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IEEE Transactions on Nuclear Science, Vol. NS-28, No. 1, February 1981
MONTE CARLO CALCULATIONS OF THE DETECTION EFFICIENCY OF ARRAYS OFNaI(Tl), BGO, CsF, Ge, AND PLASTIC DETECTORS FOR 511 keV PHOTONS
Stephen E. DerenzoDonner Laboratory
University of CaliforniaBerkeley CA, 94720
Summary
In positron emission tomographs employing arraysof detectors, the useful detection efficiency andspatial resolution depend on the size and type ofdetectors, the septa between the detectors, the angleof incidence, and the extent to which the pulse heightselection rejects multiple detector interactions. Todetermine the relative effect of these factors aMonte-Carlo computer code was developed that tracks theCompton and photoelectric interactions of photonsincident on a linear array of detectors. The photon istracked until it either escapes the array or isphotoelectrically absorbed in a detector or septum. Aphoton is successfully detected whenever theenergy-loss threshold is exceeded in only one detector.Results are tabulated for bismuth germanate (BGO),NaI(Tl), CsF, germanium, plastic and lead loadedplastic detectors from 0.1 cm X 1 cm X 1 cm to 5 cm X10 cm X 10 cm; for lead septa between detectors; forenergy loss thresholds from 0 to 511 keV; and forangles of incidence from 0 to 30 degrees. We show thatfor a given detector size, BGO has the best efficiencyand that this advantage is most pronounced for narrowdetectors.
1. Introduction
The objective of this work is the determinationof the detection efficiency of linear arrays ofdetectors with individual pulse height selection asthey are used in positron emission tomographs. AMonte-Carlo computer procedure was used to determinethe dependence of detection efficiency on pulse heightthreshold, detector size, and inter-detector lead septato permit the selection of detector design. Specialemphasis is given to narrow detectors for highresolution tomography. Other factors contributing tothe quality of the reconstructed images, such as timeresolution and1scatter rejection, are the subject of aprevious paper .
This treatment jiifers from the previous wvrk ofBerger and Doggett ' and of Anger and Davis whoseefficiency calculations do not apply to multi-detectorsystems where each detector has its own energy lossthreshold. The calculations of Parker assume that anenergy loss threshold is applied eo each recoilelectron, rather than to each detector.
Table 1 lists the basic properties of the sixdetector materials used in this work.
TABLE 1PROPERTIES OF DETECTOR MATERIALS
Material:
3Density (gm/cm )
NaI(Tl) CsF
3.67 4.61
BGO Ge Plastic Plastic(10% lead)
7.13 5.38 1.03 1.12
Index of refraction 1.78 1.48 2.15 - 1.58
Atomic Numbers 11,53 55,9 83,32,8 32 6,1 6,1 ,82
Yes Very No No No
Scintillation decaytime (nsec)
Yield in detector
Electron yieldb
230
50
2,500
5 300
1.5 4
2
300 15
75 200 150,000 800
Time resolution(nsec FWHM)c
1-5 (1 2-10 1-5 <1
Energy resolution(% FWHM)c
8 30 20
aScintillation photons per keV [NaI(Tl), CsF, BGO, plastic]bor electron-hole pairs per keV [Ge].Photoelectrons [NaI(Tl), CsF, BGO, plastic] assuming 50% light collection efficiencycand 20% quantum efficiency, or electron-hole pairs [Ge] for 511 keV energy loss.Typical values for a 511 keV energy loss.
U.S. Government work not protected by U.S. copyright.
Hygroscopic? No
2
5
250
1
1
131
2. Computer Code
The computer code written for this applicationtraces the Compton and photoelectric interactions of511 keV photons incident on an infinite linear array ofdetectors. The program assumes that one detector isuniformly illuminated along a band across its face by abeam incident at some angle. (Fig. 1) The detectionefficiencies given in this study were calculated as theprobability that an incident photon will deposit morethan the threshold energy in only one detector.
A simplified flow chart for the computer code isshown in Figure 2. Input parameters entered from akeyboard terminal describe the size and materials ofthe detectors and septa, the thickness of the photonbeam, and the number, energy and direction of theincident photons. The code tracks the specified numberof incident photons within an infinite linear array ofdetectors and inter-detector septa. Each photon istracked for as many as eight Compton interactions inthe array, or until it escapes or is absorbed by aphotoelectric interaction in a detector or septum. Forseveral energy thresholds, (0, 50, 100, 200, 300, 400,and 511 keV) the event is accumulated as a successfuldetection if the threshold is passed in only onedetector.
