IEEE Transactions on Nuclear Science, Vol. NS-32, No. 1, February 1985

MAXIMUM-LIKELIHOOD RECONSTRUCTION FOR SINGLE-PHOTON EMISSION COMPUTED-TOMOGRAPHY

Michael I. Miller*, Donald L. Snyder*, and Tom R. Miller**

* Department of Electrical Engineering,* Institute for Biomedical Computing,

and** Mallinckrodt Institute of Radiology

Washington UniversitySt. Louis, Missouri 63130

ABSTRACT

A mathematical model is formulated for a gammacamera used to observe single-photon emissions frommultiple view angles. The model accounts for thestatistics of radioactive decays, nonuniform attenua-tion, and a depth-dependent point-spread function.The maximum-likelihood method of statistics is usedwith the model to derive an algorithm for estimatingthe distribution of radioactivity.

I. INTRODUCTION

Single-photon emitters are of increasing use inemission computed-tomography. In a typical system, a

gamma-ray camera is configured so that it can berotated around a patient to acquire projection datafrom multiple view angles. These data are then pro-cessed to estimate activity distributions for viewingby a radiologist. Jaszczak, Coleman, and Lim [1] andKeyes [2] give a brief history of single-photon tomo-graphy and a bibliography to the literature on thesubject, citing Kuhl and Edwards [3] for introducingthe idea in 1963.

Other system configurations besides a rotatablegamma-camera are being contemplated for single-photonemission-tomography. These include systems consis-ting of one or more rings of detectors that surroundthe patient, systems similar in architecture to thoseused in two-photon, positron-emission tomography [4].While the algorithmic developments we shall reportare described for data acquired with a gamma camera,they can also be useful for other system configura-tions with little or no modification.

There are three fundamental problems associatedwith processing gamma-camera data which make single-photon tomography distinct from two-photon tomo-graphy. The first is that the processing must ac-

count for the attenuation of photons due to ab-sorption and scattering in tissues that intervenealong the propagation paths from the locations wherethe single photons are created to the gamma camerawhere they are sensed. Attenuation correction isrequired in two-photon tomography as well but issimpler because the attenuation does not depend onwhere along the propagation path the two photons are

created. The second problem is that the sensitivitypattern, or point-spread function, of a gamma camerais depth dependent, becoming broader with distancefrom the camera. The sensitivity pattern for two-photon tomography, on the other hand, is nearly con-stant along detector lines throughout the regioncontaining activity. And the third problem is thatwith the gamma camera, the reconstruction problem ismore appropriately treated as three dimensional; thisis because the large, circular, crystal detector in a

gamma camera forms a projection plane, so there are

no natural two-dimensional slices being imagedthrough the activity as there are with two-photontomographs having rings of small detectors sur-rounding the patient.

These three problems have been treated in variousways in the literature. Attenuation correction isthe most difficult problem and has been approached byapproximation strategies; see Jaszczak, et al. [1]and Moore [5] for a brief discussion and additionalreferences. A frequently made assumption that sim-plifies the mathematical development is that the bodysection containing activity has a uniform attenuationdensity, typically that of water. Based on thisassumption, modifications to the conventional fil-tered back-projection algorithm have been suggestedby Genna [6], Kay and Keyes [7], and Keyes, Orlandea,Heetderks, Leonard, and Rogers [8] using the arithme-tic average of opposing projection data, by Budinger,Derenzo, Gullberg, Greenberg, and Heusman [9] andJaszczak, Murphy, Huard, and Burdine [10] using thegeometric average of opposing projection data, and byTretiak and Delaney [11] and Tanaka [12] using expo-nentially weighted projection-data and exponentiallyweighted back-projection. Other strategies assume aknown, nonuniform attenuation density. Iterativereconstruction algorithms have been developed basedon this assumption. Walters, Simon, Chesler, andCorreia [13] have followed this approach using itera-tions of conventional filtered back-projection,weighting factors, and reprojections, and Budinger,et al. [9] use iterations derived from linear, least-squares theory. The algorithm we propose below is inthis second category because it is iterative andpermits a known, nonuniform attenuation density to beused. It differs from previous algorithms in thisclass because it is not based on modifications of theconventional filtered back-projection algorithm or onlinear, least-squares theory.

While the importance of the depth-dependentpoint-spread function for single-photon emission-tomography was recognized as early as 1973 by Cormack[14], most algorithms that have been developed do notcompensate for this effect. One exception is thealgorithm due to Isenberg and Simon [15]. Efforts tominimize the degradation caused by depth dependencehave been made by careful design of the collimatorsused and by the collection of projection data over afull 360-degree range of view angles, with 180-degreeopposing views being combined by weighted or geomet-ric averaging. The algorithm we propose below per-mits the depth-dependent point-spread function to beincluded. This is not done by averaging opposingviews but, rather, by using all available views,including opposing ones, in a common manner. Whilewe do not do so for brevity, it is also possible toinclude other dependencies that the point-spreadfunction may have, such as the variations that occurtoward the edges of the detector field in a gammacamera; edge effects are disregarded.

