Computer simulation of the Anger gamma camera

Computer Programs in Biomedicine 4 (1975) 189-201 © North-Holland Publishing Company

COMPUTER SIMULATION OF THE ANGER GAMMA CAMERA John B. SVEDBERG*

Department of Physical Biology, The Gustaf Werner Institute, University of Uppsala, Uppsala, Sweden

The Anger scintillation camera system is simulated on a computer making possible a study of its performance as regards coordinate distortion, intrinsic resolution and uniformity of the energy-defining pulse for optional values of the different parameters of importance in the construction of a camera. The use of a micro-film plotter permits the production of illustrative graphs and diagrams of the different properties studied. Simulation

Gamma camera

1. Introduction Since the Anger scintillation camera was presented in 1958 [1] its use has grown steadily and to-day it is one of the most popular detector systems in medical diagnosis with radioactive nuclides. It owes much of its success to the ready availability o f shortlived 99mTc-compounds, whose principal gamma energy, 140 keV, is almost ideal for the Anger camera. As there is a constant development o f new radionuclides and compounds, the demands on the detector system also change and existing systems must be modified or new principles for detection must be introduced. Because of the wide-spread use o f the Anger camera it should be of great value to know if the detector system as such, possibly after some modifications, could be used with confidence for the new applications. The computer program presented here is meant to be a tool in the study o f camera performance with regard to co-ordinate distortion, uniformity in the energy-defining pulse and intrinsic resolution due to stochastics in photomultiplier pulse production. Designers of gamma cameras, whether on a commercial basis or for use in the laboratory or in the clinic as a special-purpose detector, could easily get information about values for most of the parameters of importance in the construction of a * Present address: Radiofysikavdelningen, Regionsjukhuset, S-581 85 LinkiSping, Sweden.

camera without spending too much time and effort on experimental tests. Furthermore, the computer program could form a basis for the soft-ware in future computer-assisted camera systems. A number of investigations about the gamma camera performance in relation to accelerator-produced, positron-emitting nuclides have been made [ 2 - 5 ] with the present program.

2. The detector system A principal diagram of the Anger camera is given in fig. 1. A scintillation crystal with radius 6 F is coupled via a light guide to a number of photo-multiplier tubes (PM-tubes) positionad in a hexagonal array. Around a central PM-tube the other tubes are arranged in n concentric rings with six tubes in each (n = 1 in fig. 1). The total number of tubes is thus 1 + 6n. A spacing factor, M, is defined as the ratio between the actual distance of a tube's centre from that of the central tube and the corresponding distance for closest packing (M = 1 for closest packing, as in fig. 1). The PM-pulse, PRij from the ith tube, due to light emission at a point j, is obtained from

PRij = E X ~ X 121 × 122 × ~2ij/e

(1)

and is fed into a pulse mixing circuit where Xj, Yj and Z/-pulses are derived according to eqs. (2)

190

J.B. Svedberg, Computer simulation of the Anger gamma camera

CRYSTAL

Crystal diameter Crystal optical properties Light guide length Light guide optical properties PM-cathode diameter

LIGHT GUIDE

PMTUBES

i ................

........... ............

i

........+........... ::

I ......

\.

//'

PULSE MIXIMG CIRCUIT

Fig. 1. Schematic drawing of the Anger system.

x i G pU,Rdz

s --

l

p ,w,j l

rj :

pypR,j/z j : l

py, wis

Table 1 " Design parameters that can be varied in the program. Those in the first section determine the solid angle table to be calculated in SOLIDA before its use in ANGPLT.

(2)

I

l

ZDj= GPRis. i In these expressions, E is the energy of the absorbed gamma-ray, r/is the efficiency of the crystal for converting gamma energy into light quanta of energy e, bi1 quantum efficiency of the PM-cathode,/.t 2 the collection efficiency of the first dynode, 4nl2ij the solid angle subtended by the ith photo-cathode at light emission point j and Pxi, Pyi and Pzi are properly chosen weighting factors. The X- and Y-pulses are positioning pulses for the image point on the display while Z corresponds to the energy deposited in the crystal.

3. Computational methods The aim of the simulation program ANGPLT is to obtain information about the performance of the camera system as regards

Crystal thickness PM-tube spacing Number of PM-tubes Crystal efficiency Pkl-cathode efficiency Collection efficiency, 1st dynode Variance in electron multiplication Gamma-ray energy (i) co-ordinate distortion (ii) uniformity in the energy-defininf pulse, Z (iii) resolution component due to the stochastics in PM-pulse production for different values of the parameters of importance in the construction of a camera (table 1). Co-ordinate distortion is defined as the departure from a linear relationship between object and image co-ordinates and it is measured by DF, the rootmean-square displacement of the image points for a test pattern (fig. 2) relative to the ideal image. The quantity D F only gives an indication about the quality of the image and, earlier [2], many runs with different sizes of the test pattern (through variation o f F ) were made in order to see the variation of D F with F. After the inclusion of a subroutine DISPLY, through which the image of a test grid similar to that of fig. 2 can easily be visualized, the judgement of performance as regards co-ordinate distortion has become safer. As a measure of uniformity in Z the standard deviation dZ of Z from the mean value is taken. Isometric plots of the Z-distribution over the crystal can also be obtained via DISPLY. The resolution component, finally, is studied through the response of the detector to a point source for which a number of image points are produced. Since the type of distribution of the image points is not known and since axial symmetry cannot be assumed, the standard deviation of the radial distances of the points from their mean position is chosen as a measure of resolution, MR. There is always a weakness in using

