and appear more accurate than geometric methods (8).With count-based methods, the detected numberof countsfrom the cardiac blood-pool is correlated with the truecounts, which are then used to calculate the true volume.The major potential obstacles to these methods are attenuation effects due to surrounding body tissue and selfattenuationby the ventricle. Methods to correct for attenuation include direct measurement of its effect with atransmission source or indirect determinationby estimating the attenuation coefficient and tissue thickness(6—7,10). The use of a transmission source is limited by itsinability to account for self-attenuationeffects and limitedstrength of the source with resultant poor statistics (15).Indirectdeterminationof attenuationis hamperedby difficulty in estimating relevant parameters (16—20).

The purpose of this article is to describe a method ofvolume determination which utilizes data from an externalreference source to account for both of these attenuationeffects directly. Two steps are involved. First, two sets ofconjugate images are obtained, one with the referencesource overlying the volume of interest and the other without. Based on these data, the length perpendicularto thecamera face of a pixel (or a group of pixels with similarcount rates) in the volume is derived explicitly. Second, thevolume is obtained based on this length. These steps, aswell as potentialconfoundingfactors, are described below.The procedure is easily automatedand is quite suitable forclinical studies.

A count-basedmethodwas developedfor ventricularvolumedeterminationduring muttigatedequilibriumcardiac blood-poolimaging (MUGA).Methods: Two sets of conjugate images ofthe volumewere obtained,one withan external reference overlyingthe volume and the OtherWithOUt.The reference has knownactMty and dimension. The ratio of the geometric mean countrates over a pixel obtained from these two conjugate images,Ratio9•0,and the ratio of the specific activity in the reference tothat inthe volume,Ratio@,were used to calculatethe lengthofthe volumeoverthatpixelperpendicularto the cameraface, H,as a functionof the lengthof the referencesource, R, and theattenuation coefficient,Ub,of the volume:

e@—e@'@= Ratlo@(1—e@).Ratio@

Based on H, the volume is then calculated. This method correctsfor attenuationdirectlyand determines the volumeexplicitly.Tovalidatethismethod,phantomstudiesarecarriedoutwithknownvolumes of @Tccontaining saline solutions in situations ofvariableamount ofattenuatingmediumand backgroundactivfty.Results: Inallcases, the calculatedvolumeagrees closelywiththe actual volume.Conclusion: This is an accurate method ofvolumequantitabonthat is wellsuitedfordeterminingventricularvolumeduringMUGAstudies.

Key Words: venthcularvolume; conjugate imaging;referencesource; cardiac blood-poolimaging

J NucIMed1994;35:644-651

ft ventricularvolume is an importantprognostic factor in patients with heart disease (1—5).It can be measuredfrom multigated equilibriumblood-pool imaging (MUGA)or by geometric and count-based methods (6—14).Countbased methods are independent of geometric assumptions

ReceivedMay4, 19@;rev@ionacceptedJan. 13, 1994.For correspondenceor reprintscont@ Chi-KwanYen, MD,Deparh@entof

NuclearMedicine,Sai FranciscoGeneralHosp1t@Med@ Center,Buiding100,Room157,1001PofreroAve.,San Francisco,CA94110.

644 The Journalof NuclearMedicine•Vol.35 •No. 4 •April1994

A Count-Based Radionuclide Method forVolume Quantitation Using Conjugate Imagingand an External Reference Source: TheoreticalConsideration and Phantom StudyChi-Kwan Yen, Aileen D. Lim and Robert J. Lull

Department ofNuclear Medicine, San FmncLcco General Hospital@,Department ofLaboratoiy Medicine, Universiiy ofCalifornia,San Francirco

ThEORY

Calculating Length of the Volume Perpendicular to thCamera Face

The configurationunder consideration consists of a voltune V with uniform radioactivity C@,situated between twononradioactive attenuating mediums 1 and 2 (Fig. 1). Theattenuationcoefficient of V is Ub,while that of the attenuating mediums 1 and 2 are u1 and u2 respectively. Takingself-attenuation into account, the count rate recorded by agamma camera with a parallel-hole collimator over a pixel

/@m@\ ___ /@m@\I camera@@ camera J@ c@nsia

Iref UIref bId

I gsmn@J@9@ma\@m.mJ\@4m

Here the hottest pixel is assumed to have length H and thearea of a pixel in the imaging matrix is @2,If applieddirectly, this approach introduces a systematic underestimation of the volume due to the difference in seif-attenuation between the hottest pixel and the rest of the volume.This effect is dependent on ub and on both the shape andvolume of V. In order to correct for this error, the magnitude of the underestimationis calculated for differentvolumes of spheres in Appendix B. The results are tabulated(see Table Bi) and used to correct for the error in volumedetermination with Equation 2 above. If necessary, similarcalculations can be carriedout for differentgeometric configurations.

