three-dimensional imaging in the positron camera using fourier techniques

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Three-dimensional imaging in the positron camera using Fourier techniques

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1977 Phys. Med. Biol. 22 245

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PHYS. MED. BIOL., 1977, VOL. 22, KO. 2, 245-265. @ 1977

Three-dimensional Imaging in the Positron Camera using Fourier Techniques

GILBERT CHU Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305, U.S.A.

and

KWOK-CHEONG TAM Lawrence Berkeley Laboratory, Berkeley, California 94720, U.S.A.

Received 29 illarch 1976, infinal form 26 July 1976

ABSTRACT. A mathematical algorithm for three-dimensional reconstructions in the positron camera is described. Fourier techniques have been adapted for use in analysing the data from cameras with limited detector configuration. Noise instabilities from random fluctuations in the data are discussed and treated. The technique is tested on a computer-generated phantom and the results are presented. Ways of incorporating the effects of Compton scattering and detector response into the algorithm are discussed. It is concluded that the method is feasible and practical for obtaining accurate reconstructions.

1. Introduction

The use of radioactive tracers for diagnosis in medicine has enjoyed a long history. Radioactive isotopes which undergo positron beta decay seem to be an especially effective probe for detecting and locating medical disorders.

Typically, the radioisotope is injected into the patient’s blood stream and then allowed to assume a quasi-equilibrium distribution in the tissue. A nucleus of the tracer can emit a low energy positron. (The usual energy maximum for commonly used tracers is 1-3 MeV.) The positron passes through the tissue, eventually coming to rest and annihilating with an electron, producing two photons, which emerge back-to-back in the centre-of-mass frame, each with an energy of about 0.51 MeV, e++ e-+ y + y.

Positrons with an energy of 1 MeV, in tissue of typical densities, will usually annihilate within 1 mm of the point of emission and the photons will be antiparallel to within 10 milliradians (Wallace 1960).

The photons may then escape from the patient’s body and be detected by the positron camera. If the camera detects two photons in coincidence, then it may be inferred that a positron was emitted somewhere along the straight line defined by the detector loci.

A number of positron cameras have been conceived and built. There are systems using arrays of sodium iodide crystals (Anger 1967, Burnham and Brownell 1972, Muehllehner 1975), using multiwire proportional chambers with lead converters (Lim, Chu, Perez-Mendez, Kaufman, Hattner and Price 1975,

246 Gilbert Chu and Kwok-Cheong Tam

Hattner, Lim, Swann, Kaufman, Perez-Mendez, Chu, Huberty, Price and Wilson 1976), and using multiwire proportional chambers with liquid xenon (Zaklad, Derenzo, Muller, Smadja, Smits and Alvarez 1972). Therefore, the positron camera has been shown to be a feasible tool for medical diagnosis. A schematic diagram of a possible detector configuration is shown in fig. 1. The camera is being used to image the radioisotope density distribution in a hypothetical phantom head.

Fig. 1. Schematic diagram of a possible detector configuration in which the camera is used to image the radioisotope distribution in a hypothetical head phantom.

For the purposes of this paper we shall assume that for each electron-positron annihilation event the two photons are antiparallel and are produced at the location of the emitter. Therefore, the data may be thought of as a set of straight lines embedded in %dimensional space. For any given event, the exact location of the emitter along the line cannot be determined, except that it is somewhere inside the patient’s body.

The great value of the positron camera lies in the fact that it is capable of measuring the density distribution of the tracer in 3-dimensions. For example, a brain tumour may absorb the radioisotope at a much greater rate than the surrounding tissue, and a positron camera could then determine the exact location of the tumour. Unfortunately the data gathered by such imaging devices do not correspond in a straightforward way to the density distribution. Metaphorically, the camera is capable of photographing a patient’s internal organs, but the picture must be developed. Some mathematical analysis must be performed on the data before the density distribution can be specified.

There is a rapidly growing literature on the mathematical techniques for %dimensional reconstructions (Cho 1974, Gordon and Herman 1974). There are a number of methods which may be applied to the particular case of the positron camera. Perhaps the simplest method is the technique of back projection, The volume to be studied is imagined to be divided into a series of parallel planes. Each event corresponds to a straight line, and the intersection points of that line with each of the planes is computed. The distribution of intersection points in each plane defines the back projection. An adaptation of this technique has already been used to show pictures obtained from human

Three-dimensional Imaging in the Positron Camera 247

patients (Hattner et al. 1976). Although back projection is simple, it is not an attempt to obtain the true solution for the density distribution. A tumour in one plane will always cast a shadow on all other planes, and, conversely, background from all planes will cast shadows on the tumour planes, thus smoothing out variations in tracer density. More sophisticated methods may be employed which are capable of finding the true solution and removing the shadow effect. I n subsequent sections we describe the stages in developing an algorithm for 3-dimensional reconstruction using Fourier techniques (Chu, LBL memorandum, unpublished), and report' on our results when the technique is applied to a computer-generated phantom.

