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The determination of effective thicknesses from transmission measurements with gamma rays Goulet, Pierre 1975

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THE DETERMINATION OF EFFECTIVE THICKNESSES FROM TRANSMISSION MEASUREMENTS WITH GAMMA RAYS by P i e r r e Goulet B.Sc.A., Ecole Polytechnique, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1975 In presenting th i s thesis in par t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes i s for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t i on of th is thesis fo r f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of Physics The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada D a t e A p r i l 1/1975. ABSTRACT The usual s t a r t i n g - p o i n t f o r c l i n i c a l dosimetry i s the dose d i s t r i b u t i o n due to a r a d i a t i o n beam i n a homogeneous medium l i k e water. The m o d i f i c a t i o n of the standard d i s t r i b u t i o n f or the case of an inho-mogeneous medium (patient) requires d e t a i l e d information about i n t e r n a l s t r u c t u r e s . For heavy charged p a r t i c l e beams and for photon beams of high enough energy, the s i n g l e parameter which most nearly determines the penetration of the r a d i a t i o n i s the electron density (electrons/ 3 2 cm ). The integrated electron density along a path (electrons/cm ) i s uniquely r e l a t e d to the narrow-beam Y - r a y transmission along that path. • Hence, a measurement of Y - r a y transmission should provide useful i n f o r -mation. This work describes a new technique f o r performing these mea-60 surements. We have used a Co therapy source with a rectangular f i e l d long enough i n one d i r e c t i o n to span the width of a human cross section and narrow i n the other d i r e c t i o n to minimize the e f f e c t s of scattered r a d i a t i o n . Rather than collimate our detectors, we have devised a sim-pl e mathematical procedure to c a l c u l a t e the scatter c o n t r i b u t i o n at each point of i n t e r e s t along the width of the r a d i a t i o n f i e l d . Narrow-beam transmission values can then be evaluated from the o r i g i n a l measurements by simple substraction of t h i s s c a t t e r c o n t r i b u t i o n . An ioni z a t i o n chamber, because of i t s energy independence and accuracy, was used to acquire the basic transmission data. X-ray f i l m and s i l i c o n diodes, because of t h e i r superior r e s o l u t i o n and response speed, were selected for the measurements performed with an inhomogeneous phantom. The energy dependence of the l a t t e r dosimeters has not been a drawback when used i n the experimental conditions described here. - i i i -The method thus permits that r a p i d transmission measurements be performed along various paths of a transverse cross se c t i o n . When te s t e d with an inhomogeneous phantom, the accuracy obtained compares favourably with published values using collimated detectors. Information obtained by t h i s method i s d i r e c t l y a p p l i c a b l e to the dosimetry of treatments performed with photon beams of energy greater than 0.6 MeV and heavy charged p a r t i c l e beams l i k e protons and IT mesons. - i v -TABLE OF CONTENTS PAGE TITLE i ABSTRACT . . . . ' . . . . . . . i i TABLE OF CONTENTS . i v LIST OF TABLES . . . v i LIST OF FIGURES . v i i i ACKNOWLEDGEMENTS . x 1. INTRODUCTION . 1 2. BASIC TRANSMISSION MEASUREMENTS . . . . . . . . . . . 7 2.-1 Introduction • 7 2.2 Transmission Measurements with Square F i e l d s . . . 9 2.2.1 Measurements using a large water absorber . . . . 9 2.2.2 Measurements using t h i c k A l and Pb absorbers 12 2.2.3 C a l c u l a t i o n of f i r s t s c a t t e r c o n t r i b u t i o n 17 2.3 Transmission Measurements Using Rectangular F i e l d s . . . 1 9 2.3 .1 Central axis measurements 19 2.3.2 Off-axis measurements 25 3. TRANSMISSION MEASUREMENTS WITH AN INHOMOGENEOUS PHANTOM 29 3.1 Introduction . . . . 29 3.2 Apparatus . 29 3 .2.1 S i l i c o n diode as a dosimeter . . . . . . 2 9 3 .2.2 X-ray f i l m as a dosimeter . . . . . . . 3 0 3.3 Inhomogeneous Phantom Measurements 3U h. DISCUSSION ^ l -V-TABLE OF CONTENTS (CONTINUED) PAGE 5. CONCLUSION hk BIBLIOGRAPHY . 1+5 APPENDIX A . . . . 1+8 APPENDIX B . . 5 3 APPENDIX C 56 - v i -LIST OF TABLES PAGE I MASS ATTENUATION COEFFICIENTS OF 1.25 MeV PHOTONS IN VARIOUS HUMAN TISSUES ASSUMING THE CHEMICAL COMPOSITIONS OF KIM (5) • • • • • .' • 3 II MASS ATTENUATION COEFFICIENTS OF 1.25 MeV PHOTONS IN COMPACT BONE 3 I I I DENSITY OF VARIOUS NORMAL TISSUES k IV SYNOPSIS OF MEASUREMENTS PERFORMED . 9 V . MEASURED RELATIVE TRANSMISSION USING SQUARE BEAMS FOR A THICK WATER ABSORBER 1 0 VI • MEASURED RELATIVE TRANSMISSION USING SQUARE BEAMS FOR THICK A l AND Pb ABSORBERS . . . . . . .' . 13 VII DETAILS OF THE APPROXIMATE AND ACTUAL SOURCE •'. ' . . . l6 VIII CALCULATED NARROW-BEAM ATTENUATION COEFFICIENTS 60 OF RADIATION FROM A Co THERAPY SOURCE FOR VARIOUS MATERIALS . . • . . . l6 IX - RELATIVE TRANSMISSION INCREASE CAUSED BY SCATTERED RADIATION: CENTRAL AXIS MEASUREMENTS OF WATER SLABS IRRADIATED BY RECTANGULAR BEAMS . . . . 21 X EXTRAPOLATED NARROW-BEAM TRANSMISSION VALUES FOR WATER . 21 XI VALIDITY OF CLARKSON'S METHOD 23 XII . RELATIVE TRANSMISSION INCREASE CAUSED BY SCATTERED RADIATION: CALCULATIONS FROM OFF-AXIS MEASUREMENTS ALONG THE LONGITUDINAL AXIS OF A RECTANGULAR BEAM . . . 2 7 XIII COMPARISON OF MEASURED OPTICAL DENSITIES TO THE CORRESPONDING DIFFUSE DENSITY VALUES 3k XIV CURVE FITTING OF DENSITY VS DOSE RELATIONSHIP USING A THIRD-ORDER POLYNOMIAL FOR KODAK RP/TL FILM . . . . 36 XV EQUIVALENT THICKNESSES DERIVED FROM THE TRANSMISSION MEASUREMENTS AND THE PHANTOM'S COMPOSITION AND GEOMETRY . . . . . . . . 39 - v i i -LIST OF TABLES (CONTINUED) PAGE XVI ELECTRON DENSITIES OF THE MATERIALS COMPOSING THE INHOMOGENEOUS PHANTOM kO A l COMPARISON OF EVANS* TABULATED VALUES OF u /p (32) FOR PERSPEX TO THE VALUES e n USED IN THE CALCULATIONS . . . 5 2 - v i i i -LIST OF FIGURES PAGE 1. EXPERIMENTAL LAYOUT USED TO PERFORM TRANSMISSION MEASUREMENTS . . . . 8 2. TRANSMISSION MEASUREMENTS SHOWING THE EFFECT OF MODIFYING THE ABSORBER-TO-DETECTOR DISTANCE . . . . . 11 3. MODEL USED TO REPRESENT THE 6°Co SOURCE AND ITS SURROUNDING SLEEVE . 15 h. COMPARISON OF CALCULATED FIRST SCATTER WITH MEASURED . TOTAL SCATTER a) BY A DOSIMETER PLACED AT 30 cm FROM THE ABSORBER AND b) BY AN EXIT DOSIMETER. . . . . l8 .5-'• VARIATION OF A(0,b,t) FOR VARIOUS ABSORBER THICKNESSES AND FIELD SIZES . 22 6. -ILLUSTRATION OF CLARKSON'S METHOD • . 2k 7. GEOMETRIC DESCRIPTION OF THE QUANTITIES USED IN EQUATION (7) 2k 8. RELATIVE TRANSMISSION MEASUREMENTS ALONG THE LONGITUDINAL AXIS OF A RECTANGULAR FIELD 26 9. THE FUNCTION A(0,b,t) AS DETERMINED FROM THE MEASUREMENTS USING RECTANGULAR FIELDS 28 10. SHORT CIRCUIT CHARACTERISTICS OF CER#71 DIODE . . . . . 31 11. OPEN CIRCUIT VOLTAGE CHARACTERISTICS OF CER#71 DIODE . . 32 12. EFFECT OF THE INPUT IMPEDANCE OF THE MEASURING VOLTMETER ON THE DIODE RESPONSE PRODUCED 32 13. AVERAGE CALIBRATION CURVE OF RP/TL X-RAY FILM 137 IRRADIATED IN A 'Cs BEAM 35 Ik. TOP VIEW OF INHOMOGENEOUS PHANTOM 37 15- TRANSMISSION PROFILES a) WITH AND b) WITHOUT THE ADDITION OF A SMALL ALUMINIUM ABSORBER 1+3 A l . DIAGRAM ILLUSTRATING THE CALCULATION OF THE FIRST SCATTER REACHING A POINT P AT A DISTANCE B FROM THE SCATTERER [Adapted from Davisson and Evans (3l)] . . 50 3 A2. CALIBRATION CURVE FOR THE BALDWIN-FARMER 0.6 cm CHAMBER CORRECTED FOR THE PRESENCE OF. THE BUILDUP CAP 50 - i x -LIST OF FIGURES (CONTINUED) PAGE B l . DIAGRAM TO ILLUSTRATE THE CALCULATION OF THE FIRST SCATTER REACHING A POINT P INSIDE A SCATTERER [Adapted from Bruce and Johns (35)] CI. ILLUSTRATION OF THE GEOMETRY USED TO CALCULATE THE SCATTER REACHING THE MEASUREMENT POINTS . . . . . 