A New Method for Modeling the Spatially-Variant, Obj ect-Dependent Scatter Response Function in SPECT'

E. C. Frey and B.M.W. Tsui Department of Biomedical Engineering and Department of Radiology

The University of North Carolina at Chapel Hill

Abstract

Scatter compensation using iterative reconstruction results in improved image quality and quantitative accuracy compared to subtraction-based methods. However, this requires knowl- edge of the spatially-varying, object-dependent scatter response function (SRF). We have previously developed a method, slab derived scatter estimation (SDSE) for estimating the SRF. However, this method has reduced accuracy for nonuniform attenuators and T1-201 imaging. In this paper we present a new method which provides better modeling of the SRF for T1-201 SPECT, and should provide improved accuracy for nonuniform attenuators. The method requires 3 image space convolutions and an attenuated projection for each viewing angle. Implementation in a projector-backprojector pair for use with an iterative reconstruction algorithm would require 2 image space Fourier transforms and 6 image space inverse Fourier transforms per iteration. We observed good agreement between SRFs and projection data estimated using this new model compared to those obtained using Monte Carlo simulations.

I. INTRODUCTION In single-photon emission computed tomography (SPECT)

scatter results in a degradation of image quality and quantitative accuracy of the reconstructed images. Compensating for scatter during the reconstruction process has been demonstrated to give improved performance compared to methods that try to remove the scatter compensation prior to reconstruction. However this requires knowledge of the full, spatially varying, object-dependent scatter response function (SRF). The SRF can be estimated by integrating the transport equations using either Monte Carlo (MC) methods or by non-stochastic numerical methods. However, both of these methods require large amounts of memory to store the resulting SRFs for the case of fully three-dimensional reconstructions. As a result, it is desirable to calculate the SRFs on the fly during the reconstruction process. This estimation can be done using numerical integration techniques, but this is practically limited to estimating the single scatter component and requires a good deal of computation time. An alternative is to use approximate methods such as slab-derived scatter estimation (SDSE) [l, 21. While this method is fast and a very good a approximation for Tc-99m and for uniform objects, it does not work as well for non-uniform objects [3] and for isotopes such as T1-201. Finally, generalizing the SDSE method to converging beam geometries has proved challenging.

In this work we present an alternative method, referred to as effective source scatter estimation (ESSE), that meets some of

' This work was supported by National Cancer Institute grants R29-CA63465 and R01-CA39463. 0-7803-3534-1197 10.000 1997IEEE

these objections. The method is based on looking at the scatter estimation problem somewhat differently. Conventional methods try to estimate the scatter component in the projection data directly from the activity distribution. This new method first estimates an effective scatter source that is then projected using an attenuated projection operation. In this paper we present a derivation of a method for estimating such a scatter source. In order to do this estimation in reasonable time, the method makes several approximations. However, these approximations are less restrictive than those used in the SDSE method. We also present a verification of the model that suggests that these approximations hold quite well in uniformly attenuating objects.

The idea of an effective scatter source was previously proposed by Barrett's group at the University of Arizona [4]. In that work they proposed that the scatter component of the projection data could be estimated from the attenuated projection of an effective scatter source. This scatter source was estimated from the activity distributions first by convolving the activity distribution with a scatter kernel, then multiplying each point in this image by the electron density in the scattering medium. The scatter kernel derived was based on various approximations about the Compton scattering cross section and the probability of detection in a given energy window. One implication of these approximations is that the effective scatter source was the same in each projection view.

The new ESSE method described in this paper differs in several important aspects from the one described previously. First, there are no assumptions made about the cross section and the probability of detection in each window. As a result, there is a different effective scatter source for each projection view. In addition, several correction terms were added to account for the fact that the energy of the photon is different before and aRer scattering. The tradeoff for this increased accuracy is that there are two kernel functions that cannot be evaluated in close form and require numerical integration. In the following we present a theoretical derivation of the method, including a discussion of the underlying approximations. Subsequently we present a verification of the method based on MC simulation experiments.