The estimation of three-dimensional activitydistributions in single-photon emission-tomographyhas always been treated by estimating the activity ina series of two-dimensional slices. This is ac-complished by organizing the data acquired with arotatable gamma-camera into planar data correspondingto the activity slices being estimated; the planar

0018-9499/85/0002-0769$01.00 ©C1985 IEEE

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data are then processed independently from one planeto another. This partitioning of the data intoplanes is somewhat artificial for the gamma camerabecause of the circular shape of the camera's scin-tillation crystal. For this reason, the algorithm wedevelop does not require that the data be organizedinto independent planes. The result is an algorithmthat directly estimates the three-dimensional activi-ty distribution, which can be displayed in variousplanar sections. Since this requires substantialcomputational capability, we also include a two-dimensional version of our algorithm to be used withdata organized into planes.

The approach we take differs in another importantway from previous approaches for single-photon emis-sion-tomography in that it is based upon a Poisson-process model that accurately describes both theemission process as well as the detection process.Maximum-likelihood estimation is used with this modelto derive the reconstruction algorithm. This ap-proach parallels the one recently followed in develo-ping algorithms for two-photon, positron-emissiontomography by Shepp and Vardi [16] and for time-of-flight, positron-emission tomography by Snyder andPolitte [17]. Significant improvements in both reso-lution and signal-to-noise performance have beenobserved by Politte and Snyder [18] and Shepp, Vardi,Ra, Hilal, and Cho [19] with this two-photon algo-rithm, which make us optimistic that similar improve-ments may be seen in single-photon tomography withthe algorithm we propose.

The objectives of this paper are: (1), to formu-late in Section II a mathematical model for dataacquired with a multiview gamma-camera in single-photon emission-tomography; (2), to derive in SectionIII a reconstruction algorithm based on the model ofSection II and the assumption that the attenuationdensity is known in the body section containing acti-vity; and (3), to propose in Section IV a method forestimating the attenuation density required in Sec-tion III.

II. MATHEMATICAL MODEL

The mathematical model and approach to algorithmdevelopment that we use for single-photon tomographyclosely parallels that employed by Snyder and Politte[17] and Snyder [20] for two-photon tomography. Acontinuous model is used with the resulting con-tinuous algorithm being quantized for implementationon a digital computer.

Three-Dimensional Model

The geometry is shown in Fig. 1. A radioactivetracer is distributed three-dimensionally in a pa-tient. The intensity of radioactive decays is de-noted by X(x) counts/sec in rectangular coordinates,where x = [x1 x2x3]' is a three-dimensional column

vector. This intensity cannot be observed directly,so the purpose of making measurements and processingthe measured data is to estimate A(x) for all x or,at least, for all x in a specified volume of in-terest. Activity is measured using a rotatable gam-ma-ray camera, which consists of a relatively thinscintillation-crystal in the shape of a circle about30 centimeters in diameter. An array of closelypacked photomultipliers adjacent to the crystal per-mits the locations of scintillations caused by pho-tons incident on the crystal to be measured. Wemodel the detector as a planar circle that is perpen-dicular to a line drawn from the origin of the coor-dinate system to the center of the circle. Thedetector is shown at angle 6 in Fig. 1. We define a

detector function D(u,6) such that D(u,O) = 0 for andonly for the points u in the planar circle definingthe detector when the detector is at view angle 6.The measurements are obtained by rotating the detec-tor around the x3-axis, shown in Fig. 1, through Ndiscrete viewing angles, 6 , n = 0, 1, N-1.We assume that the detector is at a distanceL(0,P (0)) from the origin, where P (0) denotes the

perpendicular projection of the origin 0 onto thedetector face when the detector is at angle 6; thatis, P (0) is the endpoint of a line perpendicular to

the detector face and extending from 0 to the detec-tor. L(0,P (0)) = IP (0) - 01 is the length of this

line. For the large field of view, which we assume,D(u,6) defines a plane according to

D(u,6) = u cos(O) + u2sin(C) - L(0,PO(0)) = 0.

