Theory of radioisotope scanning

International Journal of Applied Radiation and Isotopes, 1958, Vol. 3, pp. 181-192. Pergamon Press Ltd., London

Theory of Radioisotope Scanning* G. L. B R O W N E L L Physics Research L a b o r a t o r y , Massachusetts General Hospital, Boston, Mass.

(Received 12 August 1957) T h e visualization of radioisotope distributions is a n increasingly i m p o r t a n t a p p l i c a t i o n of radioisotopes. A general relation is derived, valid for a n y detecting system, yielding the optimal radioisotope content as a function of resolution, detector efficiency, time d u r a t i o n of scan, a n d dimensions of the object being scanned. Various collimating systems are considered a n d isoefficiency curves calculated. T h e simplest collimator consists of a cylindrical a p e r t u r e in a lead shield. T h e use of a tapered a p e r t u r e improves the sensitivity s o m e w h a t for a given resolution. A focusing detector m a y be constructed b y means of a series of tapered apertures a i m e d at a focal spot. Finally, the detection of a n n i h i l a t i o n radiation resulting from positron detection is considered. T h e resolution a n d efficieneyof these collimating systems are compared. N u m e r o u s factors modify the general design considerations. T h e effect of p e n e t r a t i o n of lead by the ;a-rays is considered as well as the effect of finite source sizes a n d the a t t e n u a t i o n of r a d i a t i o n in the object b e i n g scanned. B a c k g r o u n d r a d i a t i o n arises from m a n y sources a n d results in a n effective limit to the a m o u n t of radioisotope which can be employed. Examples of single-channel a n d coincidence s c a n n i n g techniques are given.

THI~ORIE

DE L'EXPLORATION

RADIOISOTOPIQUE

L a perception des distributions radioisotopiques est une application des radioisotopes d o n t l ' i m p o r t a n c e devient de plus en plus grande. O n obtient u n r a p p o r t g6ndral, valable pour tous les syst~mes de ddtection, d o n n a n t le contenu radioisotopique o p t i m a l c o m m e fonction de la r6solution, de l'efficacit6 d u ddtecteur, de la durde de l'exploration et des dimensions de l'objet scrut6. Des syst~mes vari6s de collimation sont considdres et les courbes d'iso-efficacit6 sont calculdes. Le collimateur le plus simple consiste en une ouverture cylindrique dans u n 6cran de plomb. L ' e m p l o i d ' u n e ouverture conique amdliore ~ quelque point la sensibilit6 p o u r u n e rdsolution donnde. U n ddtecteur visant u n point p e u t se former d ' u n e s6rie d'ouvertures coniques d o n n a n t sur u n point focal. Enfin, la consid6ration est donnde ~ l'observation de la r a d i a t i o n d ' a n 6 a n t i s s e m e n t suivant la ddtection des positrons. O n c o m p a r e la rdsolution et l'efficacit6 de ces syst~mes de collimation. Bien des sujets sont h consid6rer dans le dessin gdndral. L'effet de la pdndtration d u p l o m b p a r les rayons 7 est tenu en vue, ainsi que l'efl~t de la g r a n d e u r ddfinie des sources et l'att6nuation de la radiation dans l'objet scrut6. L a r a d i a t i o n de fond a b e a u c o u p d'origines et impose une limite effective & la q u a n t i t 6 de radioisotope q u ' o n p e u t employer. O n d o n n e des exemples des techniques d'exploration ~ c a n n e l a t i o n u n i q u e et ~ coincidence. TEOPI4H PA~HOH3OTOYIHOFO CHAHHPOBAHFIH BH3yaJiH3aJ~Hg pacnpe~Ie~eHH~ pa,~HOH30TOHOB gpeJ~eTaB~]neT Bee 5oo~ee Bo3paeTaioiiiyio no 3HaqeH~HO O6:IaCWb gpgMeHeHI4H pajl:4on3owonon. IIpe~{Aaraeweg o6JJ~ag MaweMaTHqecteaf[ 3aB14CHMOCTb, npI~ro;~na~ XJ~ .~o6o~i perncwpHpyio~e~ eHOTeMH, HOTOpan gOSBO:IneT onpeAe~uTb OnTI4Ma~bHOe coAepmaHHe pa~HoI430TOIIa, Hall ~NHHI{1410 paapemammefi enoc06H0CTI4 yCTaHOBHI4, D~eHTHBHOCTH J{awqI4I~a, 7UII4TeJII)HOeTH eHannpoBannn ~I pa:~Mepo~ * Supported by Grant No. AT-(30-1)-1242 from the U.S. Atomic Energy Commission, an institutional grant from the American Cancer Society, and a grant from the June Rockwell Levy Foundation. 181 1

