Contributed Papers

4581

IEEE Transactions on Nuclear Science, Vol. NS-28, No. 6, December 1981

IMAGE RECONSTRUCTION AND NOISE EVALUATION IN PHOTON TIME-OF-FLIGHTASSISTED POSITRON EMISSION TOMOGRAPHY

TakehiroPhysics

National Institute of9-1, Anagawa-4-Chome,

ABSTRACT

In positron CT, the path difference of annhilationpair gamma rays can be measured by time-of-flight (TOF)difference of pair gamma rays. This TOF informationgives us rough position information along a projectionline and will reduce noise propagation in the recon-struction process. A reconstruction algorithm for TOF-positron computed tomography (PCT) based on the back-projection with 1-dimensional weight and 2-dimensionalfiltering is presented. Also a formula to evaluatethe variance of the reconstructed image and the optimalback-projection function are presented.

The advantage of TOF-PCT over conventional PCT wasinvestigated in view of noise figure. An example ofsuch noise figure evaluations for CsF and liquid Xenonscintillators is given.

Introduction

Positron computed tomography (PCT) is based on thefact that two annihilation gamma rays are radiatedessentially in opposite directions. Thus the positionof a pair of detectors recording coincidence gamma rayscan be used to determine a projection line. Positionalong the projection line cannot be determined withconventional PCT.

It is well known that time-of-flight (TOF) differ-ence between two annihilation gamma rays gives us posi-tion information along a projection line. 2 That is,the time in which a gamma ray reaches a detector is pro-portional to the flight path length of the gamma ray.Thus, flight path difference between two annihilationgamma rays can be determined by this TOF differencemultiplied by the velocity of light. Referring toFig. 1, if annihilation takes place just at the middleof the detector pair D1 and D2, then the time-of-flightof one gamma ray to D1 is just equal to that of theother gamma ray to D2. If annihilation position isdisplaced by x from the center, then the TOF differencet, is 2x/c. For example, if x=1.5cm, t is equal to3 cm/3xl010 cm/sec = 0.1 nsec.

Timing resolution of currently available gamma raydetectors is not sufficient to determine the positionalong a projection line to the accuracy required by 3-dimensional imaging.394 Yet TOF information will con-fine the position of annihilation to a finite region

TomitaniDivisionRadiological SciencesChiba-shi, 260, Japan

along the projection line in contrast to conventionalPCT, in which position along the projection line is notknown. Thus TOF information can be expected to improvesignal to noise ratio, since noise propagation is con-fined to finite region.5-9

For TOF-PCT purposes, not only good timimg but alsohigh efficiency is required in the selection of scintil-lators. Allemand et al. suggested the potential advan-tages of CsF scintillator for TOF-PCT and showed themerit of TOF-PCT with CsF scintillators over conven-tional PCT.5 Mullani et al. in a pilot experimentwith TOF-PCT studied the use of CsF.6 Ter-Pogossian etal. studied both the mathematical and experimentalbasis of TOF-PCT and are planning to construct SUPPERPET (or PET VII) to implement a TOF-PCT system uti-lizing CsF scintillators.9

The purposes of this paper are 1) to present areconstruction algorithm for TOF-PCT, 2) to develop aformula to evaluate the variance of the reconstructedimage, 3) to find the optimal back-projection functionto minimize this variance and 4) to deduce an indexto evaluate the advantage of TOF-PCT over conventionalPCT.7

Reconstruction by 2-dimensional Deconvolution

In TOF-PCT, a pair of detectors in coincidencedetermines an annihilation gamma ray pair chord whichis identical to conventional PCT. The time-of-flightinformation then determines a point along the chord.The accuracy of the position along a coincidence-pairline is considerably worse than the position infor-mation transverse to the coincidence pair line. There-fore, we have to look for a suitable reconstructionalgorithm that takes account of this physical situation.