57 ACKNOWLEDGEMENTS The author would l i k e to acknowledge the f i n a n -c i a l support of the Ministere de l'Education de l a Pro-vince de Quebec during the length of t h i s work. This project was c a r r i e d out using the f a c i l i t i e s of the Physics Department of 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 i t s head p h y s i c i s t , Dr. R.O. Kornelsen, f o r suggesting and supervising t h i s work. 1. INTRODUCTION Routine radiotherapy treatment planning involves the c a l c u -l a t i o n o f the dose d e l i v e r e d t o a tumour and the surrounding t i s s u e s . These dosimetry estimates, because of the s i m i l i t u d e between most t i s s u e s and water, are u s u a l l y made on the basis of measurements performed wi-t h i n a water phantom which i s approximately a cube of 30 cm a side. Cor-r e c t i o n s are then applied to take i n t o account the exact contour of the patient and the presence of t i s s u e inhomogeneities, p r i m a r i l y lungs and bones. In order to c a l c u l a t e the e f f e c t of these inhomogeneities on the absorbed dose d i s t r i b u t i o n , various information i s required. For each p a t i e n t , one would l i k e an anatomical cross section where the boun-daries of regions with d i f f e r i n g attenuation properties could be l o c a -l i z e d . In general, the chemical composition and density of the t i s s u e s are of i n t e r e s t . Technological c o n s t r a i n t s , i n the past, have made the acqui-. s i t i o n of accurate c r o s s - s e c t i o n data d i f f i c u l t . Recently, however, versions of t r a n s a x i a l tomography u n i t s , f e a t u r i n g h o r i z o n t a l couches, have been employed c l i n i c a l l y f o r t h i s purpose [Marinello et a l . ( l ) , Houdek et a l . ( 2 ) ] . Moreover, a complete cross-section a t l a s , produced with such an instrument, has r e c e n t l y been published by Takahashi (3). The accuracy a t t a i n a b l e has been shown (l,2) to be of the order of 0.2 cm fo r the external contour of a t e s t phantom. I f t h i s accuracy i s main-tain e d f o r i n t e r n a l s t r u c t u r e s , the method appears capable of providing a s u i t a b l e c r o s s - s e c t i o n map of a p a t i e n t . There are other methods, such as ultrasonography, which could be used t o give s i m i l a r and a d d i t i o n a l information. - 2 -Most conventional r a d i a t i o n therapy treatments make use of photons with energies greater than 0.6 MeV. Over t h i s range, f o r mate-r i a l s of low atomic number such as a l l "biological materials, Compton s c a t t e r i n g i s the predominant i n t e r a c t i o n process. Therefore, the l i n e a r attenuation c o e f f i c i e n t , u, i s proportional'to p g , the electron density, where p = p(Z/A)N (electrons/cm 3) ( l ) e a and p = density of the m a t e r i a l . Z = atomic number of the m a t e r i a l . A = mass number of the m a t e r i a l . , N = Avogadro's constant. Since the l i n e a r stopping power of heavy charged p a r t i c l e s i s a l s o , to a good approximation, proportional to p g , t h i s quantity should also be use-f u l i n the dosimetry of beams of pions and protons. Cal c u l a t i o n s of u/p, assuming incident photons of 1.25 Mev and using mass attenuation c o e f f i c i e n t s due to Hubbell (U) f o r the various elements s p e c i f i e d i n the material's chemical composition, are presented i n t a b l e s I and I I . As shown, y/p i s approximately constant f o r various t i s s u e s and f o r the d i f f e r e n t compositions of a p a r t i c u l a r one. The attenuation process of both p a r t i c l e and photon beams i s thus strongly dependent of the density of the i r r a d i a t e d t i s s u e s . Table I I I gives a range of d e n s i t i e s f o r normal b i o l o g i c a l materials. While the la r g e v a r i a t i o n s i n d i c a t e d by these f i g u r e s would seem to warrant the determination of t i s s u e electron d e n s i t i e s i n i n d i -v i d u a l cases, they p r i m a r i l y point out the necessity of defi n i n g accep-t a b l e accuracy l i m i t s of dosimetry c a l c u l a t i o n s . The American A s s o c i a t i o n of P h y s i c i s t s i n Medicine (AAPM) has defined "the c r i t e r i a f o r acceptable dosimetry" as "an agreement to ± 3% on exposure or absorbed dose measu-rement and agreement to ± 5% on f u l f i l l m e n t of tumor dose p r e s c r i p t i o n " (13). -3-TABLE I MASS ATTENUATION COEFFICIENTS OF 1.25 MeV PHOTONS IN. VARIOUS HUMAN TISSUES ASSUMING THE CHEMICAL COMPOSITIONS OF KIM (5) Tissue y/p (cm /g) Bone (compact) .0597 Bone (spongy) .0611 Brain .06ll+ Lung .0627 Muscle (lean somatic) .0625 TABLE I I MASS ATTENUATION COEFFICIENTS OF 1.25 MeV PHOTONS IN COMPACT BONE y/p Chemical composition reference (cm /g) .0585 Woodard (6) .0597 Kim (5) .0603 ICRU (7) .0608 Tipton and Cook (8) -U-TABLE I I I DENSITY OF VARIOUS NORMAL TISSUES Tissue Density, p Reference* (g/cm 3) Brain 1.07 A l l e n (9), Kim (5) Bone ( r i b s ) 1.09-1.25 Debois & de Roo (10) Bone (femur) 1.1*7-2.10 Id. Bone (vertebrae, mandible) l.lU-1.65 Id. Heart 1.05 A l l e n (9), Kim (5) Kidney 1.01+ Id. L i v e r 1.05 Id. Lung 0.25-OU " ICRU (11) Muscle (lean) 1.07 A l l e n (9), Kim (5) Muscle ( s t r i a t e d ) 1.00 Spiers (12) Spleen 1.05 A l l e n (9), Kim (5) The data of A l l e n et a l . (9) give the density of f a t - f r e e t i s s u e s . These d e n s i t i e s were corrected to that of average ti s s u e s using the biochemical compositions quoted by Kim ( 5 ) . - 5 -For a ^ C o beam, t h i s uncertainty of ± 5% corresponds to an equivalent thickness change of approximately ± 1 . 0 cm. The equivalent thickness of a material i s the thickness of water necessary to produce the same a t t e -nuation. However, because of unavoidable u n c e r t a i n t i e s such as patient movement, f o r example, a more r e a l i s t i c g u i deline f o r the u n c e r t a i n t i e s permissible was f e l t to be an equivalent thickness of ± 0.5 cm. Various workers have proposed y-ra.y transmission techniques to determine equivalent thicknesses of human t i s s u e s . The basic method i s that of Holt and Laughlin (lU) where a small collimated ^ C o source and c o l l i m a t e d . s c i n t i l l a t i o n detector are moved l a t e r a l l y across the patient by 2 cm increments. Pulse-height d i s c r i m i n a t i o n i s employed to r e j e c t the detected s c a t t e r . The transmission measurements are thus performed i n narrow-beam geometry. While b a s i c a l l y e x c e l l e n t , the method does ne-c e s s i t a t e a period of approximately 30 minutes to complete a f u l l c r o s s -s e c t i o n a l survey. This was f e l t to be not acceptable c l i n i c a l l y . The other published methods of i n t e r e s t have made use of ^°Co therapy sources. They were developed f o r the dosimetry of treatment pro-cedures where a knowledge of the equivalent thicknesses present between two opposing f i e l d s i s s u f f i c i e n t ( i . e . , complete r o t a t i o n f i e l d s , oppo- t sing pair of fixed f i e l d s ) . The measurements were performed, i n a l l cases, along the central axis of standard f i e l d s i z e s , often corresponding to those used during the actual treatment. Various techniques have been pro-posed to eliminate or take into account the scatter reaching the detector. Pfalzner (15) used an ion chamber positioned at TO cm behind the patient to minimise t h i s scatter. Fedoruk and Johns (l6) proposed a heavily c o l -limated i o n i z a t i o n chamber to be used at a c l o s e r distance to the p a t i e n t . Another method due to Woodley et a l . (IT) consisted i n p l a c i n g a thin-window ion chamber d i r e c t l y behind the p a t i e n t . From the e x i t doses recorded, equivalent thicknesses could then be i n f e r r e d from c a l i -b r a t i o n curves determined f o r s p e c i f i c f i e l d s i z e s . The methods (15,l6) in v o l v e the r e l a t i v e r o t a t i o n o f both source and detector about the pa-t i e n t and could be used to provide a complete c r o s s - s e c t i o n a l survey. The disadvantage of t h i s approach, however, i s that the information ob-tained i s too h e a v i l y weighted toward the centre while neglecting the outer edges of the body cross s e c t i o n . Because of t h i s , we f e l t an i n v e s t i g a t i o n of a d i f f e r e n t method of obtaining c l i n i c a l l y u s eful transmission data using -a standard therapy • • source was warranted. It was decided to use a rectangular r a d i a t i o n beam with i t s l o n g i t u d i n a l dimension large enough to span the transverse cross section to be measured, the other f i e l d dimension being narrow to minimize scattered r a d i a t i o n . Transmission determinations along a large number of paths could then be performed r a p i d l y i f f i l m or a light-weight scanning detector i s used. The major d i f f i c u l t y , however, i s the elimination of the produced s c a t t e r . It was f e l t that c o l l i m a t i n g our detector would be i m p r a c t i c a l . Instead, we ehose to devise a simple mathematical procedure to determine the scatter c o n t r i b u t i o n at each measurement point. It i s p o s s i b l e by t h i s technique to obtain narrow-beam transmission data from the o r i g i n a l measurements by mathematically subtracting the scat t e r and thus to determine the equivalent thickness along each y-ray path of i n t e -r e s t . Measurements of an inhomogeneous phantom were performed to t e s t the v a l i d i t y of the method and, as w i l l be shown, the agreement between the equivalent thicknesses c a l c u l a t e d from the transmission data and those c a l -culated from the phantom's geometry and composition was better than ± 0.5 cm -7-2. BASIC TRANSMISSION MEASUREMENTS 2.1 Introduction The amount of scatter reaching the detector i n a transmission measurement depends on a number of f a c t o r s . Consider the geometry used f o r such a measurement shown i n f i g u r e 1. Various parameters w i l l i n -fluence the amount of scatter measured. In p a r t i c u l a r , (a) An increase of the absorber-to-detector distance, B, w i l l decrease the scatte r reaching the detector. (b) An increase of the f i e l d s i z e , S, w i l l increase the s c a t t e r . Note that f i e l d dimensions throughout t h i s paper are quoted as measured at a distance A from the source. It was found convenient to choose A equal to 120 ± 0 . 