191

J.B. Svedberg, Computer simulation of the Anger gamma camera

of the calculations in ANGPLT are given in a relative unit of length, d. Absolute values are then obtained by letting one parameter value correspond to a certain absolute value (cf. [5] where the crystal thickness 5d, was assumed to correspond to 1.27 cm). As there is an optical interface between the emission point of the scintillation light and the PM-cathode, existing tables [6, 7] of solid angles cannot be used. Monte Carlo techniques have to be used to calculate the solid angle, when there is a difference in refractive indices between the materials containing the cathode and the point of light emission. A table with calculated solid angles will be used for a variety of values for the parameters in the lower section of table 1 and therefore, the generating program is not included in ANGPLT but constitutes a separate program, SOLIDA, producing solid angle tables on punched cards for direct use by ANGPLT.

o

o

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o

o

K

o

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Fig. 2. Test pattern consisting of 121 points in a co-ordinate system (K, L) with the origin at the crystal centre. Through variation of the distance F between the points, the size of the pattern and thus of the tested crystal are changed.

3.1. A N G P L T

a single number as a measure of properties like resolution, and M R should be used with care. Through the possibility of registering each single image point on a micro-film plotter, the examination of resolution capacity becomes much easier. The image point distribution is then visualized in the same way as in a real measurement with a camera. The plotter-output is administered directly from the main program, but the same variable, I P U T , as used in DISPLY determines if a plot should be made. In table 2 the values are given for I P U T for which the different outputs from the plotter are obtained. All the parameters in the first section of table 1 interact on the solid angle ~2# in eq. (1). Since the solid angles depend only on the relative magnitudes of the different distances incorporated, the values of the parameters given need only be relative and the results

3.1.1. Part o n e

Program ANGPLT consists of a main program with ten subroutines. The co-operation between ANGPLT and the subroutines is evident from fig. 3 and the functions of the subroutines are given in table 3. The main program can be divided into two parts, the first of which calculates optimum values for the weighting factors, Pxi, Pyi and Pzi, in eqs. (2) and determines D F and the second part calculates the measure of resolution, M R , and can be omitted if desired.

The weighting factors in eqs. (2) are optimal when the co-ordinate distortion is minimum and the energydefining pulse Z as uniform as possible over the crystal area. The test pattern (fig. 2) used for calculating the weighting factors is assumed to lie in that plane of the crystal which is most representative for the absorbed events. The exponential decrease in the number of events with increasing depth in the crystal is of minor

Table 2 Plotting options for different values of IPUT. IPUT =

0

1

2

3

4

5

6

7

Z-plot DF-plot MR-plot

NO NO NO

NO YES NO

NO NO YES

NO YES YES

YES NO NO

YES YES NO

YES NO YES

YES YES YES

192

J.B. Svedberg, Computer simulation of the Anger gamma camera

+

~IMGPLT

I

I~PUTIIPIGgM, IETg,MYI,~Y2. I P~UI:IR, IBI~S, IPUT,J04

t

IRflO, JOI

t TI~IL E OF SOLIO I:~GLES PRECI~LC. SOL IO~

I~1

C~LC. OF TOTI~ SOLID At4GLE FOR E~CH RIMG OF PM-TUBES

t

t

CflLC. OF OMI(,OML FOR

PR1PIT.OF, ~l~X, I~iY l

xIMPM*I,O,31J'

EXTRQPOL. OF

SOLIO i:~IGLE

+

15;;~;~.~,;a~ l t

302

(a) A~GPLT (CONT.)

[ IIX2UT,K,L,ffIR]AL,J031 YES PRINT ON SEP. LOGIC~L UNIT. PARAMETERS, OF GMD MR

IIK,LJ CALC.

F RYMDV(1) FROM I FOR I=],ITOT

t

IPRmT, RY.OV,I~I

t

CI~LC. GMD pRIflT IMAGE POIMT ~ITH MO STATISTICS

YES ~

Y

I

E

5

MR-I

YES

I SX;SX+Xt SY=SY*Y~ SZ=SZ+Z OX=OX*X.X,OY=OY+YoY.OZ=OZ*Z.Z

(b)

Fig. 3. Flow-charts for ANGPLT: (a) part one; (b) part two. importance for the crystal thicknesses and gamma en, ergies studied and therefore, the test pattern can be assumed to represent events in the middle plane of

the crystal. Thereby, the choice of solid angle table has also fixed the crystal thickness besides the param eters in the first section of table 1.

J.B. Svedberg, Computer simulation of the Anger gamma camera

193

Table 3 Subroutines in program ANGPLT. POISD(I) ISSON(I) VIKTER (I, J, K)

MATRIS (I) KONST(I, K) DIAG (I) RYMDTM (1) DISTRN (P, I)

DISPLY

STATd)