Definitions of operators.1) The rotational mean operator on 2-dimensional func-tion g(x,y) is defined in eq.(l) as,

,[g3(r) = 2 fg(rcosO,rsinO)dO (1)

The resultant 2-dimensional function is axially symmet-rical and r denotes the radial distance in polar coor-

dinates. If g(x,y) is sy-mmetrical with respect to theorigin, then eq. (l) becomes

M[g](r) = - g(rcoso,rsinO)dO. (1')

2) 1- and 2-dimensional Fourier transforms are definedas;

1%1[g1 (X) = f g3(x)exp(+2TrixX)dx (2)

Figure 1. Relationship between the position of posi-tron annihilation and the distance to two opposingdetectors.

Manuscript received: Feb. 2, 1981.This work was supported in part by a Grant-in-Aid forCancer Research from the Ministry of Health and Welfare.

Wrg2](XY) = ffg2(x,y)exp(+2lixX+2siyY)dxdy. (2' )

where small letter variables and functions denote param-

0018-9499/81/1200-4582$00.75( )1981 IEEE4582

eters in the spatial domain and the corresponding capi-tal letter variables and functions denote parametersin the frequency domain, e.g. G (X) in the frequencydomain corresponds to g1(x) in Ie spatial domain.

Let i (t) denote the time spread function expressedin spatiai units. When an event is detected, twodetectors in coincidence determines a projection line.Then one can draw a line with an intensity distributionf1(t) centered at the measured TOF point along theprojection line (Fig. 2).

Fourier transform of eq.(3) is calculated to be;

P(R)--P(R) = 2Sf M[i** f1(r)**h(r)1(R)= M [IFJ(R)H(R). (3')

The Fourier transform of the 2-dimensional deconvolu-tion function is,

H1(R) = P(R)REIF3(R) , (5)

Point Responseof T.O.F.

Di fb~~~~~~~~~~~~~, 1- ~ , \AD

If we choose p(r)=6(r) (Dirac's delta function),which implies a perfect reconstruction, then 2-dimen-sional deconvolution function is;

H1 (R) = - 1

,O L IF](R). (5')

Figure 2. Back-projection procedures.Poistron annihilation takes place at a point A.Two annihilation gamma rays are detected by twoopposing detectors D1 and D . Time-of-flightinformation determines point M between D1 and D2Point M may be different from point A due to finitetime resolution. On accumulation of events, pointM's may distribute as i (t) shown in the upper partof the figure. When one event is detected at pointM, a line of distribution f1(t) is drawn between Dand D2, and on accumulation of such lines, the re-sultant sum is shown in the lower part of thefigure.

Here we assume that f (t) is symmetric. Upon accumula-tion of events, a back-projected image will be obtained.Then the application of a 2-dimensional deconvolutionfilter to this back-projected image will yield a recon-structed image. Let h(r) denote a deconvolution func-tion of axial symmetry. By denoting Dirac's deltafunction as 61(t), the point spread function of thisreconstruction process is axially symmetrical and isdenoted as p(r), which may be expressed as,

p(r) JO 2EL6 *i *f ](rcose)y6 (rsinO)id**h(r)- w [6l*i*f3 (r)**h(r). (3)

where * and ** denote 1- and 2-dimensional convolutions,respectively. We define 2-dimensional functions,f(x,y) and i(x,y) for simplicity, as follows,

f(X,y) = f1(x)' (y) ; i(xy) = i(x)61(y). (4)

Then, by exchanging the order of integration, the

The denominator of the right side of eq.(5') becomes 0as R tends to infinity. Although the image is band-limited, statistical noise in the measured data extendsto infinity so that the high frequency components ofstatistical noise will be enhanced enormously when sucha filter is applied. Therefore, we have to be contentwith finite resolution images in order to suppressnoise to the level appropriate for diagnostic purposes.This situation is similar to conventional PCT.