Processing time for 20,000 incident photons on a
DEC PDP 11/34 minicomputer is 5 to 15 minutes,depending on the average number of Comptoninteractions.
Compton and photoelectric cross sections for Ge,NaI, BGO (Bi4Ge°e ), CsF, plastic (polystyrene, C nH n),10% lead loadeA piastic, and lead septa were derived
from the tabulatign of Plechaty et al. for the
individual elements .
The short range of the Compton and photoelectricrecoil electrons in the detectors and septa was
neglected for this work. From previous measurements, apositron point source with a maximum energy of 0.96 MeVwill have 75% of the annihilation7points lying within aprojected radius of 1 mm of water . In a detector suchas CsF or BGO the effective range of 511 keV recoilelectrons is less than 0.2 mm.
3. Results
3.1 Detection Efficiency vs. Detector Size
Table 2 lists the detector efficiencies fordetectors of various sizes determined from Monte-Carloruns of 20,000 incident 511 keV photons. Randomfluctuations in these results were typically 0.3% rms.The 511 keV efficiencies may not be realized inpractice due to losses caused by limited pulse heightresolution. As the photon cross sections for pure Naland NaI(T1) are virtually identical, we use the term
NaI for brevity whenever the interaction propertiesalone are being considered.
The single-crystal detection efficiencies for theNaI(Tl) and BGO crystals used in the Donner 280 crystalpositron tomograph were measured to be 45% and 67%,respectively. This is in good agreement with the 47%and 72% predicted by the program, including the crystalpacking fractions of 78% and 89%, respectively.
3.2 Effect of Energy Loss Threshold
One important limitation to the ultimate spatialresolution and sensitivity in positron tomography isthe effect of Compton scattering in very narrow
detectors. Consequently, the Monte-Carlo code was usedto investigate the detection efficiency and positionerror as a function of the energy loss threshold for
0.1 cm wide detectors, significantly narrower than the
conventional 0.5-2 cm wide detectors in common use. Theresults (Table 3) are summarized below:
Detectors Septa
/~~~~~~~~~~~~~~V J.. N- L
XBL8OI I - 3858
Figure 1: Schematic of detector, septum, and beam geometry.
132
Figure 2: Flow chart for Monte-Carlo computer code
At the lowest threshold (0 keV), there is no lossof position accuracy due to multiple detectorinteractions and the detection efficiencies are 30% forGe, 38% for NaI, 57% for BGO, 37% for CsF, 29% forplastic, and 30% for 10% lead loaded plastic.
At an intermediate threshold (200 keV), a positionerror of more than one detector distance occurs for 26%
of the successful detections for Ge and NaI, 20% for
BGO, 28% for CsF, 4% for plastic, and 10% for lead
loaded plastic. As a result of the acceptance of these
scattered photons, the efficiencies are higher: 50% for
Ge. 48% for NaI, 83% for BGO, 55% for CsF, 17% for
plastic, and 20% for lead loaded plastic. Note thatwith 0.8 cm wide, 3 cm high, 5 cm deep detectors and a200 keV threshold a position error of more than onedetector distance is less likely, and occurs for only1.4% of the successful detections for BGO and 6.0% forCsF.
At the highest energy threshold (511 keV), thereis no loss of position error caused by scattering inthe detectors, but the efficiencies are rather low: 6%for Ge, 18% for NaI, 55% for BGO, 21% for CsF, 0% forplastic, and 4% for lead loaded plastic.
