARD, J. T. LYMAN and R. K. MORTIMER, "Induction of Hetero-a l l e l i c Reversions and Lethality in Saccharomyces cerevisiae Exposed to Radiations of Various LET ( 5 0Co y Rays, Heavy Ions, andTI ~ Mesons) in Air and Nitrogen Atmospheres". Radiation Research, 47, 635-643, 1971-'43. M. R. RAJU, M. GNANAPURANI, and C. RICHMAN and B. I. MARTINS and G. W. BARENDSEN, "RBE and OER of T I ~ Mesons for Damage to Cultured T- l Cells of Human Kidney Origin". British Journal of Radiology, J+5, 178-181, 1972. 44. J. BAARLI, M. BIANCHI and A. H. SULLIVAN and M. QUINTILIANI, "An Estim-ation of the RBE of Stopped Pions from Observations of Spermato-gonia Survival in Mice". International Journal of Radiation Biology, 19, 537-546, 1971. - 129 -APPENDIX A PRODUCTION AND PROPERTIES OF PI-MESONS AND. THEIR DECAY PRODUCTS Pi-mesons are unstable p a r t i c l e s which do not normally e x i s t i n nature but n e u t r a l , p o s i t i v e l y charged and negatively charged pions can a l l be produced by bombarding a target with medium-energy or high-energy protons. The following are possible production reactions: (a) p + p » p + p + TT° (A - l a ) (b) p + n > p + n + rr° (A - lb) (c) p + p > p + n + IT + (A - l c ) (d) p + n > n + n + TT+ (A - Id) (e) p + n > p + p + ir~ • (A - l e ) The target nucleon may be (and u s u a l l y i s ) a constituent of a complex nucleus i n which case the threshold k i n e t i c energy of the protons for pion production i s the r e s t mass energy of the pion plus approximately 40 MeV to overcome the binding energy of the nucleus. In t h i s case, a continuous spectrum of pion energies i s produced from zero energy to a maximum of ( k i n e t i c energy of the protons - threshold energy). The r e s t mass energies and mean l i v e s of the pions are as follows: TT° 135.0 MeV 1 0 " 1 6 s TT+ and TT" 139-6 MeV 2.6 x 10~ 8 s Nearly 99% of the neutral pions decay to two gamma rays, TT° > y + Y ( A - 2 ) where the two gammas share equally the rest mass energy plus the k i n e t i c energy of the pion. The p r i n c i p a l i n t e r e s t i n neutral pions i n the present - 130 -context i s the fact that, as already noted i n section T5 when the high-energy gamma rays are absorbed by p a i r production they are an important source of electron contamination of the negative pion beam. The p o s i t i v e pions are not of i n t e r e s t i n the context of t h i s paper. Negative pions may decay i n f l i g h t or may d i s s i p a t e a l l t h e i r k i n e t i c energy i n a medium and come to rest before decaying. The capture by nu c l e i of the medium of the pions which come to r e s t before decay has been discussed i n section 3. For those pions which decay before capture by a nucleus, the decay process i s TT" > y~ + (A - 3) The f r a c t i o n decaying i n a distance dL is. given by dN/N = - dL/A (A - k) where A i s the mean path length of the pions. For pions t r a v e l l i n g with r e l a t i v i s t i c v e l o c i t i e s A = Y 3CT (A - 5) where x i s the mean l i f e of the pions, 8c i s t h e i r v e l o c i t y and y i s the time d i l a t i o n f a c t o r = 1/ / l - g 2 . 8 can be calculated from equation (A - l l ) — see below. The pions may decay i n the beam transport system or, a f t e r emerging from the transport system, may decay i n the medium before coming to r e s t . I t i s assumed that they are transported through the system without loss of energy — usually through an evacuated "beam pipe". Under t h i s condition, A i s constant and equation (A - k) can be integrated to y i e l d the f a m i l i a r - 131 -equation Surviving f r a c t i o n = N/NQ = e ~ L / ^ (A - 6) where L i s the path length and X i s given by equation (A - 5)- The f r a c t i o n of pions of d i f f e r e n t energies' (and d i f f e r e n t ranges i