Potential ConfoundIng FactorsThe major assumption of the proposed method is that

geometric mean count rates are independent of the locationof the radioactive source. This assumption can be evaluated by measuring camera sensitivity as a function of thedistance between the source and camera face. Other factors affectingthe proposed approachinclude the effects ofbackground activity, thickness of attenuating medium andambiguity in the assignment of regions of interest (ROIs)(15,18).

Effect of Camera SensItivity with DIstanceIn addition to its intrinsic properties, the sensitivity of

the gamma camera is dependent on the properties of thecollimator and the presence of Compton scatter. Thesefactors lead to a change in count rate with distance(7,18,21,22). They are minimized by using a high-resolution collimator, narrowingthe energy window to 5% andthe use ofbackground subtraction(15). Empirically,countrate as a function of depth will be determined and used tocorrect for this potential source of error.

Effect of Background ActivityBackground activity is usually present clinically. Al

though no perfect procedure exists to account for thiseffect, the use of backgroundsubtractionpartiallycorrectsfor this source of error. Backgroundsubtraction also partially accounts for scatter effect. In this study, ROI background is drawn manually as a crescent close to the volume. The effectiveness of this approachwill be determinedby carrying out volume determination with increasingbackground activity levels.

Effect of Thickness on Attenuating MedIumIncreasing attenuating medium thickness decreases true

count rate and increases Compton scatter count rate. Inour proposed method, the count rate from reference iscompared with those from volume. Since attenuation andCompton scattering effects are comparable for these twosources, they are not expected to significantly affect ourmethod.

ROI AssignmentThe apparentsize of a source increases with its distance

Eq. 2 from the camera face (18), which with the presence ofbackground activity, leads to ambiguity in ROI assign

FiGURE 1. SchematIc diagramof experimentalsetup used toquantitate volume (‘I).V has a specific activity;. Reference source(Rel)has hsight A and speolficactMtycr@.Conjugateimagesarefirstobtainedwiththe volumebetweenattenuatingmediumsI and2. ConjugateImagesare thenobtainedwiththe Ref placedoverthevolume.The specificactMtylnthe referenceis then relatedtothatInthe volumeby ImagIngRef witha source,bid,havingIdenticalgeometrybutcontainingspecificactMty;.

(or a group of pixels with similarcount rates) P in V withlength H perpendicularto the camera face is dependent onCb, attenuation of photons due to the external mediumsand self-attenuation, and the counting efficiency of thegamma camera. With a reference volume (ret) placed overv, thecountraterecordedalsocontainsthecontributionfrom this source. Let reference be a rectangle box withlength R parallel to H, let Ratiogeodenote the ratio of thegeometric mean count rate from P to that due to a comparable area in reference and let Ratio@,@denote the ratio ofspecific activity in the reference to that in the volume, anequation is then obtained relatingH to R:

e@—e@ =::;:(1—e“@).Eq.1Note that any attenuation effect due to the external medium is eliminated by using Ratiogeoin the equation. Theother terms in Equation 1 reflect self-attenuation effectsdue to reference and volume and an additionalterm due toreference being attenuated by the ventricle as well. Sinceall variables in this equation except H are measured, H isobtained explicitly. Derivation of Equation 1 is given inAppendix A.

Calculating VolumeThe most direct method to obtain the volume, V, is to

calculate H for each pixel, multiply by the area of the pixeland sum over V. However, this requiresmanipulatingdataon a pixel-by-pixel basis, which is not currently availableon our camera-computersystem. Instead, volume is calculated by equatingthe ratioof the count rates in V (Vcoun@)to that in the hottest pixel (Ant@,J with the ratio of thevolumes between them:

Vonunu 2Volume= *H*5.