133
TABLE 2DETECTION EFFICIENCY AS A FUNCTION OF DETECTOR SIZEa
NaI BGO CsF Plastic Plastic10% lead
Threshold (keV)= 100(511)
DetectorDepth
D (cm)b
100(511) 400(511) 100(511) 100(511) 100(511)
DetectorSpacing
C (cm)b
1 infinite52
0.80.40.20.1
3 infinite52
0.80.40.20.1
5 infinite52
0.80.40.20.1
3 infinite infinite5
20.80.40.20.1
10 10 infinite52
0.80.40.20.1
27%(26%(26%(25%(24%(23%(22%(
62%(30%)57%(25%)51%(19%)43%( 12%)38%( 8%)35%( 6%)34%( 5%)
81%(45%)72%(36%)63%(26%)51%(15%)44%(10%)41%( 7%)39%( 6%)
96%(58%)84%(45%)72%(32%)57%(18%)50%(12%)45%( 8%)43%( 6%)
97%(82%)77%(60%)58%(38%)40%(20%)31%(12%)27%( 8%)26%( 7%)
aNo septa, incidence angle 00. 20,000 incident photons.See Fig. 1 for definition of detector dimensions T, D, and C.
3.3 Effect of Lead Septa
Table 4 shows the efficiency and position errorsfor CsF and BGO when the incident annihilation photonsenter at angles of 0, 10, 20, and 30 degrees. For a
detector circle of 100 cm diameter, these are the
worst-case angles for a point source at radialpositions 0, 8.7, 17.1, and 25.0 cm, respectively.
3.4 Effect of Beam Thickness
Table 5 shows that varying the beam thickness B
(see Fig. 1) has a relatively minor effect on the
efficiency.
4. Conclusions
For all choices of detector size used in this
work, BGO had a significantly higher detection
efficiency than Ge, NaI, CsF, or lead loaded plastic.
This advantage is most pronounced when using narrow
detectors. It is shown in another treatment that
dispite its poorer time resolution, BGO provides the
best signal-to-noise in the reconstructed image of a 20
cm phantom for activity levels below 1 mCi per axial
cm. CsF may prove superior, however with its furthe_11development for time-of-flight positron tomography.
The use of an energy threshold on each detector is
an effective means of reducing position errors due to
scattering within the detectors. These position errors
can be nearly eliminated by using either a very low
134
Ge
DetectorHeight
T (cm)b1
3
3
5%)5%)5%)4%)3%)2%)2%)
0%)0%)0%)0%)0%)0%)0%)
1%)1%)1%)1%)1%)1%)1%)
8%(8%(8%(8%(7%(7%(7%(
23%(11%)22%(10%)21%( 9%)21%( 8%)20%( 8%)19%( 7%)18%( 6%)
55%(38%)50%(34%)46%(30%)40%(23%)36%(19%)33%(16%)32%(14%)
73%(54%)67%(48%)61%(41%)52%(31%)46%(25%)42%(21%)40%(18%)
96%(75%)87%(65%)77%(54%)66%(40%)59%(32%)52%( 26%)50%(23%)
94%(87%)81%(72%)68%(56%)54%(40%)46%(31%)39%(25%)37%(22%)
46%(46%)46%(46%)45%(45%)43%(43%)41%(40%)37%(35%)34%(32%)
88%(88%)87%(86%)83%(82%)78%(75%)7 1%(66%)65%(58%)60%(52%)
96%(96%)94%(93%)90%(88%)84%(79%)78%(71%)70%(62%)64%(55%)
98%(98%)96%(95%)93%(90%)86%(81%)78%(71%)71%(62%)66%(56%)
99%(99%)97%(95%)93%(91%)86%(81%)79%(71%)72%(63%)66%(56%)
28%(14%)27%(13%)26%(13%)25%(11%)24%(10%)22%( 9%)21%( 8%)
63%(47%)60%(44%)54%(37%)46%(30%)40%(23%)37%(20%)34%(17%)
81%(65%)76%(59%)68%(49%)57%(38%)50%(29%)45%(25%)41%(21%)
97%(81%)90%(71%)79%(59%)66%(43%)58%(35%)51%(28%)47%(24%)
97%(92%)86%(79%)73%(63%)58%(45%)48%(35%)41%(28%)38%(23%)