n water) s u r v i v i n g an 8-metre path i n a vacuum i s given i n table AI. TABLE AI FRACTION OF NEGATIVE PIONS OF DIFFERENT ENERGIES SURVIVING AN 8-METRE PATH WITHOUT DECAY Pion Range Mean path F r a c t i o n energy i n water 3 Y X surv i v i n g (MeV) (cm) (metres) 8-m path 0 0.0 0 1 0 0 13.9 1.0 0.416 1.10 3.54 0.104 34.9 5.0 0.600 1.25 5.80 0.254 53.0 10.0 0.689 1.38 7.42 0.340 68.2 15.0 0.741 1.49 8.60 0.394 82.2 20.0 0.777 1.59 9.64 0.436 95.2 25.0 0.804 1.68 10.52 0.467 107-5 30.0 0.825 1.77 11.39 0.495 For pions which decay i n the medium, equation (A - 6) i s no longer v a l i d since X decreases as the pions dis s i p a t e t h e i r k i n e t i c energy. In t h i s case, equation (A - 4) can be integrated as P In ( s u r v i v i n g f r a c t i o n ) = In (N/NQ) = - I (l/X) dL (A - 7) Jo where N/NQ i s the f r a c t i o n s u r v i v i n g r e s i d u a l range R without decay. Since the decay i n range R i s always small, equation (A - 7) may be wr i t t e n as Fra c t i o n decaying i n range R = (N o-N)/N Q = -ln(N/N Q) * I (l/X) dL (A - 8) - 132 -The i n t e g r a l on the r i g h t side can be integrated numerically. The zero lower l i m i t has been wr i t t e n as e, small but not zero, to avoid the i n f i n i t e integrand at zero. The f r a c t i o n s decaying i n d i f f e r e n t ranges i n water as calculated from equation (A - 8) are shown i n table A l l . TABLE A l l FRACTION OF NEGATIVE PIONS DECAYING IN RANGE R IN WATER -> Range i n water Pion energy F r a c t i o n decaying i n range (cm) (MeV) 0 0 0 1 13.9 o.oo4 5 34.9 0.013 10 53.0 0.020 15 68.2 0.026 20 82.2 0.032 25 95-2 0.038 30 107-5 0.041 The momentum of a p a r t i c l e and i t s k i n e t i c energy are r e l a t e d by the equation pc = /2m Qc 2 (KE) + (KE) 2 (A - 9) where m Qc 2 i s the r e s t mass energy of the p a r t i c l e and (KE) i s i t s k i n e t i c energy, both i n units of MeV, and p i s i t s momentum i n MeV/c. Conversely, KE = / ( m Q c 2 ) 2 + ( p c ) 2 - m o c 2 (A - 10) Also, i t may be shown that , i n terms of m Qc 2, KE and p, B = v/c = (pc)/[m 0c 2 + (KE)] = /2m Qc 2 (KE) + (KE) 2/[m Qc 2 + (KE)] (A - l l ) where v i s the v e l o c i t y of the p a r t i c l e . - 1 3 3 -Table AIII gives the k i n e t i c energies and ranges i n water of pions and muons and the k i n e t i c energy of electrons of equal momenta. In t h i s t a b l e , the momentum of pions of s p e c i f i e d energy and range was c a l c u l a t e d from equation (A - 9) and the k i n e t i c energies of muons and electrons of equal momenta were c a l c u l a t e d from equation (A - 10). The muon ranges were scaled from proton data by means of equation ( 6 ) (p. 19). TABLE AIII RANGES IN WATER AND KINETIC ENERGIES OF PIONS AND MUONS AND KINETIC ENERGY OF ELECTRONS OF EQUAL MOMENTA Pion range i n water Pion energy Momentum Muon energy Muon range i n water El e c t r o n energy (cm) (MeV) (MeV/c) (MeV) (cm) (MeV) 0 0 0 0 0 0 1 13.9 63.8 17.8 1.9 63.3 5 34.9 104.7 43.1 8.5 104.2 1 0 53.0 132.7 64.0 16.0 132.2 15 68.2 153.9 81.0 22.5 153.4 20 82.2 172.3 96.5 29.1 171.8 25 95.2 188.8 110.7 35-4 188.3 30 107.5 203-9 124.0 46.0 203.4 The muons which r e s u l t from the decay of charged pions are also unstable. The decay process f o r negative muons i s y ~ — » e ~ + v" + v (A - 12) . e y Since the re s t mass energy of the el e c t r o n , 0.511 MeV, i s small compared with that of the muon, 105-7 MeV, on the average the ele c t r o n and the two neutrinos share the a v a i l a b l e energy almost equally. Further, the muon usually d i s s i p a t e s i t s k i n e t i c energy i n the medium and comes to re s t before decaying since i t s mean l i f e , 2.2 x 10 6 seconds, i s r e l a t i v e l y long. The av a i l a b l e energy, therefore, to be shared between the ele c t r