Afltmax

645RadionuclideVolumeDetermination•Yen at al.

ment. To ensure uniformity in data processing, a 10%threshold defined as background activity plus 10%of thedifferencebetween the per pixel peak count rate in volumeand backgroundis used to determine ROIs for the volumeon the imagewith the source closest to the camera. On thesame image, the ROI for the hottest pixels is drawn. Themirror image of this ROl is then used to obtain count ratesfor the hottest pixels on the conjugate image.

MATERIALSAND METHODS

Experimental SetupA flaskis filledwith normalsalineto whichenough @“@Tcis

addedtoyielda specificactivityof around5 pCi/cc.Thecontentsof thisflaskareusedto fillphantomswhoselengthand/orvolumeare to be determined.

Theexperimentalsetupis designedto simulatethe actualsituation in the chest. A phantomsimulatingthe ventricle is viewedby thegammacamerain oppositedirections.Thechestwall issimulated by placing two attenuating mediums between the yentricleandthe camera.These mediumsconsistof books, whichhave attenuationcoefficients similarto body tissue andarereadilyavailableindifferentthicknesses.Thepresenceof lungtissuewiththe attenuatingcoefficient close to the air is simulatedby placingtheventricularphantomat a distancefromtheattenuatingmcdiurns to allow air-space between them. The schematics of the setupis shownin Figure 1.

The referencesource is requiredto have a uniformlengthperpendicularto the cameraface. In the present study, the referencesourceconsistsofa beakercontaining--10mCiof@Tc andfilledwith normal saline to a predeterminedheight.The beakermeasures 3 cm in diameter and is filled to a height of @—3—4cm. Inorder to compare the activity in the reference with that in thephantoms, a second identical beaker (BLD) is filled to the sameheightas that of the referencewith the contentof the flask.Sincethese two beakers have identical geometiy, by imaging themtogether, the specific activity of the reference can be related tothat of the phantoms. In clinical situations, the use of rectangularclosed containers with dimensions comparable to the beakers weused (i.e., 3 x 3 x 3 cm) for both referenceand BLD wouldbemoreappropriateto preventcontentleakage.The BLD contentwouldconsistof a knownvolumeof a bloodsampletakenfromthe patient. A known amountof saline would be added to fill thecontainerwhilethe referencewouldbe filledwitha knownvolumeof saline and a predeterminedamount of radioactivity.With theproposedmethod,only the lengthof the hottestpixel is to bedetermined.Thiscubicreferencevolumewouldeasilycoverthecenter of the leftventriclein patients. It can be easilypositionedover the hottestpixel by viewingthe latteron the persistencescope.

Inthepresentstudy,referenceimagingis performedwiththephantomalreadyin place.Thiscorrespondsto clinicalsituationsfor imagingthe heartafterblood-poollabeling.Moreaccurateresults are obtained if reference imagingis performed prior tohaving the phantom in place. In clinical situations, however, thiswouldresultin increasedstudytimeandpotentialmisalignmentproblems. Our approach is designed to closely approximate realistic clinical situations ratherthan try to obtain a better result byperformingreferenceimagingwithout the phantomin place.

Twotypes of phantomsare used. The firstconsistsof differentsized beakers.This type of phantomhas uniformlengthandisused to evaluate accuracy of both length and volume determina

tion. The second type of phantomconsists of differentsized balloons. This type of phantom more closely resembles the shape ofthe leftventricleandis usedto evaluatetheaccuracyof volumedetermination.

The phantomis placedbetweennonradioactiveattenuationmediums1 and 2 of differentrelativeand total thicknesses.Conjugate images are obtainedwith and without the referenceplacedabove medium 1 overlying the volume.

Thebackgroundlevel is simulatedby placinga basinon oneside of the phantom and sequentially adding predeterminedamountof radioactivity.

Data Acquisition and AnalysisImaging Parameters. Images are acquired on a large field of

view gammacameraequippedwith a low-energyhigh-resolutioncollimator.A 5%energywindowcenteringon the 140keV @“Tcpeakis used.Each imageis acquiredfor5mmina 128x 128wordmatrix using a standard computer system. A two-timezoom isused.