Produces a generator for Poisson-distributed random numbers with expected value equal to I. Function, picking random numbers from POISD (I) Administers the use of MATRIS (I), KONST (I, K) and DIAG (I) through which the matrix in eq. (5a) is inverted and the different weighting factors are obtained. I: dimension of the matrix. J4:0: the same matrix as in the preceoding call shall be used. K - l , 2, 3, determines ifPxi, Pyi or Pzi is actual. Calculates the matrix with dimension I in eq. (5a). Calculates the constant terms in eq. (5a). Solves eq. (5a) through diagonalization according to Cauchy-Jordan. Calculates the solid angles for all PM-tube - test-point combinations. If I=5, the solid angles are divided by the ZD-pulse in eq. (2). I=0: Z, ZD and dZ are calculated. I=5 : X, Y and DF are calculated. In both cases the values for the different pulses are printed out for the first quadrant of the test pattern. The formal parameter P is the two-dimensional array containing Pzi for I=0, and Pxi and Pyi when I=5, For I=5 the subroutine DISPLY is activated. Administers the output of Z-distributions and image of test pattern on the micro-film plotter. For Z, a channel width, ZPROC, is applied. For regions, where the Z-value is outside the channel, abnormally smooth areas will appear, flat peaks (cf. fig. 7) for too large Z-values and flat valleys for too small Zvalues. IZPL Z-distributions with channel widths equal to ZPROC, ZPROC[2, ZPROC/4.... can be obtained. Performs simulation of PM-pulse production in the Ith tube for all steps corresponding to the variables in eq. (1).

The effect o f crystal thickness o n camera performance can be studied in the program by examining the performance for test patterns at different crystal planes, but with weighting factors previously calculated for the middle plane. On the whole, by the proper choice of some "flags", the program can return to different positions and the calculations c o n t i n u e after changes in only some of the parameter values, thereby saving c o m p u t e r time. The co-ordinates of the PM-tube centres relative to a co-ordinate system (U, V), with its origin at the centre of the central tube, are o b t a i n e d by multiplying a pre-calculated set of co-ordinates with a factor SCALE, related to the spacing factor, M, and the cathode diameter, D, through the expression

SCALE = D X M/20

(3)

Since M c a n n o t be less t h a n 1.0, eq. (3) sets a lower linit for SCALE for each cathode diameter. As long as statistical variations are n o t included, only the relative magnitudes o f the PM-pulses are of importance. The values of the parameters in eq. (1) will n o t influence the co-ordinate distortion as long

as the pulses are above a bias level IBIAS. Too small pulses will n o t participate in the generation of the signals to the display. To o b t a i n weighting factors for m i n i m u m co-ordinate distortion and an energy-defining pulse, Z, which is as u n i f o r m as possible over the crystal area, the image p a t t e r n (X/, Yj) is fitted to the object pattern (K/, L/) through least squares techniques. The function

Rx=~_J(Xj K j ) 2 = ~ ( ~ P x i W i. K.~ 2 j j \i J /]

(4)

is at its m i n i m u m for the weighting factors, Pxi, obtained through the solution o f the following system o f equations dR x

apxi

- 0

(5)

or explicitly

~. Wij~. pxiWq= ~. K j ~ 7. 1 t 1

(5a)

If 1(7 is replaced by L i, the weighting factors, PYi' for

194

J.B. Svedberg, Computer simulation o f the Anger gamma camera

forming the Y-co-ordinates are obtained and if K/is set equal to a constant, Pzi is obtained. However, for symmetry reasons, all PM-tubes at the same distance from the central tube will have the same value, Pzi, and therefore, in this case the six tubes in a ring are handled as a single item, thereby reducing the dimension of the matrix to be inverted. When the weighting factors have been calculated, R x in eq. (4) and the corresponding quantity R y can also be calculated and finally one obtains D F from D F = ((R x + R y ) / 1 2 1 ) 1/2 .

PP

,i

/ , ,/ i

,,'" Pl

Y

,,,"

OO

,"" ,,,,"

×

E

b

D

(6)

3.1.2. Part two

The second part of the main program starts with the input of the parameters K, L and N T R I A L where ( K , L ) is the test point for which the point spread function is to be found and N T R I A L is the number of events to be run. To avoid that computer time is out before any results are printed, the time left is tested after every 100 events, and N T R I A L less than one hundred will not be run. Each step in the production of a PM-pulse according to eq. (1) is subjected to statistical variations. In addition, there is a variation in electron multiplication which is taken into account through the relative variance, a. The outcome of each step in the transfer chain is obtained through simulation using random numbers from generators giving normally or Poissondistributed numbers. If the outcome of a step becomes less than or equal to zero the simulation is terminated for that particular PM-tube. The image points are continously transferred to the plotter if an appropriate IPUT-value is chosen. When all events have been run, M R is calculated according to M R 2 = Vx + Vy

(7)

with Vx and Vy equal to the variances along the X and Y axes, respectively. 3.2. S O L I D A

Fig. 4. A light-ray, emitted from P in the direction (a,0), is refracted at E, the interface between two media with different refractive indices, and strikes a circular area at Q. The lines through (2RD, O) and (3RD, O) are tangents to the circle.

face at E a light-ray emitted in the direction (a, ¢) will have the direction (3, ¢) and strikes the circle plane at a point Q. The program simulates emission of the light-ray through the use of a random number generator RANF, producing equally distributed random numbers between zero and one. The solid angle in units of 4rr is the ratio between the number of times the point Q falls inside the circle and the total number of light-rays emitted, I M A X . To obtain full isotropy, the angles a and ~bshould be given by eq. (8) where R i are random numbers c o s a = 2(0.5 - R I ) q~= 2nR2"

(8)

For symmetry reasons 4~can be set equal to nR 2 in the present case. The relation between a and/3 is given by the following equation n 1 sina = n 2 sin3

o(9)

where n 1 and n 2 are the refractive indices of the two materials. Thus, sin a cannot exceed N n2/n I which gives the following equation for the generation of =

The program calculates the amount of light striking a circular area with radius R D (fig. 4), when light is emitted isotropically from a point P above the xaxis and when there is a difference in the refraction indices between the substances containing P and the circle, respectively. After passage of the optical inter-

COS 13/

cos~ = 1 - R l × T R N K T R N K - 1 - (1 - N 2 ) 1/2.