There are two random point-processes of impor-tance in our mathematical model. The first describesthe three-dimensional locations in R3 where photons(gamma rays) are created in the radioactive decays.We call this point process the 'decay point-process.'Photons are created at location x with intensity A(x)and propagate therefrom isotropically. The decaypoint-process will be denoted by Nd(dx); here, Nd(dx)

is the number of decays in the volume element[x,x+dx). From the physics of radioactive decay,this point process is well modeled mathematically asa time-space Poisson process with an intensity X(x)that is proportional to the tracer concentration.The total number of decays is given by

Ntotal decays R3Nd(dx) (1)

The second point process which we define modelsthe detected photons. For a detected event, thegamma camera produces a four-dimensional measurementpoint (u,6), where u = [u u2 u3]' denotes the loca-

tion of the detection event in R3, and 0 is the viewangle in -1 of the camera when the

event is detected. The collection of these detectionevents forms the second point process of interest,which we call the 'measurement point process,' de-noted by Nm(du,6). The total number of detected

events is given by

N -1

N JN (du,60).-total detected -2 |R3 m n

n=0

Using a similar argument as Snyder, Thomas, and Ter-Pergossian [21], we conclude that the measurementpoint-process is also a space-time Poisson processthe intensity of which we describe next.

As defined above, P (x) denotes the perpendicularprojection of x onto the detector when the detectoris at view angle 6, and L(x,P6(x)) = 1P6(x) - XIdenotes the length of the perpendicular line betweenx and the detector plane. The measurement point(u,6) associated with a decay point x differs fromP6(x) because of scattering of the photon along its

(2)

771

trajectory from x to the detector, because of imper-fect collimation, and because of errors in the elec-tronic detection-circuitry. The measurement error-vector e moves the perfect projection-measurementP (x) to a location u within the detector planeD(u,O) = 0. This measurement-error & = u - x ischaracterized by the conditional measurement-errordensity p (£lx,O), which is the point-spread functionof the camera normalized to unity. Because of thedepth dependence of the point-spread function, thisdensity depends upon the distance L(x,P (x)); to a

reasonable first approximation, in which effects nearthe edges of the scintillation crystal are neglected,this error density is a circularly symmelric normaldensity with mean P (x)-x and variance cf (x,6),

p (EIx,) -

[2a2(x,)]-1 exp[-Ie-P (x)+x2/2ct2(x,o)]6[D(x+c,0)].(3)

process is

ll(U,O) = p(u,Olx)X(x)m(dx),R3 (5)

where p(u,elx) is given by

p(u,Olx) = p (u-XIX,a) J(x,u,E) p(alx),(6)

where p(Olx) is the fraction of photons leaving xthat are directed towards the detector when it is atangle 0, which is determined by both the solid anglesubtended at x by the detector and the fraction ofthe total scan time that the detector is at angle 0.Here, p(u,Olx)du can be interpreted as the fractionof photons leaving x that are detected in [u,u+du)when the detector is at angle 0.

We note that the total number of decay and detec-ted events are related by

The delta-function singularity 6[.] in this densityarises because errors are confined to the detectorplane. The depth-dependent variance a (x,8) is afunction of the collimator used to 'focus' the gammacamera, the scintillation crystal and electroniccircuitry of the camera, the radioisotope beingimaged, and the scattering within the patient; seeRollo [22, pp. 388-391] and Muehllehner [23] for adiscussion of depLh-dependent gamma-camera point-spread functions. For the most widely used radio-nuclide, technetium-99m, the full-width at half-maximum resolution (FWHM = 2.355a(x,O)) roughlyvaries linearly from 0.9 cm at a depth of 5 cm to 1.3cm at a depth of 10 cm. The resolution can be deter-mined for any desired camera and collimator frommeasurements of the line-spread function of a thinsource of the desired radionuclide embedded in asoft-tissue equivalent scattering medium such aslucite or water [22, pp. 439-4421. Variations in theresolution due to scattering in other tissues or inair will be ignored.

Not all photons leaving x and propagating towardthe detector at angle 0 end up being recorded as ameasurement point. Some are scattered in anotherdirection, some arrive with such reduced energy thatthey are rejected by energy discriminators used inthe detection circuitry, and some unscattered pho-tons, due to the detector efficiency not being one-hundred percent, arrive at the detector but fail tobe converted into a light pulse in the scintillationcrystal and so are not sensed. Let ,B(x,u,J) denotethe fraction of photons that leave the decay point xheaded toward location u in the detector at angle 0and and which get to the detector. If a(z) denotesthe three-dimensional attenuation density of themedium containing activity, and if £j(x,u) denotes theline from x to u, then this survival probability isgiven by

P(x,u,O) = e(u,O) exp[-J a(z)m(dz)],

t(x,u) (4)

where e(u,O) is the detector efficiency for photonsdetected at location (u,0), and m(dz) is the volumeof [z,z+dz).