Variance

Since the advantage of TOF-PCT lies in its improve-ment of signal to noise ratio, the variance of a recon-structed image with the TOF method will be discussed.Consider a point in the reconstructed image. Projec-tion lines that cross this point will contribute posi-tively to the reconstructed image but the projectionlines that do not pass this point will contribute nega-tively. On the other hand, all projection lines con-tribute to the variance regardless of whether theycross the image point or not. Thus, to investigate thevariance of the reconstructed image, a uniform distrib-uted source of infinite extent will be considered here.

The reconstructed image intensity and its varianceare calculated in the following three steps.Step 1) Calculate the image intensity at a point (x,O)from the measured TOF events at a point (O,O) at anangle e (see Fig. 3a). Then tne TOF events are back-projected onto tne image plane with the back-projectionfunction f(xcosO+ysino,-xsin8+ycos6)=fl(xcose+ysinO).6 (-xsin0+ycose). Next, the deconvo-lution function, A(r), is applied to this back-proj-ected image.

Let v denote emission rate of coincidence pairs/sec/cm in the section to be imaged, then the emissionrate of coincidence pairs per unit angle in the sectionis v/7T, in the direction OQ0<r . Let s(x,y;e) denotethe image intensity at a point (x,y) from the measuredTOF event at a point (O,O) at an angle 0 within a timeinterval T with the detection efficiency E. The recon-structed image intensity at the point (x,O) is calcula-ted to be; (see appendix A)

s(x,u;u) -= fT- **1f-I(XCOS0,-Xsillo ) (6)

By denoting a=vet, this equation can be expressed ass(x,O;e)=a/r[f**h] (xcos ,-xsinO).

4583

Y I( x co7sf + ysil,o) 5(-X ;irOf ycsos) ** h(r)

,0 0) Directioni of Coincidenice Chord

X

rcsp,rsin 4)

V = var p(C) = f r.dr f'dP.var q(r)

= 2rra f, -LELf**hj2 (r)r.dr (8')

Since we assume a uniform source of infinite extent,the variance of any image point is equal to var p(O).

Optimization of Signal-to-Noise Ratio

Up to equation (8'), we are at liberty to selecta back-projection function, fI(t). This functionshould be so chosen that it will minimize the variancein eq.(8'). In eq.(8'), the variance is integratedin spatial domain, while the deconvolution function issolved in the frequency domain (eq.(5)). Now let usdeduce an expression of the variance, V, in the form ofintegration with respect to the frequency. In eq.(8'),f, and h are real functions and are symmetrical withrespect to the origin; thus their Fourier transformsare real functions. Now, with the aid of Parseval'stheorem in two dimensions, eq.(8') can be rewritten as,(see appendix 2)

2

E(FH) I(R)~R (8")

Figure 3. Procedures to calculate image intensityand its variance are shown. a) a source locatedat a point (0,0) and the image intensity and itsvariance at a point (x,O) are considered; the anni-hilation gamma ray is emitted in 0-direction.b) Exchange roles of a point (0,0) and a point (x,O)and sum up image intensity and its variance over

source distributions.

Since the number of measured coincidence pairs perunit direction, a/Tr, obeys a Poisson distribution, thevariance of s(x,0;0) is,

var s(x,o;e) = a[f**h]2(xcosO,-xsinO) (6')

Step 2) integrate s(x,0;0) and its variance over alldirections of coincidence pair lines. The resultantintegrations are axially symmetrical and are denoted as

q(r) and var q(r) in polar coordinates, in which r=x.

q(r) = fT s(x,0;0)dO = af T [f**h- (rcos6,-rsiroo)dO

=aqr lfa*h](r),

var q(r) = tfl var S(x,O;O)dO = a M[f***}-1]2](r)

(7)

(7' )

By replacing H(R) in eq.(5), we obtain,

V = 27ra R'dR P2(R) [F ](R)f

{f [IFJ (R) }2 (9)

Note here that H(R) in eq.(5) is axially symmetricalso that H(R) may be excluded from the rotational mean

operator.

