Cardiac Single-Photon Emission-Computed ... - Semantic Scholar

IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 52, NO. 1, FEBRUARY 2005

143

Cardiac Single-Photon Emission-Computed Tomography Using Combined Cone-Beam/ Fan-Beam Collimation Grant T. Gullberg, Fellow, IEEE, and Gengsheng L. Zeng, Senior Member, IEEE

Abstract—The objective of this work is to increase system sensitivity in cardiac single-photon emission-computed tomography (SPECT) studies without increasing patient imaging time. For imaging the heart, convergent collimation offers the potential of increased sensitivity over that of parallel-hole collimation. However, if a cone-beam collimated gamma camera is rotated in a planar orbit, the projection data obtained are not complete. Two cone-beam collimators and one fan-beam collimator are used with a three-detector SPECT system. The combined cone-beam/ fan-beam collimation provides a complete set of data for image reconstruction. The imaging geometry is evaluated using data acquired from phantom and patient studies. For the Jaszazck cardiac torso phantom experiment, the combined cone-beam/fan-beam collimation provided 1.7 times greater sensitivity than standard parallel-hole collimation (low-energy high-resolution collimators). Also, phantom and patient comparison studies showed improved image quality. The combined cone-beam/fan-beam imaging geometry with appropriate weighting of the two data sets provides improved system sensitivity while measuring sufficient data for artifact free cardiac images. Index Terms—Cardiac, convergent collimation, iterative image reconstruction, single-photon emission-computed tomography (SPECT).

I. INTRODUCTION

S

INGLE-PHOTON emission-computed tomography (SPECT) has become an important diagnostic tool in cardiovascular nuclear medicine. Currently, myocardial perfusion imaging [1], particularly SPECT [2]–[21], is the most widely employed noninvasive method used for the diagnosis and risk stratification of patients with known or suspected coronary artery disease. The scintigraphic method provides the only practical, noninvasive measure of myocardial perfusion and its hyperemic response to stress, which is the primary parameter directly affected by flow-limiting coronary stenoses and is the common basis of all stress testing for the diagnosis and risk stratification of coronary artery disease [22]–[32]. In cardiac SPECT imaging, a low-energy general purpose (LEGP) or

Manuscript received December 22, 2003; revised December 3, 2004. This work was supported in part by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health under Grant RO1-EB00121 and in part by the Director, Office of Science, Office of Biological and Environmental Research, Medical Sciences Division of the U.S. Department of Energy under Contract DE-AC03-76SF00098. G. T. Gullberg is with the E. O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720 USA (e-mail: [email protected]). G. L. Zeng is with the Department of Radiology, University of Utah, Salt Lake City, UT 84108 USA. Digital Object Identifier 10.1109/TNS.2004.843118

low-energy high-resolution (LEHR) parallel-hole collimator is normally used to acquire an adequate number of image counts in a clinically convenient imaging time. These parallel-hole collimators can limit SPECT image quality by not taking advantage of the entire crystal area. Improved collimation, such as fan-beam and cone-beam collimation, may offer the potential to improve risk stratification in cases of moderate to mild disease through improved sensitivity gained by mapping the image on a larger area of the crystal. For a small organ such as the heart, a converging geometry using a cone-beam collimator can improve the sensitivity and resolution of projection images. However, to date little has been done to take advantage of cone-beam collimation in cardiac SPECT. The major reasons for this are due to additional algorithm complexity, the truncation of projection data, the need for complex orbits (both noncircular and nonplanar orbits), and the 360 angular sampling requirements that are necessary for previously implemented cone-beam reconstruction algorithms. The 360 angular sampling particularly is not optimal for cardiac SPECT due to the attenuation artifacts that result from reconstructing posterior projections. The impetus to overcome these problems is based on our previous work, which has shown significant advantages in using converging collimators to improve sensitivity and resolution. That work has also shown significant potential for using cone-beam collimators to obtain images of the heart that are free of artifacts [33]. In that work a cone-beam filtered backprojection algorithm [34], which is an extension of the Feldkamp algorithm [35], was used to reconstruct projections from noncircular detector orbits and “short scan” acquisitions [33]. In the application to cardiac SPECT imaging presented in this paper, a fan-beam collimator is mounted to one detector and two cone-beam collimators are mounted to the other two heads of a three-detector SPECT system (Fig. 1) so that the face of the cameras remain parallel to the axis of rotation. The goal is to magnify the heart as much as possible over a large crystal area to maximize the sensitivity, while at the same time keeping the heart in the field of view through all rotation angles. The cameras rotate in a noncircular orbit with the center of the cameras always pointing to the center of rotation and closely following the patient’s body contour to optimize the projected spatial resolution. An important aspect of this detector arrangement is that it solves the data insufficiency problem of planar orbit cone-beam tomography. In order to obtain sufficient data, each plane that intersects the object to be reconstructed must contain an orbit point [36]. This condition is not satisfied by

0018-9499/$20.00 © 2005 IEEE

144

Fig. 1. Schematic of a three-detector SPECT system for combined cone-beam/ fan-beam tomography of the heart. The system consists of one fan-beam collimator and two cone-beam collimators.

planar orbit cone-beam tomography. However, emission data from the fan-beam collimated detector can be used to fill in the data missing from the planar orbit cone-beam data set in order to provide sufficient data for reconstruction. In previous work, algorithms have been derived to reconstruct the combination of parallel and cone-beam projection data. Jaszczak presented a paper that proposed combining cone-beam and parallel-beam data [37]. Three-dimensional images were reconstructed using an iterative reconstruction algorithm. Later, Defrise and Clack published a paper that described the development of an analytical filtered backprojection reconstruction algorithm for the reconstruction of combined cone-beam and parallel-beam projection data [38]. This approach though did not seem feasible for the combination of fan-beam and cone-beam data because it would be difficult to form parallel planar integrals directly from the fan-beam data. We later proposed an analytical algorithm for combined cone-beam/fan-beam projections [39]. Using that approach, planar projections were formed only after reconstruction of the fan-beam projections. The derived planar projections were then combined with the cone-beam data in Radon space. Through that process a somewhat ad hoc analytical algorithm was derived [39]. A more desirable approach to reconstructing combined cone-beam/fan-beam SPECT data is to use an iterative reconstruction algorithm to correct for attenuation, depthdependent detector response, and scatter [40]–[43]. We have done several studies of the application of iterative reconstruction of cone-beam data for circular orbits [41], [44], [45] and for noncircular orbits [46]–[49] in SPECT applications. Also, we have performed eigen analysis of several orbit configurations for cone-beam data acquisitions [50]. This analysis was extended to the reconstruction of combined cone-beam/fan-beam projection data [51]. The implication of this work to solving a system of equations for combined cone-beam/fan-beam geometry will be discussed in accordance with our work presented in this paper. In this paper, results obtained from phantom experiments and patient studies using high-resolution cone-beam (8.1-mm FWHM) and fan-beam (8.0-mm FWHM) collimators (both 65-cm focal length; both measured at 10 cm from the face