EffectofIncreasü@gDistanceon CameraSensitivity.Determinationof the effectof increasingdistanceon camerasensitivityand apparentvolume size is carriedout by imaginga sourceconsistingof 10 mCi of @Tcdilutedwith 150cc of water in a250-ccbeaker at differentdistances from the camera face. Thedistance ranges from directly over the camera face to 67.5 cmfromthecameraface.

Length and Volume Detennrnation. Length determination isobtained with the beaker phantom only, while volume determination is obtainedfor both beakerand balloonphantoms.Aftermeasuring the height of the solution in reference to obtain R, thefollowing set of images are acquired:

1. An image of the referencevolume and an identicalvolumecontainingCbis obtained.A box ROl is drawnin the centerof each, and the ratioof the count rates is obtainedasRatio@.

2. An image of V is obtained on the anterior view. Withoutsmoothing, a manual ROl encompassing V is drawn basedon 10%thresholding.P, a squareROleither3 x 3 or6 x 6in size, is drawnmanuallyon this imageover the hottestpixels. A background ROl is drawn manually as a 1-2-pixelthick crescent at 1-2 pixels away from this ROl. After background subtraction,@ and@ are obtained fromthese ROIs.

3. Without moving the camera or V. the reference is placedoverV and a secondimageisobtained.Usingthe sameROlforP. Ant@@is obtained.

4. The camera is rotated through 180°.Without moving V orthe reference, a posterior image of V with reference is obtamed.The referenceis then removedand a posteriorimageof V withoutreferenceis obtained.Mirrorimagesof theROISfor P andbackgroundon the anteriorvieware used toobtainboth Post,@ and Post@f.

5. Decay.correctionisperformedonalldata.6. Ratio@,is obtainedasgeometricmeancountratesbasedon

these data.7. H is obtained based on Equation 1.8. Volume of V is obtained based on Equation 2 after error

correctionwith Table Bi.

Effect of Increasing Attenuation and Background Level onVoiwne Determination. The effect of increasing attenuation onvolumedeterminationis evaluatedby increasingthe totalthick

646 The Journalof NuclearMedicine•Vol.35 •No. 4 •April1994

Count-rate as Function of Depth from Camera Facefor a ROlo@mrthe hottest pixel

80,000

60,000

40,000

20,000

0@@@ L t I@ @@j___@@0 10 20 30 40 50 60 70

dIstance In cm

Count-rate as Function of Depth from Camera Face

for the entire volume

2,000.@

1,500

Ca 1.000

@500

distance In cm

FIGURE 2. Count rate in thousands over a AOl covering theentire volume is plotted as a function of the distance in centimetersbetweenthe cameraface and the volume.

ness of attenuatingmediums 1 and 2 from 4.8 cm to 19.5 cm.Initially, the effect of asymmetrical attenuating medium on scattenng is evaluated to determine whether this would lead to adecrease in the accuracy of our method by varying the relativethickness of attenuation mediums on the two sides of the phantorn, i.e., initiallywith all the attenuatingmedium to one side ofthe phantom, and then redistributingthe mediums until they areallon the other side of the phantom.No significantdifferencesareseen. Therefore, only one value is given for each total thickness.

Theeffectof increasingbackgroundlevelsonvolumedetermination is evaluated by addingpredeterminedamountsof radioactivityto thebasinso that it containsfrom0 to 6.9MCiof@'Tc percc of normalsaline.

RESULTS

Effect of Source Depth on Count RateThe count rates for the hottest pixel and the total volume

change with distance from the cameraface. For the hottestpixel, there is an initial increase in count rate with depth.The count ratethen stabilizes untila depthof —50cm whena steady decrease is observed (Fig. 2). These data are usedto correct for changes in count rate with depth for thehottest-pixel. For total volume, a similar pattern is seenexcept when the effect is much smaller (Fig. 3).

There is an apparentincrease in the size of V with depth.Since the ROIs are drawn manually, this apparent changein size cannot be quantified.

Measurement of Length and Volume with Beaker andBalloon Phantoms

The calculated H correlates well with actual H (Fig. 4).In all cases, the error is 5%or less. Since these phantomshave uniformlength, the close relationshipbetween calculated H and actual H should lead to a close relationshipbetween calculated volume and actual volume. This is indeed the case and the erroris less than8%in all cases (Fig.5). The samecloserelationshipis seenfor balloonphan

FiGURE3. Countrateovera AOlpIecednearthecenterofthevolume@isplottedas a functionof the distance in centimetersbetweenthe cameraface and the volume.

toms between calculated and actualvolume, with less than8% error in all cases (Fig. 5).