(10)

195

J.B. Svedberg, Computer simulation of the Anger gamma camera

IfN>~ 1, TRNK is set equal to one. Through eq. (10) it is also taken into account that no light emitted with a > 7r/2 can reach the circular area. lZor the event illustrated in fig. 4, the co-ordinates for Q are x = x 0 - (r +s) cos y = (r + s) sin (11) r = a t g c~ s = b tg~3. If x 2 + y 2 <,RD 2

(12)

the point Q is inside the circle and a hit is registered for point P. For all values o f x 0 for which the x-coordinate for point Q lies between x l and x 2 eq. (12) is satisfied and the same holds for other values of a and/or b when, for instance, Q will be at QQ. The same (c~, 4~)-pair thus produced hits for all combinations of x0, a and b satisfying eqs. (11) and (12) and solid angle tables for many planes can be calculated simultaneously in the program. To increase the chances for hi t~ when x 0 is large, the positive x-axis is divided into segments, each with a length of RD. For each such segment, ¢ is forced to vary between zero and PI where PI = arcsin (RD/IXMIN)

(13)

with 1XMIN equal to the lower limit for x in the segment. For the first segment where IXMIN = O, PI = 7r. The artificially produced increase in the number of hits must be compensated for in the final calculations by correcting the number of trials according to the following equation IMAX = IMAX 0 X 2 × FACT/TRNK

(14)

where I M A X 0 is the number of (a, q~)-pairs tested and where the factors 2 and T R N K correct for the use of eq. (10) instead of eq. (8) and F A C T = P1/Tr corrects for the use of eq. (13). The amount of light hitting a cathode surface is in reality not directly proportional to the solid angle subtended by the cathode surface at the point of emission. Light reflected at the surfaces of the crystal and light guide will contribute. Since such effects depend on the geometrical and optical situations for each special case, they cannot be taken into account

in a general program. Only specular reflection at the front surface of the crystal will be treated. Here, the reflected light can be thought to emanate from a virtual point at the same lateral position as the original emission point. The total amount of light to a PMtube is thus proportional to the sum of the solid angles from the real and virtual emission points, a sum that can easily be calculated in the program. Some part of the light is always reflected at an optical interface. According to formulas by Fresnel, the intensity of the non-reflected part relative to the total intensity is R E D = 1 - 1 ( sin2(a-/3) + tg2(c~-j3)]

2 \ sin2(c~+13)

(15)

tg2(c~+13)]

The effect becomes increasingly important when the number of interfaces, NK, is high and is taken into account together with the reduction in intensity due to absorption of light in different materials. The contribution from each event to the total number of hits is thus reduced by a factor, CORR, defined by CORR.= exp ( - A B S A Xr/cos c0 X e x p ( - A B S B × s/cos ~3)X RED NK

(16)

where A B S A and A B S B are absorption coefficients for light in the two optical media. When one thousand (a, ¢)-pairs have been run, there is a test whether the smallest solid angle is determined to an accuracy in accordance with a pre-determined value, PRES. If not, another thousand events are run and so on, until the accuracy is obtained or a preset number of runs, 1TY, have been performed.

4. Hard-ware and soft-ware specifications All the programs are coded in FORTRAN IV for the IBM 370/155 computer at the Uppsala University Data Center (UDAC), Uppsala, Sweden. A special feature is a micro-film plotter (Benson 320) with plottingroutines produced internally at the UDAC. The isometric program used is constructed by the author. 4.1. Sub-routines to be supplied by the user

1. PL3D

Produces a three-dimensional plot of the values in a two-dimensional array.

196

J.B. Svedberg, Computer simulation o f the Anger gamma camera

Table 4 Data input requirements for SOLIDA. Program

Symbol

Meaning

Format

Sample run

EN

n 2/n 1

Ratio between refractive indices of the two media

F5.2

0.85

JA, KA, LA

a

Initial and final values and increment of a × 100

315

1050 1050 100

JB, KB, LB

b

Initial and final values and increment of b × 100

315

200 200 100

IXMAX

Largest x-value for which solid angles shall be calc.

15

60

KK

If KK is greater than the order number of the actual input, the program will call for more inputs after completion of the present run

15

1

NK

Number of interfaces for which Fresnel-correction should be made I5

ISR

ISR = 1 : specular reflection not included ISR = 2: specular reflection included

I5

ITY

Maximum number of runs for each segment

15

PRES

Maximum percentage error allowed

F5.2

Card one

5 5.00

PM-cathode radius

F5.2

15.00

ATOT

Distances between optical interface and crystal front surface (actual for ISR=2)

F5.2

13.00

IQ

Number of a-values times number of b-values

15

Abs. coeff, of light in the two media

2F10.3

RD

RD

Card two

ABSA, ABSB

ABSA ABSB

2. R A N F

R a n d o m n u m b e r generator for equally distributed numbers b e t w e e n zero and one. Generator for normally distributed r a n d o m numbers with m e a n value zero and variance equal to one. Gives the time:~eft ,for the run (in milliseconds). Plotting-routines p r o d u c e d internally at the U D A C from where specifications can be requested.

3. R A N S S

4. M S L E F T 5. AXIS CIRCLE INITAX NUMBER PLOTMP SYMBL4

0.000 0.000

5. Mode of availability Listings o f the programs, S O L I D A and A N G P L T , including the subroutines in table 3 can be obtained from the author. O f the subroutines in section 4.1, PL3D can also be requested from the author, while the programs under 2, 3 and 4 should be available at all c o m p u t e r installations o f a similar type and those under 5 have to be requested from U D A C .

