Camera deadtime : Rate-limiting factor in quantitative dynamic studies

International Journal of Nuclear Medicine and Biology, 1975, Vol. 2, pp. 107-l 12. Pergamon Press. Printed in Northern Ireland

Camera

Deadtime Quantitative

:

MOSHE

Rate-Limiting Factor Dynamic Studies

in

BEN-PORATH

The Abba Khushi School of Medicine of the Technion, The Israel Institute of Technology and Elscint, Haifa, Israel (Accepted 19 November 1974)

Camera deadtime presents a major problem in the quantitation of dynamic studies. Appropriate correction factors should be applied before quantitation is attempted. Tables, graphs and equations are presented for the demonstration of the effect and calculation of the errors introduced by systems of various deadtime. NEED OE some kind of data correction in dynamic studies performed with a gamma camera, due to the effect of the system deadtime on the results, has been emphasized by several workers.(2-7) However, in many instances, clinicians continue to perform quantitative dynamic studies with gamma cameras without being aware of the limitation of such studies determined by the system deadtime. In this paper, the deadtime limitations are analyzed, their effect explained and correction for deadtime losses discussed.

THE

ANALYSIS OF THE PROBLEM The patient camera system may be regarded as comprising several “blocks” as shown in Fig. 1. Radiation from the patient reaches the detector. As soon as an event occurs in the detector, it is processed and subsequently displayed. This processing of an event takes a certain time 7. During this time, r, the processor is unable to process additional events that may occur in the detector, because it is busy processing the previous. Therefore, this time interval is referred to as the deadtime of the system. If the events in the detector occur at an average rate of N,, then N,r events will not be processed and displayed. For a given 7, the higher the incident rate N,, the more counts will be lost and vice versa. This may be restated, that in order to minimize the countloss at high rates N,, it is necessary to use a processing system with a minimal deadtime 7. It is important to know, for a given

system with deadtime r, the displayed countrate N as a function of the true countrate N,. In other words how many counts N will be displayed from an output countrate N, by a system of deadtime r. Radioactive decay of atoms is a phenomenon occuring at random intervals, following the Poisson distribution. Therefore, the probability of k events to occur in the time intervals from an incident rate N,, over a long period of time compared to T, is:

The probability that no events will occur during r is obtained by substituting in equation (I), k = 0: p, = e--No’.

(2)

Therefore, the displayed countrate N by a system of deadtime T processing a true countrate of N0 is: N = P,N,

=

NOeVNo’.*

(3).

2, graphic representations of the function N(N,) for different T’S are shown. For In

Fig.

*N is the integral count. The right term in equation (3) should be multiplied by F,, the fraction of the integral counts passing through the single channel analyzer window, if a single channel analyzer is part of the processor. This equation defines the relation between N, N,, and 7 in paralyzable counting system. Such as gamma cameras are. r is the total system deadtime, determined by the various deadtimes ri of the system components: 7 = 2/m

107

MO& Ben-Porath

108

In order for N, to have the maximum value:

LVIE’ d&RCEDETECTOR PROCE!SSOR

every r there is a maximum value N,, so that if N, is further increased, N will decline. This occurs when the derivative of N (NJ equals zero : -Not - No7 e-Nor= e-No'( 1 - rNO).

dNo

(4).

lo’

Or

N,(max)

1 = 7.

[I - rN,(max)]

= 0 (5)

DISPLAY The value of the corresponding

FIG. 1. Block diagram of system: y radiation from the patient passing the collimator produces events in the detector at a rate N,. The processor with a deadtime 7 will produce N = N N&-NJ events. The single channel analyzer will forward to the display a fraction F, of counts, depending on the window width and energy resolution. The displayed countrate will be F,N.

dN -=e

dN = 0 dN,

N (max) is:

N(max) = N,(max)e-N~tmax)’ = f

(6)

or: N,(max) N(max) = e THE

= 0*37N,(max).

(6a)

EFFECT OF THE DEADTIME ON DYNAMIC STUDIES

Essentially, the deadtime limits the maximum measurable countrate N by equation (6). Obviously, the longer the deadtime, the lower the maximum usable countrate. Actually, I/T is a theoretical value, because, as seen in Fig. 3, the function of N approaches a plateau at much lower rates than N, (max). If the maximum acceptable deviation of N compared to N, is such, that a ratio of 50 per cent in N, will be recorded at not more than a 10 per cent deviation in N, then substituting this condition, we obtain:

(7)

-I

Solving, we obtain that the max true countrate for this condition to occur is: N,(50) = ‘$ and : 0.16 0.2 N(50) = - e-s** = _ 7 7 *

Fro. 2. Measured countrate N as a function of input countrate N,, for various deadtime 7. (log-log scale). Diagonal line indicates limit of maximum recommended countrates.

