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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 12, DECEMBER 2014

Reducing Multiplexing Artifacts in Multi-Pinhole SPECT With a Stacked Silicon-Germanium System: A Simulation Study Lindsay C. Johnson*, Student Member, IEEE, Sepideh Shokouhi, and Todd E. Peterson, Member, IEEE

Abstract—In pinhole single photon emission computed tomography (SPECT), multi-pinhole collimators can increase sensitivity but may lead to projection overlap, or multiplexing, which can cause image artifacts. In this work, we explore whether a stackeddetector configuration with a germanium and a silicon detector, used with (27–32, 159 keV), where little multiplexing occurs in the Si projections, can reduce image artifacts caused by highlymultiplexed Ge projections. Simulations are first used to determine a reconstruction method that combines the Si and Ge projections to maximize image quality. Next, simulations of different pinhole configurations (varying projection multiplexing) in conjunction with digital phantoms are used to examine whether additional Si projections mitigate artifacts from the multiplexing in the Ge projections. Reconstructed images using both Si and Ge data are compared to those using Ge data alone. Normalized mean-square error and normalized standard deviation provide a quantitative evaluation of reconstructed images’ error and noise, respectively, and are used to evaluate the impact of the additional nonmultiplexed data on image quality. For a qualitative comparison, the differential point response function is used to examine multiplexing artifacts. Results show that in cases of highly-multiplexed Ge projections, the addition of low-multiplexed Si projections helps to reduce image artifacts both quantitatively and qualitatively. Index Terms—Image artifacts, multiple pinhole, multiplexing, reconstruction algorithm, single photon emission computed tomography (SPECT).

I. INTRODUCTION

T

YPICAL small-animal single photon emission computed tomography (SPECT) studies require systems to have high spatial resolution in addition to high detection efficiency [1], [2]. One popular method of increasing system sensitivity Manuscript received May 23, 2014; revised July 03, 2014; accepted July 10, 2014. Date of publication July 17, 2014; date of current version November 25, 2014. This work was supported by U.S. Department of Energy Office of Science (BER) under Grant DE-SC0002437, Grant NIH/NIBIB R01EB013677, and Grant NIH/NIBIB R00 EB 009106. Asterisk indicates corresponding author. *L. C. Johnson is with the Institute of Imaging Science and the Department of Radiology and Radiological Sciences, Vanderbilt University, Nashville, TN 37232 USA (e-mail: [email protected]). S. Shokouhi is with the Institute of Imaging Science and the Department of Radiology and Radiological Sciences, Vanderbilt University, Nashville, TN 37232 USA (e-mail: [email protected]). T. E. Peterson is with the Institute of Imaging Science, the Department of Physics and Astronomy, and the Department of Radiology and Radiological Sciences Nashville, Vanderbilt University, TN 37232 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMI.2014.2340251

is to utilize a multi-pinhole collimator. The system’s magnification, field of view, and the detector’s surface area lead to limitations on the number and arrangement of pinholes in the collimator. When pinholes are placed close together, projections can overlap, or multiplex, on the detector. This overlap causes ambiguity as to what pinhole a photon passed through, which subsequently can lead to artifacts in reconstructed images [3]. The tradeoff between the increased sensitivity of multiple pinholes and the ambiguity of photon origin due to multiplexing has been widely studied with no clear conclusion. For instance, Mok et al. simulated a variety of pinhole configurations that either 1) fixed the magnification and sensitivity, 2) fixed the sensitivity and varied the magnification, or 3) fixed the magnification and varied the sensitivity [4]. This study design allowed for a quantitative comparison of the influences of all three factors of interest: system sensitivity, magnification, and by proxy the degree of multiplexing. They found that some degree of multiplexing increases overall image quality up to a point, after which the benefit is decreased due to image artifacts. The point at which this switch occurs varies depending on the pinhole configuration, system magnification, and the activity distribution of the object being imaged. Further investigations into the effect of an object’s activity distribution on multiplexing artifacts was studied in [5], and demonstrated that certain object distributions, such as a cool sphere inside of a hot background, have increased artifacts compared to hot-rod and cold-rod phantoms. These results are also dependent on the system design, making the clear “answer” to the optimal amount of pinhole multiplexing elusive. Another way to approach this issue is to choose a specific imaging task and then determine, through simulation, the optimal number and arrangement of pinholes for the task, such as Cao et al. performed for mouse brain imaging applications [6]. Another way to minimize the effects of multiplexing artifacts was investigated by Vunckx et al., where, in simulation, additional collimation was added such that all multiplexing was removed from projections and resulting reconstructions were compared to images without the additional collimation. They concluded that once the detector area has been completely utilized the sensitivity gained from multiplexing does not improve image contrast-to-noise ratio [7]. Lin and Meikle approached this problem from the perspective of sufficient data sampling in truncated projections [8]. They demonstrate that if the local Orlov condition is satisfied at every point in the object, even if projections are truncated, the object can be reconstructed without artifacts. This allows for novel