Effect of Increasing AttenuatIOn and Background Levelon Volume Measurement

Volume determinationis not affected by increasing thetotal thickness of the attenuatingmedium (Table 1). Basedon a 200-cc balloon phantom,the errorsarewithin 5%in allcases.

Volume determination is not affected by increasingbackground activity until it reaches veiy high levels. Usinga 200-cc balloon phantomcontaining5 @CVccof @Tc,theerror is less than 5% until the background activity increased to 6.9 @CVccwhen the error is 14% (Table 2).

FIGURE4. LengthincentimeterscalcUlatedwithEquation1 isplottedagainstthe actuallengthof beakerphantoms.These phantome have uniformlength. The error is small, within5% in all cases.

RadionudIdeVolumeDetermination•Yen at al. 647

50 100 150 am@Ijvok@. ki

Thickness ofattenuatingmedium(cm)19.515.711.68.14.8Vc196.0193.5188.1200.1196.1Va200200200200200%Error—2.0%—3.2%—5.9%0.0%—2.0%

am 350

Vc204.4199.9196.0189.4171.7V@200200200200200%Error+2.2%0.0%—2.0%—5.3%—14.0%

explicit expression forvolume is obtainableonly by assuming that this effect is small or by taking an orthogonal view

to estimate the length of the volume. Small volume assumption results in an errorwhich increases with increasing volume and is quite significant in cases of ventriculardilatation (15). An orthogonal view measures only thelongest lengthof theventricle. Since the leftventricle is notrectangular,using the longest length results in an errorthemagnitude of which depends on both the geometry (i.e.,how it differsfroma rectangle)andthe actualvolume of theleft ventricle. Our method is different in that it takes theself-attenuationeffect directly into account, thus enablingus to obtain pixel volume (or a group of pixels) directly.

The count rates from any distributedsource of radioactivity depend on the size, shape and attenuation coefficientof the source. Withcount-basedmethods, count rates froma small volume of a blood sample are compared with thosefromthe left ventricle to determineleft ventricularvolume.Since the left ventricle is differentin size and shape from asmall blood sample, this approach introduces an errorwhose magnitude depends on the geometry and attenuation coefficient of the left ventricle. With respect to thiserror,all count-basedmethods are dependenton geometricassumptions. Appendix B shows a method for calculatingthis errormagnitude.

Another consideration with count-based methods is thedependence of count rates from a source with distance dueto the physical characteristics of the gamma camera-coffimatorsystem. When the source is small, as is the case of asmall blood sample, variabilityof count rateswith distanceis considerable. For a source approximating the size of anadult left ventricle, count rates over the depth usuallyobserved in clinical situations is relatively constant (21).Therefore, correction with a single constant is probablyadequate. For smallerventricles, such as those for pediatric patients, a different correction factor (or factors) isprobably needed. However, in this case, the distance between the ventricle and the camera is much less than withan adult. Since even a pediatricventricle is far from beinga point source, the count rate variation with distance isexpected to be relatively small and correctable within thedepth usually encountered in a pediatric patient.

Additional factors affecting the use of any count-basedmethod of volume quantitationare the presence of scatterand backgroundactivities (15,18,21—23).In this study, we

TABLE 2CalculatedVolume(@4)of a 200-cc Balloon(AOl)Contalning5.2 @CVccof Activity as a Function of lncreaalng Background

(Bkgd)LevelsComparedwithActualVolume(‘4)

Bkgd@iCVcc(orcts ROVBkgd)

0.0 (33.3) 12 (11.6) 2.4 (6.0) 3.6 (4.4) 6.9 (1.7)

calc*ilated•volumevs actual volumeopenckc@: @ondosed square:bea@ phaMom350

am e@eof

I

250

FIGURE5. Calculatedvolumesof the beaker(0) and balloon(•)phantomsarecomparedwiththeactualvolume.Theerrorissmall, within8% in all cases. Bycom@ningthe two sets of data, thelinearregressionequationis ghienby Calculatedvolume= 2.66ml+ 0.98 x actual volume correlation coeffident is 0.997.

