New, Faster, Image-Based Scatter Correction for 3D PET

C.C. Watson CTI Pm Systems, Inc., 810 Innovation Dr., Knoxville, Tennessee 37932

Abstract We report on a new numerical implementation of the

single-scatter simulation scatter correction algorithm for 3D PET. Its primary advantage over the original implementation is that it is a much faster calculation, currently requiring less than 30 sec execution time per bed position for an adult thorax, thus making clinical whole-body scatter correction more practical. The new code runs on a single processor workstation CPU instead of a vector processor array, making it highly portable. It is modular and independent of any particular reconstruction code. The computed scatter contribution is now intrinsically scaled relative to the emission image and no longer requires normalization to the scatter tails in the sinogram when all activity is contained within the field of view, making it more robust against noise. The new algorithm has been verified against the original code on both phantom and human thorax studies. Initial results indicate that scatter correction may be accurately performed following, instead of prior to, either 3D reprojection or Fourier rebinning. Some evidence is presented that the single-scatter operator, when applied to an uncorrected emission image provides reasonable compensation for multiple scatter.

I. INTRODUCTION Scatter correction is necessary for quantitative accuracy

in 3D PET volume imaging. Arguably the most accurate methodologies for scatter correction currently available involve model-based simulation of the distribution of scattered events in the sinogram [ l , 2, 3, 4, 5, 61. The single-scatter simulation ( S S S ) algorithm [5, 61 offers a good compromise between accuracy and speed, yet despite its implementation on a vector processor array may still require 5 min or more per bed position to compute in the thorax or abdomen of a large patient [6] . This can be a prohibitive amount of time for clinical whole-body scans. The desire for a much faster scatter correction capability, together with the need to provide it on a single-processor workstation-class CPU, in place of a vector processor array, has led to the development of a new, more efficient, numerical implementation of the SSS algorithm. The details of this new implementation will be described.

It has further proven possible to directly compute the magnitude of the scatter component relative to the unscattered component of an image without recourse to scaling the computed scatter to the scatter “tails” in the measured sinogram. This makes the algorithm much more robust against noise in the data. The fact that the calculation can now be strictly image-based also makes scatter correction more modular and permits it to be applied post-reconstruction, at least when employed in conjunction with linear reconstruction algorithms such as filtered back-projection. However, in

Figure 1: Some of the scatter trajectories and LORs contributed to by the ray sums for BS.

the absence of scaling to the measured data, there is no compensation for scatter contributions originating outside the axial field-of-view (FOV) of the image volume being corrected. On the other hand there is no intrinsic limit to axial extent of this volume, which in the case of whole-body studies may encompass most of the patient.

The new algorithm has been verified against the original code on both phantom and human thorax studies, and these results will be discussed. The relation between Fourier rebinning (FORE) [7] and scatter correction has also been investigated, by comparing images reconstructed with scatter subtraction performed either before or after FORE. This is a significant question since the gains in reconstruction efficiency permitted by FORE could be enhanced if scatter correction of the data could be accurately performed after rebinning. A related question addressed is whether the scatter component at oblique angles is sufficiently different from that in direct sinograms to warrant individual calculation. The current implementation does not support computation of distinct scatter components for the oblique sinograms.

11. METHODS

A. SSS algorithm

coincidence rate in the detector pair (A, B) can be written as: With reference to Fig. I , the expected single-scatter

where

0-7803-5696-9/00/$10.00 (c) 2000 IEEE 1637

V, is the scatter volume, S is the scatter point, UAS is the geometrical cross section of detector A for y-rays incident along AS, RAS is the distance from S to A, E A S is the efficiency of detector A for y-rays incident along AS, p is the linear attenuation coefficient, X is the emitter density in the object, cc is the Compton interaction cross section, and R is the scattering solid angle. Primed quantities are evaluated at the scattered photon’s energy, while unprimed ones are evaluated at 51 1 keV. p and X are determined from estimated emission and transmission image volumes. Additional details can be found in [5 ,6] .