11. METHODS

A. Theory The underlying problem is to estimate the probability that

a photon emitted at a position, 2, inside an object is scattered and detected at a position, t , on the face of the detector. This probability is the scatter response function, S( 2, ). For simplicity, consider the case of a perfect collimator that only accepts photons that are perpendicular to the detector surface. In other words, we assume that detected photons must be

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incident on the detector in a direction parallel to 6 , the unit vector perpendicular to the detector surface.

In the deriving the ESSE method, we will divide the computation of the scatter response in four parts relative to the position, Z‘, of the last scattering event in the object being imaged before the photon is detected. The geometry is shown in Figure 1. The four parts of the computation of the SRF are: (1) the probability of a photon emitted at X reaching X’via any possible path (including multiple scatter); (2) the probability that a photon will be scattered at X‘ in a direction perpendicular to the detector surface; (3) the probability that the scattered photon will reach the detector without being absorbed; and (4) the probability that the scattered photon will be absorbed in the crystal and not rejected by energy discrimination. The probabilities (l), (3), and (4) can be combined to form the product, ki(.?,Zf)p(2’). In this product, p ( i ’ ) is the density of electrons relative to the density of water and ki(X,X’) is the effective source scatter kemel. This kemel is the product of the probabilities (l), (3), and (4), assuming that the material that the at the last scatter point is water.

Source -f P P

Last Scatter Mtenuator Point

Figure 1. Schematic of geometry used in derivation of the scatter model.

Using this we can write an expression for the scatter response. The probability (2) is an attenuation factor from the scattering point to the detector. As a result, the scatter response can be written as:

~(2,;) = jjj p(n’)ki(x,x’)exp(-ps,;(x,x’)t(2’,n)) x (1)

S((2‘ -2’ .hi))&’

In Eq. (l), ps,i(X,Xf), the scatter attenuation coefficient kernel, is the appropriately averaged attenuation coefficient for all photons emitted at x’ and last scattered at 2’. The function r ( i ’ , i ) is the water-equivalent distance from the point 2’ to the surface of the phantom in the 2 direction. Finally, the delta function term in Eq. (1) picks out all final scatter points X’ that lie on the line ; +ah . Note that as yet, no approximations have been made.

Calculating the effective scatter source kemel requires evaluation of high order integrals. It is also different for each object and each point in an object. So, in order to develop a practical model, we make some approximations about the effective scatter source and scatter attenuation coefficient kernels. In particular, we will assume they are spatially invariant. Note that this approximation is true for single scatter in uniform convex objects, and for multiple scatter in

the interior of uniform convex objects when the multiple scatter points lie within the object. It will not be true for nonuniform objects and it is likely to have errors near the edge of objects. However, with this approximation we have that:

(2) S(x,T)=~~jlp(w’)~~(x-x’)exp(-p~,~(n-i’)t(x’,li)) x

S((E’-?’.rii))fd5’

We now use Eq. (2) to estimate s(;) the scatter component of the projection data at f due to the activity distribution u(2):

S ( ( 2 - 2’. iG))&’}u(i)&

Next we interchange the order of integration and factor out the primary-photon attenuation factor exp(-por(2’ - G)), where po is the attenuation coefficient for water. Then, the integration over X’ becomes an attenuated projection operation acting on the effective scatter source, us(?’). That is:

S(i) = PP,$ {+’)}(i-) (4)

where the effective scatter source is given by

as(?‘) = jjj p(?‘)ki (? - 2‘) exp( -Aps,i (2 - ?’)T(?‘, n))a(?)&, (5 )

and PP,$ {I is the attenuated projection operator:

P ~ , ~ { ~ ( z ~ ) } ( ? ) = {jJf(xf)exp(-po +f,ri))6((~# - ~ ‘ . r i i ) ) & ~ , (6)

and the relative scatter attenuation kemel is given by:

A/%,;(.?) = P&)- Po. (7)

The expression for the effective scatter source is almost a convolution, except for exponential term. However, if we expand the exponential in a Taylor series and keep the first 3 terms we have:

us(.?’) jjjp(x^‘)ki(.? -?‘)(l -Aps,i( .?-.?‘)t(?‘,n)+ * (8)

1 2 A,& (2 - 2 ’) t (2 ’, n))u(?)d?