DISCUSSION

In this study, we describe a count-based method whichallows for absolute volume quantitationbased on comparing the geometric mean count rate from the volume to thatobtained from a known reference source situated above thevolume. By taking self-attenuation effect into account, anexplicit expression for volume is obtained with thismethod. It is independent of assumptions concerning attenuationeffects, size or shape of thevolume. Results fromphantom studies support its accuracy in volume quantitation. It is well suited for use in conjunction with MUGAstudies. We are currently incorporating this method into aroutine clinical protocol.

Attenuation correction is the most importantfactor in acount-based method of volume quantitation(6,7,9,10). Bydirectly measuringattenuation,transmissionimagingoffersa consistent method of accounting for attenuation due toexternal mediums (15). Our method is similarto transmission imagingin that attenuationdue to externalmediums isdirectly accounted for since the gamma rays from the reference and ventricular region travel throughthe same external attenuatingmediums. However, transmission imaging does not correct for self-attenuation effects and an

TABLE 1CalculatedVolume(‘4)of a 200-cc Balloonas a FunctionofTotalThickness of ExternalAttenuatingMediumCompared

with Actual Volume (‘4)

648 TheJournalof NuclearMedicine•Vol.35 •No. 4 •April1994

. Eq.A2

\INetMt- ref*Netp@@_ ref

@/Antheart* Post@

evaluated the effect of these factors on the accuracy of ourmethod for volume quantitation. Our results indicate thatscatter does not adversely impact our approach. This maybe due to the use of a 5% energy window and a highresolution collimator. With the 15%—20%energy windowcommonly used in clinical imaging, the errors due to thesefactors may be greater. Our preliminaryexperience withthe clinical study demonstrates that adequate statistics areobtained with a 5% energy window. We are in the processof comparingvolume calculations obtained with a 5% energy window with those obtained with a 15% window.

Background subtraction is necessaiy to account for bothbackground activity and scattering. An ROI placed aroundthe source is routinely used clinically and is the methodused here. Our results indicate this approach is adequatefor obtaining accurate results. A numberof more sophisticated procedures have been advocated to provide betterestimation of background activity (24—25).They will beevaluated to determine the best procedure for backgroundsubtraction in clinical situations. __________________

SUMMARY

We present a count-based radionuclidemethod for volume quantitation. Phantom studies were performed and _______attest that the proposed method is accurate under conditions of varying volume, attenuating medium and background levels. Our results indicate this method is potentially useful for left ventricular volume quantitation and canbe readily incorporated into routine clinical equilibriumgated cardiac studies. ________

APPENDIX A: CALCULATINGVOLUME These two ratios are combined and rearranged so that termsPERPENDICULAR LENGTh TO ThE CAMERA FACE containingH are moved to the left:

Conjugate ImagIng Without a Reference Volume RatioThis configurationconsists of a volume, V. with uniformradio- &“@—e@ = . e@a(1 e

activity C1,situated between two nonradioactiveattenuatingme- RatlOgeodiums 1 and2 (Fig. 1). The attenuationcoefficientofV is ub,while Since Uband R are measureddirectly, H is obtained explicitly.that of the attenuatingmediums 1 and 2 are u1 and u2, respectively. Taking self-attenuation into account, the count rate re- ppPEND@ B: ERROR ON VOLUME CALCULATiONcordedbythegammacameraontheanteriorviewwithaparallel- DUE TO SELF-A1@I@ENUATIONBASED ON RELATIVEholecollimatorovera pixelin V withlengthH perpendicularto COUNTSthecameraface is:

Volume Calculation Based on Relative Count Rates— (EC@,\ _ UbH - u1X Al The relative count rate method of volume calculation assumes

AfltH kT@)(l —e )e@ Eci. that volume is directly proportional to count rate, i.e., the ratio ofthe volumes equals the ratioof the count rates. This relationship

where E is the countingefficiency of the gammacamera and X is is used to calculate one volume when another volume and thethe length of attenuatingmedium 1 perpendicularto the camera ratioof theircount rates areknown. Withour method, the height,face. A similar expression (PostH)is obtained for the count rate H, of the hottest pixel is calculatedand the total counts@ asrecordedby the gammacameraafter180°rotationwith X replaced well as the counts in the hottest pixel (Anth@,,,@)are directly meabyY, thelengthof theattenuatingmedium2 perpendicularto the sured.Withthismethod,thetotalvolumeV is obtainedby mulcamera face. The geometricmeancount is: tiplyingthe area of a pixel (52)by H and the ratio of@ to