According to Schwartz's inequality,

{t [IF](R)i 2 =' f7I(RcosO)F(cos6)dO} 2

< !fI(RcosO)de 1fTF2(RcosO)dO= , [ I2 ] (R ) 7 [F2] (R )

(10)

where equality holds on the condition that I(X,Y)OcF(X,Y). Generally I(X,Y) and F(X,Y) will be normalizedso that they are equal, hence,

V 2 270af P() _R.dR = V

Jo 7[I' (R) min. (11)

Note that in the case of a uniform source of infiniteextent, the reconstructed image intensity at a point(x,y) from the measured TOF events at a point (0,O) is

equal to the intensity at the point (0,O) from the meas-

ured TOF events at the point (x,y). The same argumentholds for the variance of the image intensity. There-fore the role of the point (x,y) and the point (0,O)are interchangeable.

Step 3) Exchange the role of the point (x,O) with thatof (0,0) and integrate q(r) and var q(r) over allsource points in the slice.

p(o) r.dr f5dr-q(r) 27raf [ff**h](r)r.dr (8)

That is, the variance V is minimal for the conditionFl(T)=Il(T). In other words, the variance of the re-

constructed image is minimal when fl(t)= il(t). Thisresult is consistent with that expected from the theoryof the coded aperture imaging, if we regard il(t) as a

coded aperture and fl(t) as a back-projection function.In the optimized condition, the 2-dimensional deconvo-lution function in the frequency space is

H (R)= P(R)oP I'[I(R) -

(12)

In this equation, I(X,Y) appears twice. This may beinterpreted so that one I(X,Y) factor compensates for

time-of-flight blurring and the other compensates forthe back-projection function F(X,Y), the latter beingequal to I(X,Y) for the optimized case.

4584

0D0)1

(

( 0, 0

e /s

A

y

b

Comparison of Noise in TOF-PCTwith that of Conventional PCT

To compare noise in TOF-PCT with that in conven-tional PCT, the resolution of the reconstructed imageshould be specified as well as the shape of the timespread function of time-of-flight measurement.Here we assume the point spread function of the recon-struction as a 2-dimensional Gaussian function, andthus we specify spatial resolution. The point spreadfunction p(r) and its Fourier transform are;

p(r) = Lr2exp(2

) (13)

P(R) = exp(-27sa2Ro2) (13')

where a is the standard deviation of 1-dimensionalcross-section of 2-dimensional Gaussian function. Thestandard deviation a is related to the full width athalf maximum of the reconstructed image, r,,, by ru=2.3550. Note that the standard deviation of eq.(13) in2-dimensions is equal to 4ta. Likewise, we assume thetime spread function is a 1-dimensional Gaussian tunc-tion; then i (t) and its Fourier transform are;

i (t) 11 1-2-aT

exp(- -)

I1(T) = exp(-22oT 2T2)

Note that the FWHM time resolution, t10, is related to0T by tw=2.355a . The rotation of I1 T) in eq.(11) iscalculated to be

[I-2](R) = lf exp(-412 2R2cos2O)dO11(0 2)

= exp(-2 IT202R2 )I (2 7T2 o2 i2 )

30-

"' 20

.H

10

0*

0 0.1 0.2

(15)

0 . 3

Spatial Frequency (l/a)

Figure 4. Two-dimensional deconvolution filterexpressed in the frequency domain in units of 1/a.The parameter in the graph is a /a.

where I is the 0th order Bessel's function of thefirst kind. Therefore, H (R) in eq.(12) becomes

f op(R) = exp(-2s'2ao2R2 )

exp(-2fT2 T2R2)I 0(2.f2o 2R2) (16)

Note that if a =0, i.e. the time resolution is infini-tesimal; then Ihe denominator of the right-hand sideof eq.(16) is unity, and H (R) reduces to simple 2-dimensional Gaussian smooth°g with FWHM equal to2.355o . Note also that if v-> oo , i.e. no TOF infor-mations are available, then H(0) becomeskTr | Jexpt-21r2a2R2}; this is identical tT2alsonventional2-dimensional CT reconstruction filter. I Theconstant k is due to the difference in normalizationbetween conventional PCT and TOF-PCT. In the case ofconventional PCT, the back-projection function is ofunit height; while in the case of TOF-PCT, the back-projection function is so normalized that the inte-gration of it over entire space is unity.