The speed gain of the new implementation is achieved through more efficient coding. For each scatter sample point, four ray integrals must be computed. (p’ can be accurately scaled from p.) These integrals are estimated by summing samples of the images at points along the rays. More than 90% of the CPU time in the original SSS algorithm was spent on computing these ray sums. Most of this time was actually spent computing the positions of the ray sample points. Due to the organization of the code, it was necessary to recompute all rays sums for each sampled LOR. This was inefficient because a single ray sum may contribute to many different LORs in the same transaxial plane [SI, as illustrated in Fig. 1 . The ray sums for BS contribute to the scatter calculation for all LORs terminating at detector B. In the new implementation, ray sums are stored and reused for each LOR to which they contribute, greatly reducing the computational time. If N is the number of detectors in a ring and M is the number of scatter sample points in the image volume, then only 4 N M ray sums per ring are computed now compared to N 2 M previously.

A second inefficiency in the original code resulted from the fact the scatter sample points were chosen to be fixed relative to the object, requiring distinct ray sample points to be computed for each new axial position of the detectors. In the current algorithm, scatter and ray sample points are defined relative to the transaxial plane of the detector ring and their positions are pre-computed and stored as pointer offsets into the image arrays. The sample points for each detector ring are then obtained simply by axially shifting the location of the reference pointer into the arrays. This has the effect of axially translating the sample points relative to the image. This scheme is more complex to implement for oblique sinograms, which are not currently supported in the new implementation.

As in the previous version of the code, no attempt is made to sample every image pixel or physical detector location. An isotropic average scatter sample spacing of 2.25 cm is typically used giving about 88 sample points per liter. Scatter sample points with very low p values are rejected to avoid wasting time computing small contributions and to exclude the exterior of the body. 68 azimuthal detector positions per ring are computed, resulting in a 17 x 34 scatter sinogram, which represents somewhat finer sampling than previously. An axial detector spacing of 2.5 cm is typically used. Contributions

to a given detector ring are limited to scatter points lying within f20 cm axially in order to limit computational time for whole-body image volumes. The coarse scatter sinogram is interpolated in all three dimensions with cubic b-splines to make it commensurate with the measured data. Uniformly spaced samples are interpolated in the radial direction, effecting arc correction.

B. Scaling Neglecting multiple scatter for the moment, the measured

sinogram data, SAS, should be equal to the sum of the scatter contribution, SfB, given by Eqn. 1, and an unscattered component, s,AB,

where

The reciprocal of the pre-factor here is the normalization factor for the unscattered radiation:

(4)

such that

J A

This represents fully corrected projection data ready for reconstruction.

Previously, SI was computed then scaled to the scatter tails in S before subtraction. However it can be seen that only the product nABSfB is actually needed. The advantage of estimating n A E S f B instead of Sf” is that the former involves only the ratios of the detector efficiencies for scattered and unscattered radiation, such as (CAS E ; ~ ~ ) / E ; ~ . Such ratios can be estimated more accurately than the efficiencies themselves. Similar remarks apply to the geometric cross sections of the detectors. If these estimates are sufficiently accurate, no additional calibration of the calculated scatter is required, and access to the measured sinogram is not necessary. Attenuation correction (with the unscattered attenuation correction factors) and reconstruction of nABSfB produces a scatter image compatible with A. This is true regardless of the scale of X since the S1 estimate depends linearly on A. The current implementation of the SSS algorithm models nAB and directly estimates nABSfB , avoiding the need for scaling to S. The scaling step was problematic in the original implementation when the data were very noisy, when the patient occupied most of the FOV, or when little of the total activity was in the FOV. We expect the new code to be considerably more robust in these situations.