The intensity ,u(u,0) of the measurement Poisson-

Ntotal decays Ntotal detected Ntotal undetected '

(7)

where Nttl undetected is the number of photons lost

through scattering and imperfect detector efficiency.In snmmary, the mathematical model in three-

dimensions for single-photon emission-tomographyusing a multiple-view gamma camera consists of tworandom point-processes. One is the decay point-process Nd(dx), having an intensity X(x) proportionalto the concentration of radiotracer. The other isthe measurement point-process N (du,0), having an

mintensity ,u(u,0) that depends on decay intensityA(x), the point-spread function p (elx,0), and thesurvival probability 13(x,u,O) and p(Olx).

Two-Dimensional Model

For the mathematical model in two dimensions, weimagine that the circle defining the detector surfaceis partitioned into narrow strips parallel to the(x1,x2)-plane in Fig. 1. These strips can be exten-

ded to form slices, parallel to the (xl,x2)-plane,passing through the activity being imaged. Considera single such strip. Measurement points (u,f) fal-ling in this strip are separated from all the othermeasurement points and processed independently toestimate the activity in the slice defined by thisstrip. Some performance loss results from this par-titioning compared to the three-dimensional situationbecause there can be decay points x in the slice thatproduce measurement points (u,O) outside the stripand, conversely, decay points outside the slice canproduce measurement points in the strip; with thetwo-dimensional model, all measurement points in thestrip are treated as if they originated from decayspoints within the slice.

In the two-dimensional model, the slice istreated as an infinitesimal plane, and a two-dimen-sional coordinate system centered at the system axisis defined in that plane; that is, the (x,,x2)-planein Fig. 1 is now the slice plane. The two-dimen-

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sional decay point-process Nd(dx) now consists of the

(xl x2)-location of each decay, and the measurement

point-process N (u,0) consists of the (u,u2)-loca-tion of each detected event when the camera is atview angle 0. The relationship between the intensi-ties, X(x) and ,u(u,&), of the decay and measurementpoint-processes is the same as for the three-dimen-sional model provided x and u are now viewed as two-dimensional vectors and the error density is inter-preted as

p (£x,.) -

[27r2 (x,)]-1/ exp[-|a-P (x)+x2/2c2 (x,)]&6[D(x+e,0)],6

(8)

where D(u,G) = 0 for all u in the line defined by thedetector strip.

In summary, the mathematical model in two dimen-sions differs only in a minor way from that in threedimensions when vectors are interpreted as two dimen-sional rather than three.

III. KNOWN SURVIVAL PROBABILITY

In this section, we assume that the survivalprobability f3(x,u,G) is a known function; the estima-tion of this probability is discussed in the nextsection.

We will use the EM algorithm of Dempster, Laird,and Rubin [24] to develop an algorithm for estimatingthe intensity A(x) from single-photon data. An al-gorithm of a similar type was first used for two-photon, positron-emission tomography by Shepp andVardi [16] and then for positron-emission tomographywith time-of-flight data by Snyder and Politte [17].It provides an iterative computational-method fordetermining the maximum-likelihood estimate of someparameter A from some observed data y. In the termi-nology of [24], y is the 'incomplete data;' for thepresent consideration, y is the measurement pointprocess. These incomplete data take values in asample space Y = fall possible values of y} and havea density p(y:A) over this space; for the present

consideration, A = (X(x):xER3) and p(y:A) -exp[(k)], where

N -1

f(A) - - pJ ,u(u, )m(du)

n=O ~~~~~~~~~~~(9)N0-l

+| lnbI(u,0)]N (du,En)'

n=O

where the measure m(du) is the volume of [u,u+du).To circumvent the difficulty of maximizing p(y:A)directly, Y is embedded in a larger space Z in whichsome hypothetical data z, termed the 'complete data,'take values. A many-to-one mapping from Z to Y,defined by some function h(.) such that y = h(z), isassumed as is a density q(z:X) of z over Z withrespect to some measure p(dz). The densities of the

incomplete and complete data are related according to

p(y:A) = J q(z:X)p(dz).10n

{ z:y=h(z) }

There are two steps for each iteration of the EMalgorithm, an E-step and an M-step. In the E (for'expectation') step at stage i of the iteration, the

conditional expectation Efln[q(z:A) IY,xA 1) is de-"'(i-l)

termined, where A is the estimate of A deter-mined at stage i-l. In the M (for 'maximization')step at stage i, the result of this conditionalexpectation is maximized with respect to A to give

the estimate A . According to Shepp and Vardi [16]A(l) -(2)and Wu [25], the sequence of estimates X A

... converges to the maximum-likelihood estimate Aof X in terms of the incomplete data. Moreover, the

sequence of loglikelihoods t((1) ), (2) isnondecreasing, so estimated values of A get no worseat each iteration stage, and they generally improve.

