IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. ,41, NO. 4. AUGUST 1994 1585

Modeling the Scatter Response Function in Inhomogeneous Scattering Media for 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 We have previously developed a method for accurately

and rapidly modeling the scatter response function in uniform media. In this work we studied the extension of this method to nonuniform attenuators. Our approach was to use the water- equivalent source depth, i.e., the integral of the attenuation coefficient through the nonuniform object from the source to the surface divided by the attenuation coefficient of water. We have investigated the accuracy of three different methods based on this approximation using Monte Carlo (MC) simulation methods. Line sources were placed at the same water equivalent depth in slabs composed of water, bone, and lung. We observed that, for perfect collimation, the scatter response functions (SRFs) obtained for these slabs are comparable. To determine whether attenuation coefficients alone are sufficient to estimate the SRF or whether one must know the elemental composition, we simulated SRFs for sources in bone and lung as well as bone- and lung-equivalent materials. The bone- and lung-equivalent materials have the same attenuation coefficient as bone and lung, respectively, but the same elemental composition as water. For the lung- equivalent material, the SRFs were essentially equivalent to those from lung; for the bone-equivalent material, the SRFs were closer to those from water. We have also placed voids of various sizes in slab phantoms while keeping the same water- equivalent source depth. For these simple geometries two of the methods based on the effective depth were adequate for predicting the SRF. Finally, we have simulated the SRF for a line source in a realistic thorax phantom using several methods based on geometric and water-equivalent distances. The results indicate that one of the methods gives reasonably good agreement with direct MC simulations in terms of predicting the magnitude and shape of the SRFs and projection data for a complex distribution Simulating the heart. This method was not the same as the one that gave good agreement in the case of the simpler phantoms.

use of iterative reconstruction algorithms. This is especially true since it appears that these algorithms are the most likely candidates to be used in compensating for the effects of non- uniform attenuation. However, iterative reconstruction-based scatter compensation requires knowledge of the SRF at each point in the patient with respect to each projection angle; it is impractical both in terms of time and memory to compute and store the SRF using MC simulation methods. In this study we investigate both the factors affecting the SRF in nonuniform attenuators and methods for modeling it without the need for MC simulations for each patient.

These investigations were.carried out using MC simulation methods and phantoms and materials designed to separate the various factors. The factors investigated include the effect of attenuator elemental composition, density, and nonuniform attenuation distribution. Based on these studies, we have extended a previously developed scatter model for estimating the SRF in uniform objects to nonuniform attenuators. The methods for extending the model are based on the use of water-equivalent distances instead of geometric distances when computing the parameters needed for the scatter model.

We have investigated the accuracy of a family of methods based on this concept by comparing MC simulated projection data for both point and extended source distributions in an object with an attenuation distribution simulating that in the thorax. In the remainder of this paper we describe the motivations, methods, and results of these studies.

11. BACKGROUND A. Scatter Compensation in SPECT

In SPECT, scatter compensation methods can be divided into two classes: subtraction-based methods that attempt to estimate and subtract the scatter component of the projection data [l-81; and restoration methods that attempt to restore scattered photons to their place of origin [9-191. The former have the advantage that they are quick and simple. However, the subtraction process gives rise to increased image noise. The restoration methods include restoration filtering and iterative reconstruction methods. Filtering methods implicitly assume that the scatter response is stationary-an especially poor assumption in the thorax. On the other hand, iterative reconstruction-based methods have the potential of compensating for the spatially varying SRF without the noise increase seen with subtraction-based methods [20], [21] and give higher signal-to-noise ratios than reconstructions without scatter compensation [22].

B. Need for Scatter Models Scatter compensation with iterative methods requires

knowledge of the full SRF. In other words, we must know the probability that a photon emitted in each source voxel is scat- tered and detected in each projection bin for each projection view

I. INTRODUCTION

In cardiac SPECT scatter can give rise to loss of quantitative accuracy, contrast, and the introduction of subtle artifacts. This is especially true if there is substantial activity in nearby organs such as the liver. This has important implications for several of the new Tc-99m based myocardial perfusion agents where bridging artifacts between the heart and liver are sometimes observed. Unfortunately, compensating for scatter in the chest region is complicated by the presence of nonuniform attenuation distribution.