18%(18%(18%(18%(17%(17%(17%(
29%(28%(28%(27%(27%(26%(26%(
85%(79%(76%(75%(74%(73%(72%(
51%(45%(42%(40%(39%(39%(38%(
0%)0%)0%)0%)0%)0%)0%)
0%)0%)0%)0%)0%)0%)0%)
0%)0%)0%)0%)0%)0%)0%)
0%)0%)0%)0%)0%)0%)0%)
22%( 5%)21%( 5%)20%( 4%)19%( 3%)19%( 3%)19%( 3%)18%( 3%)
33%( 9%)31%( 7%)30%( 6%)28%( 5%)28%( 4%)27%( 4%)26%( 4%)
87%(28%)78%(21%)72%(16%)68%(12%)66%(10%)65%( 9%)64%( 9%)
57%(30%)45%(18%)39%(12%)35%( 8%)33%( 7%)32%( 6%)32%( 6%)
TABLE 3EFFECT OF ENERGY LOSS THRESHOLD ON EFFICIENCY
AND POSITION ERROR WHEN USING 0.1 cm WIDE DETECTORSa
EnergyThreshold
(keV)
0 EfficiencybError=OError=1Error>le
50 EfficiencybError=OcError=ldError>1 e
100 EfficiencybError=OcError=ldError>1 e
200 EfficiencybError=OcError=ldError>le
300 EfficiencybError=OcError=ldError>1 e
400 EfficiencybError=OcError=ldError>le
511 EfficiencybError=OcError=ldError>1 e
Ge NaI
30%100%
0%0%
35%92%2%6%
39%83%4%
13%
50%69%5%
26%
26%68%4%
28%
9%75%4%
21%
6%100%
0%0%
38%100%
0%0%
39%94%2%4%
39%86%3%
11%
48%69%5%
26%
38%69%5%
26%
23%82%3%
15%
18%100%
0%0%
BGO CsF Plastic Plastic(10% lead)
57%100%
0%0%
61%94%3%3%
66%86%5%9%
83%72%8%20%
82%73%8%19%
64%86%5%9%
55%100%
0%0%
37%100%
0%0%
39%94%2%4%
41%85%3%12%
55%66%6%
28%
45%67%6%
27%
27%81%4%15%
21%100%
0%0%
29%100%0%0%
27%98%1%1%
26%95%1%4%
17%95%1%4%
5%99%0%1%
0%
0%
30%100%
0%0%
28%97%1%2%
26%94%1%5%
20%88%2%
10%
10%86%1%
13%
4%91%1%8%
4%100%
0%0%
20,000 incident photons.aDetector height T=3cm; depth D=5 cm; width C=O.l cm (Fig. 1); No septa; Incidence angle 0
bProbability of depositing more than the threshold energy in only one detector.cError=O: Probability detector above threshold contains photon entry point.Error=l: Probability detector above threshold is next to detector containingephoton entry point.Error>l: Probability detector above threshold is more than one detector away from photonentry point. (Error > 1.5 x detector width = 0.15 cm )Error probabilities normalized to 100% of detection efficiency.
threshold or a full energy threshold. Even using 0.1 cm
wide BGO crystals and a 400 keV threshold, thedetection efficiency is 64% and in only 9% of theseevents is the detector above threshold farther than one
crystal width (0.1 cm) from the photon entry point. Thefull energy threshold is preferred in positrontomography for medical applications since it is also
effective in discriminating against tissue-scatteredphotons.
Inter-detector septa are somewhat effective inreducing the displacement errors, but theirdisadvantage is a loss of detection efficiency.
Acknowledgments
I thank T.F. Budinger and R.H. Huesman for helpfuldiscussions. This work was supported by the Div. ofBiomedical Research of the U.S.. Department of Energyand the U.S. National Institutes of Health Grants Nos.HL 21697-03 and HL 25840-01.
REFERENCES
1. Derenzo S: Method for optimizing side shieldingin positron emission tomographs and for comparingdetector materials. J Nucl Med 21: 971-977, 1980