There are many ways that the complete data can bedefined. This is done so that the E and M steps canbe readily evaluated. The complete data adopted forwhat follows is motivated by the underlying decayevents, the errors in their measurement, and the factthat some decay events are undetected. Thus, supposethat the complete data are formed by labeling eachpoint of the decay point-process with a mark (e,6,m),where m = 1 if the photon created in the decay isdetected, and m = 0 if it is undetected. The resultis a marked point-process [26, Ch. 3] wherein eachpoint (x,(e,9,m)) identifies the location x of adecay, the error a in the measurement of its locationwhen the camera is at angle 6, and the mark m de-signating whether or not the photon created in thedecay is detected. The function h(.) mapping pointsfrom Z to Y is defined by

h[x,(e,O,m)] =

{ (x + e, 8) = (u,8)

0

for m = 1

for m = 0,

(11)

where 0 is a null indicating no measurement point.

There are two counting processes that can beassociated with the complete data. One counts unde-tected points and the other detected points, forwhich m = 0 and m = 1, respectively. Let

Nc(dx,da,6,O) denote the first one; this is the

number of decays in [x,x+dx) producing undetectedphotons which would be measured with errors in[e,e+de) at angle 6n if they were detected; this is a

Poisson process with intensity t0(X,Ce,n) given by

IL0(ZX, £, n) = PCe (e£l x Oo) [ l-P (x +8 En) ]P (OI )x((12)

Similarly, let N (dx,de,On l) be the second counting

process; this is the number of decays in [x,x+dx)producing detected photons which are measured witherrors in [a,e+de) at angle e . This is also a

Poisson process with intensity I1(x,,On) given by

k 1U J

773

IL1(s, e ,n) = pa( e 1 on)P(x s+e on)P (O1s)k(x)13(13)

These two processes are independent Poisson processesbecause they result from independent deletions oracceptances of points from the complete data, whichis a Poisson process.

For the complete data, we have that q(z:A) =

exp[c(M)], where

c() t=CO () + c 1(A), (14)

whereN -1

t£ m(X) -. J J m(x,eaon)m(dx)m(de)n=O

N -1+

R3R ln[m(S' e.n)INc(dx,de,n,M),"=A

E[N(dx,l)Iy,A I f (x) m(dx), (17)

where we define

f il(x) =

N -10

p(u, I)l(Xi-l )( (i-l ) (u, -1N(du, )n-u

and where

ji (u,E) J p(u, Ix)AX( (x)m(dx)R3

(18)

is the estimate at stage i-l of the intensity i(u,n)of the measurement point process at view angle 0e.Hence, for the E-step, we define a function Q ac-

cording to

for m = 0 and m = 1. In the K-step, we will maximize

E(lnLq(z:X)]iy,AX(i1) with respect to A; consequen-tly, terms in ln[q(z:X)] that do not depend on X canbe suppressed without affecting the K-step, and wecan equally well consider

t;O)= - J A(x)m(dx)

R3

+ J ln[)(x)]N(dx,l)R3

where

N(dx,m) | c(dx,nde, m)n=0

Of course, Nd(dx) = N(dx,0) + N(dx,l).

For the E-step of the EM algorithm at iterationstage i, we need to evaluate the conditional mean of

Z'(X) given the incomplete data and A fromc

iteration stage i-l. For this, we observe that

E[N(dx,O)jIy,A(i_1) I =[1 - w(x)]X1 1)(x)m(dx),(16)

Q(XIX(il1)) _ Et

- 3 [-A(x) + ([l-i(x)]x(i l)(x)

(19)^f i-1)+ f' (s) Iln [A(s)]Im(dx) .

For the K-step, we maximize Q with respect to A

to get A(. Using the calculus of variations, wefind upon doing this that

X(x)= El - i(x)]A(i l)(x) + ^(i-l)(x).(20)

This equation defines a sequence of estimates A

.. that converges for any positive initial""(0)Aestimate A to the maximum-likelihood estimate X.

comment 1. Letting i tend to infinity, we see from(20) that the maximum-likelihood estimate of k satis-fies

A(x) =

N0-1A(x)[ {- (s)) + (8 ) A -Nll(u,En)Nm(du, )],

n=0OR

which is equivalent towhere

N0-1i( ) = |P(s,u,On)pa(u-xlsoeP)ptn Ix)m(du)

n=0

is the average survival probability for photonsemanating from location x, and that

N0-1

j()0= n, P(u.x)I1 (u,O )N (du,O )3,m= (22)

where

A(U,e ) =A n P(U,on Ix)i(x)m(dx).a

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This is a nonlinear integral-equation for A thatappears to be impossible to solve analytically. Itcan be solved numerically using the iteration equa-tion (20).

comment 2. Integrate both sides of (20) to obtain

JiA ((x)m(dx)R3 (23)

f |t 1 - 3( )]XA1( (x)m(dx) + NR3 total detected'