5 cm. We then investigated the e f f e c t of independently varying the distance B and the f i e l d area on the r e l a t i v e transmitted dose, T. This quantity i s the quotient of the dose measured with an absorber present i n the beam path by that measured with no absorber. Mathematically, the r e l a t i v e dose transmitted by an absorber of thickness, t , can be expressed, f o r a s p e c i f i c absorbing m a t e r i a l , as T = T(P,B,S,t) (2) where P describes the p o s i t i o n of the measurement point, u s u a l l y the distance o f f the c e n t r a l axis of the f i e l d . Table IV l i s t s the various experimental conditions investigated. The radiation source used was housed i n a commercial therapy unit (Atomic Energy of Canada Ltd., Eldorado Model 8) equipped with a sloping sided c o l l i m a t o r . -Most measurements were performed with a Baldwin-Farmer 0.6 cm' -8-FIGURE 1 EXPERIMENTAL LAYOUT USED TO PERFORM TRANSMISSION MEASUREMENTS - 9 -TABLE IV SYNOPSIS OF MEASUREMENTS PERFORMED F i e l d shape Mat e r i a l Absorber' thickness (cm) A (cm) B (cm) Section Square Water 18.106 120 0, 15, 30 U5, 60 2.2.1 Square A l 2.592 120 35, 55, 75 2.2.2 Square Pb 2.519 120 35, 55 2.2.2 Rectangular Water 9A36, 13AU3 18.106, 2U.136 120 30 2.3.1 Rectangular ( o f f - a x i s ) Water l 8 : i 0 6 , 2U.136 • 120 30 2.3.2 i o n i z a t i o n chamber p o l a r i z e d by 300 v o l t s . The i o n i z a t i o n current was integrated by an electrometer (Keithley, Model 6l0C) f o r one minute pe-r i o d s . A voltage proportional to t h i s charge was read with a d i g i t a l voltmeter. Both leakage and environmental conditions were monitored. Their e f f e c t s were l e s s than the s t a t i s t i c a l v a r i a t i o n of the measure-ments (± 0.15% on the average) and were, not corrected f o r . 2.2 Transmission Measurements with Square F i e l d s 2.2.1 Measurements using a large water absorber A large water phantom of uniform thickness was f i r s t used to measure T along the c e n t r a l axis f o r various beam areas. Table V shows the data obtained with a 18.106 cm t h i c k absorber. Additional measu-rements with a 2^.136 cm water thickness showed a s i m i l a r trend. The v a r i a t i o n of T with area, for a f i x e d B, i s f i t t e d to better than the experimental accuracy (± 0.3% on the average) by a quadratic -1 CI-TABLE V MEASURED RELATIVE TRANSMISSION USING SQUARE BEAMS FOR A THICK WATER ABSORBER F i e l d area at 120 cm T(0,B,S,l8.106) from the source (cm 2) B=15 cm B=30 cm B=l+5 cm B=60 cm 1+1+1 .1+029 .3631* .31+51 .3339 32k .3875 .3520 .3362 .3275 225 .3710 .31+02 .3277 .3221 Ikk .3529 .3297 • 3199 .3158 81 .3350 .3193 .3132 .311U 36 .3205 .3116 .3090 .3076 Zero extrapolation .3093 .3051+ .3050 .301+8 equation using a least-squares method. The zero-area extrapolation c a l c u -l a t e d from such a f i t i s indi c a t e d i n ta b l e V. Except f o r the value ob-tain e d with B equal to 15 cm, these extrapolated values l i e within 0.3$ of each other. From these l a t t e r f i g u r e s the zero-area l i n e a r attenuation c o e f f i c i e n t , UQ» was estimated to be .0656 ± .0002 cm \ Further discus-sion of t h i s r e s u l t w i l l be presented l a t e r . Figure 2 i l l u s t r a t e s the v a r i a t i o n of T with distance B f o r f i e l d s i z e s of 36 and 225 cm r e s p e c t i v e l y . The ex i t doses (B = 0) were measured with a thin-window i o n i z a t i o n chamber i n d i r e c t contact with the absorber. The centre of the f l a t c y l i n d r i c a l c a v i t y l i e s 0.1 cm below the t h i n mylar window. The procedure and readout c i r c u i t previously des-cri b e d were again used. A buildup cap of 0.5 cm of water-equivalent ma-t e r i a l was employed i n the absence of the absorber.. The ion chamber was p o l a r i z e d by 1+5 v o l t s and a c o r r e c t i o n was applied f o r the small leakage current present. -11-0 15 30 i+5 60 Distance (cm) FIGURE 2 TRANSMISSION MEASUREMENTS SHOWING THE EFFECT OF MODIFYING THE ABSORBER-TO-DETECTOR DISTANCE ., -12-We attempted to compare published r e s u l t s to our experimental data. Results s i m i l a r to those of f i g u r e 2 were presented by Fedoruk and Johns (l6) f o r distances extending from 5 to 20 cm. I n t e r p r e t a t i o n of t h e i r data i s made d i f f i c u l t , however, by the fact that the experi-mental layout i s not s p e c i f i e d . Moreover, they do not state where the quoted f i e l d dimensions were measured. Pfalzner (15) commented that at TO cm from the absorber "the s c a t t e r c o n t r i b u t i o n has been found expe-ri m e n t a l l y to be n e g l i g i b l e " . Since no d e t a i l s of the r a d i a t i o n beam dimensions were provided, one i s l e d to assume that the statement applies to a l l f i e l d s i z e s . The c r i t e r i o n used to define a c c e p t a b i l i t y was not stated. 2.2.2 Measurements using t h i c k A l and Pb absorbers 60 I t i s known that the spectrum emitted from a Co teletherapy source contains low energy components. This i s due to s c a t t e r from the source i t s e l f , the source housing, the c o l l i m a t o r and the interposed a i r column between the source and the detector. Published c a l c u l a t i o n s (l8), using Monte Carlo techniques, have been performed to determine the spec-60 trum from a Co source. However, because of the complexity and v a r i a -b i l i t y of the c o l l i m a t o r geometry, the scatter c o n t r i b u t i o n from i t was ignored. Since the narrow-beam attenuation c o e f f i c i e n t s derived from our measurements also exclude co l l i m a t o r s c a t t e r , a r e a l i s t i c comparison of the Monte Carlo c a l c u l a t i o n s and the experiments seems p o s s i b l e . In an e f f o r t to f u r t h e r characterise the composition of the photon spectrum, transmission measurements were performed with materials of higher atomic number. Aluminium (p = 2.TO g/cm ) and lead (p = 11.3*+ 3 g/cm ) were'used and the r e s u l t s appear i n table VI. The distance B was l a r g e r than 30 cm to f a c i l i t a t e zero-area extrapolation. Least-squares -13-TABLE VI MEASURED RELATIVE TRANSMISSION USING SQUARE BEAMS FOR THICK A l AND Pb ABSORBERS ALUMINIUM ABSORBER F i e l d area at 120 cm T(0,B,S,2.592) from the source (cm 2) B=35 cm B=55 cm B= =75 cm 380.3. .690U .6837 32U .7033 272.3 .6851 .6797 225 .6950 — 182.3 .6792 .6753 lUU .6866 — 110.3 .6755 .6726 81 .6785 -6732 .6710 • 36 .6733 .6707 .6698 Zero extrapolation .6685 -6679 .6678 LEAD ABSORBER F i e l d area at 120 cm T(0,B,S,2.519) from the source (cm 2) B=35 cm B=55 cm .1911 .1855 81 .1899 .18U5 72 .1886 .1838 56.3 .1868 .1832 36 .18U8 .1820 Zero extrapolation .1810 .1800 -ik-curve f i t t i n g techniques were again employed; a quadratic equation was used for the aluminium data while a s t r a i g h t l i n e was found s a t i s f a c t o r y f o r the lead data. C a l c u l a t i o n s of UQ f o r water, aluminium and lead were done using a published spectrum [ICRU ( l 8 ) ] based on the model of f i g u r e 3. Note that the scat t e r due to the source housing i s ignored. Table VII compares the parameters of t h i s model to those of the actual source. For c a l c u -l a t i o n purposes the following was assumed: (a) Photons emitted from the front surface of the tungsten sleeve are considered to be absorbed by the c o l l i m a t i n g system and not reach the detector. (b) Photons emitted from the front surface of the core are attenuated by a s t a i n l e s s s t e e l (type #316L) plate of 0.11 