Now, the integral over x’ can be expressed as 3D convolutions:

(9) a,(?’) 2 p(?’)(a(?)@ ki(?‘) - a(?) @ ( k ~ ( ? f ) A p ~ , ~ ( ? f ) ) ~ ( ? ~ , ~ ) +

+U(.) 8 ( kG (?)A& (?))2 (2 f , i))

where €3 represents 3D convolution. The ESSE model is described by Eqs.(4) and (9).

The steps in estimating the SRF are illustrated in Figure 2. First, The effective source scatter kemel and the difference in the effective source attenuation kemel are combined to give three kemels: ki (?), ki ( ~ ’ ) A P ~ , ~ (?), and ki(2’)A& (2’). These 3 kemels are convolved with the activity distribution and the resulting images are scaled at each point by 1, the effective source depth, and the effective source depth squared,

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respectively, at each point. Next, these three images ate combined with weights 1, -1, and 1/2, respectively, to give a single image. This image is then multiplied by the relative density, p(2') giving the effective scatter source. Note that for a uniform attenuator, this last multiplication has the effect of truncating the effective source to the region of non-zero attenuation. Finally, the attenuated projection of this effective scatter source gives the estimate of the scatter component.

Several observations can be made about this method. First, note that the effective scatter source and relative scatter attenuation coefficient kernels are different for every viewing angle. However, by rotating the attenuator and activity distribution to a common detector orientation, the same kemels can be used.

Second, the major approximation is that these two kernels are independent of source position. This is most likely to be a bad assumption near the edge of the phantom. At the edge we are using a kernel away from edges that would includes multiple scatter paths where photons are scattered out of the phantom and then back into it, a situation that can not exist in practice. As a result, we will investigate the accuracy of this approximation by verifying the method for sources extreme positions near the edges of attenuators.

Finally, the second approximation is in truncating the Taylor series of exponential. Clearly this will be worst where the source depth and the relative scatter attenuation coefficient kemel is large. However, for the photopeak windows investigated in this work, where the maximum detectable scatter angle is less than 90°, the relative scatter attenuation coefficient kemel is primarily non-zero in front of the source position. As a result, this approximation will be worst for large source depths and we will verify the accumcy of the model for such sources.

B. Computing the Kernels In order to use ESSE, we need to estimate the effective

scatter source kemel, k , and the scatter attenuation coefficient kernel, ps. This was done by modifying the SIMIND MC code [5] to make the necessary computations. For each photon reaching the detector, the photon's weight (including all interaction probabilities) and the weight divided by the attenuation factor from the last scattering point to the detector were computed. These were then summed in two 3D images, k and w , in the voxel containing the last scatter. This process was repeated for -lo8 photon histories. The phantom used in this calculation was a large water filled slab 40 cm thick and 200 cm wide. The effective scatter source kernel is the image k

and the scatter attenuation coefficient kemel, ps, obtained using:

can be

where i is the voxel index in the images and d, is the distance from the voxel to the surface of the phantom in the direction perpendicular to the detector surface.

C. Implementation of &SE Method The ESSE method was implemented by first rotating the

source and activity images so the rows are parallel to the detector surface. Next, the activity image is convolved with the three kernels using Fourier convolution. To avoid wrap-around effects, the image and kemels can be zero padded prior to convolution in both directions in the transaxial plane and independently in the axial direction. This convolution requires one Fourier transform of the activity image, and three inverse Fourier transforms, one for each term in the exponential expansion (the Fourier transform of the 3 kernels are precomputed). These three images are then multiplied by the appropriate power of the water-equivalent depth in each voxel and summed with the appropriate weights, giving the effective scatter source. An attenuated projector that models the distance- dependent collimator-detector response blurring was then applied to this effective source to give the scatter estimate.

D. Validation of ESSE The ESSE method was validated by comparing scatter

estimates computed using ESSE with those from direct MC simulations. The simulations included the effects of a low- energy high-resolution collimator and a detector with 10% energy resolution at 140 keV. The MC simulations were performed using the SIMIND MC code including up to fourth order scatter. Both Tc-99m and T1-201 isotopes were simulated. For Tc-99m, a 140 keV photopeak energy was simulated and for T1-201 five x-ray photopeaks with energies of 68.9,70.8, 79.8, 80.3, and 82.6 keV with abundances 27.2, 46.2, 5.5, 10.5, and 4.4%, respectively, were used. The higher energy gamma ray peaks were not simulated in this work.