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JOHNSON et al.: REDUCING MULTIPLEXING ARTIFACTS IN MULTI-PINHOLE SPECT WITH A STACKED SILICON-GERMANIUM SYSTEM

system configurations to be developed that have increased sensitivity due to smaller radii of rotation. An extension of this idea to multiplexing was investigated by Lin that determined that if a subset of nonmultiplexed projections exist that is independently sufficient to reconstruct an object, and they are consistent with the multiplexed projections, then reconstructions with both the multiplexed and nonmultiplexed data will yield an artifact-free image [9]. This study also concluded that given the above conditions, in some cases contrast-to-noise ratio can be improved even after the surface area of the detector is fully utilized. These results seem to contradict those of Vunckx et al. [7], but it is likely this is due to the difference in pinhole designs used in the simulations. We propose an alternative approach to allow for using an increased number of pinholes that have a significant degree of multiplexing by utilizing a stacked-detector approach based on the concept of synthetic collimation, which uses multiplexed and nonmultiplexed data to produce artifact-free reconstructions [10]. This concept has been demonstrated in previous simulation work in our lab using a limited-angle stacked Si detector design [11], [12]. This concept has also been demonstrated experimentally in a limited-angle small-animal system with 46 pinholes where the detector is moved to allow data collection at multiple focal lengths [13]. Mahmood et al. also demonstrated a variation of the concept using a slit-slat collimator both in simulation and experiment, where the addition of nonmultiplexed projections to multiplexed projections increased image quality compared to reconstructions with only the multiplexed projections [14], [15]. The particular stacked system explored in this present work is composed of a Si detector and a Ge detector. When used with , the photons emitted in the 27–32 keV range interact in the Si detector, while the 159 keV photons interact in the Ge detector. McDonald et al. demonstrated that the Si detector has minimal impact on 159 keV gamma rays [16]. One benefit of the stacked-detector design is that not only does this give the system the ability to better sample object space with less projection multiplexing in the lower-magnification detector, it also increases the system sensitivity without increasing the required imaging time as both detectors acquire simultaneously. This sensitivity increase could allow for an increase in signal-tonoise ratio, shorter scans times or a decrease in injected dose to the subject. Potential applications for this system include mouse imaging, in particular mouse-brain imaging, which would benefit from the additional sensitivity of the system. One challenge to this system configuration is determining how to combine both data sets in the reconstruction. This stacked-detector design leads to two data sets (one per detector) of different quality, having different magnification, spatial resolution, and noise properties in addition to the varied amount of multiplexing. The problem of reconstructing images with mixed-quality data has been previously investigated in other systems. A group at the University of Michigan has utilized mixed-quality data in a high-resolution small-animal PET system [17]. The system has a conventional scintillator ring with an additional insert composed of a small ring of Si detectors that is designed to Compton scatter annihilation events. This configuration gives rise to three different sinograms, one

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Fig. 1. Configuration of simulated stacked detector system, with Si placed close to the collimator to limit multiplexing of projections.

for each type of coincidence interaction (scintillator-scintillator, scintillator-Si, and Si-Si). Each event type has a different associated resolution and detection efficiency, and Clinthorne et al. proposed a combined system matrix that included these efficiency differences to properly weight each sinogram in the final reconstructed image [17]. Another variation of this idea, virtual pinhole PET, also utilized an insert that allows for higher resolution imaging [18]. This group tested the concept on a microPET system [19], and also utilized a system matrix with three system submatrices, one for each coincidence type, that accounts for differing detector dimensions, detector ring radii, and detector efficiency. In-depth system modeling techniques for a half-ring insert design were more recently published and give insight into the careful consideration required when combining data of mixed quality [20]. Because of the lack of previous work with mixed SPECT data of this type, the focus of this work is to demonstrate that image quality, specifically multiplexing artifacts, is improved with the additional Si detector when compared to images from a standard single-detector system in order to build a foundation for future system design and optimization. To do this, we first determine an appropriate reconstruction method to maximize image quality while utilizing both Si and Ge projection data sets. We then use this reconstruction method to investigate the benefit of the stacked-detector design to help reduce multiplexing artifacts. Digital phantoms exhibiting visible multiplexing artifacts are used to evaluate the difference in reconstructed image quality between a standard single-detector system and our stacked-detector system. Additionally, the differential point response function (DPRF) is used to qualitatively evaluate the presence of multiplexing artifacts [21]. II. METHODS A. System Configuration and Simulation Methods The system geometry is shown in Fig. 1. The 6-cm square silicon detector (1 mm thick) is placed between the pinhole collimator and the 12-cm square Ge detector (1 cm thick). A Si

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 12, DECEMBER 2014

TABLE I PERCENT OF MULTIPLEXING

Fig. 2. Pinhole configurations of the (a) p4LM, (b) p4HM and (c) p7 collimators.