cm thickness. The c a l c u l a t e d values of UQ are shown i n table VIII with the corresponding experimental r e s u l t s . The experimental value i n the Table f o r water was obtained from rectangular beam data and w i l l be discussed l a t e r . Attenuation c o e f f i c i e n t s from Hubbell's compilation (h) were used i n a l l cases except f o r lead where the work of Storm and I s r a e l (19) was also used. The v a r i a t i o n i n the c a l c u l a t e d urj f o r the case of lead l i e s w ithin the accuracy l i m i t s s p e c i f i e d by the l a t t e r authors: (a) Accuracy of 3% f o r Compton scatter c o e f f i c i e n t s . (b) Accuracy of 3% f o r p h o t o e l e c t r i c c o e f f i c i e n t s at photon energies hv such that .006 < hv < .200 MeV. (c) Accuracy of 10% f o r p h o t o e l e c t r i c c o e f f i c i e n t s at energies hv > .200 MeV. Agreement between c a l c u l a t e d and measured values i s good. Since -15-Plane view of c y l i n d r i c a l source Detector 7,0 cm (T) S t e e l back p l a t e (2} C y l i n d r i c a l tungsten sleeve (IT) Radioactive core (T) S t e e l front p l a t e FIGURE 3 MODEL USED TO REPRESENT THE 6°Co SOURCE AND ITS SURROUNDING SLEEVE [Adapted from reference ( l 8 ) ] -16-TABLE VII DETAILS OF THE APPROXIMATE AND ACTUAL SOURCE Parameter* Model Actual source r (cm) 1.0 1.0 s (cm) 0.75 0.86** Z (cm) 1.3 2.5 Z - (cm) 2.2 1.1 Z (cm) 0.0 •0.11 Packing density (g/cm ) 5.88 5.6l * See f i g u r e 3. ** Does not include the source's s t a i n l e s s s t e e l v a i l of 0;lU cm thickness. TABLE' VIII CALCULATED NARROW-BEAM ATTENUATION COEFFICIENTS OF RADIATION FROM A 6°Co THERAPY SOURCE FOR VARIOUS MATERIALS Ma t e r i a l Measured UQ (cm - 1) Calculated PQ (cm - 1) Origin of attenuation c o e f f i c i e n t data H 20 .0653 .06W Hubbell (U) A l .156 .15k Id. Pb . .680 .686 Id. Pb .680 .669 Storm & I s r a e l (19) -17-the s c a t t e r contributed by the source housing was neglected during the c a l c u l a t i o n , we would expect s l i g h t l y l a r g e r attenuation c o e f f i c i e n t s to be observed experimentally. While t h i s i s indeed the case, the s t a -t i s t i c a l s i g n i f i c a n c e of t h i s trend i s d i f f i c u l t to determine. We can only conclude that the actual beam probably possesses a spectrum of s i -milar shape to that of our assumed one. 2.2.3 C a l c u l a t i o n of the f i r s t s catter c o n t r i b u t i o n The c a l c u l a t i o n of the amount of r a d i a t i o n scattered from t h i c k slabs i r r a d i a t e d by f i n i t e area beams can only be adequately performed using a Monte Carlo procedure. However, the f i r s t s catter c o n t r i b u t i o n may be derived exactly from the Klein-Nishina equations. This was done 2 here f o r various water thicknesses i r r a d i a t e d by 36 and 225 cm c i r c u l a r f i e l d s of 1.25 MeV photons under two s p e c i f i c geometries. The case where the detector i s 30 cm from the absorber was inves-t i g a t e d f i r s t . Figure Ma) i l l u s t r a t e s the observed r e s u l t s . The f u l l l i n e i s drawn through the c a l c u l a t e d f i r s t s catter points derived from the general method o u t l i n e d i n appendix A. The c i r c l e s are experimental r e -s u l t s and represent the t o t a l s c a t t e r c o n t r i b u t i o n expressed as the func-t i o n A where A(0,30,S,t) = T(0,30,S,t) - T(0,30,0,t) where T(0,30,0,t) = exp (-]i0t) (3) S i m i l a r l y , Aj(0,30,S,t) i s the c a l c u l a t e d increase i n r e l a t i v e transmission due to f i r s t s c a t t e r only. The general shape observed f or A^ can be explained rather simply by the following argument. Neglecting the attenuation of the scattered photons, one would expect the f i r s t - o r d e r scatter to increase with absorber thickness since the t o t a l number of Compton i n t e r a c t i o n s increases with 1 1 I I f ^ - — i ~] j r 1 1 r 1 r FIGURE k COMPARISON OF CALCULATED FIRST SCATTER WITH MEASURED TOTAL SCATTER a) BY A DOSIMETER PLACED AT 30 cm FROM THE ABSORBER AND b) BY AN EXIT DOSIMETER -19-the number of electrons present. However, f o r t h i c k absorbers, the a t t e -nuation of the scattered photons cannot be neglected. This e f f e c t , be-cause of i t s exponential form, compensates for the production of once scattered photons f o r absorbers about 12 cm t h i c k and becomes the domi-nating process f o r t h i c k e r absorbers. As the attenuation becomes predominant, the production of mul- • t i p l e s c a t t e r a l s o increases. This i s demonstrated p a r t i c u l a r l y f o r the l a r g e r area f i e l d where the d i f f e r e n c e , (A-Aj), while being small, increases with the thickness of the water absorber considered. Since the e f f e c t of m u l t i p l e scat t e r should be greatest at the exit surface (B = 0), i t was decided to compare f i r s t s c a t t e r c a l c u l a t i o n s to t o t a l s c a t t e r measurements here. Figure Mb) presents the r e s u l t s des-c r i b i n g the f i r s t s c a t t e r reaching an e x i t dosimeter using the method des-cribed i n appendix B. Also presented are some experimental observations where the t o t a l s c a t t e r c o n t r i b u t i o n was recorded. The same general shape of the functions A and Aj i s again observed. The main d i f f e r e n c e i s due to the presence of a l a r g e m u l t i p l e scat t e r c o n t r i b u t i o n that accounts f o r a major p o r t i o n of the t o t a l s c a t t e r measured. We have thus characterized the nature of the scatter emitted under two p a r t i c u l a r conditions. The c a l c u l a t i o n s could be generalized to other s i m i l a r experimental s i t u a t i o n s . 2.3 Transmission Measurements Using Rectangular F i e l d s 2.3.1 Central axis measurements As was previously mentioned, the choice of a long and narrow f i e l d would appreciably reduce unwanted scatter while permitting trans-mission measurements to be performed along the f u l l width of a human -20-transverse cross s e c t i o n . The Eldorado 8 cobalt unit has a minimum f i e l d dimension of 6 cm at 120 cm from the source. The f i e l d length, 2b, can be v a r i e d from 6 to U8 cm. Central axis transmission data was obtained using water absorbers of 9-^36, 13.^ ^+3, .18.106 and 2^.136 cm thicknesses with f i e l d s i z e s of 6 x 2b cm; 6 < 2b < kQ cm. The detector was p o s i t i o -ned at a distance of 30 cm from the absorber. These measurements have-been tabulated i n t a b l e IX. Each entry i s the average of two determina-t i o n s . The notation has been s l i g h t l y modified as follows: A(0,30,S,t) = A(0,S,t) (k) since B remains equal to 30 cm i n a l l subsequent measurements and A(0,S,t; = A(0,b,t) (5) fo r a -rectangular f i e l d of dimension 6 x 2b. R e l a t i v e transmitted doses g r a p h i c a l l y extrapolated to zero area are presented i n t a b l e X. From these a value of UQ = .0653 ± .0002 cm was derived. This value i s consistent with that determined previously from the square f i e l d data (UQ = .0656 ± .0002 cm D i f f e r e n t workers have reported UQ = .066 cm ^ [Jones et a l . (20), Johns et a l . (21)] and UQ = .065 cm-''" [Payne et a l . (22)] f o r various therapy u n i t s . Our coef-f i c i e n t i s i n agreement with these; Figure 5 shows A(0,b,t) expressed as a function of water t h i c k -ness for various f i e l d s i z e s . We have shown at each point the range of the two experimentally derived values. The continuous l i n e s indicated w e r e used i n performing the c a l c u l a t i o n s discussed i n appendix C. For thicknesses ranging from 8 to 20 cm, the function was assumed constant within the experimental l i m i t s and numerically equal to the mean of the data f o r the 9-^36, 13.^3 and 18.106 cm t h i c k absorbers. The general shape of t h i s curve i s i n agreement with that observed previously using -21-TABLE IX RELATIVE TRANSMISSION INCREASE CAUSED BY SCATTERED RADIATION: CENTRAL AXIS MEASUREMENTS OF WATER SLABS IRRADIATED BY RECTANGULAR BEAMS F i e l d dimensions, 6 x 2b, at 120 cm from the source (cm) t=9.1+36 cm A(0,b,t) 13.1+1+3 cm 18.106 cm 2U.136 cm 6 x 6 • 0055 .ooi+9 • 0059 •°°5U 6 x 12 • 012)4 ' ' - 0 1 1 8 •010)4 6 x 18 ..0170 - . 0 l 6 2 .0170 .011+3 6 x 2k • 0212 .021 6 .021u . 0 l 8 ? 6 x 30 .02k2 .021+g .025^ .021 6 6 x 36 .021k .0270 .028u .02l+6 6 x 1*2 .029 2 .028 g .029 6 .025 6 TABLE X EXTRAPOLATED NARROW-BEAM TRANSMISSION VALUES FOR WATER Thickness (cm) Relat i v e transmission determined by-Extrapolation C a l c u l a t i o n ^ to zero area (UQ = .0653 cm ) 9.