of the collimator) are compared with those obtained with an LEHR parallel-hole collimator (7.2-mm FWHM at 10 cm). It has been found that the attenuation-corrected reconstruction algorithms that are applicable to cardiac SPECT imaging require a measure of the attenuation map because of the variability in attenuation throughout the thorax. In the work presented here attenuation correction was not implemented because the attenuation correction protocol did not allow for body contouring orbits, which are necessary to optimize the resolution and allow a better comparison between parallel collimation and combined cone-beam/fan-beam collimation. Therefore, the patient studies were implemented with noncircular orbits, whereas the phantom studies were implemented with circular orbits for reasons of some hardware difficulties. The main focus of this article is to evaluate the results of increased counts for the same imaging time associated with parallel-only, fan-beam only, cone-beam only, and combined cone-beam/fan-beam tomography. II. MATERIALS AND METHODS A. Data Acquisition An IRIX three-detector SPECT system (Marconi/Phillips Medical Systems, Cleveland, OH) with one fan-beam and two cone-beam collimators was used in the phantom and patient studies. The cone-beam and fan-beam collimators were manufactured by Nuclear Fields in Vortum-Mullen, The Netherlands with the dimensions: 1.5 mm hex hole; 0.23 mm septa; 35 mm core thickness; and 650 mm focal length measured from the face of the collimator. The dimensions of the collimator holes were chosen to obtain the best tradeoff between resolution and sensitivity using computer simulations based on analytical formulas developed previously [52], [53]. Results using the combined cone-beam/fan-beam collimators were compared with those using the LEHR parallel-hole collimator (hex hole: 1.22 mm; septa: 0.203 mm; core thickness: 27.0 mm). The FWHM for each collimator on the IRIX was measured ) of . using a capillary tube filled with 1.258 MBq (34 The capillary tube was imaged for 5 min at 10 cm from the face of the collimator along the central axis of the collimator. The FWHM at 10 cm was 8.1 mm for the cone-beam, 8.0 mm for the fan-beam, and 7.2 mm for the LEHR parallel-hole collimator, respectively. One patient study was performed on the PRISM 3000XP three-detector SPECT system (Picker/Phillips Medical System, Cleveland, OH) using the LEHR parallel-hole collimator (hex hole: 1.40 mm; septa: 0.18 mm; core thickness: 27.0 mm). The parallel-hole collimator on the PRISM has a system resolution of 8.0 mm at 10 cm from the face of the collimator. For all of the studies that follow, projection acquired from the IRIX SPECT system were corrected for nonuniformity using the system default flood image data. 1) Phantom Studies: Two phantom studies were performed using the IRIX three-detector SPECT system. Phantom Study 1. In this study the cardiac insert of a Jaszczak cardiac phantom (Data Spectrum Corporation, Hillsborough, NC) was used. The insert has two concentric chambers separated by 1.0 cm. Within the separation, one simulated 4 4-cm lesion

GULLBERG AND ZENG: CARDIAC SPECT USING COMBINED CONE-BEAM/FAN-BEAM

was located in the inferior wall. However, the lesion was not used in any of the analyzes of the results. The remaining volume between the chambers was filled with water (107 mL) and was ) of . The inner chamber injected with 3.7 MBq (100 of the cardiac insert was not filled with water. The insert was placed on the imaging bed with the long axis of the insert aligned with the rotating axis of the SPECT system. One tomographic study was performed using a circular orbit of 20.7 cm with the combined cone-beam/fan-beam collimators and the other using the same orbit with the LEHR parallel-hole collimators. The projection data were digitized into 120 projections of 128 128 matrices (pixel size of 4.67 mm) that were acquired over 360 . The acquisition time for each projection was 10 s and the total acquisition time was 20 min. Phantom Study 2. The second phantom study was performed using the same IRIX system with the cardiac insert placed inside the Jaszczak torso phantom including lungs and a spinal cord. The insert was injected with 92.5 MBq (2.5 mCi) of . The remainder of the torso phantom and the blood pool region in the cardiac insert were filled with water without radioactivity. One tomographic study was performed using a circular orbit (31.35 cm radius) with the combined cone-beam/fan-beam collimators; 120 views (120 step-and-shot positions per detector) were acquired over 360 with 2 s per position. Ten separate studies were performed. Another tomographic study was performed using a circular orbit (28.45 cm radius) with three LEHR collimators; 120 views (40 step-and-shot positions per detector) were acquired over 360 with 6 s per position. Again, this was repeated ten times. The combined fan-beam/cone-beam studies acquired 120 2 sec views for each detector and the parallel collimator studies acquired 40 6 sec views for each detector. This resulted in the same imaging time for both studies. 2) Patient Studies: Two cardiac patient studies were performed. The Institutional Review Board approved the studies and an informed consent was obtained from each subject. The patients were imaged with parallel-hole collimators using the standard clinical imaging protocol for rest followed by stress rest study was performed followed by a conditions. A -sestamibi-stress study. For one patient the clinical rest and stress study was performed on the PRISM 3000XP and for the other patient the studies were performed on the IRIX. After the stress study was completed for the clinical imaging protocol, both patients were then imaged using the combined conebeam/fan-beam collimation on the IRIX three-detector SPECT system. Patient 1. For the rest study, the patient was injected with . The SPECT study with parallel113 MBq (3.06 mCi) of hole collimators was performed on the IRIX SPECT system 20 min after injection. The patient was then removed from the imaging bed and prepared for the stress study. The patient was stressed pharmacologically using adenosine that was infused for 4 min. At 1.5 min after the start of adenosine injection 873 MBq -sestamibi was injected. Then, at four min, (23.6 mCi) of the adenosine infusion was terminated. The SPECT study was begun 25 min later on the IRIX camera using the same parallel-hole collimators as for the rest study. -sestamibi data were acquired with a 15% energy The window at 140 keV. The projection data of 120 gated projections