182

G. L. Brownell

cEaH~pyeMoro 06%eliTe. PaceMaTpnBatOTeg paaaHqHbIe THHbI HOJIat~MaTOpOB n pacqeTHble ~pnnbte ORHHa~oBoi~ o~eHTIdBHOCTH. I]pocTeiimn~ H0o-[aI4MaTOpnpe;IcTaB~eT ~I4~HH~IpH~JecHoe OTBepCTne B CBIIHI~OBOM~Kpane. IlpnMeHenne KOaagMaTOpa C I~OHII~IecI~IIMOTttepcTHe.~t HOCRO.JIbHO yBesluqHBaeT qyBCTBViTe.rIbHOCTB npII ~aHHOfi paapemamt~ei~ (~IIOC05HOCTH. q)oKycHpy~omm~ I~OaaHMaTOp MOmeT 6~ITb COCTaBaeH n3 pn;Ia KOHnqecRrxX OTBepcTHf~. HaIle~IeHHNX Ha OXHy H Ty me ToqKy. I4, HaHoHeII, paceMaTpnBa~OTCU MeTO;I~ perrtcTpaI~HH aHHnrm-IgtinOHHOrO HaayqeHn~ c yqeToM HanpaBaeHnn nOa~TpOHHOrO nNaeTa. IIp~ aTOM i)aa:i~qHNe TI4IIM~o~.~Mnpymm~x encTeM cpanHnBa~OTCg rio nx paapemammef~ enocoSHocTr~ II nO 3dp(~eRTtIBHOCTI4. I~OUCTpyHRHg HOJI~gMaTOpa 3aBHeHT OT MHo~ecTBa (~a~TOpOB. OIieHgBaeTeg pO.]b V-H3J~yqeHt4g, npoH~Ha~oulero CHBO3b CBI,IH!4OBMe CTeHHI4 HOJU~gMaTopa; yq~4T~,~BaIOTCg (~ai;TOpLI, onpe~e:~eM~m FeOMeTpgefl ilCTOqH~t~a~3~IyqeH~t, gMe~oLReroHOHeqHb~epa3~ep~L a TaHH4e 3aBgeHLqt4e OT yMeHhgleHH~i aHTIIBHOCTHCHaHgpyeMorO o6%e~Ta, t t a ~ e o6mero (potta iI32iyqeitI/iH, oSyc~oB~rleHitoro MHOFI/IMg I/ICTOqHIIHaMII, orpanHqgBaeT HOJInqeCTBO O(~)eHT~BHO IlCnO~i)3yeMOrO pa~Ilo~3oTona. 14pnBe;~eHb[ l~pHMepH I~ICIIOJIb3OBaHHH O~ItOHaHao]LHMX yCTaHOBOK I4 agriapaTypbt, ocytRecTB~mulef~ c~aHnponaHi~e no MeTO~y eoBua;~eH~fi. T H E O R I E DES R A D I O I S O T O P E N - S C A N N I G S Des Sichtbarmachen der Verteilung von Radioisotopen nimmt dauernd an Bedeutung zu. Eine allgemeine Beziehung wird abgeleitet, welche ftir jedes Detektorsystem Giiltigkeit besitzt und die optimale Menge an radioaktiver Substanz ergibt als Funktion des Aufl6sungsverm6gens, der Detektor-Empfindlichkeit, der Zeit, die zum Scanning ben6tigt wird, und der Gr6sse des Objektes, an welchem das Scanning durchgeftihrt wird. Verschiedene Kollimatorsysteme werden betrachtet und Kurven ffir gleiche Empfindlichkeiten werden brechnet. Der einfachste Kollimator besteht aus einer zylindrischen Apertur in Bleiabschirmung. Die Verwendung einer konischen Apertur verbessert, bei vergegebenem Aufl6sungsverm6gen, die Empfindlichkeit ein wenig. Die Konstruktion eines fokusierenden Detektors mit Hilfe mehrerer auf den Fokus gerichteter konischer Aperturen wiire m6glich. Schliesslich wird der Nachweis der Vernichtungsquanten von Positronenstrahlern diskutiert. Das Aufl6sungsverm6gen und die Empfindlichkeit dieser Kollimatorsysteme wird verglichen. Zahlreiche Faktoren sind zu beriicksichtigen, wenn ein allgemein verwendbarer Entwurf gemacht werden soll. Das Eindrigen der 7-Strahlung in die Bleiabsehirmung, die endliche Ausdehnung der Quelle und die Abnahme der IntensitRt der Strahlung in dem Objekt, welches dem Scanning unterworfen ist, werden beriicksichtigt. Die Hintergrundstrahlung besitzt verschiedenartigste Provenienz und bedingt schliesslich eine Begrenzung der Menge der verwendeten radioaktiven Substanz. Beispele ftir Ein-Kanal- und Koinzidenz-Scanning-Methoden werden angegeben. 1. I N T R O D U C T I O N IN m a n y b i o l o g i c a l a n d m e d i c a l p r o b l e m s , i t is d e s i r e d to v i s u a l i z e t h e d i s t r i b u t i o n o f a radioisotope in a region of the body. A complete representation would reqmre a three-dimensional portrayal, but most scann i n g d e v i c e s a r e l i m i t e d to p l a n e p o r t r a y a l . T h u s , for a c o m p l e t e r e p r e s e n t a t i o n , s c a n s in two orthogonal planes are required. In s o m e cases, a s i n g l e r e p r e s e n t a t i o n w i l l suffice o r a n a l t e r n a t i v e t e c h n i q u e m a y b e u s e d to obtain information concerning the distribution in the third dimension. I n this p a p e r , s o m e g e n e r a l c o n s i d e r a t i o n s of the problem of scanning are developed.