2. Scalar field generated by st point emitter

Consider the idealized situation of a point emitter. The emitter will generate a set of straight lines passing through the origin, each of which is randomly oriented in 3-dimensional space. In the limit, where the number of straight lines becomes large, it is possible to define an associated scalar field which is a function only of position relative to the point emitter. Thus an emitter located a t ro generates a scalar field 4,(r - ro).

In fact, the scalar field can assume any one of a large number of forms. We shall give a number of examples.

Example A Take a volume element located a distance I r - ro I from the emitter, subtending

a solid angle 6Q, with thickness 6r (see fig. 2(a)). The volume is

6V = ~ r - r o [ z S r 6 Q .

emitter

la1 i b J Fig. 2. (a ) Volume element, subtending a solid angle a i l , with thickness ar, located at a

distance lr-r,,l from the point emitter: the wiggly lines indicate two back-to- back photons. ( b ) Volume element of radius 6r located at the site of the point emitter.

The probability that one of the lines generated by the emitter will pass through the volume element is 6Q/27. The line falls inside the volume element along a segment of length 6r. Define a scalar field 40(r - ro) associated with the point emitter as the mean line length per unit volume per event for randomly oriented lines emanating from r,, which pass through a volume element located


a t r . Then, in the far field limit, j r - ro 19 Sr,

For a volume element of radius Sr located at ro, the scalar field can be easily calculated (see fig. 2(b))

Thus, the scalar field is nonsingular for a finite volume element, but depends strongly on the size of the volume element. An accurate parameterization is given by

1 1 +o(r-- ro) = - 2nlr-ro/2+a2 (2)

where a = Sr/J3. Notice that as the dimension of the volume element vanishes, a -+ 0, the scalar field assumes the asymptotic form of eqn (1).

Example B Take an area element of area SA oriented perpendicular to the z-axis, at a

distance 1 - ro j from the emitter (see fig. 3). The probability that one of the lines will pass through the area element is

SA cos 6 2n/r-ro12

/' /

/ /

/

t z - axis

Fig. 3. Area element with sides of length a located a t a distance lr-rol from the point emitter, oriented perpendicular to the z-axis, at an angle 6 with respect to the z-axis.

in the limit where [ r - ro 1 is much larger than the dimension of the area element, a. Here we can define the scalar field +(r - ro) as the average number of lines per unit area per event for lines emitted from ro which pass through the oriented area element located at r . In the far field limit, I r - ro 19 a,

For an area element located at ro,

Three-dimensional Imaging in the Positron Camera

Thus an accurate parametrization is given by

249

As the dimension of the area element vanishes, a+ 0, the scalar field assumes the asymptotic form of eqn (3).

Example C It is possible now to construct an entire class of scalar fields. Define bo(r - ro)

as the scalar field computed in Example B multiplied by COS% 0, for any integer n. For l r - ro l$a ,

An accurate parameterization is given by

Thus, we have seen that a point emitter can generate a number of scalar fields using appropriate definitions for how it is to be measured. Each scalar field is a function only of position relative to the point emitter, and independent of the position of the emitter itself. The spectrum of possibilities provides us with the luxury of choosing the particular scalar field which will optimize our reconstructions. Each example in this section illustrates a legitimate scalar field, but there may be advantages in choosing one over the others. Example A, which requires the integrated line length passing through a cube, is more difficult to compute than Examples B or C, which only measure the intersection of a line with a unit area. In Example C, different choices of the integer n correspond to a large range of possible shapes for the scalar field, some of which are much more sharply peaked than others. In Example C, the scalar fields all possess a singularity (either a zero or a pole depending on n ) at 0 = 7 1 2 .

3. The reconstruction problem as a convolution

Suppose we have chosen some definition for how the scalar field +o from a point emitter is to be measured. Then, an arbitrary distribution of radioisotope described by a density p will generate a scalar field 4 given by the convolution,

+(r ) = /d3r '$o(r - r') p(r')

The field $o appears as the Green's function or transfer function for the integral equation. If there are enough events to provide good statistics, the data from the positron camera measure the field 4. The problem is to solve eqn ( 7 ) for the density distribution p , where both + and q50 are known.

Any physically realizable density is integrable, and for an appropriately chosen definition, both +o and + are also integrable. Therefore, we may take

250 Gilbert Chu and Kwok-Cheong T a m

the Fourier transform of both sides of eqn ( 7 )

@(P) = @ O M ) ( 8 )

where 0, Q0 and R are the Fourier transforms of 4, do and p respectively. The transform is defined as

@ ( p ) = d3r4(r) exp (2nip .r ) . i’ (9) The solution of the problem is now very simple in momentum space,

m ) = @ ( P ) / @ o ( P ) . (10)

The density distribution in configuration space can then be obtained by taking the inverse transform,

p ( r ) = j’d3pR(p) exp ( - 2nip.r) . (11)

Thus, the solution of the problem in its simplest form is quite straightforward, once it can be expressed as a convolution integral equation.

However, the problem is never this simple in practice. Kote that the convolution integral in eqn ( 7 ) is to be performed over all space. Therefore, a major complication arises if the detection apparatus does not subtend the full 477 radians of solid angle. For example, the detector may lie in two parallel planes on each side of the patient’s head as in fig. 1. There will be no record of those events for which either of the two photons does not intersect the detector. Thus the scalar fields as described in section 2 cannot be determined in a large region of space.