The radial cross-sections of 2-dimensional filtersare plotted as a function of R in Fig. 4 and the radialcross-sections of 2-dimensional deconvolution functionsare plotted as a function of r in Fig. 5 for various0 /a.T

The variance is calculated to be

V . - 211sf0 exp(-47Tr2o2R2) H dRmiln exp(-2m2o 2R2)I(2112C 2R2)

T 0 -r

(17)= a rJO exp(-u) du

4 T2Jo exp(-KU)I (KU)

{ u = ( 2 TT aR ) , K = 2 ( a /Cy ) 2 )T

Note that at the smaller extreme of a /0, Vmin tendsto a/(4f1ro) which is equal to the variance with 2-dimen-sional Gaussian smoothing. Vmin/(a/(4ro2)) is shown inFig. 6 as a function of a /0 and turns out to be almost

70 - I I. I I _

-,41 0 *

70 12

300

6 0

5010

0

Radial Distance (Cy )

Figure 5. Two-dimensional deconvolution functionsexpressed in the spatial domain in units of a.The parameter shown in the graph i s alT/va.

4585

k14' )

o102

Time Resolution (cr,/a)Figure 6. Variance of TOF-PCT as a function of aT/a.The curve indicated as 'line' shows the variancewith the optimal deconvolution filter, while thecurve indicated as 'dot' shows dot-writing methodfor comparison.

proportional to a la for a l/a so that Vmin may beexpressed as eq.(17') with good approximation.

CT aCt

min 3 06-3-, (T/a> (17' )

where t. is the FWHM time resolution of TOF expressedin spatial units and r is the FWHM resolution of thethe reconstructed image. From this equation, it isevident that the variance is proportional to the timeresolution of TOF and inversely proportional to the 3rdpower of the spatial resolution of the reconstructedimage.

For comparison, let us now consider the varianceof conventional PCT. Here the variance of a uniformdisk source is proportional to the disk diameterdr 13and is calculated numerically as

con 33r

3 (18)

Table 1. Characteristics of scintillators.

Scintillator NaI(Tl) BGO CsF Liq.Xe

Decay time (ns) 230 300 5 2.2,27Wave length 410 480 390 180Maximum (nm)Density (g/cm3) 3.67 7.13 4.11 3.06Light Yield (1) 100 15 6 80-190Refractive Index 1.85 2.15 148 -

Reference 15 15 15 17-20

Note: 1. Light yield relative to NaI(Tl).

By comparing eq.(17') and eq.(18), one may define an'equivalent diameter', d , of TOF-PCT which reults inthe same amount of variaRce as the conventional PCTmethod.

deq = 1.60 t (19)

Thus, deq/d is a measure of the variance reduction withTOF-PCT compared to conventional PCT.

To compare the relative merit of TOF-PCT with con-ventional PCT for various scintillators, the detectionefficiency must be taken into account. The varianceand the mean image density are both proportional to thesystem efficiency e. Note that the system efficiencyof a coincidence positron camera is proportional to thesquare of the detection efficiency,ed. The amount ofnoise relative to the mean image density is equal tovarianTe/mean. Hence, the noise figure of TOF-PCT, Nf,

may be defined as

Nf = = 1.2 Xd dt

(20)

Likewise, Nf of conventional PCT may be calculated byinserting diameter d of the disk source (cf eq.(18)).Hence, Nf of TOF-PCT may be compared directly with thatof conventional PCT. Scintillator physical parametersof concern are listed in Table 1.