A general theory of rectilinear scanning will be derived, and the characteristics of various collimating devices discussed. Background effects w i l l b e c o n s i d e r e d . S o m e o f t h e c o n clusions o f t h e d e r i v a t i o n s p r e s e n t e d h e r e h a v e b e e n m e n t i o n e d in a p r e v i o u s r e p o r t , m R e c e n t l y , VAN DER DOES DE BYE (2) h a s p r e sented some interesting calculations which a r e s i m i l a r to p o r t i o n s o f t h e p r e s e n t d e r i vation. Methods of visualization employing lead grids and film or methods coupling y-ray apertures with light-intensifying devices w i l l n o t b e p r e s e n t e d h e r e , a l t h o u g h both show considerable promise.

Theory of radioisotope scanning

183

2. G E N E R A L D E R I V A T I O N For rectilinear scanning, it would be de- As a further simplification, the sensitivity sirable to use a collimator whose sensitivity will be considered to be % over the distance distribution is uniform in the direction nor- d and zero outside this distance. The final mal to the scanning plane. In general, picture of the sensitivity distribution is a however, the sensitivity distribution must be cylinder with uniform sensitivity, %, normal considered a function of three space variables, to the scanning plane and mid-plane, of x', y', z', measured from the collimator. diameter d and length l. This cylindrical This sensitivity distribution will be termed region bears a close resemblance to the actual ~(x', y', z',) The activity distribution in the sensitivity distribution with positron detecregion being scanned will be p(x, y, z). The tion, and is a useful approximation for other response at any position will be: collimating devices. A more detailed analysis of the sensitivity R = .fp(x, y, z)e(x', y', z') dv (1) distribution is of value in determining the In order to treat the problem analytically, required accuracy of response or counting some simplification must be made. First, rate. I f the actual response is represented by the z' dependence of the sensitivity and a normal distribution, the half-intensity activity distributions will be neglected, and points fall at 1-77 standard deviations, and the x' and y ' distribution on a plane called 76 per cent of the area under the distribution the source plane will be considered valid for lies between these two half-intensity points. any plane. This source plane will in general For a theoretical positron detection curve, be the mid-plane of the object being studied. 82 per cent of the area lies between these This analysis would be rigorously correct for half-intensity points and actual measured the case of activity actually lying on the values average nearer 85 per cent. This source plane or if the sensitivity distribution value for single and multiple aperture detectis actually independent of z. However, it is ing devices averages about 83 per cent. For a useful approximation to other cases. Fur- the present argument, a value of 80 per cent ther, the activity distribution will be con- will be taken for all types of collimators. At any instant, the detector records the sidered almost uniform. This might be analogous to the case where one wishes to activity in a cylindrical region of diameter d. detect a small region of slightly increased However, because of the actual shape of the activity in a large region of uniform activity. sensitivity distribution, a portion of the resThe region to be scanned will be considered ponse results from activity outside this region. It would be a waste of information to require to have an area A and a length l. The collimating apertures will be con- a statistical accuracy greater than the limit sidered to be circular, so that the sensitivity of accuracy imposed by the collimating distribution on any plane parallel to the system. The optimum situation would occur scanning plane will be a function of only one when the statistical uncertainty equals the variable. This sensitivity distribution can be geometrical uncertainty, which is assumed observed by passing a point source of acti- above to be 20 per cent. I f the events vity across the aperture on the source plane. recorded by the detector follow a Poisson The exact shape of this curve will be dis- distribution, this would imply that V/n-/n = cussed later, but for most collimating systems, 0-2, or n = 9-5 recorded counts for the detera curve similar to a normal distribution is mination of the activity in the cylindrical obtained. The similar shapes of these curves region of diameter d. The subtended area of suggest the use of two parameters for an this region is 7rd~/4, so that 4A/~rd 2 such areas analytical description of the sensitivity dis- must be observed in the complete scan. If tribution, the sensitivity of a point source at 25 counts are recorded for each area, 100A/Trd2 the center, e0, and the distance between the counts are required for the complete scan. two points at which the sensitivity is oneAs the radioisotope concentration, p, half the maximum value or the resolution, d. expressed as disintegration/sec per cm 3, is

184

G. L. Brownell

constant and the sensitivity, %, in counts/sec per disintegration/sec is constant, the detector counting rate R will be constant and will equal p%@rd~l/4). This counting rate times the total duration of scan T, must equal the total number of counts for the complete scan, and should equal the number previously calculated from statistical arguments: peo 7rd2 l = lOOA/Trd 2 4