6. Sample run The input-values are given in table 4 for S O L I D A and in table 5 for A N G P L T . Most o f the outputs, rep r o d u c e d in figs. 5 - 7 , are self-explanatory, but a few remarks will be given.

J.B. Svedberg, Computer simulation of the Anger gamma camera 0.85

1050 1050 tOO 200 200 200 60 1 1 2 5 5.0015.0013.00 1 0o0 0.0 0.99343 0,99343 0°99363 0.99363 0.99363 C.9q343 0.99343 0.99343 0.99343 0.99363 0,99363 0.99343 0*99343 0099363 0°99343 0°99363 0.99343 0.99343 0°99343 0.99343 0°99343 0.99363 0°99343 0,99343 0.q9363 0,99343 0.99363 0.99363 0°99363 0.99343 0°99343 0.99363 0.99343 0.99363 0°99363 0.99343 0°99363 0o99343 0°99343 0.99343 0.99363 0.99363 0.99343 0.99363 0*99343 0.99343 0.99363 0,99343 0.99363 0.99343 0,99363 0,99363 0.99343 0,99363 0.99363 0.99363 0°99363 0.99363 0.99363 0.99343 0.99343 0,988EA 0.98828 0°98780 0.98683 0.98596 0.98491 0,98370 0°98229 0o980E3 0.97866 0.97630 0.97346 0.97000 0.96573 0.96060 0.95363 O.96AE6 0.93322 0.91727 0.89448 0.85989 0.80216 0.68588 0~00375 TIME FOR EACH SEGMENT 2159 1997 8716 7996 PH-TUDE RAOIUS 18.00 ABSORPTION INCLUDED MITH AESA= 0 . 0 ANO 8 8 S 8 O.O N-O.85 A" 10.50 8" 2 . 0 0 IHAX= 2113 7575 7 3 2 9 5 173225 SPECULAR REFLECTION MITfl ATOT = 13.00 X SOLID ANGLE ABS.ERFOR PERC.ER808 HITS 0 0.275318÷00 0.541E-02 [.965 [[63°65 1 0.278098÷00 0.540E-02 1,943 1175.19 2 0.278748÷00 0.540E-02 1.938 1L¥7.Sb 3 0.27560E~00 0.541E-02 1.963 1164.88 4 0,27170E~00 0.542E-02 1.994 [148.22 5 0.26645E÷00 0.543E-02 2.037 1 L 2 b . CO 6 0.259018~00 0.5468-02 2.098 [094.58 7 0.24902E÷OD 0.544E-02 2.184 1052o]6 8 0o23833E÷00 0.543E-02 2,280 1007.17 9 0.22713E÷00 0.562E-02 2.386 959.~6 10 0.21158E*00 0.537E-02 2,540 894.14 11 0.20107E÷00 0.533E-02 2,653 849.70 12 0.19103E÷00 0.529E-02 2,767 807.28 13 O. ITqIBE÷OO 0.522E-02 2.911 TST.20 14 0.16690E,~00 0.513E-02 3,073 705.30 15 0.|557¥E÷00 0.504E-02 3,23A 658.26 X SOLID ANGLE ABS.ERPOR PERCoER808 H|IS lb 0o14071E÷00 0.258E-02 1.836 2131.~5 17 0.127DTE+OD 0.250E-02 1,968 1925°C8 18 0.11323E~00 0.240E-02 2.123 [715.48 19 0.10227E÷00 0.232E-02 2.266 1589.45 20 0.91800E-01 0.222E-02 2.623 1390o77 21 0.8LODGE-01 0.212E-02 2.613 1227.29 22 0.72121E-01 0.202E-02 2.799 1092.~4 23 0.63342E-01 0.191E-02 3,017 959.E3 26 0.56167E-01 0.181E-02 3,230 850o~6 25 0.50903E-01 0.176E-02 3.413 771.19 26 Oo66T[SE-01 0.164E-02 3.666 677.44 27 0o38672E-0[ 0.153E-02 3.968 585.E8 28 0.33634E-OL 0.146E-02 6,278 509.56 29 0.29658E-01 0.136E-02 4.576 449.32 30 0,25282E-01 0.126E-02 6.979 383.03 X SOLID ANGLE ABS.ERROR PERC,ERROR HITS 31 0.19288E-01 0.358E-03 1.866 2822,S7 32 0.16063E-01 0.326E-03 2,027 2356,~6 33 0.13430E-01 0*299E-03 2.223 1968.~7 34 0,11139E-01 0°273E-03 2.667 1632oE6 35 0o92889E-02 0.249E-03 2,685 1361.66 36 0.76425E-02 0,227E-03 2,965 1120.~2 37 0.62629E-02 0.205E-03 3.280 918.C8 38 OLSOTIOE-02 0.185E-03 3.6319 7~ . ~ 39 0.39717E-02 0.166E-03 6.128 582o21 40 0.32320E-02 0.148E-03 6.57~ 673,]8 41 0.25214E-02 0.131E-03 5.168 369.E0 42 0.20430E-02 0.118E-03 5.767 299.~8 43 0.[665[E-02 O. I 0 6 E - 0 3 6°390 244.C8 44 0.128688-02 0.~36E-04 T.272 IE8.~3 45 0.985878-03 0,819E-06 8,310 164.~2 X SOLID ANGLE ABS.ERROR PEAC.ERPOR HITS 46 0.8191~E-03 0.486E-04 5,931 283.~9 67 0.64062E--03 0.4308-04 6.708 221.~4 48 0.51066E-03 0,386E-06 7.514 176.$2 49 0,418T3E-03 0.368E-04 8.299 145,C7 50 0.331498-03 0.309E-06 9.328 ]14.E5 81 0.24438E-03 0.266E-04 10.865 84.~1 52 0.18|338-03 0.2298-04 12.616 62.E2 53 0.14126E-03 0.202E-04 [6.293 48.~4 54 0.10176E-03 0.171E-04 16.840 38.25 58 0.87701E-04 0.159E-04 18.140 30.38 56 O. b 9 4 9 3 E - 0 6 0.|42E-06 20.379 24°08 57 0,59367E-06 O.131E-06 22.069 20*~7 58 0.42392E-06 O,I[IE-06 26.093 14,~9 59 0.32215E-04 0o964E-05 29.932 11,16 60 0.257518-06 0,8628-05 33,479 8.;2 OUTPUT ON PUNCHED ~ROS: N=0.85 A= [ 0 , 5 0 B= 2 . 0 0 AESA=O.O ABSEO.O SPECULAR REFLECTION k I T H ATOT = 1 3 . 0 0 0.27531E*00 O.2TBOgE~OO 0.27874E+00 O.275bOE~CO 0,266458*00 0,25901E÷00 0,24902E÷00 0,238338~C0 0,21158E~00 0.201078~0C 0.19103E*00 O.17918E*CO 0.15577E÷00 0.14071E*00 0,12707E÷00 0.11323E÷CC 0.918008-01 C.810098-01 0.72121E-01 0.633428-01 0.509038-01 0.447158-01 0.38672E-01 0.336348-01 0.25282E-01 0.19288E-01 0.16063E-01 0.13430E-Cl 0.928898-02 0,764258-02 0.626298-02 0,50710E-02 0.32320E-02 0.252148-02 0.204308-02 0*16651E-(2 0,985878-03 0.81914E-03 0,64082E-03 0*510668-C3 0.331498-03 0.264388-03 0.181338-03 0.14126E-03 0.877018-04 0.696938-06 0.593678-04 0,42392E-C4 0.287518-04

