An evaluation of maximum likelihood reconstruction for SPECT

A reconstruction method for SPECT (single photon emission computerized tomography) that uses the maximum likelihood (ML) criterion and an iterative expectation-maximization (EM) algorithm solution is examined. The method is based on a model that incorporates the physical effects of photon statistics, nonuniform photon attenuation, and a camera-dependent point-spread response function. Reconstructions from simulation experiments are presented which illustrate the ability of the ML algorithm to correct for attenuation and point-spread. Standard filtered backprojection method reconstructions, using experimental and simulated data, are included for reference. Three studies were designed to focus on the effects of noise and point-spread, on the effect of nonuniform attenuation, and on the combined effects of all three. The last study uses a chest phantom and simulates Tl-201 imaging of the myocardium. A quantitative analysis of the reconstructed images is used to support the conclusion that the ML algorithm produces reconstructions that exhibit improved signal-to-noise ratios, improved image resolution, and image quantifiability.     

Automatic Contour Detection Using a "Fixed-Point Hachimura-Kuwahara Filter" for SPECT Attenuation Correction

Attenuation correction for single-photon emission computed tomography (SPECT) usually assumes a uniform attenuation distribution within the body surface contour. Previous methods to estimate this contour have used thresholding of a reconstructed section image. This method is often very sensitive to the selection of a threshold value, especially for nonuniform activity distributions within the body. We have proposed the "fixed-point Hachimura-Kuwahara filter" to extract contour primitives from SPECT images. The Hachimura-Kuwahara filter, which preserves edges but smoothes nonedge regions, is applied repeatedly to identify the invariant set-the fixed-point image-which is unchanged by this nonlinear, two-dimensional filtering operation. This image usually becomes a piecewise constant array. In order to detect the contour, the tracing algorithm based on the minimum distance connection criterion is applied to the extracted contour primitives. This procedure does not require choice of a threshold value in determining the contour. SPECT data from a water-filled elliptical phantom containing three sources was obtained and scattered projections were reconstructed. The automatic edge detection procedure was applied to the scattered window reconstruction, resulting in a reasonable outline of the phantom.     

Bayesian reconstruction and use of anatomical a priori information for emission tomography

A Bayesian method is presented for simultaneously segmenting and reconstructing emission computed tomography (ECT) images and for incorporating high-resolution, anatomical information into those reconstructions. The anatomical information is often available from other imaging modalities such as computed tomography (CT) or magnetic resonance imaging (MRI). The Bayesian procedure models the ECT radiopharmaceutical distribution as consisting of regions, such that radiopharmaceutical activity is similar throughout each region. It estimates the number of regions, the mean activity of each region, and the region classification and mean activity of each voxel. Anatomical information is incorporated by assigning higher prior probabilities to ECT segmentations in which each ECT region stays within a single anatomical region. This approach is effective because anatomical tissue type often strongly influences radiopharmaceutical uptake. The Bayesian procedure is evaluated using physically acquired single-photon emission computed tomography (SPECT) projection data and MRI for the three-dimensional (3-D) Hoffman brain phantom. A clinically realistic count level is used. A cold lesion within the brain phantom is created during the SPECT scan but not during the MRI to demonstrate that the estimation procedure can detect ECT structure that is not present anatomically.     

A projector/backprojector with slice-to-slice blurring forefficient three-dimensional scatter modeling

Scatter correction is an important factor in single photon emission computed tomography (SPECT). Many scatter correction techniques, such as multiple-window subtraction and intrinsic modeling with iterative algorithms, have been under study for many years. Previously, we developed an efficient slice-to-slice blurring technique to model attenuation and system geometric response in a projector/backprojector pair, which was used in an ML-EM algorithm to reconstruct SPECT data. This paper proposes a projector/backprojector that models the three-dimensional (3-D) first-order scatter in SPECT, also using an efficient slice-to-slice blurring technique. The scatter response is estimated from a known nonuniform attenuation distribution map. It is assumed that the probability of detection of a first-order scattered photon from a photon that is emitted in a given source voxel and scattered in a given scatter voxel is proportional to the attenuation coefficient value at that voxel. Monte Carlo simulations of point sources and an MCAT torso phantom were used to verify the accuracy of the proposed projector/backprojector model. An experimental Jaszczak torso/cardiac phantom SPECT study was also performed. For a 64 x 64 x 64 image volume, it took 8.7 s to perform each iteration per slice on a Sun ULTRA Enterprise 3000 (167 MHz, 1 Gbyte RAM) computer, when modeling 3-D scatter, attenuation, and system geometric response functions. The main advantage of the proposed method is its easy implementation and the possibility of performing reconstruction in clinically acceptable time.     