400A p - - 7r2d4leo T

dps/cm a

(2)

This gives the required activity per cm a to produce a significant picture of resolution d. In any practical scanning problem, p would be a function of position. Here, the value calculated fi'om equation (2) would represent an average value of radioisotope concentration. Equation (2) shows the marked dependence of the required activity on the resolution d. Indeed, for each detecting system it may be shown that the sensitivity % varies as d ~ as the detector area varies as d 2. The total

dependence of p on d therefore varies as the inverse sixth power. This clearly illustrates the necessity of comparing collimating devices at equal resolution. For any given system, a decrease in the resolution distance by a factor of 2 would require a 64-fold increase in activity. Actually, in designing a detecting system, the value of p is usually set by biological consideration of radiation dosage. As A and l are determined by the object being scanned and T is usually set as the maximum convenient time allowed for scanning, d will depend primarily on %. It is obviously desirable to select a collimating system which will give a maximum value of E0 and a corresponding minimum value of d so as to produce a scan which gives a maxim u m amount of information. The many assumptions which have been made limit the accuracy of the numerical factor of equation (2). However, the strong dependence of O on d is independent of these assumptions. Further, as these assumptions refer to all collimating systems, such systems can be compared solely on the basis of %.

3. D E T E R M I N A T I O N OF C O L L I M A T O R S E N S I T I V I T Y DISTRIBUTION

The collimator sensitivity, e, may be considered to be a function of four variables:

=fg s

(3)

w h e r e f i s the fraction of disintegrations which result in detectable radiation, g is the geometrical efficiency, ~ is the attenuation of radiation in the object being scanned, and s is the absolute sensitivity of the detector. The quantity g has been calculated for three general types of collimators, shown in Fig. 1. The first consists of a cylindrical aperture, case I, in an absorbing shield. An alternative design of a conical aperture, case II, is also treated. The second general method consists of a series of conical apertures focused on a point exterior to the shield, and the final type consists of coincidence detection. The method of calculation is illustrated in Fig. 2 for cylindrical apertures. In all cases angles will be assumed to be small and the shield will be assumed to be

opaque to radiation. Qualifications to these two assumptions will be discussed later. The geometrical efficiency is determined by projecting the circular area on the collimator plane, II, on to the detector plane, I, which is the back surface of the collimator and the front surface of the crystal of the scintillation counter. A' is the area of overlap of the detector area and the projected collimator area on plane I. The geometrical efficiency, g, is then: A' g -- 4rr(a + x) 2 (4) Fig. 3 shows the calculated geometrical efficiency for the cylindrical aperture. The isoefficiency lines have been normalized to 100 at a point a distance a from the front of the collimator; i.e. the center of the source plane. The dimensions are expressed as x / a and r/b, where a is the collimator length and b is the collimator radius. The geometrical

Theory of radioisotope scanning

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efficiency decreases rapidly as a function of distance from the aperture plane. The absolute value at the point (x/a = 1, r/b = 0), go, is b2/16a 2, and the value of d on the source plane is 4b. On any plane, the efficiency is constant for rib ~< 1, but decreases rapidly for r/b > 1. The geometrical efficiency of the conical aperture is shown in Fig. 4. The calculations have been performed as with the cylindrical aperture, and the isoefficiency lines have been normalized to 100 for x/a = 1, r/b = O. Whereas the absolute efficiency at this point, go, remains b2/16a 2, the resolution, d, is decreased to 1-75b. Thus an improvement in collimation for sources on this plane has been made. However, it should be noted that the relative values of resolution for other planes have deteriorated. Thus, this type of aperture offers little advantage over a cylin-

drical aperture unless the activity actually lies near this plane. I f a series of apertures are focused at a point on a focal plane, the geometrical efficiency at that point may be increased markedly while maintaining the resolution on the source plane. Usually one large detector is employed, and a series of apertures is placed in front of it. The case of conical apertures focused on the plane x = a will be considered. The geometrical efficiency distribution for a single conical aperture, Fig. 4, will be termed g(x/a, r'/b, b), i.e. let r of Fig. 4 now be r'. The function defined as u(x/a, r'/b) II a DETECTOR

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Fro. 3. Calculated geometrical efficiency for a cylindrical aperture. Case I. Values of geometrical efficiency are normalized to 100 at (x/a = I, rib = 0) a n d all distances are expressed in dimensionless terms.