Fig. 3. Axial central slice of the (a) Cool Sphere and (b) trans-axial central slice of the Cold Rod phantom. Cold Rod phantom has rod diameters range from 3 to 0.75 mm.

detector based on one used in our lab [22] having 1024 1024 strips (0.059 mm 0.059 mm) was simulated as having 256 256 strips (0.234 mm 0.234 mm) or 512 512 (0.117 mm 0.117 mm) to allow for faster reconstruction times and to reduce the influence of noise in reconstructions. The Ge detector is simulated to have 240 240 pixels of size 0.5 mm 0.5 mm. The simulated detector was chosen to be larger in size than our actual detector (circular 8 cm in diameter as opposed to a 12-cm square [23]) to allow for a larger range of multiplexing variation in the projections. The system has a 4-cm radius of rotation, with a focal length of 2 cm and 7 cm for Si and Ge, respectively, giving a magnification of 0.5 and 1.75 for Si and Ge, respectively. The Ge focal length and the radius of rotation were chosen based on the system configuration studied in Mok et al. [5], and the focal length of the Si detector was chosen at a distance that balanced the magnification and multiplexing tradeoff. Three different multi-pinhole collimators with varying amounts of multiplexing were used for the simulations. Nonfocusing, knife-edge pinholes with a 1-mm diameter and a 70 full opening angle placed in a 1-cm-thick tungsten plate were used for all collimators. The three pinhole configurations are shown in Fig. 2 and include: 1) a four-pinhole collimator with holes spaced on a 10-mm radius circle and a low amount of multiplexing (p4LM), 2) a four-pinhole collimator with holes spaced on a 7-mm radius circle and a higher amount of multiplexing (p4HM), and 3) a highly-multiplexing, 7-pinhole collimator with six pinholes on a 7-mm radius and one pinhole placed at the center (p7). Additionally, each configuration had random jitter added to their pinhole positions to remove potential aliasing effects from sampling at a fixed spatial frequency. The added jitter has an average of 0.3 mm difference between the exact and jittered pinhole position. Cool Sphere and Cold Rod phantoms, as shown in Fig. 3, were simulated as in [5]. Both phantoms were simulated voxel space and dimensions of with a

Fig. 4. Example noise-free projections of the Cool Sphere phantom demonstrating amount of multiplexing on the Si and Ge detectors (rows) with each collimator type (columns).

mm mm mm. The Cool Sphere phantom is cylindrical in shape with a 24-mm diameter and length and has a center sphere of 4.2-mm diameter with an activity 30% lower than the background. The Cold Rod phantom, also cylindrical in shape with outer dimensions matching the Cool Sphere phantom, has rods with diameters of: 3.0, 2.4, 1.6, 1.2, 1.0, and 0.75 mm with center-to-center spacing of twice the diameter. For all simulations, the previously described digital object phantoms were forward projected through the desired pinhole configuration for 64 projections over 360 . Projections were created using a voxel-driven forward projection that incorporated the geometric response of the pinhole. Detector resolution was not explicitly modeled, but is determined simply from the discretization of the detectors. Noise-free projections were simulated with the number of counts equivalent to a ten minute scan with 111.0 MBq and 88.8 MBq of activity in the Cool Sphere and Cold Rod phantoms, respectively. Prior to reconstruction, Poisson noise was added to the noise-free projections. Each simulation set includes reconstructions of 20 different Poisson noise realizations. The percentage of multiplexing for each data set was determined using the method of Shokouhi et al. [11], in which the sum of counts in detector pixels contributed by projections through more than one pinhole is divided by the total number of counts in the projection. The percent of multiplexing for each detector’s projection, pinhole, and object combination is shown in Table I, when the Si detector’s pixel dimensions were 512 512. Example projections of each pinhole type for the Cool Sphere phantom are shown in Fig. 4. The resulting forward projections (with or without added Poisson noise) were then used