If there is no attenuation and one-hundred percent

detection efficiency, then 13 = 1, Ntotal undetected

0, and the following normalization holds for everyiteration stage

A(i) A )

R3Ntotal decay (24)

More generally, (23) implies that

-(i) ^ ( i-1)+total decay total undetected total detected

(25)

where the left side is the estimate of the number ofdecays at stage i, and the first term on the rightside is the estimate at stage i-l of the number ofundetected photons. We see that the constraint in(7) is satisfied at every iteration stage by theestimated numbers of events.

comment 3. Ihe iterative reconstruction algorithmimplied by (20) has the following steps.

step 0. i=l, i-l) < A(O)

step 1. n=O, SUM=O

step 2. reproject A to get (i at angle 0

step 3. divide into measurements at angle 0nstep 4.

step 5.

back proj ect a t angle enadd the results to SUM

step 6. if ''last angle'' go to 7,else replace n by n+l and go to 2

step 7. add [1-13] to SUM

step 8.

step 9.

multiply SUM by A. to get A

if ''last iteration'' stop,else replace i by i+l and go to 1

Step-0 is an initialization step; A() must be selec-

ted to be positive except possibly at locations x

where the activity is known a priori to be zero.

Step-2 requires a reprojection of the estimate of Aat stage i-l onto the detector plane defined by

D(u,&n) = 0; this reprojection includes the effect of

the depth-dependent point-spread function, detectorefficiency, and attenuation. Step-4 requires a back

projection that also includes these effects. Uponthe completion of step-6 at the last angle, SUM

contains /(i-l),5(i-l)

comment 4. The reprojection and back-projectionsteps in the iteration of comment-3 can be simplified

by using an approximation that does not appear toorestrictive. If we note that the point spreadfunction p (ejx,6) reduces to zero fairly rapidly as

l al increases, then integrals involving

p(u,OIx) = p (u-xlx,&) P(x,u,0) p(OWx) (26)

will not change appreciably if p(u,Olx) is replacedby

p (u-xlx,O) (x,P0(x),O) p(OIx). (27)

In other words, it does not appear to be unreasonableto replace the measurement point (u,O) by the perfectprojection point P (x) in the survival probability

because of the narrow shape of the point-spreadfunction and the relative smoothness of P over thatnarrow region. With this approximation, the repro-jection required in step-2 can be accomplished by aseries of convolutions of the weighted estimate

P(x,P(x),O)p(OlI)XA( ) (x) with the point-spreadfunction at constant depth. Similarly, the backprojection required in step-4 can be accomplished bya series of convolutions of the weighted measurements

[A(i1) (u,0)]1N with the point-spread functionm

followed by a point-by-point weighting byf(x,P0(x) ,)p(eslx).

IV. ESTIMATING THE SURVIVAL PROBABILITY

The estimation of the survival probability,j3(x,u,O), is a calibration procedure required for theemission-tomography study discussed in the last sec-tion. In general, we envision a series of threemeasurements to estimate the survival probability;fewer may often be adequate, as we shall discuss.Two of these are performed without the patient, andone with the patient in the same place as for theemission-tomography measurement. The purpose of thefirst measurement is to determine the detector effi-ciency e(u,G) for all positions in the detectorplane; that is, for (u,G) satisfying D(u,O) = 0.This calibration step can be performed using a knownflood-source having the same energy as the radio-tracer to be used in the emission measurement; itwould typically be performed only once a day. Theflood source used for this might be a point source

located far from the camera so that it uniformlyilluminates the entire crystal of the camera. Thesecond and third measurements are to determine theattenuation density for photons having the energy ofthe radiotracer used in the emission measurement;these need to be performed with each emission-tomo-graphy study and are done as transmission studies.For these, we envision a planar flood-source parallelto the detector plane but on the opposing side of the

patient, as depicted by Maeda, et. al [27]. Thesecond measurement is performed without the patientand is to determine the possibly nonuniform distribu-tion of activity in the flood source itself. Thismeasurement would be performed when a new source is

prepared, typically once a day. Subsequently, at thetime of each patient study, the flood source distri-bution determined with this measurement can be cor-

rected to account for radioactive decay by using theknown radionuclide half-life. The third measurementis performed at multiple view-angles with the patientto determine the attenuation density. Thus, a totalof four measurements are envisioned for single-photon

emission-tomography study:

1. a single, daily measurement using a cali-brated flood-source to determine detectorefficiency;

2. a daily measurement without the patient usingan uncalibrated flood-source to determine theintensity of the flood source itself;

3. a transmission measurement with the patientusing the now calibrated flood source fromthe second measurement corrected for radio-active decay if necessary;

4. an emission measurement with the patient.