E@ \I(AutH*PostH)=—@(1—e@ ubH)e_(ujX+u@YY2 (V \U ‘aMints! 2

Volume= *H*5.(Mth@@)Conjugate Imaging with a Reference Volume

A referencevolumewith lengthR, attenuationcoefficientu@, @2dependson theparticulargammacameraandmatrixsizeusedand radioactivityCris placed above medium 1 (Fig. 1), R, u1and to acquirethe images.

C1aremeasureddirectly.WithoutmovingV. the countraterecorded by the gammacameraover P on the conjugateimages arerespectively:

Afltref AfltH*e@@@ (1 —C@ ufR)

Post@f = POStH + ()(1 —e u@R@- (u1X+ u2Y)e_

In EquationsA3 andA4, the firsttermafterthe equalsignisdue to V. Net count rate due to reference only is obtained bysubtractingthis term. Their geometric mean count rate is:

fEc@/(Net@1- ref@ NeL,@1ref)

. (1 —e UrR)@_ (u1X + u@Y@2e_@ Eq. A5

Derivation of HTheratio(Ratio@0)of thesetwogeometricmeancountratesis:

Ratiog@—

By imaging the reference and that of an identical volume containing radioactivityC,, with the gamma camera (Fig. 1C), the ratiobetween their count rates is obtained as:

Eq.A3

Eq.A4

Eq.A6

Eq.A7

Eq.A8

Eq. Bi

Ref@ (Crut@)(1_e_t1T1@@Ratio@=—= —_u@@R)@

Bld@ CbUr 1 e

Radionuclide Volume Determination •Yen at ai. 649

(C\(1_e_@@\

@@urJ@1_e@HJ

I camera 1

ActualvolumeAflenuation

coeffidents0.120.130.140.150.1650

ml53.453.653.954.254.4100ml108.4109.0109.7110.4111.0l5Oml164.3165.4166.6167.7168.8200

ml220.9222.5224.1225.7227.3250ml278.0280.2282.3284.4286.5300

ml335.6338.3341.0343.7346.3400ml451.8455.8459.7463.5467.3500

ml569.4574.6579.8584.8589.74A

sphericalgeometryis assumed.The numbers arepresented as anction ofthevolumeofthe spharesandattenuationcoeffidents.

Correction of Error Due to Self-AttenuationThe use of EquationB! results in an errorcaused by omitting

self-attenuation.The magnitudeof this errorcan be calculatedasfollows.Let G be a rodwith a unit area,specificactivityC,@,,attenuation coefficient Uband length g. Taking self-attenuationinto account,the count rate,@ recordedby the gammacamera with its face perpendicular to G is given by:

Eq. B2

where E is the countingefficiencyofthe camera.Let H be anotherrod with unit area, C,.,,attenuationcoefficient u and height h. Itscount rate,@ is given by EquationB3 with g replacedby h.Theassumptionthatcountratesaredirectlyproportionalto volumesimpliesthatthevolumeof g is obtainedas:

Substitutingthe expressionfor@ and@ intoEquationB3, thevolumeof G basedon therelativecountratemethodis:

1 —e U5h*l_e@@Uh• Eq.B4

Thedifferencebetweentheexpressionaboveandthetruevolumeof G, namelyg, is theerrorresultingfromnotaccountingforselfattenuation.

A similarargumentcan be used to calculate the errordue to anextendedsourcesuchas a spherewithradius,r. Theschemaofthiscalculationaregivenin FigureBi. Alongtheperimeterof acircle located at the center of the sphere with radius, r, and lyingparallelto thecameraface, the length,L, of the sphereperpendicularto the camera face is givenby:

L=@/r@-n2. Eq.B5

Thevolumeformedalongthisperimeterwitha smallthickness,dn, is given by:

2 x ir X n X dn )

source, calculations are made for different radii and attenuationcoefficients (Table Bl).

Ifdesired,similarcalculationscanbe donetoobtaincorrectionfactorsforgeometriesotherthanspheres.

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