One promising method for compensating for scatter is the

'This work was supported by a grant from the Whitaker Foundation and the National Cancer Institute under Grant #I ROlCA39463. Its contents are solely the responsibility of its authors and do not necessarily represeint the official views of the Whitaker Foundation or the NatiPhal Cancer Institute. 0018-949!)~4$04.00 OY994 IEEE

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One method for estimating the full SRF is by direct application of MC simulation methods [23]. However this has several difficulties. First, computing the full SRF will require extraordinary amounts of computer time, especially for three- dimensional (3-D) reconstruction. For example, suppose we are able to estimate the point source response for a single source voxel into a single projection view in 1 sec (a very optimistic assumption with present technology). Then, to estimate the full SRF for a 64x64~64 reconstruction from 64x64 projections at 64 angles would require 644 sec = 190 days. This computation must be repeated for each patient. In addition, storing the full SRF requires a great deal of memory. For the reconstruction i n the previous example we would require 646 = 64 billion elements.

A final difficulty is that the SRFs computed using MC simulations are accurate only to the extent that the MC simu- lation models the full propagation and detection process. It is difficult to determine all the parameters necessary in order to make the simulation accurate. For example, parameters such as the exact energy resolution and window have important effects on the shape and magnitude of the SRF.

As a result, in order to hope to use iterative reconstruction for scatter compensation we must be able to rapidly and accu- rately estimate the full SKF during each iteration. In addition, it would be advantageous if the parameters necessary for this estimation could be easily extracted from experimental measurements.

There has been a great deal of effort to characterize the SRF in uniform media 1124-271. However, there have been relatively few efforts to study scatter in nonuniform media [28], [29]. In this paper we have isolated and studied the factors affecting the S RF in inhomogeneous objects. In addition, we have investigated the extension to nonuniform attenuators of a previously developed method for modeling scatter in uniform media.

C. Modeling Scatter in Uniform Media A scatter model for uniform media has been independently

developed by two investigators. While these are conceptually the same, there are differences in implementation. The model described in [30], [19], slab derived scatter estimation (SDSE), is based on a p(irameterization of a line source in a tub of water with infinite lateral extent (a slab phantom) as a function of the source depth based on MC simulations. The method described in [31] is also based on slab phantom SRFs, but these were estimated using experimental measurements. We have subsequently collaborated to demonstrate that all the parameters required for the scatter models can be determined based on simple experimental measurements using a triangular phantom [32].

In both these methods the asymmetric, object-shape dependent SRF for a given point source location and point in the detection plane is estimated using: the distance (offset) from the projected source position to the point; the distance (depth) from the line pariillel to the collimator passing through the point; and a table of slab phantom SRFs. The necessary computations can be performed in 2-D (assuming the sources and attenuator are constant in the axial direction) in less than 3.5 seconds per iteration on a DEC 3000-600 for the reconstruction of a 64x64 image from 64 projection bins at 64 angles. Three dimensional reconstruction requires longer

times, on the order of 270 minutes per iteration (or 255 seconds per iteration per slice) on the same computer to reconstruct a 64x64~64 volume ,from 64x64 projection images at 64 views.

D. Modeling Scatter in Nonunifom Media In nonuniformly attenuating objects several factors make

the estimation of the SRF more difficult. In uniform objects we need only know the positions of the boundaries. However, in nonuniform objects the SRF depends on both the elemental composition and the density at each point in the object. This significantly increases the complexity of the problem. In this paper we investigate approximate methods based on water- equivalent distances.