145

(8 time frames per projection) were acquired in 64 64 matrices with a zoom factor of 1.333 (7.0 mm pixels). A step-and-shoot noncircular orbit acquisition was performed over a total imaging time of 18.6 min. After the stress study, the patient was removed and the parallel-hole collimators were replaced with one fan-beam and two cone-beam collimators. The patient was then repositioned on the imaging bed and the scan was re-initialized to acquire another -sestamibi scan with the combined cone-beam/fan-beam collimators. The lapse time between the parallel-hole collimator study and the combined cone-beam/fan-beam collimator study was approximately 1 h 30 min. The projection data of 120 views (over 360 ) per detector head were acquired into 64 64 matrices with a 1.333 zoom factor (7.0 mm pixels). A continuous noncircular orbit acquisition was performed with a total imaging time of 24 min. Patient 2. For the rest study, the patient was injected with 111 . The SPECT study with parallel-hole colMBq (3 mCi) of limators was performed on the PRISM 3000XP SPECT system five min after the injection. After the rest study the patient was removed from the imaging bed and was stressed on a treadmill. At the peak of stress (85% of maximum heart rate specified for -sestamibi the age of the patient) 814 MBq (22 mCi) of was injected. The SPECT study was begun 30 min later on the PRISM 3000XP camera using the same parallel-hole collimators as for the rest study. -sestamibi data were acquired with a 15% energy The window at 140 keV. The projection data of 120 gated projections (8 time frames per projection) were acquired in 64 64 matrices with a pixel size of 5.34 mm pixels. A step-and-shoot noncircular orbit acquisition was performed over a total imaging time of 21 min. After the stress study was performed on the PRISM 3000XP, the patient was repositioned on the imaging bed of the IRIX -sestamibi scan utilizing the system to acquire another combined cone-beam/fan-beam collimators. The lapse time between the parallel-hole collimator study on the PRISM 3000XP and the combined cone-beam/fan-beam collimator study on IRIX was approximately 10 min. The projection data of 120 views (over 360 ) per detector head were acquired into 128 128 matrices with a pixel size of 4.67 mm. A continuous noncircular orbit acquisition was performed over a total imaging time of 24 min. 3) Data Processing: The transaxial images of the heart were 64 and 128 128 sampled projecreconstructed from 64 tions of the cone-beam, fan-beam, and parallel-beam data sets. The reconstructed voxel size was the same as the projection pixel size. For the phantom, all images were displayed and analyzed utilizing a voxel size of 4.67 mm. The statistical measures (described below) were accomplished using these voxel dimensions. For Patient 1, all images were reconstructed and displayed utilizing a voxel size of 7.0 mm. For Patient 2, the combined cone-beam/fan-beam data were reconstructed using a voxel size of 4.67 mm and the parallel projection data were reconstructed using a voxel size of 5.34 mm. The parallel reconstructions were then interpolated to 4.67-mm voxels to enable visual comparisons with the combined cone-beam/fan-beam reconstructions.

146


An iterative ML-EM algorithm [54] was used to reconstruct all projection data sets. All projector/backprojectors used line length weighting. Attenuation and depth-dependent detector response were not modeled. The fan-beam projector/backprojector was from the RECLBL Library [55]. The reconstruction of the parallel-beam projections were obtained using the same fan-beam projector/backprojector with an extremely long focal length. The cone-beam code was similar to that given in a previous publication [40] but without modeling for attenuation. Filtering was not performed on either the projections or the final reconstructed result. The ML-EM algorithm was implemented with an additional weighting factor, , for each projection ray (1)

where is the result at the th iteration of the th voxel in is the the image, is the value of the th projection bin, and projection weighting from the th image voxel to the th projection bin. In this paper, only line-length weighting [40] is used for . If for all , (1) represents the standard ML-EM algorithm. This was used in the reconstruction of cone-beam only, fan-beam only, and parallel-beam only data sets. If the projection data and image model are exactly matched (i.e. the projections are consistent), the weighting factors, , do not affect the solution of the algorithm. When the projections are inconsistent, which is usually the case, even in most “noise-free” computer simulations, the weighting factors, , determine which projections are more reliable than others. When (1) was used to reconstruct the combined cone-beam/fan-beam data in this paper, the if are fan-beam projecweighting factors were set to if are cone-beam projection data. tion data, and In Phantom Study 1, a long axis slice of the cardiac insert was selected (Because of the placement on the imaging table, this corresponded to a transaxial slice.) and a normalized standard deviation was calculated for a region of interest (ROI) of , where 11 voxels using (2) was the mean and (3) was the standard deviation and was the mean sampled over and the error the 11 voxels in the ROI. The error of the mean were also calculated using of the variance

Fig. 2. Phantom study (isolated cardiac insert). (Reconstructions, left to right: combined cone-beam/fan-beam, cone-beam, fan-beam, parallel). The top row shows 64 64-transaxial images along the long axis of the cardiac insert. The voxel size is 4.67 mm. The bottom row shows short axis views. The results were obtained with 10 iterations of the ML-EM algorithm with a = 0:5. In Figs. 3–5, the combined cone-beam/fan-beam results include data from two cone-beam collimated detectors and one fan-beam collimated detector; the cone-beam results include data from two cone-beam collimated detectors; the fan-beam results include data from one fan-beam collimated detector; and the parallel results include data from three parallel-hole collimated detectors. The total number of counts in each reconstruction correlates with the noise texture seen in each image.

2

where was the sampled reconstructed voxel value [56]. All calculations were performed by sampling voxels of 4.67 mm in both the parallel and the combined cone-beam/fan-beam reconstructions. In Phantom Study 2, ten acquisitions were performed for each collimator configuration. Each image was reconstructed independently. The ensemble mean and variance were calculated in the myocardium region, which contained 2000 voxels. The mean, variance and standard deviation of the mean, and standard deviation of the variance were calculated as follows: from 10 difTen images were reconstructed ferent projection data sets. The 10 images were summed and divided by 10, resulting in a mean image . An ROI in the image was selected by setting a threshold, which gave 2000 image voxels in the ROI. were calcuThe mean-value and standard deviations of . lated using the 2000-voxel values in this ROI of the image were evaluated using Then 10 new images , , where the expression in parenthesis means evaluating the difference and squaring the , was deterresult voxel-by-voxel. Next the variance image where the summations mined by were performed pixel-by-pixel. In , the mean-value and the standard deviation of the mean value were cal. Finally, culated for the same ROI determined in the image error was calculated using . the III. RESULTS

(4) and (5)

A. Phantom Study 1 The results are shown in Fig. 2 for the first phantom study consisting of only the cardiac insert. The total counts for the ; for the fan-beam parallel-hole collimator study was ; and for the cone-beam only the count only the count was was . Keep in mind that the fan-beam count is the sum


147

TABLE I PHANTOM STUDY (ISOLATED CARDIAC INSERT) WITH THREE PARALLEL COLLIMATED DETECTORS COMPARED WITH THE COMBINATION OF TWO CONE-BEAM PLUS ONE FAN-BEAM COLLIMATED DETECTORS. THE RESULTS OF PARALLEL RECONSTRUCTION ARE COMPARED WITH COMBINED CONE-BEAM/FAN-BEAM RECONSTRUCTION FOR DIFFERENT WEIGHTING FACTORS . THE STATISTICAL MEASURES WERE OBTAINED FROM RECONSTRUCTIONS OF THE PHANTOM DATA USING 10 ITERATIONS OF THE ML-EM ALGORITHM. ALL STATISTICAL MEASURES ARE DEFINED IN THE TEXT AND WERE CALCULATED FOR AN ROI OF 11 VOXELS LOCATED ALONG A CONTINUOUS STRIP OF THE WALL OF THE INSERT. EACH VOXEL HAD A DIMENSION OF 4.6 mm. THE BEST RESULTS WERE OBTAINED FOR = 0:1, WHEREAS THE RESULT FOR = 0:9 SEEMS TO BE AN OUTLIER