Fig. 5. The o u t p u t list from the sample run o f SOLIDA.

0.27170E~00 0.22713E÷00 0,16690E~00 0.10227E÷00 0.56167E~01 0.29658E-01 0,11139E-01 0*39717E~02 D*128688~02 0,41873E-03 0.101768~03 0.32215E~04

197

511KEV TEST POINT 15 S C L I [ ANGLES 0,12191E-01 0,61118E-01 O*LS13ZE-G6

ETA= 0 . 1 5 30

qVl = 0.25 MY2- 0 . 7 5 PMVAR 0 . 5 0 500 POINTS EACH CAUSED BY 5 1 I KEV

TOTALLY 0,4261~E+C0 0.12191E-01 0.271608600 0,23668E-01 O°24974E-OA ~.10267E-06 O.76663F-OT 0.581998-08 0.74663E-07

0.I0059E-03

0.10709E-06

0. I0709E-06 0,10059E-03

0,69863E-06 O.41118F-OI

0.23667E-01 O,3ZSI3E-03

kCRPALIZATIGN POINTS IPIAS=

Xb= 500

14,82~45

Y5 =

I 29.28461

Z0 =

ANALYZEDEYE'ITS GAVE A MEAN NUMBER OF

Z040.20876

Z=

4824.17578

1 0 8 8 0 REGISTRATE9 LIGHT PHOTnNSWITH A SoD. DE

I16

MEAk VALUES AND STANDARD DEVIATIONS IPIAS" 1 MX ~Y RMSOEX 14.84904 29,28165 0,46924 CORRECTED Z-PULSE HAS MV= 4783,~0

RMSQEY 0,56659

RMSQE

AND RMSQE=

0.73413 IRO. RT

1 20,ODD 2,000 0.025 32 0,278318~00 0,278C9E~00 0,278768600 0.27560E#00 O.ZTI70E÷O0 0,26645E~00 O,2BqOlE~O0 0.249028+00 0.238338÷00 0.227138#00 O*211SEE+CC 0,2OlCTE~OO O,19103E÷OO 0,179188÷00 O.L669OE~OO 0,15577E÷00 0,16071E÷00 0,127C78~00 0,113238*00 O.lO227E~OO C,BI80OE-Ol 0,81009E-01 0.72121E-01 0.633~2F-01 0.56167E-01 0.509038-01 0,64715E-01 0,386728-01 0,336348-01 0.296588-01 0,25282E-C1 0,1S258E-01 O.16063E-Ol O*L3~3OE-01 0.11139E-01 0,$2889E-02 0.766ZBE-O2 0.62629E-02 O.50TIOE-02 0,39TITE-O2 0,32320E-C2 0.282168-02 0,206308-02 0.16651E-02 0.12868E-02 0,98587E-03 0,81914E-03 0,64062E-03 0.51066E-08 0.61873E-03 0,33169E-03 0,26438E-03 O,IBL33E-08 0,14126E-03 0,10176E-03 0,87701F-04 0.694938-06 0.59367E-04 0.423928-04 0,822158-06 0.25751E-C6 CMK- - 0 . 2 5 0 6 7 E ÷ 0 0 OHL= 0,66762E÷01 0.25830E-04 0,20103E-04 SOLID ANGLE TABLE~ S R I O . S ÷ 2 . APLSI3 PM-RADIUS~ 15 NUMBER OF PM-TUBES= 19 F " 10 SCJLE= 1.800 8ORDER= 0 . 2 0 9 PRO~ILLE CORRECTED Z-PULSE HAS A MEAN VALUE OF 0 , 9 9 5 6 0 AND A SoO, OF 0 , 0 6 6 1 6 WHICH MEANS 6 , 6 6 RERCENE 0 1 2 3 6 5 K= Z Z0 Z ZD Z 20 Z L ~ ZD ~ ZD Z ZD 6 0 76 0.67 O 71 0.64 C.98 0.86 5 1.00 0.92 0.93 0.86 0,92 0.86 0,94 0.90 0.87 0,80 6 C,96 0.93 0,96 0.95 0,95 0.95 0.95 0.94 0,97 0.90 0.82 0.76 3 0*88 0,96 0,89 0.98 0.91 0*98 0.91 0,94 0,99 0,9~ O,ql 0,83 2 0,86 0,95 ~,87 0,97 0,88 O,9T 0.86 0.94 0,93 0,96 0,94 0.88 0,78 0.92 0.96 ~.90 0.96 0,86 0,96 0.89 0.90 1.00 0.91 0.95 0.95 0.91 l.O0 1.DO 0.93 0.96 OoR7 0.96 3 4 5 K= 0 1 2 X X Y X Y X Y L X Y X V X Y 6 O* OO 5,76 0,91 5,65 2,96 4.97 S 0,00 5.32 0.95 5.21 2.15 5.03 3.04 6,16 8=88 3.93 4 -0.00 6.33 0.