In. METHODS A. Monte Carlo Simulation

In order to allow separation of scattered and primary pho- tons we performed this investigation using MC simulation techniques. In particular, we used the SIMIND MC code [33] . The simulations were performed using a primary photon energy of 140 keV, an energy window of 126-154 keV, and an energy resolution of 10%. Multiple scatter was included in the simulation up to an order of 6. For simulations of the slab phantoms (including the vertically and horizontally layered slab phantoms described below), a collimator/detector with essentially perfect resolution was simulated in order to eliminate the influence of the geometric response on the shape of the SRFs. For the simulations in the thorax phantom described below, a low energy, high resolution (LEHR) collimator and an intrinsic resolution of 4 mm were used. This collimator was used for this phantom in order to more realistically simulate clinically realistic imaging situation. In order to improve the efficiency of the MC simulations, we performed these studies in two-dimensions (2-D). That is, all source and attenuation distributions had infinite axial extent.

B. Water-Equivalent Distance and Material The methods used to extend SDSE to nonuniform objects

are based on the water-equivalent distance between two points in an image. This is defined by: water-equivalent distance =

integral of attenuation distribution w.r.t. distance attenuation coefficient of water

To investigate the effects of elemental composition of materials on the SRF we used materials having the same elemental composition as bone and lung. The elemental composition of these materials approximating bone and lung were taken from [34]. We also used bone- and lung-equivalent materials. These materials have the same elemental composition as water, but have a density such that the attenuation coefficient is the same as bone and lung, respectively. For each of these materials the density is given by the ratio of the attenuation coefficient for the material divided by that for water. The densilies for the bone- and lung-equivalent materials at 140 keV were 1.85 cm-3 and 0.26 ~ m - ~ , respectively.

C. Phantoms Used

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D. Scatter Models Investigated

Slab phantoms composed of a line source imbedded in a slab of bone, bone-equivalent, lung, lung-equivalent and water were used to investigate the effects of attenuator composition and density. For these phantoms water-equivalent source depths of 2, 5, 10, 15, and 20 cm were simulated.

In order to investigate the effect of nonuniform density distribution, we used water-filled slab phantoms with voids introduced in two ways. The first phantom, referred to as a horizontally layered slab phantom, consists of two slabs of water 6 cm thick separated by a void (zero density) with thicknesses of 0, 1, 2, 4, and 8 cm. A schematic of the horizontally layered slab phantom is shown in Figure 1 (a). The second phantom, referred to as a vertically layered slab phantom, is a slab phantom with a void introduced by cutting the phantom and offsetting it in a direction parallel )to the collimator face. Void widths of 0, 1, 2, 4, and 8 cni at a distance 2 cm from the line source were investigated in a slab with a source depth of 12 crn. A schematic of the vertically layered slab phantom is shown in Figure 1 (b).

Finally, to investigate the effects in complex nonuniform attenuation distributions we used a thorax phantom. The attenuation distribution for this phantom was derived from a patient CT study. In this attenuation distribution we simulated both line sources and an extended source distribution simulating a myocardial perfusion study. Both the source and attenuation distributions are depicted in the images in Figure 2. The line sources were chosen to be in approximately the center of the object (position A), mediastinum (position B), and left lung (position C). The phantom was simulated assuming that it was composed of water with varying densities (the elemental composition was not used).

"& Detector (b'& Detector

Figure 1 . Schematics of the (a) horizontally and (b) vertically layered slab phantoms.

Figure 2. Images of cardiac-chest phantom. The left image shows the attenuation distribution and the right image shows the activity distribution simulating a myocardial perfusion study. The three line source positions used in this sludy are labeled in the attenuation image as A, B, and C.