^36 .5383 .51*00 13.1+1+3 .1+156 .1+157 18.106 .3061+ .3066 2U.136 .2077 .2068 FIGURE 5 VARIATION OF A(0,b,t) FOR VARIOUS ABSORBER THICKNESSES AND FIELD SIZES -23-square f i e l d s of s i m i l a r areas. As stated p r e v i o u s l y , the method of measurement out l i n e d i n t h i s paper r e l i e s on the development of a mathematical procedure to quan-t i f y the sc a t t e r reaching the measuring instrument. A convenient proce-dure i s one analogous to the method proposed by Clarkson (23) for depth-dose c a l c u l a t i o n s of i r r e g u l a r f i e l d s . Consider f i g u r e 6. The scatter reaching a detector at point 0 can be expressed, assuming the v a l i d i t y of Clarkson's method, as A(0,b,t) = Z i A ( 0 , S . , t ) (6) 1=1 n 1 2 where . S ^ = area of square f i e l d of equivalent radius = irr^ . The v a l i d i t y of equation (6) was v e r i f i e d by c a l c u l a t i n g A(0,b,t) f o r a range o f f i e l d s i z e s . The comparison with experiment i s presented i n t a b l e XI. The c a l c u l a t i o n i s r e s t r i c t e d to a maximum f i e l d of 6 x 30 cm because the square f i e l d data could not be further extrapolated with TABLE XI VALIDITY OF CLARKSON'S METHOD F i e l d dimensions 6 x 2b, at 120 cm from the source (cm) A(0,b,18.106) Measured Calculated 6 x .12 .OllQ . 012 6 x 18 .0170 .0169 6 x 2h .021u .021U 6 x 30 • 025U • 02UT FIGURE 6 ILLUSTRATION OF CLARKSON' METHOD • xi- • x2-JL Longitudinal axis FIGURE 7 GEOI4ETRIC DESCRIPTION OF THE QUANTITIES USED IN EQUATION (7) -25-confidence. The agreement i s good and proves the a p p l i c a b i l i t y of a Clarkson-type c a l c u l a t i o n i n t h i s s i t u a t i o n . 2.3.2 O f f - a x i s measurements We now wish to compute the s c a t t e r reaching a detector p o s i -tioned at an a r b i t r a r y distance X along the l o n g i t u d i n a l axis of a r e c -tangular f i e l d . Figure 7 i l l u s t r a t e s the relevant geometry. Then, A(x,b,t) = [A (0,xi,t) + A ( 0,x 2,t)]/2 (7) Assuming A(X,b,t) and A ( 0 ,X2,t) to be known, i t i s possible to determine A(0,X!,t) = 2A(X,b,t) - A(0,x 2,t) (8) I t i s thus p o s s i b l e , from the above equation, to extend the data of the previous section using the r e s u l t s of o f f - a x i s measurements. These measurements were performed with a f i e l d s i z e of 6 x 1+5 cm and with water thicknesses of 18.106 and 2^.136 cm. It was assumed that the r e s u l t s obtained with the former absorber would apply to thicknesses ranging from 8 to 20 cm. The data i s presented i n f i g u r e 8 where the open symbols r e f e r to the o r i g i n a l measurements. Each'of these was then cor-rected f o r the increased attenuation caused by the oblique path t r a v e l l e d by the y rays through the phantom. These corrected points, shown as f u l l symbols, then represent the r e l a t i v e transmission through an oblique path .of 18.106 and 2U.136 cm r e s p e c t i v e l y . By using equation (8) and subse-quently f i t t i n g the obtained r e s u l t s , the values of table XII were deter-mined. Figure 9 c o l l e c t s the data of tables X and XII f o r thicknesses of I 8 . I O 6 and 21+.136 cm r e s p e c t i v e l y . The dots are the values from the tables while the f u l l l i n e was employed to perform the c a l c u l a t i o n s of appendix C. -26-• 3U o •H CO W • •H 6 CO tsi f-i -P > •H -P cd H CU .33 .32 h .21+ .23 .22 r ~ i l8.106 cm of water \ Corrected Uncorrected 1 0 2 U 6 8 10 Off-axis angle (degrees) FIGURE 8 RELATIVE TRANSMISSION MEASUREMENTS ALONG THE LONGITUDINAL AXIS OF A RECTANGULAR FIELD -27-TABLE XII RELATIVE TRANSMISSION INCREASE CAUSED BY SCATTERED RADIATION: CALCULATIONS FROM OFF-AXIS MEASUREMENTS ALONG THE LONGITUDINAL AXIS OF A RECTANGULAR BEAM Field dimensions, 6.x 2b, at 120 cm . from the source (cm) A(0,b,t) t=l8.106 cm t=2l+.136 cm 6 x W .030k . 026g 6 x 5l+ • 0312 .0276 6 x 60 .032^ ^ .028 3 6 x 66 .0325 .028 g 6 x 72 • . 032Q • 0292 6 x 78 .032 9 • 0295 6 x 81+ .0330 .029 ? FIGURE 9 THE FUNCTION A(0,b,t) AS DETERMINED FROM THE MEASUREMENTS USING RECTANGULAR FIELDS -29-3. TRANSMISSION MEASUREMENTS WITH AN INHOMOGENEOUS PHANTOM 3.1 Introduction The basic measurements discussed i n the previous chapter have been applied here to the determination of equivalent thicknesses along various paths i n an inhomogeneous phantom. Two new detecting systems ^ were used and a d e s c r i p t i o n of these w i l l precede the presentation of the r e s u l t s . 3.2 Apparatus 3.2.1 S i l i c o n diode as a dosimeter S i l i c o n diodes have been employed extensively i n dosimetry ..applications f o r over a decade [Jones (2U)]. Our work was performed using a commercial device ( S o l i t r o n CER #7l). Both short c i r c u i t current and open c i r c u i t voltage modes of operation were i n i t i a l l y i nvestigated. For these preliminary experiments, the diode and a Baldwin-Farmer 0.6 cm ion chamber .were placed side by side i n a Co beam. The diode was positioned i n a closed end aluminium tube which served both as buildup l a y e r and electromagnetic s h i e l d . No p o l a r i z a t i o n voltage was applied to the PN junction and the leakage current was found to be n e g l i -g i b l e . The ion chamber was again p o l a r i z e d with a 300 v o l t s battery. A v a r i a t i o n of dose rate was achieved by r a i s i n g and lowering the source from a s p e c i f i c point of known dose r a t e . A Keithley electrometer (Model 6 l 0 C ) was used to measure the voltage and current from the diode and the current from the i o n i z a t i o n chamber. A d i g i t a l voltmeter again monitored the electrometer output. Environmental conditions were not recorded but v a r i a t i o n s of 0.5 °C were t y p i c a l and should not have a f f e c t e d the diode -30-r e s u l t s . The short c i r c u i t r e s u l t s of the diode are presented i n f i g u r e 10. As expected, the current v a r i a t i o n i s l i n e a r with dose r a t e . F i -gure 11 presents the voltage measurements obtained with an input impe-dance of about 1 0 ^ ohms. This input impedance, as f i g u r e 12 i n d i c a t e s , i s s u f f i c i e n t l y l a r ge to guarantee that the junction was operating i n the • open c i r c u i t mode. Jones (2k) showed that open c i r c u i t measurements should be proportional to the logarithm of the dose r a t e . However, t h i s i s only approximately true f o r our device as f i g u r e 11 demonstrates. The l i n e a r i t y with dose rate shown previously makes the short c i r c u i t mode of operation the l o g i c a l choice. Moreover the time constant and the temperature dependence of the detector [Parker and Morley (25)] are also minimized by t h i s choice. For the r e s u l t s to be presented i n Section 3.3, the diode was moved at constant v e l o c i t y along the detector plane of f i g u r e 1. A recor-ding of the diode current as a function of time was obtained by replacing the voltmeter by a s t r j p - c h a r t recorder. During each c y c l e , the diode crossed the f i e l d twice, thus producing two complete transmission scans. The time required for one cycle was about 0.7 minutes. By t r i g g e r i n g the recorder's event marker at f i v e known p o s i t i o n s , i t was possible to r e -l a t e these points on the trace to the object. Some t y p i c a l r e s u l t s w i l l be presented. 3.2.2 X-ray f i l m as a dosimeter The use of f i l m f o r dosimetric measurements i s well recognized and i s widespread i n s p i t e of the associated d i f f i c u l t i e s [ E h r l i c h (26)]. Moreover, the advent of automatic processors has reduced the amount of Dose rate (rads/minutes) FIGURE 10. SHORT CIRCUIT CHARACTERISTICS OF CER #71 DIODE - 3 2 -20 30 50 75 100 200 300 Dose rate (rads/min) FIGURE 11 OPEN CIRCUIT VOLTAGE CHARACTERISTICS OF CER #71 DIODE i o 2 ioh i o 6 i o 8 i o 1 0 i o 1 2 i o l U Input impedance (ohms) FIGURE 12 EFFECT OF THE INPUT IMPEDANCE OF THE MEASURING VOLTMETER ON THE DIODE RESPONSE PRODUCED -33-work necessary to obtain acceptable r e s u l t s . We have used Kodak type RP/TL ( i n "ready pack" form) i n a l l our f i l m measurements. This f i l m i s r o u t i n e l y used f o r therapy v e r i f i c a t i o n a p p l i c a t i o n s and i s processible i n standard medical processors. A scanning densitometer (Kipp & Zonen, Model DD 691-D) was used for the o p t i c a l density measurements. The output was displayed by a s t r i p -chart recorder. An approximately rectangular s l i t of 0.01 by O.k cm was employed to i l l u m i n a t e the sample. Densities were determined with base and fog co n t r i b u t i o n s included. It has been recommended by E h r l i c h (26) that d i f f u s e