To test the ability of the model to estimate the SRF we used cubic voxel sized sources with a side length of 0.64 cm at two locations in a 32x22 cm elliptical phantom 40 cm long. The first source position used was at a position (0.32,0.32,0), where the origin of the coordinate system was at the center of the phantom, the distances are measured in cm, the z-axis is along the phantom's axis, and the x and y axes are along the major and minor axes of the ellipse, respectively. The second

-. -. position was at ' Scatter (7.36,9.92,0) in same k ... - Estimate coordinate system, a

position very near the edge of the phantom. To test the overall accuracy of the model, we simulated an

Effective elliptical phantom that consisted of a 32~23.04 cm elliptical cylindrical attenuator 40 cm long and a 30.72x21.76 cm

A

P Scatter Source

Figure 2. Illustration of steps used in the ESSE scatter estimation method.

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voxelized elliptical emitter 0.64 cm long. Projection data were simulated into a 64x64 projection

matrix with pixels having a side length of 0.64 cm. For the ESSE model, the activity and source distributions were digitized into a 64x64~64 volume with size length 0.64, while for the MC simulation an analytically defined attenuator was used. In both cases the projection data were collapsed parallel to the phantom's axis to reduce the noise in the MC projection data. These data are equivalent to 2D simulations using a line- source geometry, but were performed in 3D. These simulations were repeated with projection angles of 0, 90, 180, and 270". For the T1-201 simulations, a dataset was generated using the SDSE method to investigate the accuracy and computational requirements of the two scatter models. These projection datasets were used to assess the accuracy of the ESSE method by comparing projection data normalized to account for the geometric efficiency of the detector (which is not modeled in the ESSE code) and scatter-primary rations (SPRs).

111. RESULTS

A. Comparison of Projection Data A comparison of the scatter component of the projection

data for Tc-99m and for the two voxel source positions and one view of the extended source is shown in Figure 3. Note that the only normalization that was done was to account for the eficiency of the detector system (i.e., no arbitrary normalization to the same total area was performed). In all cases, the ESSE model produces an estimate of the scatter component that agrees very well with the MC estimate. Figure 4 shows a comparison of the MC, ESSE, and SDSE estimates of the projection data for the voxel and extended sources for T1-201. While the agreement for ESSE is not as good for extreme source positions such as the one shown in the center graph of Figure 4, it is better than for SDSE; the agreement is very good for the extended source and the source near the center of the phantom. Table 1 shows a comparison of the SPRs for Tc-99m and TI-201 for the various source positions. The agreement in the SPRs correlates well with the agreement in the shapes. Especially for the extreme positions for TI-201, the ESSE method produces superior results to SDSE. In all cases, the ESSE method produces SPRs which are within 4.1% of the MC estimates, and the agreement is typically better than 1%.

B. Computation Time The time to compute the scatter estimate, excluding the

projection step and the collimator response blurring, was determined for both the SDSE and ESSE methods. The time for SDSE depends on both the pixel size, extent of the activity distribution, and isotope being modeled. For an elliptical cylinder 32x22~40 cm in a 64x64~64 volume with 0.64 cm pixels into 128 projection 64x64 projection images requires 2.1 minutes of processor time on a single CPU of a Sun Ultra 2200 workstation.

The computation time for the ESSE method depends only on the size of the activity image, the number of terms used in the exponential expansion, and whether or not zero padding is used. For the case of using all 3 terms and zero padding in all three directions, the ESSE method requires 130 minutes on the

same computer to produce the scatter estimate for the same number of projection views. The vast majority of the computation time is spent in the Fourier and inverse Fourier transforms. The computation time can be reduced by a factor of 29 if zero padding is not done, though this has the potential of introducing wraparound effects. In addition, the computation time can be reduced by a third if only two terms are used in the exponential expansion. It should be pointed out that little effort has been made to optimize the ESSE code, while we the SDSE code has previously been highly optimized. In particular, the fast Fourier transform code used to perform the Fourier Transforms is written is not highly optimized.