JOHNSON et al.: REDUCING MULTIPLEXING ARTIFACTS IN MULTI-PINHOLE SPECT WITH A STACKED SILICON-GERMANIUM SYSTEM

as an input into the reconstruction algorithms. All reconstructions were based on voxel-driven forward and backward projections that utilize a central ray in conjunction with a geometrically-modeled PSF. This algorithm did not include penetration or scatter near the pinhole edges, scatter within the detector, nor scatter or attenuation in the object. The Si and Ge intrinsic detection efficiencies were modeled for their appropriate energy ranges. The Si detector efficiency was determined using the XCOM database [24] to obtain the mass attenuation coefficients for the 27–32 keV emission range. Each mass attenuation coefficient was weighted by the probability of emission at the specified energy to create a single . To determine the linear attenuation coefficient of 4.794 linear attenuation coefficient for Ge at 159 keV, an MCNP 5 [25] simulation modeled a 1-cm-thick Ge crystal irradiated with a mono-directional source of one million 159 keV photons. The linear attenuation was estimated from simulation results, and . The intrinsic efficiency was inwas found to be 0.7029 corporated into both the forward projections and the reconstruction algorithms. All reconstructed images consisted of voxels of size mm mm mm. B. Quantitative Evaluation Two different quantitative metrics are used to evaluate and compare image quality between reconstructions [5]. The average normalized mean square error (NMSE) is used to evaluate error in image reconstructions, while the normalized standard deviation (NSD) is used as a measure of noise. The average NMSE was calculated as (1) where is the number of voxels in the reconstructed volume, is the voxel value in the original phantom, is the mean voxel value of the original phantom, is the voxel value in the reconstructed image, and is the voxel index. The reported value is the average over all 20 noisy reconstructions. The NSD is based on the mean and variance of the 20 noisy reconstructed images. The NSD was calculated over a region of 2100 voxels, which was drawn in 3-D regions of uniform activity in each object. The NSD was calculated as (2) where is the number of noise realizations (20 in this case), is the number of voxels in the selected region of interest, is the voxel value in the noisy image, is the mean voxel value of the noisy reconstructions, is the noise realization index, and is the voxel index. C. Differential Point Response Function (DPRF) Evaluation In addition to NSD and NMSE, we also used the DPRF to qualitatively evaluate multiplexing artifacts [21]. First, a lumpy background object was created using an altered form of The University of Arizona’s Center for Gamma-Ray

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Imaging’s Gaussian Lump MATLAB code (http://radiology.arizona.edu/cgri/image-quality/software/image-quality-toolbox). voxels with dimensions of A 3-D object with mm mm mm was first initialized to a value of 10 in all voxels, and then the lumpy background was created by adding to the image space 310 Gaussian spheres having a of 20 maximum magnitude of 1 and a standard deviation voxels, with each sphere’s centroid randomly placed throughout the object space. Next, an object of the same size as the lumpy background was created with a single voxel of low activity (500) at its center and zero elsewhere. To give perspective, the sum of the intensity of all voxels in the lumpy object was and had a mean voxel intensity of 40.25. These objects were then forward-projected separately using the same process described previously (Section II-A). No noise was added to any of the projections used to determine the DPRF. The forward projections of both the lumpy object and the small signal object were summed together and reconstructed to form a combined lumpy and signal object. The original lumpy projections were also reconstructed separately. Finally, the original lumpy-only image was subtracted from the combined lumpy and signal image, and the resulting image—the DPRF—was visually assessed for artifacts due to multiplexing. D. Reconstruction Algorithm Investigation The first task of this work was to determine an appropriate method of combining the mixed-quality projections. Five different reconstruction algorithms were tested with the Cool Sphere phantom and p7 collimator, including: 1) MLEM with all projections from both Si and Ge used at each iteration, 2) OSEM with eight subsets, with the first four subsets using all of the Si projections and the next four subsets using all of the Ge projections, 3) OSEM with two phases, where a number of iterations are first completed with four subsets of only Si projections, followed by a second phase of four subsets of only Ge projections, 4) another OSEM with two phases, where the first phase uses four subsets of only Si projections and the second phase uses eight subsets, four with Si followed by four with Ge, and 5) OSEM with four Ge subsets (Ge-Only), which is used for comparison to the stacked-detector configurations. Twenty noisy sets of projection data were reconstructed using each of the five reconstruction algorithms, with all Si projec256 (0.234 mm 0.234 mm) pixels. As tions having 256 the convergence point was unknown at the time of running the simulations, reconstructions were run to the point of over-iteration (300 iterations for MLEM and 150 iterations for OSEM) to ensure reconstructions reached convergence. In order to implement the two-phase reconstructions, the update number (or sub-iteration) at which the reconstruction changes from using the first phase projection set to the second phase projection set, subsequently referred to as the switch-point, must be determined. For this study, we chose the switch-point to occur when the first phase image reaches a balance between the amount of noise in the reconstruction and its accuracy. This tradeoff can be visualized by plotting the NSD versus NMSE for every iteration, where the origin of the plot represents a perfect reconstruction—no noise and