There is an elegant way to circumvent t’his difficulty simply by redefining the scalar fields. Usually the region to be scanned is characterizable by well- defined boundaries. I n a brain scan, the observed radioisotope density distribution may be confined to the head area by arranging lead shielding to absorb any photons which originate below the patient’s neck. The observed annihilation events will come only from that part of the density distribution p located within the head region. Events originating from other parts of the patient’s body will simply go unobserved. If the detectors are large enough to extend beyond the boundaries of the head, then for each point r in the head there is a ‘local cone of detection’, Those back-to-back photons originating from r which fall within the local cone will be detected. Those which fall outside t’he local cone will not be detect’ed. The size of t’he local cone of detection is clearly a funct’ion of position r . Define the universal ‘cone of detection’ to be the smallest’ loca,l cone of detection for all the points r in the head. Any photons emit’ted within the universal cone will be detected, and the size of the universal cone is obviously independent of position.

A schematic illustration is given by fig. 4. The universal cones at the points rl and r2 are indicated by the shaded areas. At point r2 the local cone is significantly larger than the universal cone, and the boundaries of the local cone are shown. At point rl the universal and local cones are identical.


Redefine the scalar field as follows. For all r , consider only those events which fall within the universal cone. For the present, we shall disregard all the other events, which may in fact include a substantial fraction of the total number of events recorded by the detector. The scalar field is computed by

Fig. 4. Two-dimensional schematic illustration of an oval head phantom placed between two parallel detector plates: at the point r2, we have drawn both the universal cone of detection (shaded) and the local cone of detection (unshaded, and containing the universal cone); at the point rl, the universal and local cones happen to be identical.

any of the methods described in section 2, except that only those events which fall within the universal cone are used. The convolution equation will still hold,

4 ( r ) = ~ ' c ~ r t + ~ ( r - rt) p ( r t ) .

Here, the Green's function is the scalar field of a point emitter restricted to the universal cone. The integration is still performed over all space. However the field 4 may vanish over some region of configuration space, because of the redefinition of the Green's function y50, and because of the restriction of the measured density distribution p to the head region. The need to measure 4 over all 4 r solid angle is abrogated simply because it vanishes in the regions not subtended by the detectors.

Notice that if the detector consists of two parallel plates, for example, then the scalar field will vanish €or large regions of configuration space. Define a polar coordinate system in which the z-axis is perpendicular to the plane of the detectors and passes through their centre. The angle 0 is the angle between a line and the z-axis. Then, the poles in the scalar fields described in Example C (for n < 0 ) will never appear since there can be no events oriented with 0 = ~ 1 2 .

4. Computation done on a lattice

Obviously, any numerical reconstruction of the density distribution of radioisotope must be done by dividing space into a lattice of discrete points and then using discrete Fourier transforms to obtain the solution. The continuous integrals of section 3 must be replaced by series representations with some finite cut-off.

10

252 Gilbert Chu and Kowk-Cheong T a m

Divide the region of space in the vicinity of the density distribution into a 3-dimensional lattice, N, x N, x NZ, with spacing between the lattice sites,

The scalar field can be computed by placing a volume element or an area element a t each of the lattice sites and then using any of the methods described in section 2.

The discrete Fourier transform Q, of the field C$ is expressed as a finite series,

S,, S,, 8,.

@ ( p ) = V 2 $(r)exp(2nip.r) N I - l

~

mr=O

where the cell volume V = S, S, S, and the

r = (m, S,, m, S,, m, 6,). The inverse transform is

variables are placed on the lattice,

nd = 0, Ni- 1 (13)

where N = N, NUNz. Of course, Q, and 4 are now both periodic, repeating in each dimension after each Ni lattice sites.

The lattice spacing is determined chiefly by four considerations: (a) the resolution desired for the reconstruction, (b) the need to minimize ‘aliasing’ (Brigham 1974) in the Fourier transform, (c) the detector configuration, and (d) noise problems arising from the fact that only a finite number of events are detected. The problem of noise will be considered in the next section.

In (a), it is clear that the lattice spacing must be smaller than the desired resolution for the reconstruction.

In (b), aliasing is completely eliminated if the lattice spacings S, satisfy the relations,

1 - >p(max), i = x, y, z 28,

where p(max) is the magnitude of the largest momentum for which @ ( p ) is non-vanishing. If the conditions (15) are not satisfied, then portions of @.(p) from adjacent periods will begin to interfere with each other, resulting in what is known as aliasing, In practice, it may be the case that @ ( p ) does not have a limiting momentum, and the best that can be done is to choose the lattice spacing small enough to minimize the effects of aliasing.