We investigated three methods for extending the SDSE method. Consider Figure 3 which illustrates the parameters used by the various methods to estimate the number of scattered photons measured at point P on the projection image for a source at the position indicated. One parameter required for SDSE is the perpendicular distance to the surface of the phatom, d , from the line, A?, passing through the source parallel to the collimator and the surface of the phantom. All three methods are based on substituting the water-equivalent distance for this geometric distance. The methods differ in how the offset, defined as the distance from the projected source position, d t o the point of interest, P, was computed: method 1 used the geometric distance; method 2 used the water-equivalent distance along k, and method 3 used the water equivalent distance averaged over the shaded region in Figure 3. This region is a rectangle bounded on the sides by the lines which are perpendicular to the detector and passing through the source and the point where the value of the response function is to be determined. The top and bottom are bounded by the lines parallel to the collimator and passing through the source and the edge of the phantom at the line projecting to the point of interest.

In method 2 the offset distance, t?, is determined based solely on the attenuation distribution on Line 1. With this method or method 1, the presence of a lung in the shaded region in Figure 3 would have no effect on the estimated SRF. Method 3 was investigated because the attenuation distribution inside the shaded region should have some effect on the SRF. The shaded region in Figure 3 completely contains the region through which all scattered photons with detectable energies must travel. When the number of electrons (integrated electron density) in this region decreases, the probability of scatter or absorption in the region will decrease. Thus, more photons will pass through the region and have the possibility of scattering on the line d in Figure 3. Thus there should be an increase in the SRF for this offset. Method 3 represents a simple method of attempting to incorporate this effect into the extended SDSE method.

1 Point 6 Paint s

nuator

Figure 3. Schematic showing region used to compute the average water-equivalent distance for use as the offset in method 3.

E. Criteria for Comparing Scatter Models In the following we compare both the magnitude and shape

of the SRFs for the various phantoms and methods for extend- ing SDSE. Magnitude comparisons are made in terms of the values of the scatter-to-primary ratios (SPRs). For the line source objects in the slab and layered slab phantoms, shape - comparisons were made using scatter and total line source

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response functions (SLSRF and TLSRF, respectively). In these comparisons the response functions were normalized SO they represent the same number of emitted photons.

In the simple phantoms we compared MC SLSRFs in phantoms with different compositions and densities. Finally, when the SLSRFs for the slab phantoms are plotted the horizontal coordinates were scaled by multiplying the geometric distance times the ratio of the attenuation coefficient for the matenal in the phantom divided by that for water. This has the effect of scaling the horizontal coordinate so that it represents the water-equivalent offset from the projected source position. Agreement in SLSRFs compared in this way indicates that the SLSRFs could be predicted for these phantoms using SDSE extended using methods 2 and 3.

Using the cardiac-chest phantom shown in Figure 2, we compared the total line source response functions (TLSRFs) and projection data computed using MC simulation with that from the SDSE projector extended for nonuniform attenuators using each of the three: methods described above. In both cases data from the projector and MC simulation were normalized to have equal area under the unscattered component (the total areas were not equal). This was done to account for the sensitivity of the imaging system which is not modeled by the projector, but still show differences resulting from errors in estimating the scatter portion of the data.

rv. RESULTS

A. EfSect of Attenuator Composition The effect of attenuator composition was studied by com-

paring SRFs for line sources at the same water equivalent source depth in bone and bone-equivalent and lung and lung- equivalent. For these mlaterials the physical thickness is the same, but the composition is different. Figure 4 shows a com- parison of the SPRs as a1 function of depth. Note that for bone there is approximately a 5% discrepancy in the SPR at all depths as compared to bone-equivalent. For lung and lung- equivalent the SPRs are almost identical. As demonstrated in Figures 5 and 6, the shapes of the SLSRFs followed a similar trend: there is a slight discrepancy for bone versus bone- equivalent but good agireement for lung versus lung equiva- lent. This result is not surprising in light of the fact that the el- emental composition of lung is close to that of water and the relative importance of Compton scatter and photoelectric ab- sorption should be the same. However, for bone the presence of higher atomic number elements such as calcium results in increased probability of photoelectric absorption of the scat- tered photon in bone compared to bone-equivalent. This is because the photon has a lower energy after scattering and, because of the higher atomic average atomic number, the chance of photoelectric absorption after scattering is higher in bone.