density determinations be made. However, the geometry of-our instrument does not permit a measurement of a l l the o p t i c a l photons scat-tered by the sample. Consequently, the o p t i c a l d e n s i t i e s reported i n t h i s paper are l a r g e r than the corresponding d i f f u s e d e n s i t i e s . A c a l i b r a t i o n s t r i p provided with a d i f f u s e densitometer (Macbeth Corporation, Model TD-100A) was measured with our densitometer and the r e s u l t s are presented i n t a b l e XIII. The r a t i o c a l c u l a t e d i s known i n sensitometry as C a l l i e r ' s c o e f f i c i e n t and the trend observed here i s t y p i c a l [Label and Dubois (27)]. The influence of t h i s e f f e c t must be considered when comparisons to pu-b l i s h e d r e s u l t s are attempted. When selected points of a s p e c i f i c sheet of f i l m were i r r a d i a t e d by the same dose, the density uniformity achieved was ± 0.1% on the ave-rage. It was found advantageous to c a l i b r a t e each f i l m i n order to e l i -minate the e f f e c t s of v a r i a t i o n s i n processing. The following procedure was used. About h a l f of the sheet's surface was used f o r the measurement of i n t e r e s t . Then, over the remaining area, a number of points were i r r a -diated by a Cs unit (Picker X-ray Corporation, Model'Cs-600) f o r d i f -ferent exposure times. These were measured from the u n i t ' s timer, cor-r e c t i n g f or any timer error present and are thus proportional to the dose. -3U-TABLE XIII COMPARISON OF MEASURED OPTICAL DENSITIES TO THE CORRESPONDING DIFFUSE DENSITY VALUES Diffu&e density* (1) Measured density (2) Ratio ( 2)v(l) .20 .284 1.29 .38 .532 1.33 .58 .801+ 1.31+ .78 1.05 1.31 .98 1.28 1.29 .1.17 1.1+8 1.21+ 1.37 1.67 1.20 1.57 1.81 l . l U * From c a l i b r a t i o n s t r i p densitometer ind i c a t e d included •with i n t e x t . d i f f u s e A source-to-film distance of about 1+0 cm was used and 0.5 cm of water-equivalent material provided a buildup l a y e r . Figure 13 shows a c a l i -b r a t i o n curve obtained by averaging the r e s u l t s of 13 f i l m s (from the same production batch) processed over a period of 1+5 days. Note that both scales are l i n e a r . The data from one of these f i l m s i s presented i n t a b l e XIV. A least-squares f i t of the measured points to a t h i r d -order polynomial [Price (28)] was performed and the r e s u l t s of t h i s f i t are also presented i n ta b l e XIV. 3.3 Inhomogeneous Phantom Measurements These measurements were performed with a h o r i z o n t a l ^ C o beam (Eldorado 8). A rectangular f i e l d measuring 6 x 1+5 cm at 120 cm from -36-T A B L E X I V C U R V E F I T T I N G O F D E N S I T Y V S D O S E R E L A T I O N S H I P U S I N G A T H I R D - O R D E R P O L Y N O M I A L F O R K O D A K R P / T L F I L M O p t i c a l ' d e n s i t y D I r r a d i a t i o n t i m e ( s ) C a l c u l a t e d t i m e f ( D ) * ( s ) • 578 6.61 6.58 .962 12.61 12.75 1.29k 18.61 18 . 3 3 1.572 2U.61 2U.89 1.746 30.61 30.50 * f(D) = - 9.3U52 d e t e r m i n e d b y a + 1 + 0 . 0 8 3 D - 27.H96D2 + 10 . 0 8 5 D 3 : a s l e a s t - s q u a r e s p o l y n o m i a l f i t . the source was employed throughout; the absorber-to-detector distance B was 30 cm. Figure ik shows a top view of the inhomogeneous phantom that was b u i l t . I t s walls are v e r t i c a l and are made of 3/l6 inch perspex. They extend 10 cm above a bottom p l a t e of the same m a t e r i a l . Inside are positioned a 1.90 cm diameter aluminium tube and a hollow polystyrene tube of 2.53 cm i n s i d e diameter and 0.18 cm w a l l . The dimensions of the wood block can be determined from the f i g u r e . L i n c o l n s h i r e bolus (29), a tissue-equivalent m a t e r i a l , was used to f i l l the remaining volume of the phantom. Diode scans were taken with and without the phantom i n place. From these scans, the r e l a t i v e transmitted doses were determined at points across the f i e l d . The unattenuated beam p r o f i l e was also used as a base l i n e f o r the f i l m measurements. The same geometry was used to obtain the attenuated beam p r o f i l e with f i l m . The time required to expose the f i l m was about 0.3 minutes. To source I — l 1 cm To detector FIGURE Ik TOP VIEW OF INHOMOGENEOUS PHANTOM -38-The r e s u l t s obtained from the diode and f i l m observations are presented i n t a b l e XV using the c a l c u l a t i o n method o u t l i n e d i n appendix C. Also shown are the equivalent thicknesses determined from the geome-t r y of f i g u r e ih and from the electron d e n s i t i e s r e l a t i v e to water of t a b l e XVI. The RMS d i f f e r e n c e between the equivalent thicknesses obtained from transmission data and from c a l c u l a t i o n s based on the phantom's geo-metry and composition was estimated using the expression RMS = ,2 V i = l i I ( t . - T . ) ' n . , 1 1 2 (9) where t . = equivalent thickness along path i , determined from transmission measurements T. = equivalent thickness along path i , computed from the geometry and composition of the phantom. The obtained RMS f o r the average of both diode scans was 0.29 cm while i t was 0.1+3 cm f o r the f i l m observations. This i s within the l i m i t of ± 0.5 cm set at the beginning of t h i s paper Most workers who have performed equivalent thickness determi-nations have neglected to mention the accuracy attai n a b l e by t h e i r me-thods. Woodley et a l . (17) compared the equivalent thicknesses measured t using t h e i r e x i t dosimeter to those obtained from narrow-beam measurements, The RMS d i f f e r e n c e that they obtained at a v a r i e t y of s i t e s i n humans was s l i g h t l y l e s s than 1 cm. We f e e l our method advantageously compares with t h i s while o f f e r i n g the added c a p a b i l i t y of measuring a large number of paths simultaneously. -39-TABLE XV EQUIVALENT THICKNESSES DERIVED FROM THE TRANSMISSION MEASUREMENTS AND THE PHANTOM'S COMPOSITION'AND GEOMETRY Equivalent thicknesses P o s i t i o n Diode scan* Diode scan* Film* Phantom #1 #2 (cm) (cm) (cm) (cm) 1 21.35 21.35 21.3k 20.75 2 20.26 20.79 20.73 20.29 1+ 20.22 20.1+2 21.00. 20.31+ 6 20.37 20.6k 20.82 20.39 8 20.81+ 20.81 ' 20.61 20.1+5 10 20.18 20.38 20.1+8 20.35 12 19-95 19.62 . 19.78 19.90 l U 18.72 18.36 ' 18.52 18.87 16 17.11+ 16.83 16.67 17.13 18 15.02 11+.76 11+.18 11+.80 20 11.88 11.06 10.86 11.17 3 20.60 20.60 20.88 20.20 5 20.30 20.1+7 20.91 20.22 7 17.61+ 18.69 18.01 17.80 9 17-23 17.26 17.52 16.83 11 16.67 16.69 16.77 16.78 13 16.32 • 16.1+7 16.68 16.1+0 15 15.78 15.72 16.13 15.95 17 16.53 16.1+6 17.00 17.07 19 11+.09 15.02 Ik.5k lU.75 21 10.U6 11.75 10.62 11.00 From transmission measurements performed with the s p e c i f i e d detector. * From the phantom dimensions and the electron d e n s i t i e s of tabl e XVI. -RO-T A B L E X V I ELECTRON DENSITIES . OF THE MATERIALS COMPOSING THE INHOMOGENEOUS PHANTOM Mat e r i a l Electron density r e l a t i v e to that of water Aluminium 2.35* Perspex 1.15* Polystyrene 1.03* Wat er 1.000 Bolus ( 2 9 ) •957 2** Wood .l+905** * From the mass attenuation c o e f f i c i e n t s f o r 1 . 0 MeV photons of Hubbell (U) and'density values quoted by Trent et a l . (30). ** From y-ray transmission measurements using 6 0 Co photons. - U l -k. DISCUSSION The measurements presented i n t h i s paper were performed with three detection systems. A Baldwin-Farmer 0.6 cm ion chamber, because of i t s approximate energy independence was used to acquire the basic i n f o r -mation necessary to quantify the produced s c a t t e r . The chamber does not, however, possess the s p a t i a l r e s o l u t i o n and response speed that are r e -quired during c l i n i c a l measurements. S i l i c o n diodes were i n i t i a l l y investigated as p o s s i b l e a l t e r -natives. For s i l i c o n , the a c t i v e volume per unit of i o n i z a t i o n current produced i s approximately 3,000 times smaller than that of an ion chamber. Consequently, the s i z e needed f o r a p a r t i c u l a r s i g n a l l e v e l i s small, r e s u l t i n g i n an increased s p a t i a l r e s o l u t i o n . Moreover, when operated i n the short c i r c u i t current mode, they possess a f a s t e r response time than i o n i z a t i o n chambers. Prolonged r a d i a t i o n exposure w i l l produce a decrease i n s e n s i t i v i t y of these devices and, f o r t h i s reason, we have used them p r i m a r i l y 1 t o perform r e l a t i v e measurements. From the point of view of r e s o l u t i o n and response speed, how-ever, f i l m i s a superior a l t e r n a t i v