There is no explicit treatment of multiple scatter in the SSS algorithm. Previously, multiple scatter was largely

0-7803-5696-9/00/$10.00 (c) 2000 IEEE 1638

Figure 2: Phantom images. Top: axial slices. Bottom: transaxial slices. Left: transmission. Right: emission.

compensated by the sinogram scaling. However there may be substantial compensation for multiple scatter even in the absence of scaling when the scatter contribution is computed from an uncorrected estimated image. The uncorrected image, A, may be thought of as the sum an unscattered, XO, and a scattered component A,. Application of the single-scatter operator defined by Eqn. 1 , SI, to this image gives

&(A) = s i ( A o ) + &(A,) = Si +&(As) (6)

where SI is the true single-scatter contribution. It is plausible, although certainly not physically rigorous, that SI (A,) approximates higher order scatter, at least in scale.

C. Phantom studies A phantom study was performed on a SiemensKTI

HR+ tomograph. The phantom was a water filled cylinder, 21.2 cm in diameter and 12.5 cm long, with a 6 cm diameter cylindrical water-filled internal chamber. The main chamber was uniformly activated with lSF, and the smaller chamber was cold. Two cold arms consisting of 8.7 cm diameter water-filled cylinders were placed beside the phantom. The emission data were acquired in standard 3D mode, with a maximum ring difference of 22, resulting in one direct and four oblique data segments. A transmission scan was also performed. Axial and transaxial slices through the emission and transmission images are shown in Fig. 2. Note that the axial FOV of the HR+ is 15.5 cm and that the activity in this study was fully contained within this FOV.

Images were reconstructed with and without scatter correction, employing the original SSS implementation, and with both 3D reprojection (3DRP) and FORE followed by filtered back-projection (FORE+FBP) reconstruction algorithms. Scatter images were obtained by subtracting the corrected from the uncorrected images. For comparison, the new SSS algorithm was applied to the uncorrected images and the resulting scatter estimates were attenuation corrected and reconstructed with ramp-filtered back-projection. The attenuation correction factors in this case were obtained by forward projecting the transmission images.

To determine the effect of scatter from activity outside the FOV, a solid 68Ge cylindrical phantom, 21.2 cm in diameter, 22 cm long and centered in the FOV of the HR+, was scanned. Calculated attenuation correction was used in this case.

Table 1 Scatter Correction Timing Summary

LOR Scatter Time Relative samples points (sec) timekample

4 x i860 2184 1384 293 28.5 UltraSparc 4046 1580 25 1 . I PentiumII 4046 1580 22 1 .o

D. Human thorax study The two implementations have been compared on a

whole-body study acquired on an ECAT ART with single- photon attenuation correction. The data were single-slice rebinned during acquisition, so only 2D FBP reconstruction was employed. Six bed positions were acquired with a 12 plane overlap. This patient had some activity in the heart, but exhibited much higher uptake in two large lesions immediately below the heart, as well as the bladder. Scatter estimates were performed in two ways. First, only the data from a single bed position were employed to form the preliminary image. Then the scatter was estimated from an image also encompassing the two adjacent bed positions. These calculations were compared to the standard algorithm, which computes each bed position individually, but employs scaling.

111. RESULTS

A. Timing Although the two algorithms are difficult to compare directly

due to the differences in sampling schemes and platforms, we believe that the intrinsic speed-up with the new implementation is a factor of 10-20. A comparison was made for a single bed of the thorax study. The original algorithm was run on an array of four i860 vector processors. The new algorithm was run on a 300 Mhz UltraSparc processor, and a 450 Mhz Pentium I1 with a 100 MHz bus. The results for the sampling and execution times are given in Table 1. Although the new implementation estimates over twice as many samples, it is more than ten times faster.

B. Phantom studies Sinogram profiles for the short cylinder study are shown in

Fig. 3. The data represent a vertical projection in a plane near the center of the phantom. The solid curve is the normalized measured data (rebinned to 128 elements); the dashed curve is the scatter component estimated from the uncorrected 3DRP image using the new algorithm and no scaling; the dotted curve is the scatter estimated from a scatter corrected image obtained by subtracting the first estimated scatter image from the uncorrected image. The dotted curve should more accurately represent the single-scatter component of the data and is 25% lower in magnitude than the initial estimate. The agreement of the initial estimate with the measured data thus suggests that the multiple scatter component in these data is also approximately 25% of the total scatter. In this case at least, no additional correction for multiple scatter appears necessary.