We have found that the detector configuration (c) is an important consider- ation in adjusting the relative size of the lattice in different dimensions. The ratio of lattice spacings S, : S, : S, should be chosen to minimize the effect of edge effects at the boundaries of the universal cone. Clearly, edge effects become less important as the lattice spacings are made smaller. But for a given size for the unit cell in the lattice, the optimum choice for the ratio of lattice spacings would be that for which the boundaries of the universal cone


are closely approximated by major diagonal planes in the lattice. In the 2-dimensional example of fig. 4, the optimal choice would be

S,/S, = tan ( Q / 2 ) (16)

Once the lattice spacing is fixed, the size of the array is determined by where 0 is the opening angle in the universal cone.

choices for N,, N,, NB. The array dimensions are given by

Ai = Nisi. ( 1 7 )

For a periodic function, if Ai are chosen equal to the period of the function, then the discrete Fourier transform will be free of ‘leakage’ (Brigham 1974). However, the field C$ is of non-compact support and manifestly aperiodic. Leakage must occur. The problem may be minimized by choosing Ni large enough such that for some small E ,

+(r)/C$max < E (18) for I ri 1 >Ai. Here, one is actually choosing some finite cut-off on what is formally an infinite integral in the continuous Fourier transform.

I n our computer simulations, we have experimented with detectors arranged in two planes as in fig. 1 where the opening angle Q is such that tan ( Q / 2 ) 1. We have found that for a given total number of lattice points, the reconstructions are best when the array dimension A, (along the line perpendicular to the detector planes) is considerably larger than A, and A,. Such a result is reasonable since the scalar field C$ from some finite density p falls most slowly in the direction of the detectors. I n many cases it may be necessary to choose A, to be larger than the distance between the detector planes.

It is possible to improve on a simple truncation by multiplying the field + by an appropriate ‘window function’ before computing the Fourier transform. Window functions can reduce the leakage significantly, with the result that spurious oscillations in the solution p are significantly reduced. We have experimented with a variety of choices for the window function, In each case, the net effect is to make the cut-off in the Fourier transform relatively smooth. One example is known as the Hanning function (Blackman and Tukey 1958).

W(.) = 0.5 + 0.5 COS (nx/A,), 0 6 X A, (19) w(x) = 0, x<O and x>A,.

A second example, the Hamming function (Blackman and Tukey 1958, Hamming and Tukey, unpublished), provides somewhat better results,

W ( X ) = 0.54 + 0.46 COS (nz/A,), 0 6 X Ax. (20) A third example, suggested by Blackman and Tukey (1958), yields even better results,

W ( X ) = 0.42 + 0.50 COS (rz/A,) + 0.08 COS (2nx/AX). (21)

After some investigation, we have obtained the best results using window functions in the shape of a Gaussian

w(z) = exp [ - (x - A,/2)2/~,2] (22)


with ax in the range 0.25A, to 0.40Ax. The full window function in 3-dimensions must be

W ( r ) = w(x) w(y) w(2). (23)

I n our experience, the choice of window function makes a noticeable contribution to the quality of the reconstruction, and the Gaussian of eqn (22) worked substantially better than the other examples mentioned.

5. Unstable solutions arising from noise in the data In the ideal situation where the number of counts is infinite, the scalar fielp + may be determined exactly and the problem can be solved as illustrated in

the previous sections, where the only 1imitat.ion on accuracy is the computation time required by the size of the lattice. In the limit, where the lattice extends to infinity and the lattice spacing shrinks to zero, the solution becomes exact. However, the number of counts must be limited by both physical and medical considerations. The dose of radioisotope administered to the patient should be small enough to minimize adverse side effects and large enough to provide enough data for a reasonable reconstruction. I n any case, there will be statistical variations in the measurement of the scalar field + at the lattice sites.

Although the presence of noise is inevitable, a careful examination of the problem may minimize its effects, and even provide reasonable reconstructions where none was possible with a naive approach. The first step is to note that the convolution equation we must solve ( 7 ) is essentially a Fredholm equation of the first kind:

+ ( r ) = f d3r’K(r, r ’ )p(r ’ ) (24)

where tho kernel K ( r , r ’ ) is given by the special form +,,(r - r’) . The equation is non-singular, since on a lattice, the integration limits are finite and the kernel is bounded.

It is well known that such an equation is unstable. This can be seen in the following way (Phillips 1962). Suppose p ( r ) is a solution to eqn (24). Consider a finite fluctuation in the solution, p,(r) = sin (mx). Then as m-tco,

+m(r) = 1 d3r’K(r, r ’ ) p,(r‘) + 0. (25)

Hence, an infinitesimal fluctuation in the scalar field + may produce a finite, non-vanishing fluctuation in the solution p . That is, the equation is unstable with respect to noise.

The instability of the equation places a limit on the resolution of the reconstruction. As the lattice distance is decreased, the accuracy of the reconstruction begins to improve, but at some point the accuracy begins to worsen. The critical point occurs when the lattice spacing grows small enough to admit terms in the Fourier series with frequencies high enough for eqn (25) to become valid.

The instability in eqn (25) will occur for fairly small values of m if the kernel K ( r , r’) is a smooth function of r‘. Conversely, the stability may be


improved by choosing a kernel which is sharply peaked. In the extreme case in which the kernel is a delta function, K ( r , r’) = 6(r- r’), there is no instability.