$7 A.15 2,07 4.05 3.10 3,01 4,17 3*02 4.74 2.H8 3 -C*O0 2.95 1,04 2,93 1.95 2,94 3.08 2,06 6.23 2,18 4,96 2.09 2 0,00 1.92 1,el 1,96 1,92 1.99 2.97 0.96 4.04 1906 5.16 0.99 5.86 1 0.00 0.86 0.90 0.93 1.97 0.96 5.96 2,98 O,O0 3,99 O,DO 5.22 O.OO O -C,O0 -O,OO 0.88 -0.00 1,95 -0,00

SnLl[

INGLE TABLEI S R I O . 5 * 2 . PA 5

EF 1.?37

APLSI3

IEIAS~ NnFMX

PM-RADIUS= 15

NUMBER OF PM-TUBES= 19

0.999

Sell[ ANGLES 0,16232E-01 0,67681E-GL 0.22689E-06

ETA= 0.15 30

60.00

30,00 -30.00

0.0

51,96

51.96

2.84

2.84

2*84

e.44

3.17

-3.17

- O , OO

5,41

5,6

MY2= 0.75 PMVAR 0,80 POINTS EACIt CAbSED BY 511 KEV

500

TOTALLY 0.62682E~00 0,26913E+00 0,32~93E-01 0,14741E-06 0,81238E-07 0.678258-07

V5"

1 29,31200

ZD"

ANALYZE9 EVENTS GAVE Z MEAN NUMBER OR

0,16406E-03 0,83998E-05

0.13~868-04 O,7~892E-OA

2083°87695

10847

Zs

0.69326E-04 0,3~781E-01

4789.355~T

REGISTRATEO LIGHT PHOTONS WITH 6 S . O .

~EAE VALUES ~NO STANDARD DEVIATIGNS IPIAS= I MX MY RMSOEX 12,96466 29.2963T 0.~8267 CORRECTED Z-PULSF HAS MY= 4T64.89

RMSGEY RHSQE 0.55857 0,73822 AND RMSQE= 179061

0UTPUT ON PUNCHED CARDS: E l O SRIO. 5 # 2 . FIO S R I C , 5 t 2 ,

0.66 -0,00

MYI- 0.25

IBIAS=

500

Y

0,998

0,10247E-01 0.6C977~-06 0,737728-EB

12,91778

0,72 0,82 6

pR 10

NORMJLIZATION POINTS

XS-

Z0

1 NflRMY

511 KEV ETA" 0*15 MVI= 0 . 2 5 MVZ" 0 . 7 5 PMVAR 0 . 5 0 X= 0,0 30,00 18,00 -15,00 -30,00 -15,00 15,00 45.00 0,00 -45.00 -45,00 -0,00 45,00 X= - 6 0 . 0 0 -3O,OG 3 0 . 0 0 y1 0*0 O,O 25,98 25,98 O,OO -25,98 -25,98 25,98 51,96 25,98 --25,98 -51,96 --25,98 Y= 0,00 -51,96 -51,96 WEIGHTS FOR Z PZ = 2,82 2,16 2,16 2.16 2.16 2.16 2.16 2.95 Z,g5 2*95 2.95 2,95 2*95 PZ" 2.84 2.84 2.84 WEIGHTS FOR PA 9 PR= - 0 , 0 0 2.78 1.41 -1,61 -2,76 -1.61 1.61 4.96 DoOR - k , 9 6 -6.96 -G.OO 6.96 PX-6.66 -3.17 3,17 PY= - 0 . 0 0 -O,OO 2,39 2.39 0.00 -2.39 -2,39 2,90 5.98 2.90 -2,90 -5,98 -2,~0 PY= 0,00 -5.61 -5.41 511KEV TEST PCIET L3

6

APLS13 033 ~ ~ S L . 5 0 A P L S | 3 030 S1,50

81L 511

15 15

25 25

75 0 , 8 0 75 0 , 5 0

1,737 1,737

15 13

Fig. 6. The printed output from the sample run of ANGPLT.