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perfect accuracy. Si-Only reconstructions were first run beyond a reasonable optimal number. NSD versus NMSE was then plotted for each update, where the ideal switch-point would be the update number that minimizes the distance to the origin. However, the scales of the two axes—the NSD and NMSE—cannot be converted into units that allow them to be comparable. To determine a reasonable switch-point, the Si-Only curve was first shifted relative to its own minimum value, such that the minimum NMSE value was set to zero and the rest of the NMSE values were linearly shifted by the minimum. The same method was applied to the NSD, resulting in a plot that has “zero-shifted” axes, which helps to correct for large differences between the scales of the NSD and NMSE values. The update number of the nearest completed iteration that has the minimum distance to the zero-shifted origin was then used as the switch-point between the Si projection subsets and the Ge projection subsets. For reconstruction algorithm comparison, first the stopping point of the Ge-Only reconstruction was determined by finding the iteration at which the minimum of the zero-shifted axis was closest to the origin. The other reconstruction methods’ stopping points were then chosen to have an equal amount of noise to the chosen Ge-Only image. Although the amount of acceptable noise in a reconstructed image is highly dependent on the specific application, choosing a single level facilitates comparison across reconstruction algorithms. The DPRF for each reconstruction type and projection combination was calculated, including: MLEM (Si + Ge), OSEM (4 Si + 4 Ge), OSEM two-phase (4 Si + 4 Ge), OSEM two-phase (4 Si + (4Si, 4Ge)), and Ge-Only OSEM (4 Ge). The switch-point for the two-phase reconstructions and update number comparison points were the same as used in the noise-equivalent Cool Sphere reconstruction comparison. Both the DPRF and the quantitative and qualitative image evaluation were used to determine which reconstruction algorithm to use in the multiplexing artifact comparison study. E. Multiplexing Artifact Comparison Based on the results of Section II-D, the two-phase OSEM reconstruction with four Si subsets followed by four Ge subsets was expected to yield the best results in the stacked detector system and was therefore was used for the reconstructions in the multiplexing artifact comparison study. Twenty noise realizations each were reconstructed for the Cool Sphere and Cold Rod phantoms for each of the three pinhole collimators (p4LM, p4HM, p7) using the Si detector with 512 512 pixels. The same object and pinhole combinations were then reconstructed for the Ge-Only OSEM (four subsets) case as well. The switchpoint of the two-phase reconstruction for each object and pinhole combination was determined using the same methods as in Section II-D, with the iteration and update number for each switch-point shown in Table II. NMSE and NSD were determined for each reconstruction iteration and were plotted. As in Section II-D, the stopping point for the Ge-Only reconstruction was determined using the minimum distance to the zero-shifted origin, and the Si + Ge reconstruction was then compared at the noise-equivalent iteration.

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 12, DECEMBER 2014

TABLE II PHANTOM SWITCH-POINTS

TABLE III DPRF SWITCH-POINT AND STOPPING POINTS

In addition, the DPRF was determined for each pinhole combination, but as the switch-point for each pinhole type is highly dependent on the object type, a slightly different approach for determining the switch-point was utilized. Both the Cool Sphere and Cold Rod phantoms have a large amount of hot background, which differs from the single-point object that is being recovered in the DPRF images. To better judge an appropriate switchpoint and stopping point, in data not shown, a high-contrast Hot Rod phantom similar to that described in [5], was simulated and used with each pinhole aperture to estimate appropriate switch-points and stopping points for the DPRF simulations. The switch-points and stopping-points used are shown in Table III. III. RESULTS A. Reconstruction Algorithm Investigation Quantitative Evaluation: First, to determine the switchpoint, reconstruction of four subsets of Si-Only projections were determined using the Cool Sphere phantom and the p7 collimator. The resulting NSD versus NMSE curve is shown in Fig. 5, where the “x” indicates the point on the curve that is the minimal distance to the zero-shifted origin. The closest point to the zero-shifted origin at a completed iteration occurred at iteration 15 (update 60), which was subsequently used as the switch-point between the Si-projection-only first phase and the second phase of the two-phase reconstructions. The reconstructed images were used to calculate the NSD and NMSE, and the resulting NSD versus NMSE curves are shown in Fig. 6. The sharp turn of the curve occurs when the reconstruction changes from the Si OSEM phase to the Ge OSEM phase. The “x” indicates the location of the zero-shifted minimum, rounded to the nearest complete iteration for the Ge-Only OSEM case. The black horizontal line represents the noise-equivalent comparison point. In the case of the OSEM two-phase (4 Si + 4 Ge), where multiple noise-equivalent comparison points were present, the point with the lowest NMSE was used. The iteration and update number of this point on each curve are shown in Table IV. It can be seen that at the noise-equivalent line, the OSEM two-phase (4 Si + 4 Ge) has the lowest NMSE.

JOHNSON et al.: REDUCING MULTIPLEXING ARTIFACTS IN MULTI-PINHOLE SPECT WITH A STACKED SILICON-GERMANIUM SYSTEM

Fig. 5. NSD versus NMSE for an OSEM reconstruction with four Si subsets of the Cool Sphere phantom with the p7 collimator, showing the update number (four updates per iteration), and the “x” indicates the minimum distance from the origin to the curve.

Fig. 6. NSD versus NMSE curves for the Si + Ge reconstructions of the p7 collimator and Cool Sphere phantom indicate the overall image quality trends, with the inset indicating the minimal (zero-shifted) distance to the origin for the Ge-Only case at “x.” The black horizontal line indicates the noise level chosen for comparison.