B. Efiect of Attenuator Density The effect of attenuator density was studied by comparing

SRFs for sources at the same water-equivalent depth in bone- equivalent, lung-equivalent and water-filled slabs. In this case the composition is the same and the different densities result in a different physical thickness. One would expect, for a perfect collimator, that the SRFs are the same for the same water equivalent source depth if the horizontal axis is also

scaled by the ratio of the attenuation coefficients. This is because the depth and offset are, in this case, the only geometric parameters that affect the SRF. Thus, the change to a new density in the attenuator combined with rescaling the horizontal and vertical axes is simply a change of scale which should have no change on the SRF. This is just what we observed. In Figure 7, we show that the SPRs for bone- and lung-equivalent and water are essentially identical. In Figures 8 and 9 we show a comparison of the SLSRFs for source depths of 5 and 15 cm. Again, they are essentially identical. The results of these experiments would be predicted by methods 2 and 3, but not by method 1. Also note that, these

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Figure 4. Comparison of SPRs as a function of water-equivalent source depth in bone, bone-equivalent, lung, and lung-equivalent to illustrate the effect of attenuator composition. The curves for lung and lung-equivalent are essentially identical.

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Figure 5. Comparison of SLSRFs for a water-equivalent source depth of 10 cm in bone and bone-equivalent materials.

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Water-Equivalent Offset (cm)

Figure 6. Comparison of SLSRFs for a water-equivalent source depth of 10 cm in lung and lung-equivalent materials.

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results can also be used in the case of non-ideal collimators. In this case we would compute the SRF for the ideal case using the water-equivalent depth and offset, then convolve the SRF with the collimator-detector response function computed based on the geometric (not water-equivalent) distances.

........- Bone-Equiv. - - - Lung-Equiv.

0.8 0 0.6 a

0 5 10 15 20 Water-Equivalent Source Depth (cm)

Figure 7. Comparison of SF'Rs as a function of water-ecluivalent source depth in water, bone-equivalent, lung-equivalent to illustrate the effect of attenuator density. -

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Figure 8. Comparison of slab SLSRFs for a water-equivalent source depth of 5 cm in water, bone- and lung-equivalent.

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Figure 9. Comparison of slab SLSRFs for a water-equivalen,t source depth of 15 cm in water, bone- and lung-equivalent,

C. Eflect of Simple Nonuniform Attenuator Distributions The next factor that affects the SRF is nonunifonmity of

the attenuation distribution. To investigate this for a simple distribution we used the horizontally layered slab plhantom described previously for void thickness of 1, 2, 4, and 8 cm. As shown in Figure IO, we observed that the SRFs for void thicknesses less than or (:qual to 4 cm are very similar.

Similarly, the SPRs, as shown in Table I , differ by less than 2%. However, for the 8 cm void thickness there is a larger discrepancy in the shape and the SPRs differ by 5%. In particular. at 8 cm the SRF is slightly smaller in the central region and slightly larger in the tails. All three methods for extending SDSE to nonuniform objects would predict that the SPRs are the same. Thus we see that the accuracy of these methods will be poorer when there are relatively large regions where the attenuation coefficient is significantly different than that for water.

The second simple nonuniform attenuation distribution investigated is the vertically layered slab phantom. The SLSRFs for this phantom with void widths of 0, 1 , 2,4 , and 8 cm are shown in Figure I 1 .

The first method for extending SDSE to nonuniformly attenuating objects (water-equivalent depth, geometric distance fix offset) predicts that the SRFs measured for the vertically layered slab phantom be equal to the SRFs for a void width of 0 cm, except that they would be zero in regions over the void. Figure 11 shows that this is not quite the case: in the region to the right of the void the SLSRFs are slightly higher than the curve for zero void thickness. There is also some disagreement in the SPRs, as shown in Tdble 2. In that table, the third column represcnts the SPR predicted by method I . Again, there is some discrepancy with the largest being approximately 6% for a 2 cni void width.