of projection counts for the complete revolution of one detector data set, whereas the parallel count is the sum for the complete revolution of three detector data sets, and the cone-beam count is the sum for the complete revolution of two detector data sets. The sum of the one fan-beam data set and the two cone-beam data sets represents an increase in counts 2.7 times greater than that of the total counts from the sum of the three parallel-beam data sets. The top row in Fig. 2 compares the reconstructed long axis slices (no interpolation) of combined cone-beam/fan-beam, cone-beam only, fan-beam only, and parallel-beam only data. The bottom row shows the short axis slices. Since the cardiac insert was placed on the imaging bed with the long axis of the insert near to and aligned with the rotation axis of the SPECT system, the images display coronal sections through transaxial slices such as those shown in the top row. The results were obtained with 10 iterations of the ML-EM algorithm with the . factor It appears qualitatively that all reconstructions give equivalent results, that is, there do not appear to be any artifacts. However, as expected, upon closer inspection the most noise appears in the fan-beam only results followed by less noise in the parallel-beam only, still less in the cone-beam only, and finally the least in the combined cone-beam/fan-beam results. The combined cone-beam/fan-beam results included data from two cone-beam collimated detectors and one fan-beam collimated detector whereas the cone-beam results included data from two cone-beam collimated detectors, the fan-beam results included data from one fan-beam collimated detector, and the parallel-beam results included data from three parallel-hole collimated detectors. Therefore, the total number of counts in each reconstruction correlates with the quality seen in the images. Table I gives the quantitative comparison for the phantom study as a function of the weighting factor . The mean and variance was computed for an ROI of 11 voxels located along a continuous strip of the wall of the insert. The ROI was fixed for all images. By comparing the normalized standard deviation it

is immediately obvious that for all (except the outlier for ) the combined cone-beam/fan-beam had a lower normalized standard deviation than did the parallel-beam only; as well, the cone-beam only had a lower normalized standard deviation than . the fan-beam only. The best result was obtained for In Table II, the parallel collimation results are compared with the sum of one fan-beam data set and one cone-beam data set. Whereas in Table I the parallel collimation results, which are the sum of three data sets, are compared with the results of the combination of one fan-beam data set and two cone-beam data sets. In Table I the best result (lower normalized standard deviation) was obtained at whereas in Table II the best result was obtained for . This indicates that as the noise increases in the cone-beam data, the cone-beam data become less important; alternatively it becomes more important to weight the good fan-beam data, that is, data that satisfy the data consistency condition. B. Phantom Study 2 The results are shown in Fig. 3 for the Jaszczak cardiac torso phantom (activity only in the outer chamber of the cardiac insert). The total counts for the parallel-hole collimator study was ; for the fan-beam only the count was ; and for the cone-beam only the count was . The sum of the one fan-beam data set and the two cone-beam data sets represents an increase in counts 1.7 times greater than that of the total counts from the sum of the three parallel-beam data sets alone. The top row in Fig. 3 compares the reconstructed transaxial slices of combined cone-beam/fan-beam, cone-beam only, fanbeam only, and parallel-beam only data. The line of the profile is indicated which is shown in the next row. The bottom row shows the transaxial slices without the profile location indicated. The results were obtained with 10 iterations of the ML-EM algo. rithm with the factor As with the first phantom study, the most noise appears in the fan-beam only results followed by less noise in the par-

148


TABLE II PHANTOM STUDY (ISOLATED CARDIAC INSERT) WITH THREE PARALLEL-HOLE COLLIMATED DETECTORS COMPARED WITH THE COMBINATION OF ONE CONE-BEAM PLUS ONE FAN-BEAM COLLIMATED DETECTORS. THE RESULTS OF PARALLEL RECONSTRUCTION ARE COMPARED WITH COMBINED CONE-BEAM/FAN-BEAM RECONSTRUCTION FOR DIFFERENT WEIGHTING FACTORS . THESE RESULTS DIFFER FROM THOSE IN TABLE I IN THAT THE COMBINED CONE-BEAM/FAN-BEAM RECONSTRUCTION INCLUDES ONLY ONE CONE-BEAM DETECTOR DATA SET PLUS ONE FAN-BEAM DATA SET. THE STATISTICAL MEASURES WERE OBTAINED FROM RECONSTRUCTIONS OF THE PHANTOM DATA USING 10 ITERATIONS OF THE ML-EM ALGORITHM. THE STATISTICAL MEASURES WERE CALCULATED FOR THE SAME ROI OF 11 VOXELS AS IN TABLE I. EACH VOXEL WAS 4.6 mm. THE BEST RESULTS WERE OBTAINED FOR = 0:3 AS COMPARED WITH = 0:1 FOR THE RESULTS IN TABLE I

Fig. 3. Phantom study (Jaszczak torso phantom with cardiac insert). (Reconstructions, left to right: combined cone-beam/fan-beam, cone-beam, fan-beam, parallel). The top row shows 64 64-transaxial images of the cardiac insert in the Jaszazck cardiac torso phantom. The voxel size is 4.67 mm. The profiles in the next row are plots of intensities along the corresponding line fiducial in the upper row of images. The bottom row is the same set of images as the top row but without the line fiducial. The results were obtained with 10 iterations of the ML-EM algorithm with a = 0:5.

2

allel-beam only, still less in the cone-beam only, and finally the least in the combined cone-beam/fan-beam results. Therefore, the total number of counts in each reconstruction correlates with the quality seen in the images. This is also corroborated in Table III. By comparing the normalized standard deviation it is immediately obvious that for all the combined cone-beam/fan-beam had a lower normalized standard deviation than did the parallel-beam only; as well, the cone-beam only had a lower normalized standard deviation than the fan-beam only. . The best result was obtained for In Table IV the parallel collimation results are compared with the sum of one fan-beam data set and one cone-beam data set. In

Table III the best result (lower normalized standard deviation) whereas in Table IV the best result was obtained at . As with the first phantom study, the is obtained for noise increases in the cone-beam data, the cone-beam data become less important; alternatively it becomes more important to weight the good fan-beam data, that is, data that satisfy the data consistency condition. However, in this study it is even more important to weight the fan-beam data, because with fewer counts than in the first phantom study the cone-beam data becomes even less important. In analyzing the data in Tables I–IV, there is one point of are not given. These caution. The errors for the term calculations are complicated because the term is a function of two random variables that are correlated. Also, since the term is not normally distributed, any error probability, such as a 0.667 probability for a one confidence interval in a normal distribution, usually corresponds to asymmetric limits about the mean of a nonnormal probability distribution. Thus, it is easier from Monte Carlo to calculate the errors of the term simulations or to obtain them from many more realizations than the 10 that we acquired. A measure of the errors would make our results more meaningful since large errors in the term would lessen our confidence in the trends shown in Tables I–IV. C. Patient Studies Figs. 4 and 5, show representative scans from two patients. For the first patient study, the total counts for the parallel- hole ; for the combined cone-beam/ collimator study was , giving a gain in sensitivity of 1.75 fan-beam was (a gain of 2.2 if we correct for decay). For the second patient study, the total counts for the parallel-hole collimator ; for the combined cone-beam/fan-beam study was , giving a gain in sensitivity of 2.73. study was The images demonstrate the improved image quality of combined cone-beam/fan-beam cardiac SPECT. For both patients the background in the parallel collimation study is much