2. Berger MJ and Doggett J: Response function ofNaI(Tl) scintillation counters. Rev Sci Instr 27:269-270, 1956.
3. Berger MJ and Doggett J: Response function ofthallium-activated sodium-iodide scintillationcounters. J Rsh Nat Bureau of Standards 56:355-366, 1956
4. Anger HO and Davis DH: Gamma-ray detectionefficiency and image resolution in sodium iodide.Rev Sci Instr 35: 693, 1964
5. Parker RP: Degradation of spatial resolution ingamma cameras employing sodium iodide orgermanium detectors. Phys Med Biol 15: 493-502,1970
135
TABLE 4.EFFECT OF LEAD SEPTA AND INCIDENCE ANGLE ON
EFFICIENCY AND POSITION ERROR a
Beam LeadAngle Septuma
fraction
Detection PositionEfficiency b Error=O c
Position Position PositionError-1 c Error=2C Error>2 c
BGO CsF BGO CsF BGO CsF BGO CsF BGO CsF
00 0.00.10.20.3
10° 0.00.10.20.3
200 0.0
0.1
0.20.3
30° 0.00.10.20.3
aDetector height T-3 cm, depth D-5 cm. For septum fraction f, septa are f x 0.8 cm wideand detectors are (1-f) x 0.8 cm wide. 20,000 incident photons.Efficiency for depositing mote than the0threshold energy [400 keV for BGO; 100 keV for CsF]in only one detector. Incidence angle 00.Normalized to 100% of detection efficiency.Position error-N: Detector above threshold is N detectors away from photon entry point.
TABLE 5EFFECT OF BEAM THICKNESS ON EFFICIENCY
Ge Nal BGO CsF Plastic Plastic(10% lead)
Threshold (keV)-
Beam thickness (cm)0
123
100(511) 100(511) 400(511) 100(511) 100(511) 100(511)
51%(15%)51%(15%)53%(15%)54%(15%)
52X(31%)52%(31%)53%(30%)53%(29%)
84%(79X)84%(79%)83%(79%)81%(76%)
57%(38%)57%(38%)58%(36%)59%(35%)
27%( 0%)28%( 0%)27%( 0%)27%( 0%)
28%( 5%)28%( 5%)29%( 5%)29%( 5%)
a
Detector height 3 cm, width 0.8 cm, depth 5 cm, No septa. 20,000 incident photons.
6. Plechaty EF, Cullen DE, and Howerton RJ: Tablesand graphs of photon-interaction cross sectionsfrom 0.1 keV to 100 MeV derived from the LLLtvaluated-Nuclear-Data Library. LawrenceLivermore National Laboratory Report No. UCRL50,400, Vol 6, Rev 2, 1978. Magnetic data tapesupplied by R.J. Howerton, Lawrence LivermoreNational Lab.
7. Derenzo S: Precision measurement of annihilationpoint spread distributions for medicallyimportant positron emitters. Proceedings 5th
International Conference on PositronAnnihilation, (Lake Yamanaka, Japan) pp 819-823,1979
8. Derenzo SE, Budinger TF, Huesman, RH, et al:Imaging properties of a positron tomograph with
136
280 BGO crystals. IEEE Trans Nucl Sci NS28: No 1,
1981 (this volume)
9. Allemand R, Gresset C, and Vacher J: Potentialadvantages of a Cesium Fluoride scintillator for
a time-of-flight positron camera. J Nucl Med 21:153-155, 1980
10. Mullani NA, Ficke DC, and Ter-Pogossian MM:Cesium Fluoride: a new detector for positronemission tomography. IEEE Trans Nucl Sci NS-27:No.1, 572-S75, 1980
11. Mullani NA, Markham J, and Ter-Pogossian MM:
Feasibility of time-of-flight reconstruction in
positron emission tomography. J Nucl Med 21:1095-1097, 1980
93%94%96%96%
5%4%3%2%
84%74%64%55%
84%70%59%49%
84%70%57%46%
84%69%56%45%
57%63%60%54%
57%57%49%40%
58%57%47%38%
58%57%46%37%
95%96%97%98%
74%81%86%90%
57%63%67%72%
46%50%55%58%
6%5%4%3%
38%28%21%16%
43%42%38%32%
38%39%39%38%
60%70%78%84%
40%46%54%62%
31%36%42%48%
0x5%0x2%0.2%0.2%
0.5%0.4%0.3%0.2%
4.6%3*5%2.5%2*1%
9.9%9.2%7.0%6.0%
1 . 0%0.7%0.6%0.5%
1.4%1.5%1. 0%0.7%
16%11%
7.8%5.0%
20%17%14%12%
25%19%14Z10%
39%34%31%26%
41%39%37%35%
0. 0%0.0%0.0%0.0%
0.1%0.1%0.0%0.0%
0.2%0.2%0.1%0.1%
2.8%2.1%1.4%1.0%
0.5%0.3%0.1%0.1%
0.4%0.4%0.3%0.1%
0.8%0.8%0.5%0.4%
12%8.1%5*3%3.2%