Methods 2 and 3 for extending SDSE both predict that the SRFs for the vertically layered slab phantom are those obtained by offsetting the SRF for zero void width by the thickness of the void. Figure 12 depicts the resulting SLSRFs compared with the MC simulated S U R F S . Notice that for small void widths the discrepancy is small, hut for larger widths the discrepancy is rather large. A comparison of the SPRs, shown in the fourth column of Table 2, makes this clear. The discrepancy at 8 cm is 15%, while the discrepancy at 2 cm is about 2%.

._

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0.006 g 9 0.004

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0 - -20 -15 -10 -'5 0 5 10 15 20

Offset (cm)

Figure IO. Comparison of SLSRFs for line sources i n a water filled horizontally layered slab phantom with various void thicknesses. In all cases the water equivltlent depth was 12 cm

Table 1. Scatter/Primarv ratios for various void thickness in horizontally layered slab-phantom.

Void thickness (cm) I SPR for Layered Slab Phantom 0 I 0.75 1

I 8 I 0.7 1 1 I

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Offset (cm)

Figure 1 1 . Comparison of half of the SLSRF for line sources in the water filled vertically layered slab phantom for various void widths. In all cases the water equivalent source depth was 12 cm.

0 5 10 15 20 Offset (cm)

Figure 12. Comparison of SLSRFs for vertically layered slab phantom with those that would he predicted using the water- equivalent distance as the offset (SLSRFs labeled with w.e.d.) for various void thicknesses.

Table 2. Scatter/primary ratios for various void widths in the vertically layered slab phantom compared to those that would he predicted by the geometr c (method I ) and watcr-equivalent (methods 2 and 3) distance offset models.

i methods 2 & 3, SPR from MC SPR predicted by SPR predicted by

method I

D. Evaluation of Methods for Extending the SDSE Scatter Model The above results suggest that, for small lung thickness, it

should be possible to extend SDSE based on one of the methods described previously. However, it is unclear which of the methods we would cxpect to give the best results. The good agreement in Figures 5-9 might lead us to expect that methods 2 or 3 would give the best result since thcy would compute the offset, for a uniform phantom, using the water- equivalent distance (recall that the horizontal coordinates of the SLSRFs are scaled by the attenuation coefficient). Method 1, on the other hand, would not predict these results since it

Table 3. Errors (in percent) in the SPR for the projector using method 1 (geometric distance offset) relative to the MC simulation

5 . 0 0 4 p il o o o L , 3 2 c f f l : . . . : ' . ' : ' . . : . .- ?+lzmErT"~x

B 0 8 1 6 24 3 2 40 48 5 6 6 4

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Figure 13. Comparison of total projection for the thorax phantom and the extended source distributed computed using MC simulation and method 1 (prjl), method 2 (prj2), and method 3 (prj3).

uses the physical distance as the offset. For the horizontally layered slab phantom all the methods are equivalent and would predict that all the SLSRFs in Figure 10 and the SPRs in Table 1 are the same. While this is not exactly true, it is a reasonable approximation for small, clinically relevant void thicknesses. However, for the vertically layered slab phantom we see from Figures 1 1 and 12 that method 1 works better for large vertical void widths than methods 2 and 3.

To investigate which if any of the methods are more accurate for complex attenuation distributions we computed projections for both the line sources and the extended distribution in the thorax phantom using MC simulation and projectors based on the three methods. We observed large discrepancies in both the SPR and the shape of the projection data for the projectors based on methods 2 and 3. Method 1, which uses geometric distance as the offset, gave the best results. These observations are illustrated by the errors in the SPRs for method 1 in Table 3 and the comparisons for all three methods of the projection data (Figure 13) and the TLSRFs (Figures 14-16). This is just the opposite of what we would expect based on the results for the slab phantom geometries.