Absolute detection efficiency data are lackingexcept for NaIATl) and Bismuth Germanate (BGO) calculat-ed by Derenzo. Using his data, a scintillator sizeis assumed as 2 cm wide x 5 cm long x 3 cm high, andthe discrimination level at 100 keV. Efficiency datafor liquid Xenon and CsF were estimated from NaI(Tl)data assuming that photo-fractions of these scintilla-tors are equal to that of NaI(Tl), since the effectiveatomic number of these scintillators are close to NaI(Tl). In Table 2, noise figures for a uniform disksource are shown. Noise figures for conventional PCTwith NaI(Tl) and BGO are shown for comparison; a uni-form disk source of 30 cm diameter is assumed. In thetable, noise figures relative to conventional PCT withNaI(Tl) are shown on the bottom row.

Since efficiency depends on the crystal size andon the energy discrimination level, validity of noisefigures in Table 2 is limited to the particular condi-tions described above. Note that in these calcula-tions, the variances with the TOF-PCT method are summedover the entire space, which will overestimate thenoise of the TOF-PCT for sources comparable to the TOFspatial resolution.

Comparison of Dot Writing Methodwith Line Writing method

It is possible to record a TOF detected event as a

dot on the back-projection plane rather than a finitedistribution function. As before, the 2-dimensionalfilter may be applied to this image to obtain the finalimage. Here we assume the same functional forms fori1(t) and p(r) as described in the previous section.By putting fl(t)=6l(t) and hence F(X,Y)=1 in eq.(9),the variance of the dot writing method, Vd, is calcu-lated to be

v = 2ira f P2(R) RdRd Jo{ E[I]I(R)} I

a r0 exp(-u)4r2fO0{exp(-Ku/2 )I (Ku/2) }2

du (21)

4586

Table 2. Noise figures of scintillators. Nf's of NaI(T1) andBGO are those in the conventional PCT mode, while Nf's of CsFand liq. Xenon are those in the TOF-PCT mode. In the conven-tional PCT mode, a 30 cm dia. uniform disk source is assumed.

Note: 1. Private communication from M.E.Ter-Pogos sian.

2. Estimated value from reference17-20.

3. Cited from ref.(21) for crystalof 2 cm wide x 3 cm high x 5 cmlong.

4. Estimated values from NaI(Tl)data in ref. (21) on the assump-tion that photo-fractions ofboth CsF and liq. Xe are equalto that of NaI(T1).

By comparison of the denominators in eq.(11) and eq.(21),Vd is apparently larger than Vmin. Vd/(a/(4fra2)) isplotted in Fig. 6 as a function of OTr/O. It is seen tobe almost proportional to (OT/a)2 for values of o.,lo >2.This is in contrast to the line writing method of eq.(17'). Vd may be approximately expressed as

2a

V =a TX V T--

d m4 MilVl

The merit of the line writing method over the dotwriting method is apparent from Fig. 6. In practice,however, three-dimensional data acquisition is requiredfor the line writing method, since the two ray coordi-nates and the time-of-flight position along the raymust be recorded. In principle, the dot writing methodwould require only two-dimensional data acquisition,since only the coordinates corresponding to the loca-tion of the event in the back-projection plane arerequired. In practice, however, difficulties inprecise time adjustment of all the detectors willrequire three-dimensional data acquisition so thatdifferences among detector pairs may be compensated forwith software.

Discussion

From the noise figures in Table 2, the liquid Xenonscintillator is the most promising. The merits ofliquid Xenon lie in the flexibility of size and theshape of scintillator(s) and the possibility of adopt-ing the Anger camera principle. This will allow infi-nitely fine sampling in contrast to discrete crystalsystems. In the latter case,, either a wobbling motion23-25 or a random arrangemgnt9of crystals along withthe rotation of the gantry is necesary to makelinear sampling fine enough to make full use of thespatial resolution set by crystal width. However,when using liquid Xenon, there are technical diffi-culties such as purification of Xenon, cooling of thescintillator(s), optical coupling between the scintil-lator(s) and photo-multipiers, choice of a goodreflector and a suitable wave length shifter sinceliquid Xenon scintillates in the uv.17-20 Thereare a number of studies needing attention in thedevelopment of liquid Xenon scintillator TOF systems.