I n section 2, we discussed various possibilities for the kernel (or Green’s function) in the convolution integral. In the absence of noise, all the choices are equally valid. However, in the presence of noise, it is clear that greater stability for the solution can be achieved by choosing a scalar field with sharp peaks. Such peaks occup in the scalar fields discussed in Example C in section 2 . If the detector consists of two parallel plates, the coordinate system is oriented so that the x-axis is perpendicular to the plates. Then the scalar field will be peaked for large negative values of n. That is, events near the edges of the universal cone are given the greatest weight. (Large positive values of n produce a sharply peaked kernel in the x and y directions, but not in the x direction, thus degrading resolution in the z direction. In fact, as n- tm, we are left with a simple 2-dimensional projection down the x-axis.)

At first it may appear that as n is made more and more negative, the solutions will become more and more stable. However, there is a second balancing effect. As n becomes more negative, the cone of detection becomes effectively more restricted, and less of the data is used for the reconstruction. Events with small B are given less weight, and there is some point a t which the construction must begin to suffer. We have found that the choice n = - 3 seems to yield the best reconstructions. The quality of the reconstructions is not a strong function of the choice for n. For example, n = - 2, - 4, - 5 seem to work about as well as n = - 3 .

6. Removal of noise instabilities

Phillips (1962) has suggested a technique for dealing with the problem of noise which removes the instability in eqn ( 2 4 ) . His discussion applies to any non-singular Fredholm equation of the 6rst kind and therefore may be applied to our problem. However, we shall recast his treatment, since the convolution integral permits a particularly elegant realization of the technique. I n particular, it will be convenient to work in 3-dimensions and in the continuum rather than on a lattice.

Suppose the scalar field + ( r ) is known only up to some error ~ ( r ) . Then, the equation to be solved is

+(r)+E(r) =jd3r’+o(r-r’)p(r’) ( 2 6 )

where the error ~ ( r ) is an arbitrary function except for some condition on its magnitude, such as 1 ~ ( r ) I < N . Eqn ( 2 6 ) may be solved for p ( r ) by taking the Fourier transform, as outlined in section 3,

The functional derivative of p(r) with respect to &(r‘) is a function of r - r‘,

- = d3p 8s(r’) @o( P ) aP(r) exp [ - 2rip * ( r - r’)] = a ( r - r’) .


The error tends to be random from point to point, and it generates instabilities in the solution p(r) which are manifested as sharp fluctuations. Thus, a reasonable condition on p(r) is a requirement for smoothness, in which a solution is sought such that

/d3 r[V2 p(r)I2

is minimized. Suppose the total error is some fixed number e where

e2 = d3r[&(r)I2.

Then the smoothness condition may be re-expressed by introducing a Lagrange multiplier l /y and minimizing the following expression with respect to variations in &(r) :

[ d3 r[V2 p(r)I2 + - d3 r &(r)2.

It is clear that y must be non-negative if there is to be a meaningful solution. The functional derivative of this expression with respect to &(r’) gives an expression for the smoothness condition on the solution p(r),

.r (29)

Y ’1

subject to the constraint of eqn (29).

which is proportional to y , Integration by parts and use of eqn (28) yields an expression for the error

~ ( r ) = - y d3r’p(r’) V ’ 4 4 r ‘ - r) S S @&P)

(31)

where

V’40l(r‘-r)= ( 2 7 ~ ) ~ d 3 p L e x p [ - 2 x i p . ( r ’ - r ) ] . ( 32)

Substituting eqn (31) into eqn (26) gives a new convolution equation

d3r’[+,(r - r’) + Y V ’ ~ ( ~ ’ - r)] p(r’) = +(r ) . (33)

The solution for the density distribution in momentum space is simply

while the error is expressed in terms of the Lagrange multiplier

The ideal case in which there is no error in the determination of the scalar field corresponds to setting y = 0. Then eqn (34) reduces to the canonical


solution of the convolution equation given in eqn (10). Increasing the value of y increases the amount of smoothing in the solution p( r ) .

In the case where there is a finite error, an estimate for the appropriate value for y may be obtained iteratively. First choose some small value for y and compute R @ ) . Then using eqn (35) compute the corresponding error function and compare the result to the total error e expected from statistical fluctuations in the data. Only a few iterations are needed, since by eqn ( 3 5 ) , the total error e is roughly proportional to y .

From the form of the solution in eqn (34), it can be seen that the incorporation of this technique into the mathematical analysis is rather simple and straightforward. No extra Fourier transforms or scalar fields need to be computed for eqn (34). All that is needed is a simple recombination of the scalar field

We have found that the use of this technique is the single most important $0.

factor in improving the quality of our reconstructions.

7. Summary of the algorithm

1. From the configuration of the detectors and the dimensions of the region to be imaged, compute the universal cone of detection.

2. Choose the lattice spacings, a,, 6,, Ss, taking into account the desired resolution, aliasing, detector configuration and noise in the data.

3. Choose the array dimensions, Ai = Ni6,, to be large enough to make leakage unimportant.

4. Compute the scalar field from a point emitter, $o(r) , choosing the form which minimizes instabilities in the solution ; e.g. take n = - 3 in the definition for $o in eqn (6).

5 . Choose a window function, W ( r ) = w(x)w(y) w(z), which minimizes leakage; e.g. take w(x) to be a Gaussian as in eqn (22).