149

TABLE III PHANTOM STUDY (JASZCZAK TORSO PHANTOM WITH CARDIAC INSERT) WITH THREE PARALLEL COLLIMATED DETECTORS COMPARED WITH THE COMBINATION OF TWO CONE-BEAM PLUS ONE FAN-BEAM COLLIMATED DETECTORS. THE RESULTS OF PARALLEL RECONSTRUCTION ARE COMPARED WITH COMBINED CONE-BEAM/FAN-BEAM RECONSTRUCTION FOR DIFFERENT WEIGHTING FACTORS . THE STATISTICAL MEASURES WERE OBTAINED FROM RECONSTRUCTIONS OF THE PHANTOM DATA USING 10 ITERATIONS OF THE ML-EM ALGORITHM. ALL STATISTICAL MEASURES ARE DEFINED IN THE TEXT AND WERE CALCULATED FROM AN ENSEMBLE OF TEN SEPARATE STUDIES WHEREAS THE RESULTS PRESENTED IN TABLES I AND II WERE OBTAINED FROM ONE REALIZATION. EACH VOXEL HAD A DIMENSION OF 4.6 mm. THE BEST RESULTS WERE OBTAINED FOR = 0:2

TABLE IV PHANTOM STUDY (JASZCZAK TORSO PHANTOM WITH CARDIAC INSERT) WITH THREE PARALLEL-HOLE COLLIMATED DETECTORS COMPARED WITH THE COMBINATION OF ONE CONE-BEAM PLUS ONE FAN-BEAM COLLIMATED DETECTORS. THE RESULTS OF PARALLEL RECONSTRUCTION ARE COMPARED WITH COMBINED CONE-BEAM/FAN-BEAM RECONSTRUCTION FOR DIFFERENT WEIGHTING FACTORS . THESE RESULTS DIFFER FROM THOSE IN TABLE III IN THAT THE COMBINED CONE-BEAM/FAN-BEAM RECONSTRUCTION INCLUDES ONLY ONE CONE-BEAM DETECTOR DATA SET PLUS ONE FAN-BEAM DATA SET. THE STATISTICAL MEASURES WERE OBTAINED FROM RECONSTRUCTIONS OF THE PHANTOM DATA USING 10 ITERATIONS OF THE ML-EM ALGORITHM. THE STATISTICAL MEASURES WERE CALCULATED THE SAME AS IN TABLE III. EACH VOXEL WAS 4.6 mm. THE BEST RESULTS WERE OBTAINED FOR = 0:4 AS COMPARED WITH = 0:2 FOR THE RESULTS IN TABLE III

higher than in the convergent beam study. For Patient 1, 1 h 30 min elapsed between the two scans, during which time the -sestamibi washed out from the background tissue. For Patient 2 only 10 min elapsed between the two scans but this -sestamibi was sufficient to show significant washout of from the background tissue. Upon close observation, it is qualitatively apparent that the noise in the fan-beam reconstructions is greater than the noise in the parallel reconstructions, which is still greater than the noise in the cone-beam and the combined fan-beam/cone-beam reconstructions. It is recognized that there can be differences, especially for Patient 1, in the time of imaging after cessation of stress-induced exercise for the parallel-hole and convergent-beam collimator studies (a difference of 1 h 30 min between the parallel study and the combined fan-beam/cone-beam study). This dif-

ference was less for Patient 2 with a difference of only 10 min. Since redistribution occurs, it is expected that there will be isotope distribution differences. Therefore, the cone-beam and parallel-hole collimator images might be expected to be somewhat discordant for Patient 1. IV. DISCUSSION This work was originally intended to be accomplished using combined cone-beam/fan-beam collimators with the STEP system [57] (Picker, International, Cleveland, OH) for simultaneous transmission-emission imaging [19], [41], [58], [59]. The fan-beam collimator was to be secured to the simultaneous transmission-emission detector and the cone-beam collimators were to be secured to the other two detectors that acquire

150


Fig. 4. Patient 1 (reconstructions, left to right: combined cone-beam/ 64fan-beam, cone-beam, fan-beam, parallel). The top row shows 64 transaxial images of 7.0 mm voxels. The profiles in the next row are plots of intensities along the corresponding line fiducial in the upper row of images. The next row is the same set of images as the top row but without the line fiducial. The bottom row shows short axis views zoomed to focus more on the heart itself. The results were obtained with 10 iterations of the ML-EM algorithm with a = 0:5.

2

Fig. 5. Patient 2 (reconstructions, left to right: combined cone-beam/ fan-beam, cone-beam, fan-beam, parallel). The top row shows 64 64transaxial images of 4.67 mm voxels. The profiles in the next row are plots of intensities along the corresponding line fiducial in the upper row of images. The next row is the same set of images as the top row but without the line fiducial. The bottom row are short axis views zoomed to better visualize the heart. The results were obtained with 10 iterations of the ML-EM algorithm with a = 0:5.

2

emission data only. The attenuation of the emission distribution was to be corrected using information derived from the reconstruction of the transmission data. Combining cone-beam collimation with simultaneous transmission-emission imaging on a three-detector SPECT system offers the advantage of improved geometric efficiency when compared with fan-beam collimated three-detector SPECT systems and two-detector (orthogonally oriented) SPECT systems with parallel-hole col-

limators. In addition this detector arrangement solves the data insufficiency problem of planar orbit cone-beam tomography. However, the focus of this work changed with the development of the IRIX three-detector SPECT system. With the availability of this system the decision was made to test the combined cone-beam/fan-beam collimation concept utilizing the IRIX’s advanced capabilities. With this system attenuation correction is accomplished using the Beacon device, which uses Ba-133 scanning point sources (with moving electronic windows) to penetrate both the patient body and collimators. We intended to use this attenuation correction capability in our work, however we were not able to accomplish body contouring orbits for the emission study when we selected the Beacon protocol. Therefore, attenuation correction was not included in the reconstruction results. Nevertheless, the results of our work do have important implications not only for the IRIX SPECT system but also for the older PRISM 3000XP STEP system. Such a collimator configuration will be useful in performing simultaneous transmission/emission studies in animals, as well as in SPECT imaging of the brain. Combining cone-beam and fan-beam collimators in a SPECT system produces high detection sensitivity (due to cone-beam collimation) and complete tomographic measurements (due to fan-beam collimation). The images show that the noise structure correlates with the counts obtained regardless of what collimator combination is employed. That is, a 360 rotation of a single fan-beam collimated detector acquire fewer counts than three parallel-hole collimated detectors, which acquire fewer counts than two cone-beam collimated detectors, which in turn acquire fewer counts than combined cone-beam/fan-beam collimated detectors. However, the statistics in nuclear medicine imaging can often mask reconstruction artifacts [60]. One can see ring artifacts in noiseless cone-beam reconstructions (transaxial views) [44]. These are due to line-length weighting that intersects each voxel with the same line length for each projection view. The ring artifacts may not be visible in reconstructions with noisy data because they are buried under the noise. With the addition of completed data from fan-beam projections it can be seen from the measured normalized standard deviation that the artifacts are reduced. Even though the combined use of cone-beam and fan-beam collimation increases photon detection efficiency compared with parallel-beam collimation with similar spatial resolution characteristics, the determination of the sensitivity gain for imaging the heart is complicated by other factors, such as the distance of the collimators from the heart (varies with patient size and positioning of the heart in the chest) and tissue attenuation. Scanning an isolated heart insert in air gave a gain of 2.7 whereas scanning the cardiac insert in the torso phantom (more realistic of patient imaging) resulted in a gain of 1.7. For one patient study, a gain of 2.2 was obtained, whereas for the other patient study a gain of 2.7 was obtained. Though care was taken to obtain counts in the projections of just the heart in the field of view, the measurements in patients are probably biased by activity in the background, liver, and other organs. Since the heart does not sit in the center of the body it can be a difficult organ to image and the gain in sensitivity with converging collimation will vary with patient size and the positioning of the heart in the chest.