TABLE IV COMPARISON STOPPING POINT FOR EACH RECONSTRUCTION ALGORITHM WITH THE COOL SPHERE PHANTOM AND P7 COLLIMATOR

Fig. 7 shows slices of the reconstructed images at the determined noise-equivalent stopping point. For the OSEM twophase 4 Si + (4 Si, 4 Ge) results, the stopping point actually occurs during the first Si-Only phase of the results, eliminating

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Fig. 7. Slices of the Cool Sphere phantom with the p7 collimator reconstructed with MLEM Si + Ge at 60th iteration (a), OSEM one-phase Si + Ge at 64th update (b), OSEM two-phase (4 Si + 4 Ge) at 96th update (c), OSEM two-phase (4 Si + (4 Si, 4Ge)) at 56th update (d), and Ge-Only OSEM at 36th update (e).

the inclusion of the second detector and therefore eliminating the purpose of the two-detector multiplexing study. The image slices show that the Ge-Only reconstruction has visible artifacts, with a dark band through the center of the slice appearing in Fig. 7(e). Comparing the reconstructed image slices shown in Fig. 7, visual inspection shows fairly similar image quality at the equal-noise comparison point for all of the Si + Ge images. Although Fig. 7(c) seems to most clearly resolve the cool sphere at the center of the object, as previously discussed this reconstruction is not considered a viable solution due to the lack of including Ge projections. When comparing the remaining reconstructions, (d) seems to have slightly less count pile up at the top and bottom of the phantom, with a marginal improvement in resolving the cool sphere portion. Differential Point Response Function Evaluation: The DPRF coronal image slices (Fig. 8) and line profiles (Fig. 9) for each reconstruction show the effect of the highly-multiplexed Ge projections with the p7 collimator. In a multiplexing- artifactfree reconstruction only a single central point would be present. The DPRF image slices for each case display the same type of artifacts, the high-activity ring of counts around the central signal point, except they vary in magnitude. This ring-type artifact has been previously associated with multiplexing [26]. The magnitude of these artifacts is better seen in the line profiles (Fig. 9), where the two smaller side peaks are the multiplexing artifacts. The Si + Ge reconstructions show a decrease in magnitude relative to the Ge-Only reconstruction, and the magnitude of the two-phase (4 Si + 4 Ge) peaks appear to be slightly lower than the other reconstruction methods. At the chosen noise-equivalent point, the NSD versus NMSE (Fig. 6) curve has the lowest NMSE in the two-phase OSEM (4 Si + 4 Ge) algorithm, and visual inspection of images shows this reconstruction equal to or slightly improved from the other algorithms compared. Additionally, the DPRF showed improvement when using two-phase (4 Si + 4 Ge) reconstruction as compared to using Ge alone, and appeared to reduce, if only marginally,

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Fig. 8. Coronal image slices of the DPRF for each reconstruction type, including Si+Ge MLEM (a), one-phase OSEM (b), the two-phase OSEM (4 Si + 4 Ge) (c), and the Ge-Only OSEM (d). The line in (a) represents the line profiles through each image shown in Fig. 9.

Fig. 9. Normalized line profiles through the DPRF slices of the varying reconstructions, including the Ge-Only OSEM, Si+Ge MLEM, one-phase OSEM, the two-phase OSEM (4 Si + 4 Ge), and the single-voxel signal object. The inset shows a zoomed view of the far right multiplexing artifact peak. Line profile acquired for each is indicated by the arrow in Fig. 8(a).

the magnitude of the multiplexing artifact peaks relative to the other reconstruction algorithms tested. Because of this, simulations in Section III-B utilize the two-phase (4 Si + 4 Ge) reconstruction.

Fig. 10. NSD versus NMSE for (a) Cool Sphere, (b) Cold Rod, for each pinhole combination and for Ge-Only and the combined Si + Ge reconstruction, where asterisks indicate the stopping-point iteration.

TABLE V STOPPING POINT OF ALL PHANTOM AND PINHOLE COMBINATIONS

B. Multiplexing Artifact Comparison Quantitative Evaluation: The NSD versus NMSE curves for each object are shown in Fig. 10. It can be seen that, even in the low multiplexing p4LM case, the addition of the Si projections into the reconstructions results in more accurate, less noisy images than with reconstructions based on Ge projections alone. The resulting equal-noise stopping points for each pinhole and object combination are shown in Table V and are indicated in Fig. 10 as asterisks. The Cool Sphere reconstructions show significant artifacts, as shown in Fig. 11. The low-multiplexing case of p4LM shows little difference among images but in the p4HM and p7 cases where higher amounts of multiplexing are present, it can be seen that the addition of the Si projections compared to Ge-Only reconstructions helps to significantly reduce artifacts. This is especially true in the p7 case, where the artifact causes the cool

sphere at the center of the object to have roughly the same intensity as the hot background, while in the Si + Ge case, the cool sphere is better resolved. Image slices of the Cold Rod phantom are shown in Fig. 12. The addition of the Si projections appears to increase the ability to resolve the cold regions in the lowest multiplexing case, Fig. 12(a) compared to (d). The highest multiplexing case, Fig. 12(c) compared to (f), shows that the addition of the Si projections helps to lower the intensity of the hot artifact at

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Fig. 11. Cool Sphere reconstructions at the stopping point, showing (a) p4LM, (b) p4HM, and (c) p7 Ge-Only slices and (d) p4LM, (e) p4HM, (f) and p7 Si + Ge slices. Fig. 13. p7 DPRF results are shown, with (a) and (b) showing coronal slices of Ge-Only and Si + Ge reconstructions, respectively, and (c) showing the resulting line profiles through the center of the image slices similar to that indicated in Fig. 8(a).