0-7803-5696-9/00/$10.00 (c) 2000 IEEE 1639

I I i

bln number

Figure 3: Sinogram profiles for the short cylinder.

Figure 4: Short cylinder central images. Top: SDRP. Bottom: FORE+FBP. From left to right: uncorrected, scatter image x3, new correction, original correction.

Central slices through the uncorrected, scatter, and scatter corrected emission images of the short cylinder are shown in Fig. 4. There is no visible difference between the 3DRP and FORE+FBP images in this plane, or between the original and new scatter corrections. The estimated scatter fraction for the volume (scatter/total) was 28%. Image profiles are shown in Fig. 5. These are 5 pixel wide horizontal profiles through the cold spot near the axial center of the scanner and phantom (the same plane shown in Fig. 4). The profiles for the four combinations of reconstruction and scatter correction algorithms are virtually identical. Planes near the axial ends of the cylinders (not shown) do exhibit some significant differences between FORE+FBP and 3 D W reconstruction however, but not between the scatter algorithms. Recall that in the original implementation, distinct scatter components were computed and subtracted from each oblique angular data set prior to either FORE or filtering and back-projection in 3DRP. In the new approach, the scatter is in effect computed for the direct segment only and subtracted following the axial rebinning step. This would appear to be a very good approximation for these data.

Results for the longer, uniform 68Ge phantom are shown in Fig. 6. 3DRP reconstruction was used. These are horizontal profiles 5 pixels wide through the center of the phantom. The top curve is the uncorrected image, the middle curve is the corrected image, and the lower curves are from the two scatter images. The estimated scatter fraction for this image volume was 33%. The dashed curve is from the

\- ~

/ , . , / .

0 20 40 60 80 1W 120 Plxel number

Figure 5: Short cylinder central image profiles. The two higher curves are uncorrected. the four lower ones are scatter corrected.

1.5 i

o.oLTJ L j

Figure 6: 68Ge phantom central image profiles.

new SSS implementation, without rescaling. It appears to underestimate the true scatter by about 28%. Scaling the estimate up by this amount (as shown in the figure) gives very good agreement with the image produced by the original algorithm. It is plausible that this discrepancy is due mainly to the approximately 30% of the phantom lying outside the FOV of the scanner, and thus not accounted for in the absolute scatter calculation.

C. Thorax study Transaxial images slices through the heart for the patient

study are shown in Fig. 7, comparing the old (top) and new (bottom) scatter correction algorithms. In this plane the scatter component is nearly 80% of the total uncorrected image due to the presence of an extensive region of enhanced uptake in the abdomen. For the new calculation, the two bed positions adjacent to the one of interest were included in the simulation, for a total axial extent of 39.5 cm. The previous implementation did not support scatter estimation from an extended image volume, compensating by scaling to the data.

Profiles along a horizontal line through the center of the heart in the uncorrected and scatter images are shown in Fig. 8. Note the large regions of negative apparent activity in both the

0-7803-5696-9/00/$10.00 (c) 2000 IEEE 1640

Figure 7: Thorax images. Top: Original algorithm. Bottom: New algorithm. Left: Uncorrected. Center: Scatter. Right: Corrected.

0 2 0 4 0 60 80 100 120 pixel number

Figure 8: Thorax uncorrected and scatter image profiles.

uncorrected and scatter image profiles. Such negative regions are also easily visible in the images of Fig. 7 above and below the arms, and above the center of the chest. We believe they are caused by the inconsistency resulting from the application of 51 1 keV attenuation correction factors to the scattered data component.