e . I t s greater energy dependence does c o n s t i t u t e a major drawback and one would expect t h i s detector to be of l i m i t e d value i n s i t u a t i o n s where scatter i s present. In s p i t e of t h i s , however, f i l m has proven to be an adequate dosimeter as the r e s u l t s of t a b l e XV c l e a r l y i n d i c a t e . The reasons for t h i s are i l l u s t r a t e d i n f i g u r e k where i t i s shown that the major scatter component reaching the f i l m i s due to photons scattered only once and scattered predominantly i n the forward d i r e c t i o n with minimum energy degra-dation. F i l m , because of i t s ease of handling, i s surely the detector of choice i f routine measurements i n v o l v i n g patients are to be performed. We i n i t i a l l y required that our method permit us to measure the equivalent thicknesses of an inhomogeneous phantom with a maximum inaccu-racy of ± 0.5 cm. The RMS deviations that we have experimentally ob-served are smaller than t h i s but we wished to confirm these estimates of the accuracy by an independent evaluation. Figure 15 shows transmission p r o f i l e s measured using the inho-mogeneous phantom ( f i g u r e l U ) . The only d i f f e r e n c e between case (a) and case (b) i s that i n the l a t t e r a small aluminium c y l i n d e r of 0.31 cm d i a -meter i s present i n the mid-plane of the .phantom. This a d d i t i o n a l 0.h2 cm of water-equivalent material modifies v i s i b l y the transmission pattern as the scan i n d i c a t e s . Similar modifications were observed when the c y l i n -der was moved to other p o s i t i o n s i n the phantom. The a b i l i t y to detect such an object i s i n agreement with the previous estimate of accuracy. We have performed no measurements of patients at t h i s time and a complete a p p r a i s a l of the method cannot be attempted before these measu-rements are made. We f e e l , however, that the method o f f e r s decided advan-tages over s i m i l a r procedures as described i n the l i t e r a t u r e , with compa-rable or better accuracy. m o T-) i) • p - p •rH 6 cd E H 44 -MX III! if iii M bit ijli Relative p o s i t i o n (a) w o x! -o <U -p -p •H CO a E H lit! •i: if': •ETH i l l ! ; I:I ; ! i::. ~f : {!: : -T I i: • ; i irrr iii ' ; [:!: f :::: THT : \ \ : \ I M i'lj Tit? .fr: /l I •i iii in;.; •.? i.: 1 • : • -{ \ -/ v I?: •rrH-:n.tt + -LLll l. iltt . i S S i j i i i i i t lift "i-'f^ f HE! iSt !:h.i lill tin •[:!:,':'•• itt: ]:!:!:[ • Relative p o s i t i o n (b) FIGURE 15 TRANSMISSION PROFILES a) WITH AND b) WITHOUT THE ADDITION OF A SMALL ALUMINIUM ABSORBER i U) I 5. CONCLUSION A method has "been devised f o r the determination of water-equi-valent thicknesses along the f u l l width of a human transverse cross sec-t i o n . Both f i l m and commercial s i l i c o n diodes were employed to perform the y-ray transmission measurements. The scatter reaching the detector was then determined using a Clarkson-type mathematical procedure and used to c a l c u l a t e narrow-beam transmission data. Water-equivalent thicknesses were then deduced using a measured l i n e a r attenuation c o e f f i c i e n t , up, of .0653 ± .0002 cm which i s i n agreement with published values. With eit h e r detector it. was p o s s i b l e to deduce, from transmission measurement, the equivalent thickness along a path i n an inhomogeneous phantom to better than ± 0 . 5 cm. The exposure time needed to obtain the information, 0.3 and 0.7 minutes f o r f i l m and diode r e s p e c t i v e l y , i s c l i n i c a l l y acceptable from the point of view of patient movement. The simpler procedure with f i l m makes i t the choice f o r routine measurements i n v o l v i n g p a t i e n t s . When used with a t r a n s a x i a l tomography unit or any instrument capable of producing an image of a transverse cross section, the method w i l l permit the determination of the electron density, ( r e l a t i v e to water) of the various t i s s u e s present. I t w i l l then be p o s s i b l e to account f o r the presence of these t i s s u e s when dosimetry c a l c u l a t i o n s are performed during treatment planning with beams of X rays, y rays or heavy charged p a r t i c l e s . -1+5-BIBLIOGRAPHY 1. G. MARINELLO, C. MELLE, J . VILCOQ, ET AL. , "Interest de l a tomogra-phie a x i a l e en p o s i t i o n couch£e pour l a preparation des malades a l a radiotherapie". Journal de Radiologic, de 1'Electrologie et-de-Mgdecine Nucleaire, 5j+_, 81+1-81+7, 1973. 2. P.V. HOUDEK, K.K. CHARYULU, A. SUDARSANAM, ET AL., "Role of t r a n s -verse a x i a l tomography i n three dimensional treatment planning". Radiology, 112, 1+09-1+12, 191k. 3. .S. 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PRICE, "A method of determining the sensitometric properties of non-screen X-ray films".. B r i t i s h Journal of Radiology, h6, 719-723, 1973. -1*7-29- D. D. LINDSAY, B.E. STERN, "A new t i s s u e - l i k e material f o r use as bolus". Radiology, 60, 355-362, 1953. 30. H.M. TRENT, D.E. STONE, R.B. LINDSAY, "Density of s o l i d s " . In "American I n s t i t u t e of Physics Handbook", D.E. Gray, coordinating e d i t o r (McGraw-Hill, Toronto, 1957), pp. 2-17, 2-3*+. . 31. C.M. DAVISSON, R.B. EVANS, "Gamma-ray absorption c o e f f i c i e n t s " . Reviews of" Modern Physics, 2l+_, 79-107, 1952. 32. R.D. EVANS, "X-ray and y-ray i n t e r a c t i o n " . In "Radiation Dosimetry, volume 1", second e d i t i o n , F.H. A t t i x andW.C. Roesch, editors (Aca-demis Press, New York, 1968), t a b l e XXIII, p. 136. 33. C.G.A. AIRD, F.T. FARMER, "The design of a thimble chamber f o r the Farmer dosimeter". Physics i n Medecine and Biology, 17, l69-17^ » 1972. 3k. G.P. BARNARD, E.J. AXTON, A.R.S. MARSH, "A study of c a v i t y ion cham-bers . f o r use with 2 MV X-rays: Equilibrium wall'thickness: Wall . absorption c o r r e c t i o n " . Physics i n Medecine and Biology, 3_, 366-39^, 1959-35. W.R. BRUCE, H.E. JOHNS, "The spectra of X rays scattered i n low atomic number mater i a l s " , B r i t i s h Journal of Radiology Supplement #9 ( B r i t i s h I n s t i t u t e of Radiology, London, i960), pp. 6-7--US-APPENDIX A . . CALCULATION OF FIRST SCATTER FOR THE GENERAL CASE Figure A l i l l u s t r a t e s the conditions under which the c a l c u -l a t i o n i s performed. We assume a monoenergetic source of 1.25 MeV photons f a l l i n g on an annular r i n g of volume 2irydydx. We wish to evaluate the num-ber of photons dNi scattered only once at an angle 9 from t h i s annular r i n g and reaching the detector. From Davisson and Evans ( 3 l ) , 2 2 dN 2 = [N 0(cos <)>o/(A-x) )2?rydy exp(-p 0(x 0-x)sec<)> 0) ] [K ( e)p dx] 2 2 [expC-yjxsec^)] [ s( sec<j>! ) e(8 )/((B+x) sec (A-l) where Nq = number of photons emitted by the source per u n i t time. s = area of plane detector. K(e) = Compton cross section of the number of photons scattered 2 per u n i t s o l i d angle i n the d i r e c t i o n 8. (cm /electron) K(9) = r 0 2 [ l + c t ( l - c o s 8 ) ) " 2 (l+cos 28+(a 2 ( l - c o s 9 ) 2 ) / ( l + a ( l - c o s 8 ) )]/2. K(8) = r 0 2 k ( 8 ) / 2 . e ( e ) = energy dependence of detector r e l a t i v e to 1.25 MeV photons, rg = c l a s s i c a l radius of the electron r 0 = 2.8l8- 1 0 " 1 3 cm. = electron density of the s c a t t e r e r . 23 3 = 3.3^^*10 electrons/cm f o r water. e narrow-beam attenuation c o e f f i c i e n t of the primary 1.25 MeV photons. 2 u 0 = .0653 cm /g for-water. Hi = attenuation c o e f f i c i e n t of the scattered photons neglecting p h o t o e l e c t r i c absorption. Ui = u 0 f ( a ' ) / f ( o ) . . f(a) = 2Trr 0 2 ((l+a)/a 2) ( (2(l+a)/(l+2a)) - cx _ 1ln(l+2a)) + (2 a ) - 1 l n ( l + 2 a ) - (l+3cO/(l+2a) 2). -1*9-a = energy of primary photons i n uni t s of electron mass. a = 1.25/.5110. a' = energy of scattered photons i n uni t s of electron mass, a' = a/ ( l + a ( l - c o s 8 ) ) . The d e s c r i p t i o n of the equation i s reproduced from Davisson and Evans: "The f i r s t bracket i n the equation i s the number of photons incident on the.annular r i n g per second; the second bracket i s the f r a c t i o n of the photons scattered per unit s o l i d angle i n the d i r e c t i o n 6; the t h i r d represents the lo s s i n the num-ber of scattered photons due to absorption i n the absorber; and the l a s t bracket i s e s s e n t i a l l y the s o l i d angle subtended by the detector at any point of the sca t t e r i n g volume." The number of primary photons measured by the detector without 2 the absorber i s Nos/(A+B) . Then, from equation ( A - l ) , the r e l a t i v e num-ber of photons s i n g l y scattered dFi from the volume element i s dFi = .083 1+3[(A+B)/((A-X)(B+X))] 2COS 2<)) 0COS<{) 1 exp[-y 0(x 0-x)sec<j> 0-y 1xsec<j> 1]k(6)e(e)dxdy ( A - 2 ) 2 where . 