o n and the - 134 -neutrinos i s u s u a l l y j u s t the r e s t mass energy of the muon. On t h i s has i t i s assumed that the average k i n e t i c energy of the electrons r e s u l t i n g from muon decay i n approximately 35 MeV. - 1 3 5 -APPENDIX B USE OF SIMPSON'S RULE FOR NUMERICAL INTEGRATIONS In using Simpson's r u l e f o r numerical evaluation of a d e f i n i t e i n t e g r a l such as (B - 1 ) the range of i n t e g r a t i o n , A to B, must be div i d e d i n t o n i n t e r v a l s where n i s EVEN. The coordinates of the curve at the end of each i n t e r v a l may be denoted by x^,y^ where i takes a l l i n t e g r a l values from o to n and x Q = A, x n = B. I t i s customary to divide the range, A to B, in t o n EQUAL i n t e r v a l s but equal i n t e r v a l s are not e s s e n t i a l . The e s s e n t i a l requirement f or use of Simpson's r u l e i s that the i n t e r v a l s are equal by p a i r s such that 1 + 2 1+ 1 1+ 1 1 f o r every even i ( i n c l u d i n g zero). I f t h i s condition i s s a t i s f i e d , f o r even l * Xi+2 y dx = (y. + 4 y i + 1 + y i + 2 ) ( x i + 2 - (B - 2 ) The use of pa i r e d i n t e r v a l s , not a l l equal, allows choice of small i n t e r v a l s where y and/or i t s d e r i v a t i v e s are changing r a p i d l y and l a r g e r i n t e r v a l s where the changes are slower. A furt h e r condition i s placed on the i n t e r v a l s i n t o which the range of i n t e g r a t i o n , A to B, must be divided i f two integrations by Simpson's r u l e are required i n sequence. For example, l e t U 9 = / y dx, U u = / y dx, e t c . , (B - 3) - 136 -and l e t ra (B - 4) Then, the integrand U of equation (B - 4) can be evaluated at X q , x 2 , x^, etc., by means of equations (B - 3)• The second i n t e g r a t i o n by Simpson's rul e i s permissible i f , f o r every i which i s a multiple of 4, x. , - x. , = x. , - x. 1+1+ 1 + 2 1+ 2 i where i i s the o r i g i n a l running number. To s a t i s f y t h i s condition, the t o t a l number of i n t e r v a l s i n t o which the range, A to B, i s divided must be a multiple of four, and the i n t e r v a l s must be selected i n groups of four equal i n t e r v a l s . By extension of t h i s argument, i f three Simpson's r u l e i n t e g r a -tions are required i n sequence, the number of i n t e r v a l s into which A to B i s divided i n i t i a l l y must be a multiple of eight — i n groups of eight equal i n t e r v a l s . Table IV i s given i n f u l l i n t h i s paper as an example. The ranges i n t h i s t a b l e were selected i n groups of eight equal i n t e r v a l s and, there-f o r e , three Simpson's r u l e integrations with respect to range were permissible i n sequence. Only abridged versions of the f u l l tables used f o r c a l c u l a t i o n s have been included i n t h i s t h e s i s for tables V to VIII i n c l u s i v e and f o r table XI. In these abridged t a b l e s , the i n t e r v a l s of the abscissa, as shown i n column 1, are paired and the degree of abridgment i s stated i n a footnote. I t i s to be understood i n a l l cases that the values of the abscissa which have been omitted were equally spaced within the i n t e r v a l s shown. For example, i n table VI every second value i s shown; i n the f u l l t a b l e , the ranges were se l e c t e d i n groups of four equal i n t e r v a l s and two Simpson's r u l e integrations i n sequence were permissible. - 137 -No err o r t e s t was included i n the subroutine used for the Simpson's ru l e integrations but several checks to t e s t the accuracy of the integrations w i l l have been noted i n the c a l c u l a t i o n s — e.g., the comparison of columns 5 and 6 of t a b l e IX (p. 3 2 ) . 

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