30 30

0,734 0.T38

OF

116

O,15401E-Ol 0,48459E-03

199

J.B. Svedberg, Computer simulation o f the Anger gamma camera

Table 5 Data input requirements for ANGPLT. Meaning

Format

IZPL

Number of Z-displays with different Z-channels

I5

ZPROC

Value of Z-channel in % for first Z-display

F10.3

20.0

ZFK

Magnification factor of Z-plot

F10.3

2.0

SYHJD

Size of point in MR-plot

F10.3

0.025

IZ

Intensity in MR-plot (max=32)

15

32

Initiation value for RANSS (KK, RR), odd integer

I5

17

IROMAX

Number of solid angles in the table

15

61

IRAD

PM-cathode radius

15

15

JO1

JOI:~0: an additional run with a diff. table of solid angles will be made. JO1=2: no weighting factors will be calculated

15

1

Code-name for the solid angle table used

1X,4A4

SRI0 .5+2 AP LS13

Table of solid angles

5(E14.5, 1X)

see output SOL1DA

Number of PM-tube rings

I5

Program

Symbol

Sample run

Card one 1

N e w card

KK N e w card

N e w card

ITEXT(I) !=1,4

N e w card

OMEGA(I) I=1, IROMAX

N e w card

NPM

n

3

MVAL

If =0: SCALE=D X M/20 else SCALE from next card

15

1

JO2

JO2~0: an additional run starting with this card

15

0

F

Spacing between test points

15

10

SCALE

Needed only if MVAL:~0

F5.2

1.5

INGAM

E

Gamma-ray energy, keV

15

511

lETA

r~

Crystal efficiency (%)

15

15

MY1

~1

PM-cathode eff. (%)

15

25

MY2

~2

Coll. eft. 1st dynode (%)

15

75

PMVAR

o

Rel. variance, PM-tube ampl.

F5.2

0.5

JF New card

SCALE N e w card

200

,1.tl. Svedberg, Computer simulation of the Anger gamma camera

Table 5 (Cont.) Program

Symbol

Meaning

Format

Sample run

IBIAS

Lower limit for PM-pulses

15

1

IPUT

Plotter options (cf. table 2)

15

7

JO4

JO44:0: an additional run starting with this card

I5

0

Test point for which MR shall be calculated

215

15, 30 (13.30)

NTRIAL

Number of test events for object point (K, L)

I5

J03

JO3:~0: an additional run starting with this card. MR-plot will be in the same figure as the present one

New card K,L

K,L

6.1. Output f r o m SOLIDA Input data are printed and then there is a table over the RED-values due to Fresnel's formula. The time in milliseconds for the different segments in the calculations is given as well as the calculated number of events, IMAX, for each segment. The solid angles are printed and punched on cards in a format suitable for the input to ANGPLT. Memory space required: 48K bytes CPU-time for sample run: 21 seconds. 6.2. Output from A N G P L T OMK and OML given just below the input table of the solid angles are factors used in the extrapolation procedure outside the range o f the table. Values, corresponding to the last table-value and the one just outside the range, are printed out. BORDER is equal to the lower limit of the solid angles taken into account, i.e., that for which the PM-pulse is IBIAS. Co-ordinates for the PM-tubes are given just above the weighting factors and the normalization point is the position of the image point in case there are no stochastics in pulse production. The MR-value, finally, is found under the heading "RMSQE". An important part o f the output is that from the micro-film-plotter (fig. 7). To identify the solid-angle table used, a label of not more than 16 characters must be given. It can be formed individually by the

50O (500) l

(0)

user and appears on all lists and figures. The intensity of the dots in the MR-plot must be chosen according to the number o f events in order to avoid under- or overexposure. Memory space required: 224K bytes CPU-time for sample run: 45 seconds.

Z-DISTRIBUTION Z-CHAMMEL(x)=20.O SRI0.5+2. APLS13 D=30 n=3 M=I.O0 KEU=511 F=IO

(7a)

J.B. Svedberg, Computer simulation o f the Anger gamma camera

~

SRI0.5+2. APLSI3 D:30 n=3

~

~

I

~ ~

~

4 i i

o

~

KEU=511 g=lO

iilll lill

\

II Ii i i i ii 0

l0

20

30

40

.50

60

X-COORDINATES (7b) ~-

"

~

~ ~o-

-

8u

:~-

SRI0.5+2. APLS13

~ x ~

D=30 n=3 M=I . 0 0 KEU:511 =10

i

c:~ 10

20

30

40

X-COORDIMRTES

50

60

(7c) Fig. 7. Plotter output from sample run of ANGPLT: (a) isometric plot of Z-distribution; (b) image of fine-mesh test pattern; (c) point spread functions for two test points.

20]

References [1] H.O. Anger, Rev. Sci. Instr. 29 (1958) 27. [2] J.B. Svedberg, Phys. Med. Biol. 13 (1968) 597. [3] J.B. Svedberg, in: Medical radioisotope scintigraphy, Vol. I (IAEA, Vienna, 1969) 125. [4] J.B. Svedberg, Acta Radiologica, Suppl. 313 (1972) 242 [5] J.B. Svedberg, Phys. Med. Biol. 17 (1972) 514. [6] A.H. Jaffey, Rev. Sci. Instr. 25 (1954) 349. 17] A.V.H. Masket and Wm. C. Rodgers, U.S.A.E.C. Report TID-14975 (1962).