6. Compute the Fourier transform of $,,(r) W ( r ) , which we shall denote as Qo(p) . (All the steps up to this point need be performed only once for a given positron camera, with Q0(p) stored on a tape.)

7. Using the data collected by the camera, compute the scalar field $ ( r ) due to the density distribution using the same definition employed in step 4. Only data falling inside the universal cone are used.

8. Compute the Fourier transform of $(r) W ( r ) which we shall denote as @ ( p ) . 9. Compute the solution in momentum space,

where the parameter y is determined by the estimated level of noise in the data (see section 6). The point a is the central point in the lattice array, and the phase shift is necessary since the functions are defined on a lattice where r = 0 is located a t a corner and r = a is located in the centre of the array.

10. Compute the inverse Fourier transform of R @ ) , and finally obtain the reconstruction of the radioisotope density p(r) .


8. Computer solutions

In this section we shall discuss the results obtained by applying our algorithm to a computer-generated phantom. For convenience, we have chosen a relatively modest lattice of dimensions 32 x 32 x 32 to illustrate our results. In principle, a somewhat larger lattice can be chosen to obtain better resolution, if one is willing to pay for the extra computation time.

The Fourier transforms are performed using a Fast Fourier Transform program written by Singleton (1969), and the actual computing was done on a CDC 7600 at L.B.L. Two Fourier transforms are required for a reconstruction, and we found that for our lattice dimensions, the time required was 2-5 S per transform. In general, the time required for a lattice with M sites is roughly proportional to Mlog, M, so the time required for larger lattices may be estimated from our results.

For a given detector configuration, the scalar field for a point emitter +o(r) and its Pourier transform ( D o @ ) need be computed only once and then stored. Reconstructions for arbitrary objects can then be made using the stored values for O 0 ( p ) . Therefore, there is no need to economize on time spent in computing Q 0 ( p ) . We have found that the quality of the reconstructions depends strongly on the accuracy with which $o(r) and @,(p) are computed. In particular, for a detector configuration of limited solid angle, the scalar field $,,(r) has large variations near the origin and effects at the edge of the universal cone are important there. Since time spent computing @ , ( p ) is rather unimportant, it is an advantage to do it with care.

We have generated a set of data in the computer by Monte Carlo methods corresponding to a phantom, consisting of a ‘skull region’ corresponding to the blood supply for the brain, a ‘brain’ and a hypothetical ‘tumour’. The ‘skull region’ was a spherical shell 3 cm thick, with an inner radius of 18 cm and an outer radius of 21 cm, and with a relative radioisotope density, p = 1.0. The ‘brain’ was located inside the skull, with density p = 0.2. The ‘tumour’ was a sphere located off-centre in the ‘brain’, with a radius of 3.6 cm and a density p = 2.0. The relative radioisotope densities are similar to those found in actual patients (D. Chu, private communication). The phantom was permitted to generate a total of 818 785 positron emissions.

We chose a detector configuration of two parallel square plates. Relatively modest increases in the size of the plates were capable of producing significant improvement in the quality of the reconstructions. Alternatively, if the head is surrounded on four sides with detectors, there would be an even more marked improvement, while only doubling the detector area.

In this paper we shall show our results for a detector where the plates are 84.86 x 84.86 cm and 50 cm apart. For such a configuration 229 429 of the total events generated were detected by the camera.

The lattice distances were 2-5 x 2-5 x 5.0 cm, thus making the array dimensions 80 x 80 x 160 cm. We found it necessary to make the array considerably larger in the z-direction to optimize the reconstructions. Gaussian window functions were used, and we found that the best values for the


parameters ai in eqn (22) were 32, 32 and 40 cm. The scalar field was defined as in eqn (6) with n = - 3.

In fig. 5 we show the results of our reconstruction for two different values of the parameter y . In each case, only the central plane, passing through the

a

i

i

2

D

lo ) (6) Pig. 5. Reconstruction of the computer-generated phantom with noise for two different

values of the parameter y (only the central plane perpendicular to the z-axis passing through the centre of the tumour is shown) : (a ) y = 0, where no noise filter is used and the instability has left no trace of the phantom; ( b ) y = 2 x 10-8, where the tumour and skull region are now visible.

centre of the tumour is shown. The numbers indicate relative densities, where zero is indicated by a blank space. Fig. 5(a) corresponds to the application of our algorithm with y = 0, where no noise filter is used. It is clear that noise in the data is important enough to create such large instabilities that the reconstruction is useless. In fact, there are large regions in which the densities are negative. Fig. 5(b) shows the results for y = 2 x The improvement is dramatic, and the reconstruction is a reasonable facsimile of the actual density. Negative densities still appear, but they are relatively insignificant.

In figs. 6, 7 and 8, we show reconstructions displayed on a cathode ray tube, using a program which can differentiate 64 grey levels in the display. The density of dots is proportional to the density in the reconstruction. In each figure, we have shown a series of planes in the reconstruction, starting from the central plane and working outward in one direction.

Fig. 6 shows the actual density distribution being used to generate the data. This gives the reader a basis of comparison for the quality of the reconstructions.