G. L. Brownell

186

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detector is available to the apertures, this would correspond to a b o u t 12 conical apertures. T h e curves have been n o r m a l i z e d to 100 at the point x/a = 1, rib = 0, and it is found that the actual m a x i m u m occurs at a point x/a = 0"85, r/b = 0, and has a value of 108. T h u s an isoefficiency curve at 100 exists. T h e value of d on the source plane is the same as in Fig. 4, or 1.75 b, but the efficiency go has been increased to (25/32) (b2/a 2) if one-half of the detector area is used. Further, a true focal point exists as the efficiency decreases t o w a r d the a p e r t u r e plane. However, it should be noted that the resolution on planes off the source plane increases markedly, and this effect increases with increasing n u m b e r of apertures. T h e focusing collimator is ideally suited to detection of activity distributions lying on or n e a r the source plane, b u t its advantages diminish with extended sources. T h e detection of the annihilation radiation resulting from positron emission provides a third m e t h o d of achieving a desirable efficiency p'attern. T h e two annihilation q u a n t a are emitted at 180 ° , or " b a c k to b a c k " , from the site of the positron-electron annihilation, and these two q u a n t a are detected b y observing coincident pulses from two scintillation detectors placed on opposite sides of the object being scanned. T h e geometrical efficiency of this array m a y be

Theory

~

radioisotope scanning

187

TABLE I Collimator

Efficiency, go

Resolution, d

Single I (cylinder)

b2116a 2

4-00b

1-00

Single I I (taper)

b~ll6a 2

1.75b

5.23

Focus (bo = 5b)

25b2/32a ~

1-75b

65.4

b~/4a 2

0.82b

95"5

Coincidence

determined in an entirely analogous manner to the single-aperture case, although here the area of one detector is projected through the point of observation on to the plane of the second detector. If A' is the area of overlap, the geometrical efficiency is: g

-

A' 0 ~< x ~< a 4rrx 2

-

(6)

The efficiency is obviously symmetrical about

the plane x/a

=

1.

Fig. 6 shows the geometrical efficiency for the case of circular apertures. The values have been normalized to 100 at the center of the source plane (x/a = 1, r/b = 0). This distribution most nearly produces a cylindrical efficiency distribution. The value of d on the mid-plane is 0.82b, while the value of go is b2/4a 2. The symmetrical nature of the distribution is clearly of advantage in certain applications.

Relative efficiency normalized to equal d

Table 1 summarizes the values of go and d for the fou r collimating systems considered. In Section 2 it was pointed out that the sensitivity comparison was meaningful only for equal values of resolution. Consequently, the relative efficiencies normalized to equal values of d are also presented. The conical aperture has a normalized geometrical efficiency 5.23 times greater than the cylindrical aperture, while the focus collimator has a normalized efficiency of 65.4 times the cylindrical aperture. However, the coincidence detector has a normalized efficiency of nearly 100 times the equivalent cylindrical collimator. This high efficiency coupled with the spatial distribution of the coincidence detector makes it highly desirable for visualizing activity distributions lying deep within an object and particularly for "

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FIG. 7. Experimental resolution curve for cylindrical aperture collimator. T w o rnm source moved across aperture on plane at x/a ~ 0 " 6 6 . b = 8 ram. a 152 ram. Curves obtained in air and with I0 cm water moderator between source plane and detector.

188

G. L . B r o w n e l l

scanning symmetrical distributions. The focused detector is particularly advantageous for scanning activity distributions lying on or near a plane and for superficial distributions. The geometrical efficiency distribution for cylindrical apertures, (3-5) focus apertures, (6-81 and coincidence detection, (9,~°) have been studied by a number of investigators. Our measurements indicate a good agreement with the theoretical curves. Figs. 7 and 8 present typical experimental data for point sources moved across the aperture on the source plane for the case of a cylindrical aperture and a coincidence detector. A positron emitter, As v4, was used for the coincidence detector, and I ~al w a s employed for the cylindrical aperture. Pulse height discrimination was used in the latter case.

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oWithout

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" × With LT]~[ f~i~ moderator~

i -3

-2

-1

O

1

2

3

4 crn

FIG. 8. E x p e r i m e n t a l resolution curve for coincidence detector. T w o m m source moved across aperture on source p l a n e at x/a : 1, b ~ 9.5 mm. Curves obtained in air a n d with 10 cm w a t e r m o d e r a t o r placed on each side of source plane.

Three experimental factors will tend to modify the theoretical curves. First, the detector has a finite depth, and the assumption of a detector plane is obviously an approximation. The general effect of finite detector depth is to increase the apparent collimator length. In the case of the coincidence detection, this has the advantage of making the sensitivity distribution somewhat more insensitive to distance from the midplane, but maintaining the theoretical resolution.