V. DISCUSSION

We have observed that, at 140 keV, we can predict the SRFs with reasonable agreement based only on knowledge of

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Figure 14. Comparison of the 'TLSRFs plotted on a log scale for a line source at position A (as shown above) in the thorax phantom computed using MC: simulation and projectors based on SDSE extended by method 1 (prjl), method 2 (prj2), and method 3 (prj3).

Bin Number

Figure 15. Comparison of the TLSRFs plotted on a log scale for a line source at position B (:as shown above) in the thorax phantom computed using MC simulation and projectors based on SDSE extended by method 1 (prjl), method 2 (prj2), and method 3 (prj3).

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Bin Number

Figure 16. Comparison of the 'I'LSRFs plotted on a log scale for a line source at position C (as shown above) in the thorax phantom computed using MC simulation and projectors based on SDSE extended by method 1 (prjl), method 2 (prj2), and method 3 (prj3).

the attenuation coefficients. This is because at this energy Compton scattering, which depends only on density and not

I I I I I

Figure 17. Schematic showing the geometric effect of approximating the SLSRF for the horizontally (A) and vertically layered (C) slab phantoms based on extended SDSE (B and D).

on elemental composition, dominates both the scattering and attenuation of photons.

Second, we have observed that, for objects with clinically relevant sizes, we can predict the SRFs in uniform objects of different densities simply by scaling the horizontal distances and thickness of the object based on the attenuation coefficient. Again this is reasonable in light of the physics of the situation. Changing the density combined with scaling the horizontal and vertical distances is equivalent to a change of scale and should not change the SRFs.

From the horizontally layered slab phantom we observed that, for small void thicknesses, the SRFs can be approximated with good accuracy by a slab phantom, equivalent to using SDSE extended by methods 2 and 3. For large thicknesses this approximation is not as good. The reason for this change can be understood by considering the geometric factors affecting the SRF. Figure 17 (A) and (B) shows the underlying assumption made when approximating the horizontally layered slab SRFs using methods 2 and 3. This approximation is equivalent to assuming that the scatter probability along photon paths a and b in Figure 17 (A) are the same as along a' and b', respectively, in Figure 17 (B), respectively. There are several reasons why this approximation is not exact. First, by moving the interaction point closer to the source we increase the flux of photons at the scattering point. Second, we increase the scattering angle. This changes both the scattering cross section and the energy of the scattered photon. Since the scattering angle is increased in the slab phantom, the energy of the scattered photon is greater so the attenuation on the outward path is smaller. In addition, the higher energy photon has a smaller chance of being rejected by the energy discrimination circuitry. Finally, the photon in the slab phantom (without the void) has to travel through a larger thickness of scattering medium in before it is scattered, thus decreasing the probability that a photon will reach the scattering site.

These geometric factors can help explain the observations made about the accuracy of the methods for extending SDSE. For large offsets from the projected source position (e.g. the photon paths labeled b and b' in Figures 17 (A) and (B)), elim- inating the void results in a larger increase in the path in the attenuating medium than for small offsets (e.g., the photon paths labeled a and a'). Thus we would expect that at large

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offsets and void thicknesses the slab phantom SRF would underestimate that of the layered phantom. Indeed this is what is observed in Figure 10. However, for small offsets and large void thicknesses, the first factor must dominate and we see that the slab SRF overestimates that of the horizontally layered slab.

In the vertically layered slab phantom we saw that the SRF was better predicted by the water-equivalent offset for small void widths and the geometric offset for large widths. Again we consider the geometric factors resulting in this observation. Figure 17 (C) and (D) shows the effect of the approximations used to extend SDSE. Figure 17 (C) shows the path of a scattered photon, c, in the vertically layered slab phantom. In method 1 we approximate this using path d’ in the slab phantom (same water-equivalent depth, same geometric offset) shown in Figure 17 (D). In methods 2 and 3 we approximate path c by path c’ (same water-equivalent depth, same water-equivalent offset) in Figure 17 (D). There are several factors affecting the accuracy of these approximations. First, the flux of photoris at the scattering point is increased for path c’ since the point is closer to the source. However, for path d’, the flux is unchanged. Second, for path c‘, the scattering angle is decreased, resulting in a higher energy and less attenuation after scattering. The scattering angle stays the same for path d‘. Third, as the void thickness is increased the amount of scattering material through which the photon must travel increases for both path c‘ and d’, but the increase is much larger for path d‘ since the geometric distance stays the same, but the void is now filled with attenuating medium.