To design a collimator for a particular imaging task the imaging time, resolution, and sensitivity must be optimized. The optimizations are best determined through human observer studies using receiver operating characteristic analysis [45]. Alternatively, they can be estimated using a mathematical observer such as a channelized Hotelling-trace mathematical observer [61], [62]. Also, it is important to note that most of the cone-beam heart slices are located near the central plane, which is reconstructed with a more volume-symmetric point response than slices located further from the central plane. Understanding the implications of using an asymmetric point response, as well as achieving any better understanding of the reconstructed texture [63] in terms of the effects these factors have on lesion detectability, will require further study. Finally, we have found that the 65-cm focal length cone-beam collimator with the large field of view camera is adequate for truncation-free imaging for most heart sizes in the patient population. The reconstruction of combined cone-beam/fan-beam data involves the solution to two large systems of equations, one for the cone-beam projection data and one for the fan-beam projection data. The condition number for the fan-beam system is much lower than that for the cone-beam system [51]. Therefore, the fan-beam reconstructions are less sensitive to model mismatch errors than the cone-beam reconstructions. It has also been shown that the bias for a fan-beam-only system is less than that for a cone-beam-only system. However, a properly combined fan-beam and cone-beam system has less overall bias (even when the projection data are noiseless) and better conditioning than either the fan-beam or cone-beam system alone. It has been observed that the bias is different for varied distances between the combined cone-beam/fan-beam detector and the center of rotation (the assumed location of where the object is being imaged). This indicates that the best combination of the two systems depends upon the location of the rotation axis. When the distance is increased the object is closer to the focal point, and more projection rays intersect the voxels of the object. A finer sampling of the voxel is equivalent to having more equations for the same number of unknowns, resulting in a better conditioning. On the other hand, when the distance is decreased the cone-beam “incomplete data” problem becomes less severe, resulting in less bias and a better condition number. If the distance is decreased further, the cone-beam begins to resemble the parallel-beam geometry. As a result the “incomplete data” problem becomes less dominant, but the advantages of using the convergent imaging geometry also vanish. As well, the projected image is smaller on the detector and the sampling intervals relative to the object become larger. Even though the “incomplete data” problem becomes less dominant, the overall condition number still is worse. If one were to study the bias and system condition numbers for different weighting schemes of cone-beam and fan-beam data, it could be observed that neither the minimum bias nor the minimum system condition number can be reached for the same weighting of cone-beam and fan-beam data [51]. The disagreement between the bias and condition number results implies that the condition number is not the only indicator for choosing an optimal weighting, as there are other factors that affect the results such as the object, projection data, system perturbation, the

151

shape of the singular value spectrum, systematic and random noise, the choice of an iterative algorithm, the number of iterations, and regularization. These comments and observations are based upon bias and condition numbers whereas the results we observed in this paper were based upon varying to obtain optimum noise variance, which is not the same. It is recognized that improvements in image quality can be achieved by carefully choosing the appropriate weighting of the fan-beam and cone-beam data. The weighting can be chosen to either emphasize the fan-beam data to reduce the cone-beam artifacts or to emphasize the cone-beam data to improve the signal to noise. In fact, a more effective weighting scheme is to weight each individual projection ray separately. This scheme however is complex and is beyond the scope of this paper. Finally, we must point out that when the image model in the projector/backprojector pair is sufficiently accurate, the weighting factor becomes unnecessary and it is then only necessary to consider the noise in the projection data as in the development of the ML-EM algorithm. Also, when the Poisson noise in the projection data dominates the model mismatch errors the “naïve” weighting is appropriate, as was done in this paper. Cone-beam collimation has the potential to provide a significant increase in sensitivity over that obtained with present cardiac SPECT imaging with parallel-hole collimators. A multi-detector SPECT system combining fan-beam and cone-beam collimators has the potential to improve geometric efficiency and provide sufficient data for accurate cone-beam reconstruction when performing cardiac imaging with myocardial radiopharmaceuticals. Today, the rotating scintillation camera is the most widely used SPECT system in clinical nuclear medicine because of its ability to perform body and brain SPECT imaging as well as conventional planar imaging. That being the case, it is essential that improvements be made in collimation in effort to improve clinical nuclear cardiology. ACKNOWLEDGMENT The authors thank S. Webb for careful editing of the manuscript. REFERENCES [1] E. Botvinick, Ed., Nuclear Medicine Self Study Program III—Nuclear Cardiology Topic 6—Myocardial Perfusion Scintigraphy—Clinical Aspects. Reston, MD: Society Nuclear Medicine, 2001. [2] K. A. McKusick, B. L. Holman, A. G. Jones, A. Davison, P. Rigo, L. Guemes, A. H. Vosberg, and J. Moretti, “Comparison of three Tc-99m isonitriles for detection of ischemic heart disease in humans,” J. Nucl. Med., vol. 27, p. 878, 1986. [3] S. J. McPhee and D. W. Garnick, “Imaging the heart: cardiac scintigraphy and echocardiography in U.S. hospitals (1983),” J. Nucl. Med., vol. 27, pp. 1635–1641, 1986. [4] H. W. Strauss and E. L. Palmer, “Cardiovascular nuclear medicine— training for the future,” J. Nucl. Med., vol. 27, pp. 1642–1643, 1986. [5] K. A. Narahar, J. Villanueva-Meyer, C. J. Thompson, M. Brizendine, and I. Mena, “Comparison of thallium-201 and technetium-99m hexakis 2-methoxyisobutyl isonitrile single-photon emission computed tomography for estimating the extent of myocardial ischemia and infarction in coronary artery disease,” Amer. J. Cardiol., vol. 66, pp. 1438–1444, 1990. [6] J. A. Leppo, E. G. DePuey, and L. L. Johnson, “Cardiac imaging with sestamibi and teboroxime,” J. Nuc. Med., vol. 32, pp. 2012–2022, 1991. [7] F. L. Datz, G. T. Gullberg, F. V. Gabor, and K. A. Morton, “SPECT myocardial perfusion imaging update,” Seminars in Ultrasonic, CT, and MR, vol. 12, pp. 28–44, 1991.