Also shown in Fig. 8 is the profile from a scatter image simulated from a single bed position with the new algorithm, scaled by a factor of 1.55 to match the scatter estimated by the rescaling algorithm. The shapes of the profiles are quite similar. The absolute estimate using three beds is a factor of 1.46 higher than the single bed estimate in the image plane considered here. It underestimates the rescaling calculation by only 3%. It would appear from these results that at this position in the patient approximately 32% of the scatter component arises from outside the axial FOV of a single bed acquisition, and that the use of two adjacent bed positions in the simulation accounts for most of this effect.

IV. SUMMARY A new implementation of the SSS algorithm has been

described which is 10-20 time faster than the original implementation, with finer sampling, and which provides intrinsic scaling relative to the initial image data. The estimated scatter agrees very well with the original calculation. The new algorithm does not support computing scatter in oblique sinograms, which does not have an observable effect for either 3DRP or FORE for the cases examined so far. This question will need further evaluation for larger axial FOV cameras. The new algorithm does not explicitly treat multiple

scatter, but some compensation appears to be afforded by the presence of scatter in the initial image estimate. Scatter contributions originating from activity not included in the approximately 15 cm axial FOV of a typical camera can be substantial. Including adjacent regions of the body in the initial image can compensate for this effect, although a limitation of the current implementation is that it does not incorporate a model of the side shields of the tomograph. Such side shields reduce scatter and cause dips in the apparent scatter fraction in the bed overlap regions for multi-bed acquisitions. The results presented in this paper pertain only to linear, FBP type reconstruction algorithms. The non-linearity of iterative reconstructions algorithms raise some questions regarding the optimality of post-reconstruction scatter correction, although it is likely that the image based approach described here would work reasonably well as long as the computed scatter component is reconstructed consistently.

ACKNOWLEDGEMENTS I would like to thank Dr. A. Schaefer and Dr. C.M. Kirsch

for providing the patient data, and Dr. D. Newport and Dr. M.E. Casey for helpful discussions.

V. REFERENCES J. Barney, J. Rogers, R. Harrop, and H. Hoverath, “Object shape dependent scatter simulations,” IEEE Trans. NUC. Sci.,vol. 38, pp. 719-725, 1991. L. Hiltz and B. McKee, “Scatter correction for three- dimensional PET based on an analytic model dependent on source and attenuating object,” Phys. Med. Biol., vol. 39,

C. Levin, M. Dahlbom, and E. Hoffman, “A Monte Carlo correction for the effect of comptom scattering in 3-D PET brain imaging,” IEEE Trans. NUC. Sci., vol. 42, pp. I 181- 1 185, Aug. 1995. J. Ollinger, “Model-based scatter correction for fully 3D PET,” Phys. Med. Biol., vol. 41, pp. 153-176,1996. C. Watson, D. Newport, and M. Casey, “A single scatter simulation technique for scatter correction in 3D PET,” in Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (p. Grangeat and J.-L. Amans, eds.), pp. 255-268, Dordrecht: Kluwer Academic Publishers, 1996. C. Watson, D. Newport, M. Casey, R. deKernp, R. Beanlands, and M. Schmand, “Evaluation of simulation- based scatter correction for 3-D PET cardiac imaging,” IEEE Trans. NUC. Sci., vol. 44, pp. 90-97, Feb. 1997. M. Defrise, P. Kinahan, D. Townsend, C. Michel, M. Sibomana, and D. Newport, “Exact and approximate rebinning algorithms for 3-D PET data,” IEEE Trans. Med. Img, vol. 16, pp. 145-158, Apr. 1997.

pp. 2059-207 I , 1994.

[8] E.-Mumcuo~U, R. Leahy,- and S . Cherry, “Bayesian reconstruction of PET images: Methodology and performance analysis,” Phys. Med. Biol., vol. 4 I , pp. 1777- 1807, Sept. 1996.

0-7803-5696-9/00/$10.00 (c) 2000 IEEE 1641