Fig. 7 shows the results obtained by a simple back projection of the data, as described in section 1. Only those events lying inside the universal cone are used, to remove any bias as one moves out toward the edges. The tumour is clearly visible in the central plane, but the contrast has been severely degraded by shadows from other planes. Furthermore, the tumour has cast a recognizable shadow on at least two other planes, where it does not exist.

Fig. 8 shows the results obtained using our algorithm. Here we have set y = 5 x 10-5 which seems to give the best reconstructions. Computing with eqns (29) and (354, we find the corresponding value for the root mean squared


deviation to be e = 237. One may estimate e corresponding to the statistical fluctuations to be 2461. Thus the optimal value for y appears to be smaller than would have been expected, a phenomenon also observed by Phillips.

. . , ........ ........... ............. ............... ............... ............... ................. ................. ................. ............... ............... ............... ............. ........... .,..... .. .

...... ............. ............... ............... ................. ................. ................. ................. ................. ................ ................. .............. ............... ............ .. W......,. ......

, .amm+*we

. I x . ..*.*,. .......... ............. ............ ............. ............. ............. ............. ............. .***.*.**U. ......... .......

~ ' " " ' ' " ' " ' ' ~ " " " ' ' ' " ' ~ ( f l

Fig. 6. Actualnoise-free density distribution used t o generate the phantom by Monte Carlo methods: (a) shows the central plane passing through the centre of the tumour; (b ) , (c), (d), ( e ) , ( f ) show parallel planes, each successively displaced from the central plane by 5 cm; the 'skull region' is 3 cm thick, with inner radius of 18 cm, and relative radiosiotope density p = 1.0; the 'brain' has density p = 0.2; the 'tumour' has a radius of 3.6 cm and density p = 2.0.

The image of the tumour appears quite clearly, and shadowing onto other planes has disappeared. The improvement over the back projection is obvious, and comparison with the actual density is very favourable.


We found that, apart from the determination of the Green's function y$,(r), by far the most important factor in determining the quality of the reconstruction was the presence of noise. For example, when we applied our techniques to a

-4 . . . . . . .

. . . . . . . . . . . . . . . . . ( d l . . . . . . . . . ................. ................... ................... ..................... ...................... ~ . . , r**........sL~.*....'

.,.,I.*.. I~....Y*,*..L.. . . .......................

. . . , . .SO.O ............... ........................ ."ri*.onoa.m.~mos.4.-.-. ........................ . , ....................... ...................... ....................... . ?*."YIB..iD** *. . , .

. , " D a . * t l " . r . . . . . . . . I' " I . . r r Y * * m l l ? l . . r ' .

. . . . . . . . . ' : , ' : rl

. . . . ... . _ . , ... .... .... ....

. I .. ...

. . . . . . . . . . . . . . . . . . .. , . .

Fig. 7. Reconstruction of the phantom with noise by back projection; the planes in (a) to (f) correspond to those in fig. 6.

perfectly noise-free computer phantom, the reconstruction proved to be very close to the actual solution. The sharp discontinuities in the phantom were somewhat smoothed, but the fluctuations apparent in fig. 8 were not present. The choices of the particular window function, lattice size and lattice spacing were relatively unimportant as long as these choices were reasonable according to the criteria in section 4. On the other hand. once noise 'was introduced into


the phantom, it became necessary to employ the noise-filtering techniques of section 6 and even then spurious fluctuations as shown in fig. 8 arose in the reconstructions.

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

......... ............ ............. ............... L . . , ,

................ ................. ................ ................. ................. ................ ............... ............... .............. ............ .......... .... i ..I .... ..-.. . I ' I . . ..... . . . . . . ...... ...... ..... .".. .*. , . l

Fig, 8. Reconstruction of the phantom with noise by Fourier techniques setting 7 = 5 x to optimize the quality: the planes (a ) to ( f ) correspond to those in fig. 6 ; negative densities were insignificant and have been set equal to zero; the results are a marked improvement over the back projection in fig. 7.

9. Compton scattering

So far we have not mentioned experimental sources of background other than statistical fluctuations in the data. It has been found that a serious source of background arises from the possibility that a photon may undergo Compton scattering before emerging from the patient's body (D, Chu and


C. B. Lim, private communication). A second source arises from the possibility that a photon may fail to trigger a response at the point where it enters the detector, undergo Compton scattering inside the detector, and finally trigger a response a t some other point (Chu and Lim, private communication).

The significance of Compton scattering may be estimated by noting that the attenuation coefficient for 500 keV photons in water is about 0.097 cm2g-l. (The attenuation coefficient a t 500 keV is roughly the same in bone. It is 0.173 for hydrogen, 0.087 for carbon, nitrogen and oxygen, and 0.088 for calcium.) At 500 keV, almost all of the attenuation is due to Compton scattering. Thus, for 15 cm of water (a typical head size), the probability for Compton scattering is about 77%.

The problem of obtaining reconstructions from the real data which include Compton scattering is currently under investigation. We are optimistic about the prospects because the method of back projections applied to real data is capable of imaging brain tumours in human patients (Hattner et al. 1976). Thus all the problems associated with Compton scattering and a practical detector are not so severe that the information from the radioisotope distribution is lost.