152

[8] F. L. Datz, F. V. Gabor, P. E. Christian, G. T. Gullberg, C. E. Menzel, and K. A. Morton, “The use of computer-assisted diagnosis in cardiac perfusion nuclear medicine studies: a review (part I),” J. Digital. Imag., vol. 5, pp. 209–222, 1992. [9] J. C. Maublant, X. Marcaggi, J.-R. Lusson, J.-Y. Boire, J.-C. Cauvin, P. Jacob, A. Veyre, and J. Cassagnes, “Comparison between thallium-201 and technetium-99m methoxyisobutyl isonitrile defect size in singlephoton emission computed tomography at rest, exercise and redistribution in coronary artery disease,” Amer. J. Cardiol., vol. 69, pp. 183–187, 1992. [10] A. Cuocolo, L. Pace, B. Ricciardelli, M. Chiariello, B. Trimarco, and M. Salvatore, “Identification of viable myocardium in patients with chronic coronary artery disease: comparison of thallium-201 scintigraphy with reinjection and technetium-99m-methoxyisobutyl isonitrile,” J. Nucl. Med., vol. 33, pp. 505–511, 1992. [11] A. N. Serafini, S. Topchik, H. Jimenez, A. Friden, W. I. Ganz, and G. N. Sfakianakis, “Clinical comparison of technetium-99m-teboroxime and thallium-201 utilizing a continuous SPECT imaging protocol,” J. Nucl. Med., vol. 3, pp. 1304–1311, 1992. [12] K. C. Allman, J. Berry, L. A. Sucharski, K. A. Stafford, N. A. Petry, W. Wysor, and M. Schwaiger, “Determination of extent and location of coronary artery disease in patients without prior myocardial infarction by thallium-201 tomography with pharmacologic stress,” J. Nucl. Med., vol. 33, pp. 2067–2073, 1992. [13] F. L. Datz, F. V. Gabor, P. E. Christian, G. T. Gullberg, C. E. Menzel, and K. A. Morton, “The use of computer-assisted diagnosis in cardiac perfusion nuclear medicine studies: a review (part II),” J. Digital. Imag., vol. 6, pp. 1–15, 1993. [14] F. L. Datz, C. Rosenberg, F. V. Gabor, P. E. Christian, G. T. Gullberg, R. Ahluwalia, and K. A. Morton, “The use of computer-assisted diagnosis in cardiac perfusion nuclear medicine studies: a review (part III),” J. Digital. Imag., vol. 6, pp. 67–80, 1993. [15] B. Samman, S. D. Hahn, D. E. Messinger, and G. V. Heller, “Spontaneous silent myocardial ischemia assessed by technetium-99m-sestamibi imaging,” J. Nucl. Med., vol. 34, pp. 134–136, 1993. [16] F. Mannting and M. G. Morgan-Mannting, “Gated SPECT with technetium-99m-sestamibi for assessment of myocardial perfusion abnormalities,” J. Nucl. Med., vol. 34, pp. 601–608, 1993. [17] B. Gunalp, B. Dokumaci, C. Uyan, E. Vardareli, E. Isik, H. Bayhan, M. Ozguven, and E. Ozturk, “Value of dobutamine technetium-99m-sestamibi SPECT and echocardiography in the detection of coronary artery disease compared with coronary angiography,” J. Nucl. Med., vol. 34, pp. 889–894, 1993. [18] K. F. Van Train, J. Areeda, E. V. Garcia, D. Cooke, J. Maddahi, H. Kiat, G. Germano, G. Silagan, R. Folks, and D. S. Berman, “Quantitative same-day rest-stress technetium-99m-sestamibi SPECT: definition and validation of stress normal limits and criteria for abnormality,” J. Nucl. Med., vol. 34, pp. 1494–1502, 1993. [19] F. L. Datz, G. T. Gullberg, G. L. Zeng, C.-H. Tung, P. E. Christian, A. Welch, and R. Clack, “Application of convergent-beam collimation and simultaneous transmission emission tomography to cardiac SPECT imaging,” Semin. Nucl. Med., vol. 24, pp. 17–37, 1994. [20] G. Gioia, E. Milan, R. Giubbini, N. DePace, J. Heo, and A. S. Iskandrian, “Prognostic value of tomographic rest-redistribution thallium 201 imaging in medically treated patients with ventricular dysfunction,” J. Nucl. Cardiol., vol. 3, pp. 150–156, 1996. [21] C. Maunoury, C. C. Chen, K. B. Chua, and C. J. Thompson, “Quantification of left ventricular function with thallium-201 and technetium-99msestamibi myocardial gated SPECT,” J. Nucl. Med., vol. 38, pp. 958–961, 1996. [22] D. S. Berman, R. Hachamovitch, and H. Kiat et al., “Incremental value of prognostic testing in patients with known or suspected ischemic heart disease: a basis for optimal utilization of exercise technetium-99m sestamibi myocardial perfusion single-photon emission computed tomography,” J. Amer. Coll. Cardiol., vol. 26, pp. 639–647, 1995. [23] B. Canhasi, M. Dae, and E. Botvinick et al., “Interaction of “supplementary” scintigraphic indicators and stress electrocardiography in the diagnosis of multi-vessel coronary disease,” J. Amer. Coll. Cardiol., vol. 6, pp. 581–587, 1985. [24] S. T. Dahlberg and J. A. Leppo, “Physiologic properties of myocardial perfusion tracers,” Cardiol. Clin., vol. 12, pp. 169–186, 1994. [25] R. Hachamovitch, D. S. Berman, and H. Kiat et al., “Comparison of incremental prognostic value, risk stratification and cost effectiveness of rest/exercise Tl-201/Tc-99m sestamibi SPECT in women and men,” J. Amer. Coll. Cardiol., vol. 28, pp. 34–44, 1996.


[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35] [36] [37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