IV. DISCUSSION AND CONCLUSIONS

This paper has presented a theoretical derivation of the ESSE method for estimating the scatter component. This method differs from previous methods such as SDSE in that it first estimates an effective scatter source, then estimates the scatter component of the projection data via an attenuated projection. This method has several potential advantages. First, it is likely to be more accurate than SDSE for use in nonuniformly attenuating objects. Like SDSE, it can not model the effects of nonuniform attenuation distribution on the probability of reaching the last scatter point. However, it can accurately model the effects of nonuniform attenuation distributions at the last scatter position and along the outward path toward the detector. The SDSE method uses only approximate methods for these latter effects.

Similarly, for non-parallel geometries such as cone or f8n beam collimators, this method could be directly applied simply by replacing the parallel-beam attenuating projector with a suitable projector for the desired geometry. This would not provide an exact estimate, as it would not account for the change in the Compton cross section or post-scattering attenuation coefficient due to the difference in scattering angle for the parallel and non-parallel geometries. However, it does allow modeling the spatially varying sensitivity of the collimator at the position of the last scattering event. Previous studies have suggested that this latter effect is important in modeling the SRF for these collimators.

Finally, ESSE is likely to perform better for scattering to lower energy windows. This is because SDSE can not account for changes in the shape or magnitude due to attenuating material behind a source. While this is not a major limitation for Tc-99m, it is somewhat important for T1-201, where typical photopeak window can detect some backscattered photons, and for downscattering into lower, non-photopeak, energy windows. The ESSE method can, on the other hand, can account for these effects.

The validation of ESSE presented in this paper demonstrates that ESSE performs very well in simple uniform attenuators for both T1-20 1 and Tc-99m energies. Particularly for T1-201 and extreme source positions, ESSE performs better than SDSE. However, at present, ESSE carries a significantly larger computation burden. It is possible that, by a combination of approximations (such as reducing the number of terms in the exponential expansion) and improved convolution algorithms, The ESSE method could become fast enough to use as the scatter model in an iterative reconstruction algorithm.

108 15

- Y IUS .- s 4105 b

3 1u5 z 2105

3 1 1 0 5

9 0 100

a 9

.-

0 8 16 24 32 40 48 56 64 Projection Bin Number

Figure 3. Comparison of scatter component of in Tc-99m projection data for 2 voxel source positions and an extended source. For the voxel sources, the position of the source is indicated by a small white disk and the position of the attenuator by a gray ellipse.

-1.6 .; 1.4 $1.2

2 1 $ 8

.G 4

E 2

‘ 6

2 - 0

0 8 16 24 32 40 48 56 64

Table 1. Comnarison of Errors in SPRs for ESSE akd SDSE relative to MC

Source for ESSE for ESSE for SDSE Errors Errors Errors

V. REFERENCES

[l] F. Beekman, E. Eijkman, M. Viergever, G. Borm, and E. Slijpen, “Object shape dependent PSF model for SPECT imaging,” IEEE Trans Nucl Sci, vol. NS-40, pp. 31-39, 1993.

[2] E. C. Frey and B. M. W. Tsui, “A practical method for incorporating scatter in a projector-backprojector for accurate scatter compensation in SPECT,” IEEE Trans Nucl Sci, vol. NS-40, pp. 1107-1 116, 1993.

[3] E. C. Frey and B. M. W. Tsui, “Modeling the effects of composition and inhomogeneity of scattering media on the scatter response function in SPECT,” in Joumal of Nuclear Medicine, vol. 34, 1993, pp. 72P.

[4] A. V. Clough, “A Mathematical Model of Single-Photon Emission Computed Tomography,” : The University of Arizona, 1986.

[5] M. Ljungberg and S.-E. Strand, “A Monte Carlo program for the simulation of scintillation camera characteristics,” Comp Meth Prog Biomed, vol. 29, pp. 257-272, 1989.

Projection Bin Number Figure 4. Comparison of scatter component of in T1-201 x-ray photopeak projection data for 2 voxel source positions and an extended source. For the voxel sources, the position of the source is indicated by a small white circle and the position of the attenuator by a gray ellipse.

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