IV. DISCUSSION

Fig. 12. Cold Rod reconstructions at the stopping point, showing (a) p4LM, (b) p4HM, and (c) p7 Ge-Only slices and (d) p4LM, (e) p4HM, and (f) p7 Si + Ge slices.

the center of the object, and subsequently gives better contrast between the hot background and cold rods. Differential Point Response Function Evaluation: The DPRFs were determined for Ge-Only and Si + Ge reconstructions for each pinhole combination. In the lowest multiplexing case, p4LM, no multiplexing artifacts are seen and therefore the DPRF is not shown, but the normalized line profiles show a slight worsening of the FWTM between the Ge-Only and the Si + Ge, from 2.27 to 2.71 mm. The DPRF for pinhole p4HM (also not shown) had similar multiplexing artifacts compared to p7 and showed a limited improvement in the Si + Ge case. The DPRF of the highly multiplexed p7 case is shown in Fig. 13, where the line profiles through the multiplexing artifacts show a decrease in the artifact peak’s normalized intensity in the Si + Ge case relative to the Ge-Only reconstruction. This decrease demonstrates the advantage of the addition of the Si projections to highly multiplexed Ge projections.

The benefit to moving from a single-pinhole collimator to a multi-pinhole collimator is the increased sensitivity, but artifacts due to pinhole multiplexing can degrade image quality. Adding a Si detector to detect the low-energy photons from decay shows the potential to reduce multiplexing artifacts in addition to increasing the system’s sensitivity. The first challenge to this stacked-detector configuration is to determine a method to combine projections of mixed quality in order to improve image quality. Although some work has previously utilized mixed-quality data, to the authors’ knowledge no other groups have previously investigated reconstruction methods for multi-pinhole SPECT in a stacked-detector system. Our work demonstrates that the way in which projections are combined in the reconstruction algorithm can lead to differences in image quality; out of the tested reconstruction approaches, the best image quality was obtained by using the low-multiplexing, low-magnification Si projections for a small number of iterations followed by the higher-magnification and high-multiplexing Ge projections. Other studies using mixed-data projections have focused on more traditional MLEM and OSEM, such as Mahmood et al. [15], where OSEM was utilized with both multiplexed and nonmultiplexed projections in each subset, which is similar to the OSEM one-phase reconstruction tested in this work. Lin [9] also used a typical OSEM algorithm, as projections in that study are not easily separable into low or highly-multiplexed groups. In Havelin et al. [13], where limited-angle tomography was performed with 6 different geometric configurations, the quality of their reconstructed images might improve if, instead of the published MLEM reconstruction, an alternative reconstruction scheme were used that altered how projections were combined. After determining the reconstruction algorithm, three different pinhole configurations were used to investigate the image

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quality with and without the additional Si projections. Results presented focused on phantoms that Mok et al. demonstrated to have obvious multiplexing artifacts [5]. Qualitatively, the additional Si projections either eliminated or reduced the visibility of artifacts in both phantoms. Quantitatively, the NSD versus NMSE curves show that at the equal-noise comparison point, the Si + Ge reconstructions provided improved NMSE when compared to their Ge-Only counterparts. The DPRF results also show that the additional Si projections decreased the severity of the multiplexing artifacts. These results show that a combined Si-Ge system has the potential to improve image quality over a highly multiplexing Ge system alone. The DPRF method is typically used to measure system spatial resolution independent of the reconstruction algorithm, and is quantified by measuring the FWHM of the main peak. The spatial resolution of the two-phase (4Si + 4Ge) in Fig. 9 is worse than that in the Si + Ge shown in Fig. 13. This difference demonstrates that more work is necessary in the future to understand how the two-phase reconstruction algorithm affects image spatial resolution. One limitation to utilizing the DPRF for studying multiplexing artifacts is that multiplexing is spatially-dependent, yet this method only investigates the response to a single position in object space. If the DPRF were to be the only measure of image quality and multiplexing artifacts, then the source position would be required to be moved through all of object space to fully capture all multiplexing artifacts. In this study, however we have used the DPRF with a single position in object space in conjunction with other qualitative and quantitative methods to demonstrate improvement between the stand-alone Ge system and the stacked Si + Ge reconstruction. Although this study investigates one particular system configuration (Si and Ge focal lengths, and radius of rotation), the different collimators examined provide varying degrees of multiplexing on the detectors. Changing the focal lengths and radius of rotation in this system would alter the sensitivity and spatial resolution as well as the magnification and multiplexing, but the different collimators used in this study demonstrate the merit of the stacked detector system design, and the next step of this project will focus on designing an optimal system configuration. Given the challenge in determining switch-points and stopping-points of the two-phase reconstruction algorithms in real data, and that in Fig. 6 all Si+Ge reconstruction algorithms tested provide an improvement in image quality, using the one-phase OSEM (8:4Si4Ge) algorithm would help to simplify the optimization process while still providing improved image quality and reduced multiplexing artifacts in comparison to a single-detector configuration. However, the reconstruction algorithm would require a new investigation if the focal length and radius of rotation were changed after determining the optimal system configuration. Additional work in the future may also be to investigate alternative reconstruction algorithms, such as a two-phase MLEM, or using differing number of subsets for OSEM. Given the increased sensitivity of the dual-detector system, this system would be useful for applications such as mouse brain imaging, as there should be benefits in the sensitivity-resolution tradeoff. Although this system requires the use of in order to utilize the stacked-detector geometry, a variety of radio-