The effect of penetration of y-radiation through the walls of the collimator has been studied. (n-131 In general, the effect is equivalent to an apparent increase in the radius of the collimator. It can be shown that the fractional increase in the resolution and in the apparent aperture diameter is given by the approximate expression: Ab

Ad --

d

1 --ffa

(7)

where # is the absorption coefficient of the collimator material--usually lead. For 1131 detection with a collimator of length 10 cm, this increase is only about 5 per cent. However, for higher energy gamma-rays, the effect will be greater. It is of interest to note that the fractional increase in resolution depends only on the collimator length. This correction does not apply to coincidence detection, as shielding is not directly involved in this case. I n any experimental efficiency determination, a finite source must be used, and this will tend to increase the resolution. A simple correction may readily be made for this effect. However, in the case of positron detection, a more fundamental effect occurs. The positrons have an average range of the order of a millimetre before annihilation. This range will increase the resolution distance slightly. To compare overall sensitivities of the various collimating systems, the remaining three factors of equation (3) must be included. The quantity f will vary for each isotope, but is usually between 0.5 and 1.0 for both y- and positron-emitters. ~ will in general be smaller for positron detection, because two quanta must penetrate the object being scanned, s will also be somewhat smaller, but the absolute efficiency of scintillation detectors to y-rays can be made to approach 1, so that the requirement that two quanta be detected does not greatly decrease s. Although all these factors will effect the overall sensitivity, go still remains the most important factor in the comparison. The use of square crystals for both singlechannel detection and coincidence detection

Theory of radioisotopescanning has been investigated both theoretically and experimentally. An efficiency distribution having the form of a square on a source plane would obviously be ideal for rectilinear scanning, as a more uniform coverage of the isotope distribution would result. The complete portrayal of such efficiency distributions requires sets of diagrams, as three dimensions are involved. However, Figs. 3-6 for circular detectors are adequate approximations 4. E F F E C T

OF

FINITE

In the preceding section, the constant represented the attenuation of radiation originating from the source plane in the object being scanned. It is obvious that this attenuation will also affect the sensitivity distribution if the activity is distributed throughout a source of finite thickness. This effect may best be considered as a modification to the thin-source calculations. It is of interest in this connection to examine the response of a detector to thin slabs of activity located at various depths in a finite object. If absorption in the object is neglected and if the slabs are of large area, the response of any detector will be independent of depth. This can be seen by direct integration of the isoefficiency curves or from the following argument. Consider a small solid angle segment constructed at the front surface of the detector. The area on the thin slab of activity defined by this solid angle will increase as the square of the depth. The efficiency of detection per unit area of slab will decrease with the inverse square of depth. Thus, these two effects will cancel and the overall response will be independent of depth. An object of finite depth may be considered as a series of thin slabs of large area. The total sensitivity for each slab, neglecting source attenuation, will be equal. It is clear that the concept of a focal region in the isoefficiency distribution of a collimator may be somewhat misleading when considering an extended source. Although activity on the source plane will be detected at high resolution, any activity on planes other than

189

to the distributions on the central plane of square detectors whose side dimensions equal the circular diameter, b. In general, square detectors and circular detectors of equal width will provide approximately equal resolution in the direction of scanning, but the square detector will yield an increase in sensitivity both because of the larger detector area and because of the larger source area. This increase is approximately 60 per cent. SOURCE

THICKNESS

the source plane will be detected at lower efficiency but over a larger area. The resultant scan will show the activity on the source plane at high resolution, but superimposed on this will be the portrayal of activity on planes other than the source plane at lower resolution. The optimum situation will occur when the depth of the focal region covers most of the activity to be observed. This situation is somewhat easier to produce with positron detection than with single detector methods, because the absorption does not affect the shape of the coincidence geometrical efficiency curves, i.e. Fig. 6 is valid for finite source thicknesses. The total distance traversed by the two annihilation quanta is independent of the source depth, and so the relative shapes of the curves are unchanged. The constant, % will be the same for each plane. In the case of singledetector techniques, the effect of source attenuation may be included by simple exponential attenuation, providing the buildup of scattered radiation is neglected. This may be a reasonable approximation if y-ray spectroscopy is employed. The total response of slabs of large area will now decrease as e-~x, where # is the total absorption coefficient of the y-rays in the object and x is the depth. The overall effect will be to decrease the efficiency values of Figs. 3, 4 and 5 at large x/a, and to compress the isoefficiency curves towards the detector. The magnitude of this effect will depend on the y-ray energy and the dimensions of the object being scanned. For large objects, this may limit somewhat the usefulness of single-channel techniques.

190

G. L. Brownell 5. B A C K G R O U N D

The response of any detecting system may be written: 7rd2 R =p.%.~-I (8) Using the optimized value of p found in Section 2, equation (8) becomes: R-

100A 7rd2

(9)

Thus the response varies as the inverse square of d. Any detector response other than that obtained with the resolution d will be considered background. This background can usually be expressed in terms of a power series of (l/d2), i.e.: B = B o @ B1/d 2 @ B2/d 4 + B3/d 6 + ' ' '

(10) where B0, B1, B2, etc. are constants of the system independent of d. B" will refer to the three single-detector methods and B c to coincidence detection. The following approximate analysis is applicable to the three single-detector systems considered, i.e. single aperture I and II and the focusing aperture. In equation (10), B0" is the response of the crystal to cosmic radiation and v-radiation other than that in the object being studied. This quantity can be reduced by pulse spectroscopy, but is independent of d. The second term, B1~, arises from the scattering of v-radiation. This term may also be reduced by spectroscopy, but the finite resolution of v-ray spectrometers means that some scattered radiation will be detected. In the object being studied, each volume may be considered a source of scattered radiation as well as primary radiation. Let the ratio between scattered radiation flux within an energy range which can be detected and the primary radiation flux be the constant, u. This constant will be a function of the size and geometry of the object and the energy of the v-ray, but will be independent of the

EFFECTS

resolution d.