Among readily available scintillators, CsF is themost promising.5,6,9 Prom the example shown inTable 2, a variance reduction of a factor of 2 may be

expected with CsF TOF-PCT over conventional PCT withBGO. It should be noted that there is room forimproving the light yield from CsF. This wouldimprove the timing and consequently increase the signal-

(22)

-to-noise ratio. On the other hand, no such improve-ment can be expected in the case of conventional PCTwith NaT (Ti) or BGO. Another advantage of TOF-PCTwith CsF lies in the higher couting rate capabilityover conventional PCT with BGO.

Acknowledgements

The author would like to gratefully thank toLiichi Tanaka for his most valuable suggestions through-out this work. The author wishes to acknowledge toNorinasa Nohara

,Mikio Yamanoto and Hideo Murayama for

their most helpful discussions; Tadashi Hashizume andHiroshi Tsunemoto for their interest in this work andMitsue Chiba for her assistance in preparation of thismanuscript. The author is pleased to thank to NizarA. Mulllani for his most valuable discussions and toMichael M. Ter-Pogossian for inf ormation on CsF timing.

Appendix 1. Proof for equation (6)

From the definition of s(x y6), we have the followingexpression, where r=fx+y

s(x,y; 0) = f(xcosO+ysinO,-xsin0+ycoso)**h(r)

fYff((X-x')cos0+(y-y')sinO,-(x-x')sinO+(y-y')coso)h(x' 2+y'2)dx'-dy' (A-l)

Exchange variables from x' and y' to s=x'cos8+y'sineand t=-x'sin6+y'cose, then

s(x,y;0) =f f(xcosO+ysin0-s.-xsine+ycosO-t)-_ o -00

h (/s2+t 2 )ds*dt (A-2)

By putting y=O, then we have equation (6).

Appendix 2. Proof for equation (8")From eq.(8') and eq.(6), we have the following

expression for V.

V = 2saf r.dr dO [f**h]2 (rcosO,-rsinO) (A-3)

4587

Scintillator NaI (Tl) BGO CsF Liq.Xe

Time resolution (ns) 1.4 3.5 o.43 #1 0. #2Equivalent disk diameter (cm) (30) (30) 10 4.8

Efficiency 0.59#3 0.90#3 0.71 0 68#4Noise figure 9.3 6.1 4.5 3.2

Relative noise figure 1 0.66 O.49 0.35

Let s=rcosO and t=-rsin6, then we have;

V = affs dt [f**h]2 (s, ) (A-4)co0 -00

By application of Parseval's theorem in 2-dimensions toeq.(A-4), then we have the following expression for V,where f(x,y) is assumed to be symmetrical with respectto the origin.

v = af'f'dS.dT (FHI)2(S,T)-co -Mc

= af R.dRf do. (FH)2 (Rcos6,Rsino)0 0

= 2Traf R-dR if de(FH)2(RcosO,Rsine)

= 2lTaf R.dRR[(FH)2 ](R). (A-5)

Hence, we have eq.(8").

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9. M.M. Ter-Pogossian, N.A. Mullani, D.C. Ficke, J.Markham and D.L. Snyder, Photon Time-of-Flight(TOF) Assisted Positron Emission Tomography (PET).J. Comput. Tomogr., to be published.

10. S.W.Rowland, "Computer Implementation of ImageReconstruction", p.p.9-79 in "Image Reconstructionfrom Projection" G. Herman, ed; Springer Verlag,Berlin, 1979.

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Optimizing the Reconstructed Image in TransverseSection Scan, Phys. Med. Biol. 20:789, 1975.

13. E. Tanaka and T.A. Iinuma, Correction Functions andStatistical Noises in Transverse Section PictureReconstruction, Comput. Biol. Med. 6:295, 1976.