Let A1(x), for x in R3, denote the known intensi-

ty of the flood source used in measurement-l. Thecorresponding measurement-l point-process is a Pois-son process with intensity

Al(u.) = e(u,G) P|P,air 1lt'6)Al(x)

(28)

where p .air(u-xlx,O) is the point-spread function of

the camera at the depth of the flood source and inair. If this measurement is performed over a longtime-interval so that many detection events are ob-served, then, to a good approximation, measurement-lyields (1u,)M. Since p 8.air (u-xlxx)XA (x) may beassumed to be known, measurement-l then yieldse(u,O). If a point source of strength 1 is used,

and if this point source is placed sufficiently farfrom the detector that the detector is uniformlyilluminated, then (28) implies that e(u,e) is propor-tional to 1N(u,M). This calibration measurement only

needs to be performed at one view angle if small,second-order effects are neglected.

Let X 2(x), for x in Rs, denote the unknown inten-

sity of the flood source used in measurement-2. Thecorresponding measurement-2 point-process is a Pois-

son process with intensity

112(u,) = e(u,&) P air (u-xlx 0)2(x)m(dx).3

The result of measurement-2 may be taken to a goodapproximation to be p 2(u,O) because a long observa-

tion-interval can be used. Here, e(u,e) and

P, air(u-xlx,O) are known, and X2(x) must be deter-mined. This can be accomplished if p .(u-xlx,O)

F.airis an invertible kernel or if k2(x) is assumed to be

spatially invariant in the plane of the flood-source.Data for measurement-2 only needs to be acquired andprocessed at one angle.

Let X3(x) = X2(x) be the known intensity of the

flood source used in the transmission measurement-3.With the patient now present, the Poisson statisticsof the measurement-3 point-process become importantbecause the radiation dose must be limited. The

intensity is

IL3(u, ) =

e(u,O) Pe,trans (u-xt, )3(x,u,0)p(6Ix))3 (x)m(dx),

(30)

where pe, trans (U-xlx,e) is the point-spread functionof the camera for the flood source with the patientlying between the source and camera, and P(x,u,0) isunknown and to be estimated. We presume thatPetrans (u-xIx,) is known. Here, the survival pro-bability P is given in (4) with the line t(x,u) beingfrom a location x in the flood source, through theattenuating medium of the patient, to the detectorlocation (u,O). What is needed for the emissionreconstruction described by (20) is P for all loca-tions x in the three-dimensional region containingactivity and not just those x in the flood-sourceplane. Our approach for extending the estimatedvalue of P from the restricted set of locations toall locations is to: (i), estimate A(x,u,O) for x inthe flood-source plane from the data acquired inmeasurement-3 at angle 8; (ii), divide this estimateby the detector efficiency e(u,O) obtained frommeasurement-l; and (iii), note from (4) that thenegative logarithm of this ratio is a line integralthrough the estimated attenuation-density and, so,reconstruct the attenuation density a(z) using fil-tered back-projections or any other algorithm suit-able for inverting multiple view-angle, line-integralprojections. The estimated survival-probability forall locations x can then be determined from theestimated attenuation-density using (4).

We use the EM algorithm to estimate P3(x,u,O) fromdata acquired in measurement-3. For the completedata, we augment each decay point x in the floodsource by a mark (a,O,m) indicating the error a inthe measurement of its location on the camera atangle e and whether or not the photon created in thedecay is detected, m = 1 or m = 0, respectively. Weparallel the development of (12) to (19). Thecounting process for undetected photons is a Poissonprocess with intensity

IL0(x,8e,ID) =

0~~~~~ nnP£,trans(eIS09n)[1-i(X,X+e.O )]P(e ni,))A3(X)(

( 31)

29) For detected photons, the intensity is

IL1(x'O U) pa,trans (Elx,o )n)(x,x+e,0n)p( nlx)X3(x).(32)

The loglikelihood tc(o) of the complete data, corres-

ponding to that in (15), becomes

N0-1V(fi) = ln[i(x,x+p.,( )]N (dx,de, ,l)c n c no1

(33)N-l

+ > fR3JR ln-l (x,x+g,e n)]Nc(dx,de,On8,O).n=0

775

776

For the E-step, we need to determine

(i-i)] (i-l) (34)

We note for this that

E[Nc(dx,ds,O,,o)IY,05(i 1)] _ 1(x,8,0 )m(dx)m(de),

(35)

where s(i-1) is tL of (31) with j replaced by j i).We also note that

E[N0 (dx,da,On,l) Iy, )(36)

IL x,(t)se,O )[IL )(U,@A lN (du,Otn m(dx),

where N3(du,& ) is the measurement-3 point-process,

ii) (x,e,0n) is ,u(x,e,n) of (32) with P3 replaced

by (i-l) and

1A ( u, On) = A1 ) (xs, U-X, 0n)m( dx)n

is the estimate at iteration stage i-I of the inten-sity of the measurement-3 point-process. Conse-quently,

Qtl(i-i) =

N -l

(i-1)(x,u-s,9 )[A l)U )-ISn=0R3R

N -1

n=O iRJR

x lnp(x, u, n)]N3 (du, 0t) m(dx)

(37)

(i-l)x, w-x, )

x ln[l - ,B(X,u, n)]m(du) m(dx).