The value of B1* will then be: BI" =

100Au

(11)

7r

The second term refers to primary radiation which has penetrated through the shield. I f this penetration is termed ~1, B~' will be: B~~ --

400A2~ 1 rr~

(12)

Higher terms originate from second-order effects such as the penetration of scattered radiation. These coefficients are small, however. In the case of coincidence detection, two terms predominate. Although the coincidence detector has a response essentially zero in the absence of a source of positrons, there is a term which is independent of d, B0c. This term results from the detection of simultaneous but randomly orientated v-radiation and annihilation radiation which occur in certain decay schemes. A second term is proportional to l i d 4, Be ~, and results from the finite resolving time of the coincidence circuit. The detailed analysis of these two factors requires a more elaborate consideration of the decay schemes, and this analysis has been presented elsewhere. (1°1 It is important to note that even though the amount of activity was not limited by other considerations, the background effects result in an optimum amount of activity and value of d. This is primarily because of the background terms varying as l i d 4. As the resolution is decreased and activity increased, the background effects will tend to mask the desired true response. Design considerations will modify the coefficients of the background terms, but an ultimate limit must be reached. In the case of a single detector, an increase in shield thickness and pulse spectroscopy will decrease the background effects, while in coincidence detection, choice of radioisotope and decrease in resolving time serve this purpose. Even with NaI detectors, resolving times of 0" 1 #sec may be employed, and with plastic crystals, resolving times one-tenth this value are attainable.

L.M. Q/20/56

FIG. 10. Positron scan (PCG) on patient with tumor in Scan of normal patient shows increased frontal area. activity over face muscle and around periphery of head but a low, uniform content over area of brain.

FIG. 11. Unbalance scan (AGG) of patient shown in Fig. 10. Cluster of straight marks over frontal area indicates that tumor is slightly to left of midline. Curved marks indicate a right unbalance.

FIG. 12. Positron scan (PCG) in AP plane of patient shown in Figs. 10 and 11. Tumor is seen to lie near midline but slightly to left.

p. 190

Theory of radioisotopescanning

191

6. I N F O R M A T I O N A N A L O G Y Certain aspects of information theory are two forms, background subtraction and the applicable to the scanning problem. Each use of a recording device with non-linear radioactive disintegration within the object response such as photographic film. being scanned during the scanning period Although many of these serve the purpose ot may be considered a bit of primary infor- making good scans more spectacular, it mation. Thus, during the scanning period should be noted that any such technique 5°, there occur pAl50 bits. Because of the results in a loss of information as well as efficiency of detection, only the fraction increasing the critical dependence on the %~rd2/4A are actually recorded by the de- various parameters. However, in certain tectors. This low overall efficiency of the cases such techniques may serve to emphasize order of 10 3 to 10 -a appears to be necessary the most useful information. This is particuto obtain the resolution. A further loss of larly true in cases where it is desired to information occurs in the electronic circuitry delineate an area of fairly high activity. and in the scaling of pulses before pr!nting Perhaps the soundest method is the direct or otherwise recording. The maximum recording of the primary data and subsequent available information is presented on the treatment by electronic computation. It record if each pulse produces a print. How- should also be remembered that the brain ever, subjectively, the information appears of the record reader is an efficient information more interpretable to the eye if some scaling processing device. is performed before printing. Scale factors Counts received from all background between 4 and 10 are often used, and the sources may be considered as noise. In latter value is used with the positron scanner. general, it is desirable to keep the total Very little information is lost so long as the noise counts below 20 per cent of that from distance between prints is less than the the desired signal: As indicated in Section 4, resolution distance. the signal-to-noise ratio depends on the There recently has been a number of amount of activity and on the collimating reports of non-linear recording. These take system used. 7. E X A M P L E S

OF S C A N N I N G

Fig. 9 illustrates the use of a cylindrical aperture to delineate a region of fairly high concentration of activity. In this study performed by Dr. PHILIPPE BENDA, Au 19s adsorbed on colloidal carbon particles was infused into the internal carotid artery of a dog as part of a study of the capillary filtration in the brain. Scans were obtained of the brain area and of the liver area. In this case no activity was found in the liver, indicating complete filtration in the brain. The scan adequately outlines the portion of the brain supplied by the internal carotid artery as well as several small nodes of activity resulting in tissues supplied by bifurcations of the internal carotid. In this rather simple scanning problem, where large amounts of activity can be employed, the single-aperture technique is probably as satisfactory as any other.