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 12, DECEMBER 2014

tracers are used for many different applications [27], in particular and , which target dopamine transporters and receptors, respectively, have been shown useful in mouse models [28]. can also be utilized for imaging of amyloid plaque in studies of Alzheimer’s disease [29]. In cases where another radiotracer is required and a more typical SPECT system is unavailable, it would be possible to utilize this stacked-system with other radiotracers, such as . The Si data would have very few counts and would likely be dominated by noise and, therefore, would not be usable, but the Ge detector data would be able to be reconstructed independently, assuming a system configuration with minimal multiplexing was used. V. CONCLUSION We have determined an appropriate reconstruction algorithm to employ with a stacked Si-Ge multi-pinhole SPECT system. The simulated system and reconstruction algorithm were used to demonstrate the potential to improve image quality of highlymultiplexed Ge projections when the Si projections are added into the reconstruction. This configuration allows for both an increase in system sensitivity and improvement in image quality due to reduced multiplexing artifacts. Given the improvements seen in image quality in this study, future work will be to build a stacked Si-Ge system. This will allow for experimental validation of these simulation results in a similar system design. Our lab is currently developing a small-animal Ge-SPECT system [23], which will be modified by adding a Si detector between the Ge detector and the collimator. Future work will also involve determining optimal collimator designs, as this simulation study did not explore pinhole design in detail and only provides proof-of-principle that additional information from low multiplexed Si projections can improve the reconstructions of highly multiplexed Ge projections. ACKNOWLEDGMENT The authors would like to thank the Ionizing Imaging group at the VUIIS: Dr. M. N. Tantawy, D. Campbell, O. Ochinnikov, and R. Perea for their support and assistance. REFERENCES [1] B. L. Franc, P. D. Acton, C. Mari, and B. H. Hasegawa, “Small-animal SPECT and SPECT/CT: Important tools for preclinical investigation,” J. Nucl. Med., vol. 49, no. 10, pp. 1651–1663, Oct. 2008, 2008. [2] S. R. Meikle, P. Kench, M. Kassiou, and R. B. Banati, “Small animal SPECT and its place in the matrix of molecular imaging technologies,” Phys. Med. Biol., vol. 50, no. 22, p. R45, 2005. [3] N. U. Schramm, G. Ebel, U. Engeland, T. Schurrat, M. Behe, and T. M. Behr, “High-resolution SPECT using multipinhole collimation,” Nucl. Sci., vol. 50, no. 3, pp. 315–320, 2003. [4] G. S. P. Mok, W. Yuchuan, and B. Tsui, “Quantification of the multiplexing effects in multi-pinhole small animal SPECT: A simulation study,” IEEE Trans. Nucl. Sci., vol. 56, no. 5, pp. 2636–2643, Oct. 2009. [5] G. S. P. Mok, B. M. W. Tsui, and F. J. Beekman, “The effects of object activity distribution on multiplexing multi-pinhole SPECT,” Phys. Med. Biol., vol. 56, no. 8, p. 2635, 2011. [6] Z. Cao, G. Bal, R. Accorsi, and P. D. Acton, “Optimal number of pinholes in multi-pinhole SPECT for mouse brain imaging—A simulation study,” Phys. Med. Biol., vol. 50, no. 19, p. 4609, 2005. [7] K. Vunckx, P. Suetens, and J. Nuyts, “Effect of overlapping projections on reconstruction image quality in multipinhole SPECT,” IEEE Trans. Med. Imag., vol. 27, no. 7, pp. 972–983, Jul. 2008.

JOHNSON et al.: REDUCING MULTIPLEXING ARTIFACTS IN MULTI-PINHOLE SPECT WITH A STACKED SILICON-GERMANIUM SYSTEM

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