In the M-step, we maximize Q[w3 1 )] with respect

to A3 to obtain A . The result is

" ( i)A3 (X,u,0 )

s.(i-l) u, )N3(du, n)

)(x,u,e N(du,e )+j (u,Oe (x,u,e )]n3 n nn

(38)

where N3(du,On) = N3(du,On)/m(du) is the counts per

unit area observed in measurement-3 in pixel [u,u+du)at angle 0n

Eq'n. (38) defines an iteration sequence "(1)such that the sequence of loglikelihoods

t(A()), t(p (2)), ... is nondecreasing, so the esti-

mated values of , get no worse at each stage, an,(,)they generally improve for any initial estimate ,(

(0)satisfying 0 <f3 < 1. We note that for any ini-tial estimate such that 0 < p (O) < 1, there will

hold 0 < M(i) < 1 for every subsequent iterationstage. Also, in the limit as i tends to infinity, wesee after rearranging (38) that the limiting estimate

of the iteration , satisfies

(s,u, On)t- P(x,u,e )][N3 (du,OAn)-( n

(39)

subject to the constraint 0 < P(x,u, ) < 1, where

,u (u, o)(40)

JRPa,transa(unsn P(x,u,On)P(0nWX)33(x)m(dx).Eq'n. (39) indicates that j3(x,u,0 ) = 0, or P((x,u,0n

1, or P(x,u,o ) is the solution to the linear

integral-equation

JPe,trans (u-x Ix,On)P(X,U,On)P(OnIx)X3(x)m(dx)= N3(du,O ), (41)

subject to the constraint that 0 < P(x,u,0 ) < 1.n-

The estimate of the survival probability can bedetermined by solving the constrained, linear inte-gral-equation (41); one way to do this numerically isto implement the iteration defined by (38).

comment 5. It is possible to make an approxima-tion, similar to that in comment-4 above, that great-ly simplifies the problem of estimating the survivalprobability. For this, let P0(u) denote the perpen-

dicular projection of the location (u,O) in the de-tector plane at angle 0 onto the flood-source plane.We observe that the integral in (41) will not changeappreciably if O(x,u,O) is replaced by 13(Pa(u),u,O)for the point-spread function p trans relatively

narrow and , nearly constant over the narrow part ofthe point-spread function. With this approximation,we then have

u(PU)J,) = minfn

[I Pe,traus(uu-lx,On)p(enIx)X3(x)m(dx)] N3(du,On)).(42)

In summary, the survival probability 13(x,u,J)required for the reconstruction algorithm of SectionIII for single-photon emission-tomography can beestimated from three measurements using a floodsource. Measurement-3 yields i by solving (41),which can be accomplished iteratively using (38).Provided the performance degradation is not great,the approximation of comment-5 directly yields 1

along projection lines without the need to solve an

integral equation. Once A is determined from mea-

777

surement-3, we use the detector efficiency from mea-surement-2 to form the projections

-ln[P(P (u),u,On)/e(u, Onn

through the attenuation density. These projectionscan be inverted to determine the attenuation densitya(z), for example, by using the filtered back-projec-tion algorithm. Then 1(x,u,e), for x in R3 as re-quired for the emission-tomography reconstruction canbe determined from (4).

V. CONCLUSIONS

The algorithm we have developed for single-photonemission computed-tomography should provide a morequantitative estimate of radioactivity distributionsthan other algorithms derived by modifications of thefiltered back-projection algorithm of x-ray tomo-graphy and the use of ad hoc methods for accommoda-ting nonuniform attenuation and a depth-dependentpoint-spread function. The algorithm is computa-tionally demanding but has a structure that may permit specialized hardware to be designed for its prac-tical implementation. We are presently collectingphantom data and implementing our algorithm so thatits performance can be evaluated and compared toalternative algorithms in current use.

VI. ACKNOWLEDGEMENT

This work was supported by the National Institutes ofHealth under grants RR01379 and RR01380 from theDivision of Research Resources and HL17646, Spe-cialized Center of Research in Ischemic Heart Di-seases, and by the National Science Foundation undergrant ECS-8215181.

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detector

X2 ( sseAx-/t<" i/ ~~~~system axis

Figure 1. Geometry for Single-PhotonEmission-Tomography