These factors can again be used to explain the observed results. For small void widths we see that the water-equivalent distance works best. The change in the scattering angle is relatively small and the inaccuracy resulting from assuming that the void is filled with attenuating medium, as we do when we use the geometric offset, results in errors. However, for larger void widths the change in the scattering angle and photon flux at the scattering point that results from using the water-equivalent offset results in a significant error, apparently larger than the error resulting from assuming the void is filled with scattering material.

The above discussion helps to understand why method 1 works best in the thorax. phantom. In the thorax phantom the lungs represent large areas of low density. In the horizontal direction the maximum size of the lungs is approximately 11 cm. This corresponds to a water equivalent thickness of approximately 3 cm. Thus the difference between the geometric and water equivalent distance is 8 cm. From the data for the vertically layered slab phantom we see that, for voids corresponding to a difference of 8 cm in the water- equivalent and geometric distances, using the water- equivalent distance for the offset will result in substantial errors. Further, for such a large difference we would expect the geometric offset based approximation to give better results. This is what was observed.

We observed that method 1 was most accurate in the thorax phantom for both the point and extended sources. The error in the SPR was on the order of 5%. It should be remembered that an error in the SPR of 5% translates into an error in the total counts o n the order of 1.5% assuming 30% of the total counts are from scatter. Thus method 1 may provide reasonable accuracy. However, from looking at the shapes of the tails of the TLSRFs in Figures 14-16, we see there is

substantial disagreement in the tails. How these deviations would affect the reconstructed image quality and quantitative accuracy is an area requiring further study. From these TLSRFs we also see that the true TLSRF is always between that predicted by the geometric and water-equivalent distances. This suggests that an extended SDSE method could be developed based on a function that maps the water- equivalent and geometric offset into a single distance.

VI. CONCLUSIONS

In this study we have investigated the factors affecting the SRF in nonuniform attenuators. We conclude that, for 140 keV photons, the actual elemental composition of the attenuator is a relatively minor effect. In other words, we will be able to estimate the scatter component of the projection data with reasonable accuracy by knowing only the attenuation coefficient distribution; we do not need to know the detailed elemental composition.

We have investigated three methods for extending SDSE to uniform and nonuniform attenuators based the use of water- equivalent distances. We have shown that, at 140 keV and for reasonable water-equivalent source depths, the second and third methods can predict with high accuracy (less than 1% error in the SPR) the SRF in uniform attenuators composed of lung and bone using the water-equivalent source depth and the SRF for the source in water. Thus, the SDSE method should work well in these uniform media based on a parameterization of the slab phantom SRF measured in water.

To study the performance of the three methods in nonuniform attenuators we used both simple and realistic attenuation distributions. We observed that all three of the methods adequately predict the SRF when the attenuator is composed of a horizontal layers of water separated by void. However, for a phantom composed of vertical layers the first method worked best for larger air thicknesses and the second and third methods were best for small thicknesses. In a phantom simulating the attenuation distribution in the thorax the first method achieved an error in the SPR on the order of 5% but there were some discrepancies in the tails of the TLSRFs. The agreement in the SPRs is good, and would result in an error of less than 2% in the total counts. However, the effect of the error in the SPR and the shape of the tails of the TLSRF on reconstructed images has not been determined. The fact that the first method did not work well for the uniform phantoms composed of lung and bone but worked well for the thorax phantom indicates that further work is needed. The results for the thorax phantom also suggest it might be possible to develop an improved method for extending SDSE by combining first and second methods.

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