1)4. E. Tanaka and T.A. Iinuma, "Image Processing forCoded Aperture Imaging" in "Proceedings of theIV-th International Conference on InformationProcessing in Scintigraphy", Orsay, 1975,C. Raynaud and A. Todd-Pokropek, editors.

15. "Harshaw Scintillation Phosphors", The HarshawChemicals Co., Ohio.

16. J.A. Northrop, J.M. Gursky and A.E. Johnsrud,Further Work with Noble Element Scintillators, IRETrans. Nucl. Sci. NS-5(3):81,1958.

17. K. Monahan, V. Rehn, E. Matthias and E. Poliakoff,Photoluminescence Decay in Solid Xenon, J. Chem.Phys. 67:1784, 1977.

18. S. Kubota, M. Hishida and J. Ruan(Gen), Evidencefor a Triplet State of the Self-Trapped ExcitonStates in Liquid Argon, Krypton and Xenon, J. Phys.C 11:2645, 1978.

19. S. Kubota, M. Hishida, M. Suzuki and J. Ruan(Gen),Dynamical Behavior of Free Electrons in the Recombi-nation Process in Liquid Argon, Krypton and Xenon,Phys. Rev. B 20:3)486, 1979.

20. S. Kubota,, M. Suzuki and J. Ruan(Gen), Specific-Ionization-Density Effect on the Time Dependenceof Luminescence in Liquid Xenon, Phys. Rev. B 21:2632, 1980.

21. S.E.Derenzo, Positron Ring Cameras for Emission-Computed Tomography, IEEE Trans. Nucl. Sci. NS-24:881, 1977.

22. M.M. Ter-Pogossian, private communication.

23. M.M. Ter-Pogossian, N.A. Mullani, J.T. Hood, C.S.Higgins and D.C. Ficke, Design Considerations for aPositron Emission Transverse Tomograph (PET V) forthe Imaging of the Brain, J. Comput. Assist. Tomogr.2:539-637, 1978.

24. C. Bohm, L. Eriksson, M. Bergstrom, T. Litton,R. Sundman and H. Singh, A Computer Assisted RingDetector Positron Camera System for ReconstructionTomography of the Brain, IEEE Trans. Nucl. Sci.NS-25:624-637, 1978.

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I Tomography, Phys. Med. Biol. 26:)4S89-499, 1981.

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29. N. Nohara, E. Tanaka, T. Tomitani, M. Yamaa to,H. Murayama, Y. Suda, M. Endo, T.A. Iinuma, Y.Tateno, F. Shishido, K. Ishimatsu, K. Ueda and K.Takami, Positologica: a Positron ECT Device witha Continuously Rotating Detector Ring, IEEE Trans.Nucl. Sci. NS-27:1128-1136, 1980.

Takehiro Tonitani was born in Tokyo, Japan in1940. He received the B.S. degree and the M.S.degree in nuclear engineering from the University ofTokyo. Since 1966, he has been working at the Na-tional Institute of Radiological Sciences, where heengaged in the development of the monitor for theassessment of Plutonium lung burden. Meanwhile hestudied the potential analysis of proportionalcounters of rectangular cross-section with the aid ofthe theory of elliptic functions. Since 1969, he hasbeen working in the field of instrumentation in

nuclear medicine, including an analog image processor, a single photon CTdevice in 1972, a focal plan type positron camera with multi-crystal focal planedetector in combination with a conventional gamma camera in 1973, gammacamera position arithmetics, a circular ring positron CT for head studies called"Positologica" in 1978 and a 3-layer whole body positron CT. He stayed inthe Lawrence Berkeley Laboratory, the University of California, Berkeley in1974 and studied the liquid rare gas chambers. He is a member of the JapanSociety of Applied Physics, the Japan Society of Nuclear Medicine, NipponSocietas Radiologica and the Japan Society of Medical Electronics andBiological Engineering.

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