TECHNIQUES

The localization of brain tumors in humans presents a more difficult problem because of the lower permissible amount of activity and

: : - - - 2 2 : - : : : . - : ; - -. _- : . - - : : - -

"; - 2 : : i - L :

COIL/~,U ~emal

' ~ -caro'hd

FIG. 9. Single-channel scan w ittl cylindrical aperture of Au 19s labeled carbon particles infused into the brain of a dog by intracarotid injection. Aperture dimensions b = 6 m m , a = 152 mm. Symbol size i s 5 m m .

192

G. L. Brownell

the smaller c o n c e n t r a t i o n of isotope relative to s u r r o u n d i n g tissue. Figs. 10-12 illustrate a positron t e c h n i q u e for such localization. (1'1°,) Fig. 10 is a lateral view of the distribution o f As 7~ in the h e a d of a p a t i e n t w h o has a frontal t u m o r . T h e n o r m a l distribution of this radioisotope in the h e a d shows a higher c o n c e n t r a t i o n over the muscle a r e a of the face a n d a r o u n d the p e r i p h e r y of the h e a d , reflecting the higher c o n c e n t r a t i o n in muscle, scalp, a n d b o n e relative to n o r m a l brain. T h e a r e a of the t u m o r is i n d i c a t e d b y the dense c o n c e n t r a t i o n in the frontal region. This scan is called a p o s i t r o c e p h a l o g r a m , or P C G for short. T w o techniques are possible for d e t e r m i n i n g the lateralization of the t u m o r . T h e first m e t h o d is to c o m p a r e the gross c o u n t i n g rates on the two sides of the h e a d . Fig. 11 shows a scan of the detector u n b a l a n c e for this patient. This scan is obtained routinely at the s a m e time as the P C G scan, a n d two symbols indicate a n y u n b a l a n c e in c o u n t i n g r a t e ; a c u r v e d symbol for a right u n b a l a n c e a n d a straight symbol for a left u n b a l a n c e . T h e cluster of straight symbols o v e r the frontal a r e a shows the t u m o r is lightly to the left of midline,

a l t h o u g h the degree of u n b a l a n c e is v e r y slight. This scan is called an a s y m m e t r o g a m m a g r a m , or A G G for short. F o r m o r e precise lateralization, a P C G m a y be perf o r m e d in the o r t h o g o n a l plane, or an A P view. Fig. 12 shows an A P scan on this patient, a n d the t u m o r is seen to be n e a r l y on the midline, a l t h o u g h slightly to the left. These three scans p r o v i d e fairly precise inf o r m a t i o n for a n y t h e r a p e u t i c technique. This t e c h n i q u e for b r a i n t u m o r localization is n o w in routine use at the M a s s a c h u setts G e n e r a l H o s p i t a l a n d elsewhere. I n o u r l a b o r a t o r y , over 2000 patients h a v e b e e n scanned with this a p p a r a t u s .

A c k n o w l e d g e m e n t s - - T h i s project has b e e n car-

ried out in c o l l a b o r a t i o n with Dr. WILLIAM SWEET of the N e u r o s u r g i c a l Service a n d Dr. SAUL ARONOW of the Physics R e s e a r c h L a b o r a t o r y of the M G H . M u c h of the calculation was p e r f o r m e d b y M r . WILLIAM QUIGLEY. T h e single-channel scans w e r e o b t a i n e d in c o l l a b o r a t i o n with Dr. PHILIPPE BENDA, a n d Miss PAULINE MONTGOMERY p e r f o r m e d the positron scans.

REFERENCES 7. STEYER W. A., JR. Gamma lens .for resolving 1. BROWNELLG. L. and SWEETW. H. Proc. Geneva isotope distributions in tissue. B.S. Thesis, M I T Conf. Vol. 10, 249 (1956); Acta Radiol. 46, 425

(1956). 2. VAN DER DOES DE BYE J. A. ~¥. Nucleonics 14, No. 11, 128 (1956). 3. CORBETTB. D. and HONOURA.J. Nucleonics 9, No. 5, 43 (1951). 4. MAYNEORD W. V. and SINCLAIRW . K. Advances in Biological and Medical Physics. No. 3, p. 1. Academic Press, New York (1953). 5. ALLEN H . C., JR. and RlSSER J. R. Nucleonics 13, No. 1, 28 (1955). 6. NEWELL R. R., SAONBERS W. and MILLER E. Nucleonics 10, No. 7, 36 (1952).

(1954). 8. FP.ANClSJ. E., BELL P. R., and HAaalS C. C. Nucleonics 13, No. I1, 82 (1955). 9. WRENNF. W., JR., GOODM. L. and HANDLEaP. Science 113, 525 (1951). 10. BROWNELLG. L. and SWEET W. H. Nucleonics 11, No. 11, 40 (1953). 11. MATHER R. L. Bull. Amer. Phys. Soc. l, 378 (1956). 12. COOKC. Bull. Amer. Phys. Soc. 1, 378 (1956). 13. TOMNOVEC F. M. and MATHER R. L. Bull. Amer. Phys. Soc. 1, 378 (1956).