The Green’s function for Compton scattering inside the patient is generally a function of position. However, consider a roughly spherical region like the head. If the point of annihilation occurs a t the centre of the head, both photons will traverse equal paths before emerging, with the sum of path lengths equal to the diameter of the head. If annihilation occurs toward the edge of the head, the path lengths of the two photons will be different. However, almost all annihilation events will occur inside the skull, and both photons will have to pass through the skull before emerging. Thus, the sum of the path lengths will still be roughly the diameter of the head, and the probability for Compton scattering will be the same in both cases.

Expressing this quantitatively, we assume that the two photons must travel distances x1 and x2, respectively, in the head, while the distance d = x1 +x2 is a fixed number independent of the annihilation point. Suppose that the effective attenuation coefficient for head tissue is given by a fixed number p. (The probability that a photon passes through tissue of thickness x is given by P = exp (-px).) Then the probability that both of the photons will emerge from the head without scattering or absorption is

P1P2 = exp(-px,)exp(-px2) = exp(-pd)

independent of the position of annihilation. The probability that either photon undergoes Compton scattering is therefore proportional to 1 - exp ( - pd), also independent of position.

We conjecture that the Green’s function will prove to be roughly independent of position in the head, when measured experimentally.

Similarly, if the detectors give a uniform response over their entire surface area, then the Green’s function for false triggering in the detector will also be independent of position. Edge effects may be removed by sufficient restriction of the universal cone.

264 Gilbert Chu and Kwok-Cheong T a m

We are then left with the familiar convolution in eqn (7) except that now the Green’s function $o(r) is the scalar field due to a point emitter in a patient’s head, as measured by a real detector. The subsequent algorithm is identical. The only difference is that the scalar field $,,(r) must be measured experimentally rather than calculated for the ideal case.

10. Conclusion We conclude that Fourier techniques afford an extremely attractive

means of obtaining 3-dimensional reconstructions with the positron camera. Experimental problems such as limited detector configuration, and statistical fluctuations in the data, may all be treated by the technique in a natural and elegant manner. The method appears to be simple, feasible and practical.

We are optimistic about the possibility of obtaining reconstructions from data with Compton scattering. Many other questions await investigation: the way in which various detector configurations affect the quality of the reconstruction; whether there are more effective means for dealing with noise in the data; whether iterative schemes utilizing all the data obtained by the camera are feasible; whether convolution methods instead of Fourier transforms are more efficient. There may be room for great improvement in obtaining reconstructions in the positron camera.

We wish to thank V. Perez-Mendez for suggesting this problem t o us and for his advice and enthusiasm during the course of our work. We extend our thanks to P. Concus for a discussion concerning the problem of noise, to B. Macdonald for generously making available the program for the Fast Fourier Transform and to R. Huesman who wrote the program for the CRT display, to D. Chu, C. B. Lim and V. Perez-Mendez for discussions concerning the experimental problems, and to 0. Buneman, R. Bracewell, L. Meissner and R. Huesman for discussions of mathematical problems. One of us (G. C.) thanks J. D. Jackson of the Theoretical Physics Group a t LBL and S. Drell of the Theoretical Physics Group a t SLAG for their support and encouragement while much of this work was being done.

The work was supported by the United States Energy Research and Develop- ment Administration.

R ~ S U M ~ Visualisation en relief dans la camera B positons utilisant les techniques de Fourier

positons. Les techniques de Fourier ont Btb adapt6es B l’emploi d‘analyse de donnees de cameras L’expose decrit un algorithme mathematique pour la reconstruction en relief dans la camera A

ayant une configuration de dbtecteur limitbe. On discute et on traite des instabilites de souffle provenant de fluctuations aleatoires dans les donnees. La technique est Qprouvee sur un fantBme engendre par ordinateur et l’expos6 en prbsente les resultats. On discute de divers moyens d‘incorporer les effets de la dispersion de Compton et de la reponse du detecteur dans les algorithmes. On conclut que la m6thode est faisable et pratique pour l’obtention de reconstructions prbcises.


ZUSAMMENFASSUNG Dreidimensionale Abbildung in der Positronenkamera aufgrund der Fouriertechniken

Beschrieben wird ein mathematischer Algorithmus fur dreidimensionale Rekonstruktionen in der Positronenkamera. Fur die Analyse von Daten aus Kameras mit begrenzten Detektor- konfigurationen werden Fouriertechniken herangezogen. Geriiuschschwankungen aufgrund von Zufallsfluktuationen in den Daten werden erortert und beseitigt. Das Verfahren wird an einem Komputerphantom uberpruft, und die Ergebnisse werden vorgelegt. Die Autoren erortern Wege, auf denen die Ausworkungen von Comptonstreuung und Detektorverhalten in den Algorithmus einbezogen werden konnen. Die Schlussfolgerung besteht darin, dass das Verfahren durchfuhrbar und praktisch ist, um akkurate Rekonstruktionen zu erzielen.

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three-dimensional imaging in the positron camera using fourier techniques

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