(9)

is the maximum measured countrate for this condition to take place. The values of N,(max) , N(max), NJ50) and N(50) for various deadtimes T are presented in Table 1. From this table we see that to process a countrate of 200,000 counts/set without correction for deadtime losses, a camera with a deadtime of 1-O pet is required. Countrates N, of 200,000 counts/set are today reality. For example, a cardiac study may be considered. If the effective detector

Cameradeadtime TABLE.

i

(lls)

1 .o ,1.5

No ha%)

109

1

Nhax)

No(5@)

N(50)

1.000

368

200

160

667

245

133

106 80

2.0

500

184

100

;2.5

400

147

80

64

3.0

333

122

67

53 40

4.0

250

92

50

5.0

200

74

40

32

8.0

125

46

25

20

LO.0

100

37

20

16

12.0

a3

31

11

13

15.0

67

25

14

10

2,T.O

50

18

10

8

radius is R = 15 cm, the injected bolus is b = 10 mCi 99rnTc and exposes the entire field, there is a 90 per cent absorption (A) in 15 cm tissue, which is assumed to be the average bolus-to-collimator distance, the crystal elIiciency is 7 = 0.9 and a collimator of efficiency E = 0.03, then the expected countrate N, will be: N,, = b x 3.7 x IO’ x rr x R* x (1 -A)

2w

N.‘ :CPSx Id, FIG. 3. Measured countrate N as a function of input countrate N, for various deadtimes 7 (linear display). Diagonal line indicates limit of Maximum recommended countrates. CORRECTING

x 17 x E

N,r I

4 x 7T x 12 N 2 X lo5 count/set Fig. 3 shows the N (NJ curves for N, up to 200,000 counts/set. The heavy diagonal line (also in Fig. 1) divides the region of N, and respective N’s. Only the values on the left of this line are permissible for a dynamic range not exceeding the N(50) values. Figure 4 shows graphically, how dynamic curves will be affected if N, is too high for a given 7. From Table 1 and Figs. 2-4, it is evident that no quantitation of dynamic studies should be attempted with initial true countrates of 100,000 counts/set if 7 > 2 psec. If 7 > 4 ,usec, 50,000 counts/set is the limit. For r > 8 psec 25,000 counts/set and for r = 20 psec, 10,000 counts/set. For higher countrates, correction factors should be applied.

FOR

DEADTIME

As mentioned above, deadtime corrections may be applied only in the range of N, I 11~. It is theoretically impossible to calculate N, from N if N, > l/7, unless we know if an ascending or descending curve is being measured. Therefore, the first condition to even attempt deadtime correction is: 1

or

NT f = 0.368.

The straight way to calculate N, from N = NOeJJ, if N and r are known is by iteration, which is a numeric procedure. This can be done for any N, < 1. Another approach is by developing X = N,e-NJ to a MC Laurin series: N zzzQ+

N,r < 1

= NOT

then

N = N,T(~ -

or: N,,=-.

1 + NOr (11) N,T) = N 0 N I-NT

(12)

110

MO& Ben-Porath B.HI,!A,.

W, N.=aopoO

CPS

FIG. 4. Relative change of countrate with time for various deadtimes T. (a) initial countrate N, = 200,000 countjsec. (b) N,, = 100,000 count/set. (c) N, = 50,000 counts, and (d) N, + 20,000 countslsec.

This is the classic expression which, as pointed out, is valid in Poisson distribution if N,, Q 1.0. For every countrate N measured with a system of deadtime r, there is a correction factor f, so that the corresponding true counttrate N, may be calculated: N, = The correction factor

f to be applied to N is:

and : Calculating we obtain:

f

on the basis of equation fa =

These

(13a)

: f = t+*. Ex-

fEeNor= 1

+

NOT

+

y

+

ii%$?+ . . .

(16)

correction factor and f is Let us determine the maximum value of NT for which the correction as expressed in equation (12)) may be used without introducing errors larger than 0.1 in the computation off,. This may be written:

f, is the approximate

(13) the accurate correction factor.

fN.

N, = Ne*J.

However, from equation panding, we obtain:

(13b) (12))

1 + Nr.

are the first two terms of the series:

(f - O-1)sf,

If

or

N,T < (NT + 0.1)

(17)

and

(18)

which occurs when: (14)

Nor 5 O-32*)

NT 5 O-23.

* The approximation eNo = (1 + N,T) for N, S 0.32 is accurate within 0.05 therefore the overallaccuracy will be 2/OC,Z2 + 0.052 = 0.11.

111

Camera deadtime

N”i

= fN?

Calculated correction factorsf (equation (13a) ) for various values of NT (measured countrate multiplied by deadtime). N, = f N: simply multiply measured CR by corresponding f to obtain real CR. FIG. 5.

This means, that as long as the input countrate is kept below the N, = 0*32/r limit, the measured countrate N can be corrected by N, = N(1 + Nr). For 0.32 < N,, < 1.0, N, should be computed from N = N,e-No’by iteration, or by preloading the computer with a correction curve, such as shown, for example in Fig. 5.