0831+3 = Trrn p . e The measured i o n i z a t i o n current i s assumed proportional to the energy, dAj, absorbed i n the perspex cap of the chamber. For dFi photons dAi = d F i a ' y (a')/v (a) (A-3) x en en where u e n(a)/p = mass energy absorption c o e f f i c i e n t of perspex at energy a. By i n t e g r a t i n g equation ( A - 3 ) , i t i s possible to determine the f i r s t scatter reaching P, A : = \ \ dAj (A-1+) where x n = t o t a l absorber thickness. R = f i e l d radius of c i r c u l a r f i e l d measured at depth (XQ-X). It i s , f o r a diverging beam, a function of the thickness x. Absorber v - — - — " 1 * 0 r J 6 A Isotropic source of 1.25 MeV photons h < A Isotropic detector < : — B + \ -*-x 0 -> 9» p FIGURE A l DIAGRAM ILLUSTRATING THE CALCULATION OF THE FIRST SCATTER REACHING A POINT P POSITIONED AT A DISTANCE B FROM THE SCATTERER [Adapted from Davisson and Evans (3l)] E f f e c t i v e photon energy (MeV) .082 .098 .112 .165 -375 -538 1.19 r 1 1 r - r — i 1 1 1.5 2 k 8 10 15 Half-value l a y e r (mm Cu) FIGURE A2 CALIBRATION CURVE FOR THE BALDWIN-FARMER 0.6 cm' CHAMBER CORRECTED FOR THE PRESENCE OF THE BUILDUP CAP . -51-R = R 0 ( A - x ) / A . RQ = f i e l d radius of c i r c u l a r f i e l d at distance A from the source. Equation (A-U) was evaluated "by numerical i n t e g r a t i o n . The following r e l a t i o n s were used to express equation (A-3) i n terms of the v a r i a b l e s x and y only, cj>0 = Arc tan(y / ( A-x)). • = Arc tan(y/(B+x)) 6 = <fr0 + <fr» . . (A-5) Evans (32) has tabulated values of y /p f o r perspex. His data en can be f i t t e d , f o r photon energies between 0.10 and 1.25 MeV, by the empi-r i c a l expression, y (ct)/p = ( .0352-.0026a)/(l+.U55 exp(-l+.29(a-.196))). (A-6) en Table A l shows the q u a l i t y of the f i t . : The energy dependence, e ( e ) , of the Baldwin-Farmer 0.6 cm ion chamber used i n our experiments i s not known. Our chamber i s of an older design f o r which t y p i c a l c a l i b r a t i o n curves have been r e c e n t l y published (33). An average of these c a l i b r a t i o n curves i s presented i n f i g u r e A2. The c a l i b r a t i o n was made r e l a t i v e to 2 MV X rays at the National Physical Laboratory. A d i f f e r e n c e between c a l i b r a t i o n and experimental conditions does e x i s t , however. During our measurements, the ion chamber was perma-60 nently f i t t e d with a buildup cap s u i t a b l e for i r r a d i a t i o n i n a Co beam. The presence of t h i s a d d i t i o n a l absorbing material must be accounted f o r and s u i t a b l e c o r r e ctions applied to the c a l i b r a t i o n curve before e(6) can be i n f e r r e d from i t . Figure A2 shows the e f f e c t o f . t h i s c o r r e c t i o n using some r e s u l t s of Barnard et a l . (3^). The energy v a r i a t i o n i s then of the order of 0.1% over the range of half-value layers extending from -52-TABLE A l COMPARISON OF EVANS' TABULATED VALUES OF y /p (32) FOR e n PERSPEX TO THE VALUES USED IN THE CALCULATIONS a y en Tabulated Calculated* (cm 2/g) (cm 2/g) .1957 .0238 .0238 • 2935 .0265 .0265 .391*+ .0286 .0287 • 5871 .0310 . 0310 .7828 .0320 .0318 .9785 .0321 .0322 1.17h .0319 . 0319 I.566 .0311 .0311 -1.957 .0301 .0301 2.935 .0275 .0275 * From equation (A-6). 1 to 12 mm of copper. We thus f e e l j u s t i f i e d i n neglecting t h i s e f f e c t and e(6) = 1 was used throughout the c a l c u l a t i o n s . -53-APPENDIX B CALCULATION OF FIRST SCATTER AT THE EXIT SURFACE OF AN ABSORBER Bruce and Johns (35) have described a method to c a l c u l a t e the f i r s t s c a t t e r c o n t r i b u t i o n i n s i d e a water phantom i r r a d i a t e d by c i r c u l a r photon beams of various energies. The method can be applied to the exit dose s i t u a t i o n by simply neglecting the backscatter c o n t r i b u t i o n . Figure B l i s adapted from t h e i r paper and defines some of the qua n t i t i e s used. The r e l a t i v e energy fluence d l i f o r photons of energy between hv' and hv' + dhv' scattered at an angle 9 i s d l i = p ehv'/hv(da/dhv')[(exp(-y 0xo))/(yi-yocos9)] [ l - e x p ( - ( y i - y o c o s 0 ) x )] dhv' (B-l) u m where dc/dhv 1 = Compton cross section of the number of photons scattered per u n i t energy i n t e r v a l at energy hv 1. = - r r r 0 2 ( a h v ) ~ 1 [ ( l + c o s 2 e ) + a 2 ( l - c o s e ) 2 / ( l + a ( l - c o s 9 ) ) ] . hv = incident photon energy = 1.25 MeV hv 1 = scattered photon.energy = hv/(l+a(l-cos9))• x = maximum absorber thickness between the edge of the beam and the point P. The meaning of the other symbols i s s p e c i f i e d i n appendix A. Ai can then be determined by i n t e g r a t i o n with respect to the energy of the scattered photon, ,hv A 2 = \ dixy (a')/y (a) .(3-2) ), en en hv . min where hv . = minimum energy of the photons scattered i n the forward min d i r e c t i o n reaching P. = hv/(l+a), -5k-FIGURE B l DIAGRAM TO ILLUSTRATE THE CALCULATION OF THE FIRST SCATTER REACHING A POINT P INSIDE AN HOMOGENEOUS SCATTERER [Adapted from Bruce and Johns (35)] -55-In t h i s paper, equation (B-2) was evaluated as a summation tending from 0.35 to 1.25 MeV by 0.10 MeV i n t e r v a l s . -56-APPENDIX C CALCULATION OF THICKNESSES FROM TRANSMISSION MEASUREMENTS The mathematical method employed to c a l c u l a t e water-equivalent thicknesses from transmission r e s u l t s i s described here. The measurements y i e l d values of r e l a t i v e transmitted dose T at points d i s t r i b u t e d along a l i n e i n the detector plane. This l i n e corres- . ponds to a geometrical p r o j e c t i o n of the l o n g i t u d i n a l axis of the rectan-60 gular Co beam. The f i e l d dimensions, at 120 cm from the source, are 6 x 2b where 2b i s the length of the l o n g i t u d i n a l side. Mathematically, T = T ( P i , b , t i ) (C-l) where P. = distance of point i o f f the c e n t r a l axis of the f i e l d , 1 (i= 1,...,n). t. = water-equivalent thickness along the path j o i n i n g the source and point i . Since t ^ i s r e l a t e d to the primary transmitted dose, exp(-uot^), we can writ e , from equation (3), t± = [ln[T(P ,t>,t ) - A ( P i , b , t i ) ] ] / - y 0 (C-2) The quantity A(P^,b,t_^) must then be evaluated in'the general case where the absorber i s an inhomogeneous body. We have assumed f o r t h i s evalu-a t i o n that the primary beam-is uniform over the length of the f i e l d . We consider the case where the points, i , are equally d i s t r i -buted along the measurement l i n e . This is. shown i n f i g u r e CI f o r the case where n = 5. S i m i l a r l y , we segment the r a d i a t i o n beam into n smaller beams indexed as shown. At each point i , we then write n A(P.,b,t.)= E A(P.,b/n,t) (C-3) 1 1 j = l 1 -57-Point k 2 1 3 5 Detector plane FIGURE CI ILLUSTRATION OF THE GEOMETRY USED TO CALCULATE THE. SCATTER REACHING THE MEASUREMENT POINTS -58-where A(P.,b/n,t ) = r e l a t i v e scatter reaching point i from beam j . t = water-equivalent thickness along the path j o i n i n g the source and the point j . In the p a r t i c u l a r case of points on the c e n t r a l axis of the rectangular beam ( i . e . , point 1 i n f i g u r e C l ) , the determination of the terms of equation (C-3) follows from the r e s u l t s presented i n section 2.3. Thus, A. . (P.,b/n,t.) = A(0,b/n,t.) i=j l l ' l A j odd ( P i ' b / n ' V = U ( 0 , j b / n , t j ) - A ( 0 , ( j - l ) b / n , t j ) ] / 2 A (P ,b/n,t ) = [A(0,(j+l)b/n,t!) - A(0,jb/n,t!)]/2 (C-k) j even ± j j J . where t'. = t . cos8 . J J J G = Arc tan[(j-2)b/n/120], j=3,5,... = Arc tan[ ( j - l ) b / n / l 2 0 ] , j=2,l+,... For the general case of points o f f the axis of the o r i g i n a l beam, the c a l c u l a t i o n i s performed i n a s i m i l a r fashion assuming the v a l i d i t y of the procedure described i n the t e x t . The c a l c u l a t i o n s performed i n section 3-3 were based on an i t e -r a t i v e procedure. We i n i t i a l l y assumed s p e c i f i c thicknesses t ^ and solved equations (C-2) and (C-3). The newly determined thickness values were then incorporated i n equations (C-2) and (C-3) and a new s o l u t i o n obtained. The procedure was continued u n t i l a s p e c i f i c c r i t e r i o n of convergence was s a t i s f i e d . Usually a few i t e r a t i o n s were s u f f i c i e n t to ensure that the v a r i a t i o n of A(P ,b,t ) was l e s s than 1% and thus w e l l within the expe-rimental accuracy. 

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