, “Exercise myocardial perfusion SPECT in patients without known coronary artery disease: incremental prognostic value and impact on subsequent patient management,” Circulation, vol. 93, pp. 905–914, 1996. R. Hachamovitch, D. S. Berman, and L. J. Shaw et al., “Incremental prognostic value of myocardial perfusion single photon emission computed tomography for the prediction of cardiac death: differential stratification for risk of cardiac death and myocardial infarction,” Circulation, vol. 97, pp. 535–543, 1998. G. V. Heller, S. D. Herman, and M. I. Travin et al., “Independent prognostic value of intravenous dipyridamole with technetium-99m sestamibi tomographic imaging in predicting cardiac events and cardiac-related hospital admissions,” J. Amer. Coll. Cardiol., vol. 26, pp. 1202–1208, 1995. R. Krishnan, J. Lu, M. W. Dae, and E. H. Botvinick, “Does myocardial perfusion scintigraphy demonstrate clinical utility in patients with markedly positive exercise test?—an assessment of the method in a highrisk subset,” Amer. Heart J., vol. 127, pp. 804–815, 1993. M. Ladenheim, T. Kotler, and B. Pollock et al., “Incremental prognostic power of clinical history, exercise electrocardiography and myocardial perfusion scintigraphy in suspected coronary artery disease,” Amer. J. Cardiol., vol. 59, pp. 270–277, 1987. T. Sharir, G. Germano, and P. B. Kavanagh et al., “Incremental prognostic value of post-stress left ventricular ejection fraction and volume by gated myocardial perfusion single photon emission computed tomography,” Circulation, vol. 100, pp. 1035–1042, 1999. L. J. Shaw, R. Hachamovitch, and D. S. Berman et al., “The economic consequences of available diagnostic and prognostic strategies for the evaluation of stable angina patients: an observation assessment of the value of precatherization ischemia. For the Economics of Noninvasive Diagnosis (END) multicenter study group,” J. Amer. Cardiol., vol. 33, pp. 661–669, 1999. G. T. Gullberg, P. E. Christian, G. L. Zeng, F. L. Datz, and H. T. Morgan, “Cone beam tomography of the heart using single-photon emission-computed tomography,” Invest. Radiol., vol. 26, pp. 681–688, 1991. G. T. Gullberg and G. L. Zeng, “A cone beam filtered backprojection reconstruction algorithm for cardiac single photon emission computed tomography,” IEEE Trans. Med. Imag., vol. 11, pp. 91–101, 1992. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Amer. [A], vol. 1, pp. 612–619, 1984. H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math., vol. 43, pp. 546–552, 1983. R. J. Jaszczak, J. Li, H. Wang, and R. E. Coleman, “Three-dimensional SPECT reconstruction of combined cone beam and parallel beam data,” Phys. Med. Biol., vol. 37, pp. 535–548, 1992. M. Defrise and R. Clack, “Filtered-backprojection reconstruction of combined parallel beam and cone beam SPECT data,” Phys. Med. Biol., vol. 40, pp. 1517–1537, 1995. G. T. Gullberg and G. L. Zeng, “Three-dimensional SPECT reconstruction of combined cone-beam and fan-beam data acquired using a three-detector SPECT system,” in Proc. Int. Meet. Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, P. Grangeat, Ed., Aix-les-Bains, Savoie, France, Jul. 4–6, 1995, pp. 329–332. G. T. Gullberg, G. L. Zeng, B. M. W. Tsui, and J. T. Hagius, “An iterative reconstruction algorithm for single photon emission computed tomography with cone beam geometry,” Int. J. Imag. Sys. Technol., vol. 1, pp. 169–186, 1989. G. T. Gullberg, G. L. Zeng, F. L. Datz, P. E. Christian, C.-H. Tung, and H. T. Morgan, “Review of convergent beam tomography in single photon emission computed tomography,” Phys. Med. Biol., vol. 37, pp. 507–534, 1992. G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, and J. A. Terry, “Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci., vol. 38, pp. 693–702, 1991. G. L. Zeng, Y.-L. Hsieh, and G. T. Gullberg, “A rotating and warping projector-backprojector pair for fan-beam and cone-beam iterative algorithms,” IEEE Trans. Nucl. Sci., vol. 41, pp. 2807–2811, 1994. G. L. Zeng and G. T. Gullberg, “A study of reconstruction artifacts in cone beam tomography using filtered backprojection and iterative EM algorithms,” IEEE Trans. Nucl. Sci., vol. 37, pp. 759–767, 1990. B. M. W. Tsui, J. A. Terry, and G. T. Gullberg, “An observer evaluation study of defect detection in cardiac SPECT using cone beam collimation,” Invest. Radiol., vol. 28, pp. 1101–1112, 1993.


[46] G. L. Zeng, Y. Weng, and G. T. Gullberg, “Iterative reconstruction with attenuation compensation from cone-beam projections acquired via nonplanar orbits,” IEEE Trans. Nucl. Sci., vol. 44, pp. 98–106, 1997. [47] G. L. Zeng and G. T. Gullberg, “Iterative and analytical reconstruction algorithms for varying-focal-length cone-beam projections,” Phys. Med. Biol., vol. 43, pp. 811–821, 1998. , “Helical SPECT using axially truncated data,” IEEE Trans. Nucl. [48] Sci., vol. 46, pp. 2111–2118, 1999. [49] G. L. Zeng, G. T. Gullberg, P. Christian, and D. Gagnon, “Cone-beam iterative reconstruction of a segment of a long object,” IEEE Trans. Nucl. Sci., vol. 49, pp. 37–41, 2002. [50] G. L. Zeng, G. T. Gullberg, and S. A. Foresti, “Eigen analysis of cone-beam scanning geometries,” in Computational Imaging and Vision Series: Three-Dimensional Imaging Reconstruction in Radiology and Nuclear Medicine, P. Grangeat and J.-L. Amans, Eds. Norwell, MA: Kluwer, 1996, pp. 75–86. [51] G. T. Gullberg and G. L. Zeng, “On combination of cone-beam and fanbeam projections in solving a linear system of equations,” in Conf. Rec. Int. Meet. Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine, P. Kinahan and D. Townsend, Eds., Nemacolin Woodlands, PA, Jun. 25–28, 1997, pp. 113–116. [52] B. M. W. Tsui and G. T. Gullberg, “The geometric transfer function for cone and fan beam collimators,” Phy. Med. Biol., vol. 35, pp. 81–93, 1990. [53] E. C. Frey, B. M. W. Tsui, and G. T. Gullberg, “Improved estimation of the detector response function for converging beam collimators,” Phys. Med. Biol., vol. 43, pp. 941–950, 1998. [54] K. Lang and R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr., vol. 8, pp. 306–316, 1984.

153

[55] R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, Users Manual—Donner Algorithms for Reconstruction Tomography. Berkeley, CA: Lawrence Berkeley Lab., 1977. [56] H. Stark and J. W. Woods, Probability and Random Processes With Applications to Signal Processing, 3 ed, NJ: Prentice-Hall, 2002, p. 286. [57] G. T. Gullberg, H. T. Morgan, G. L. Zeng, P. E. Christian, E. V. R. Di Bella, C.-H. Tung, P. J. Maniawski, Y.-L. Hsieh, and F. L. Datz, “The design and performance of a simultaneous transmission and emission tomography system,” IEEE Trans. Nucl. Sci., vol. 45, pp. 1676–1698, 1998. [58] C.-H. Tung, G. T. Gullberg, G. L. Zeng, P. E. Christian, F. L. Datz, and H. T. Morgan, “Nonuniform attenuation correction using simultaneous transmission and emission converging tomography,” IEEE Trans. Nucl. Sci., vol. 39, pp. 1134–1143, 1992. [59] E. P. Ficaro, J. A. Fessler, P. D. Shreve, J. N. Kritzman, P. A. Rose, and J. R. Corbett, “Simultaneous transmission/emission myocardial perfusion tomography,” Circulation, vol. 93, pp. 463–473, 1996. [60] G. T. Gullberg, “An analytical approach to quantify uniformity artifacts for circular and noncircular detector motion in SPECT imaging,” Med. Phys., vol. 14, pp. 105–114, 1987. [61] S. D. Wollenweber, B. M. W. Tsui, D. S. Lalush, E. C. Frey, and G. T. Gullberg, “Evaluation of myocardial defect detection between parallelhole and fan-beam SPECT using the Hotelling trace,” IEEE Trans. Nucl. Sci., vol. 45, pp. 2205–2210, 1998. [62] G. L. Zeng and G. T. Gullberg, “A channelized Hotelling-trace collimator design method based on reconstruction rather than projections,” IEEE Trans. Nucl. Sci., vol. 49, pp. 2155–2158, 2002. [63] L. J. Hahn, R. J. Jaszczak, G. T. Gullberg, C. E. Floyd, S. H. Manglos, K. L. Greer, and R. E. Coleman, “Noise characteristics for cone beam collimators: a comparison with parallel hole collimator,” Phy. Med. Biol., vol. 33, pp. 541–555, 1988.