DISCUSSION High countrates are essential in fast dynamic studies in order to obtain adequate image quality for short frame-times. In a study carried out at a frame rate of IO/set, if we assume a region of interest of 10 ems, then in order to achieve an information density of 800 count/cm2, 8000 counts have to be displayed (F,N) inside the region of interest. If we assume that F, = 0.5, then N = 160,000. By equation (18), NT = 0.23 is the limit for correctability by the classical method, which means to display counts at a rate of 160,000 countsjsec, r has to be not longer than 1.5 ,usec. If 7 is longer, lower countrates have to be used, and subsequently poorer image quality is obtained, or lower frame rates, are to be used. The analysis of the deadtime effect described here is based on the relation of equation (3),

COUNTS /sec.

EXPECTED

FIG. 6. Data adapted from Anger (7) N-W. N, for four cameras of different brands. The curves follow the calculated patterns of Fig. 2.

which was derived from theoretical considerations. However, experimental results published by ANGER@)and TANAKA et aZ.,fg) support this relation. Figure 6 represents N(N,) curves adapted from ANGER’S publication,(*) where commercial cameras of four manufacturers were tested. All four curves follow the pattern of equation (3) and Fig. 2. TANAKA(~) derived equation (3), empirically from experiments with his delay line camera. It is therefore evident that the deadtime limitations, as described are supported both by theoretical consideration as well as experimental results. On this basis, the limitations imposed by system deadtime upon the maximum usable countrate may be summarized as follows: If NT I O.l6(N,r) I O-2), no correction is required. If 0.16 < NT < 0*23(0*2 I N,T I 0.32), N may be corrected by N, = N( 1 + Nr). If O-23 < NT < O-368(0*32 I N,T I I) N, should be calculated by N, = NeNJ. REFERENCES 1. BEN-P• RATH M. Radioactive Isotope (in press). 2. HARRIS C. C., JONES R. H., BUFFALOE T. S. et al. J. nucl. Med. 11, 325 (1970).

112

MO& Ben-Porath

3. ROBERT H. J. and BRYCE B. B. J. nucl. Med. 14,413 (1973). 4. WILSONG. A., KEYESJ. W. and WEBER D. A. J. nucl. Med. 14, 642 (1973). 5. JONESR. H., GRENIER D. C. and SABUTAND. C. Medical Isotope Scintigraphy Vol. 1, pp. 299-310. IAEA, Vienna (1973). 6. BITTER F. and ADAM W. E. Mea&al Isotope Scintigraphy Vol. 1, pp. 443-457 IAEA, Vienna (1973). 7. BUDDINGERT. F. Medical Isotope Scintigrafihy Vol. 1, pp. 501-555 IAEA, Vienna (1973). 8. ANGER H. 0. LBL report 2027, UC-37 (1973). 9. TANAKA E., NOHARA N., KUMAHO N. et al. Medical Isotope Scintigraphy, Vol. 1, pp. 169-180. IAEA, Vienna (1973). D. J. nucl. Med. 14, 10. ADAMSR. and ZIMMERMAN 496 (1973).

APPENDIX Mtasurement and calculation of deadtim

(1)

The classical method to measure deadtime is by the two source method, which is based on the N, = N/l + NT approximation, and therefore, limited by the condition N, < 1. Ten methods to calculate deadtime from such measurements are described in detail by ADAMSand Z~MMER-_(lO)

(2) The

constant geometry method, described by ANGER(~) may be used to calculate deadtime. A source of known activity is placed at a distance from the detector. N, is changed by placing absorbers between the source and detector, and N measured for each absorber. Knowing the absorption of the radiation from the source by the absorbers, the geometric parameters of the system and the source strength, N, may be calculated. The deadtime T may be calculated from N = FeNOe-Nor. It is recommended to calculate 7 for several N f N, relations and compute the average value. (3) The short halflife-source method is probably the most direct method as it involves no changes in the experimental set-up, except the decay of the source. If a 2 mCi source of a Qstn7?is placed 100 cm from a 30 cm diameter camera, then N,, = 4 x IO6 count/set and N = Fe x 4 x 1(-ye-4t106

FIG. 7. Deadtime calculation by the short halflife isotope method. 2 mCi ~Tc placed 100 cm from the detector. The FeNe line is extrapolated from the linear (log) part of the experimental curves. N is measured and recorded as the source decays. N, decreases with time, and subsequently N approaches N, with time. The experiment is continued until N follows the exponential pattern of the source decay, with time. Figure 7 shows the results of such two experiments, one with a T - 1.5 ,LAWC system, the second with a 7 = 10 psec system. In both instances Fe = O-6. After reaching the exponential part of the function N vs. time, this part is extrapolated to t = 0. This curve will be the F,Ne curve. By applying N = NOesNor at several t and takmg the average value, r can be calculated quite accurately. With this method, it is not necessary to know the initial value of N, in order to calculate 7, as the value of N, are calculated by extrapolation. (4) A fast, but rough method, is the paralysis method. The maximum count rate N(max) that a system of deadtime r can measure is l/re. N(max) can be determined experimentally within minutes and r = Fe/eN(max) estimated.