Modeling the axial extension of a transmission line source within iterative reconstruction via multiple transmission sources

Reconstruction algorithms for transmission tomography have generally assumed that the photons reaching a particular detector bin at a particular angle originate from a single point source. In this paper, we highlight several cases of extended transmission sources, in which it may be useful to approach the estimation of attenuation coefficients as a problem involving multiple transmission point sources. Examined in detail is the case of a fixed transmission line source with a fan-beam collimator. This geometry can result in attenuation images that have significant axial blur. Herein it is also shown, empirically, that extended transmission sources can result in biased estimates of the average attenuation, and an explanation is proposed. The finite axial resolution of the transmission line source configuration is modeled within iterative reconstruction using an expectation-maximization algorithm that was previously derived for estimating attenuation coefficients from single photon emission computed tomography (SPECT) emission data. The same algorithm is applicable to both problems because both can be thought of as involving multiple transmission sources. It is shown that modeling axial blur within reconstruction removes the bias in the average estimated attenuation and substantially improves the axial resolution of attenuation images.     

Attenuation Correction for SPECT: An Evaluation of Hybrid Approaches

Most methods that have been proposed for attenuation compensation in single-photon emission computed tomography (SPECT) either rely on simplifying assumptions, or use slow iteration to achieve accuracy. Recently, hybrid methods which combine iteration with simple multiplicative correction have been proposed by Chang and by Moore et al. In this study we evaluated these methods using both simulated and real phantom data from a rotating gamma camera. Of special concern were the effects of assuming constant attenuation distributions for correction and of using only 180 degrees of projection data in the reconstructions. Results were compared by means of image contrast, %RMS error, and a chi-square error statistic. Simulations showed the hybrid methods to converge after 1-2 iterations when 360 degrees data were used, less rapidly for 180 degrees data. The Moore method was more accurate than our modified Chang method for 180 degrees data. Phantom data indicated the importance of using an accurate attenuation map for both methods. The speed of convergence of the hybrid algorithms compared to traditional iterative techniques, and their accuracy in reconstructing photon activity, even with 180 degrees data, makes them attractive for use in quantitative analysis of SPECT reconstructions.     

Estimation of images and nonrigid deformations in gated emission CT

In this paper, we propose and test a new iterative algorithm to simultaneously estimate the nonrigid motion vector fields and the emission images for a complete cardiac cycle in gated cardiac emission tomography. We model the myocardium as an elastic material whose motion does not generate large amounts of strain. As a result, our method is based on minimizing an objective function consisting of the negative logarithm of a maximum likelihood image reconstruction term, the standard biomechanical model of strain energy, and an image matching term that ensures a measure of agreement of intensities between frames. Simulations are obtained using data for the four-dimensional (4-D) NCAT phantom. The data models realistic noise levels in a typical gated myocardial perfusion SPECT study. We show that our simultaneous algorithm produces images with improved spatial resolution characteristics and noise properties compared with those obtained from postsmoothed 4-D maximum likelihood methods. The simulations also demonstrate improved motion estimates over motion estimation using independently reconstructed images.     

Direct least-squares estimation of spatiotemporal distributions from dynamic SPECT projections using a spatial segmentation and temporal B-splines

Artifacts can result when reconstructing a dynamic image sequence from inconsistent, as well as insufficient and truncated, cone beam single photon emission computed tomography (SPECT) projection data acquired by a slowly rotating gantry. The artifacts can lead to biases in kinetic model parameters estimated from time-activity curves generated by overlaying volumes of interest on the images. However, the biases in time-activity curve estimates and subsequent kinetic parameter estimates can be reduced significantly by first modeling the spatial and temporal distribution of the radiopharmaceutical throughout the projected field of view, and then estimating the time-activity curves directly from the projections. This approach is potentially useful for clinical SPECT studies involving slowly rotating gantries, particularly those using a single-detector system or body contouring orbits with a multidetector system. We have implemented computationally efficient methods for fully four-dimensional (4-D) direct estimation of spatiotemporal distributions from dynamic SPECT projection data. Temporal B-splines providing various orders of temporal continuity, as well as various time samplings, were used to model the time-activity curves for segmented blood pool and tissue volumes in simulated cone beam and parallel beam cardiac data acquisitions. Least-squares estimates of time-activity curves were obtained quickly using a workstation. Given faithful spatial modeling, accurate curve estimates were obtained using cubic, quadratic, or linear B-splines and a relatively rapid time sampling during initial tracer uptake. From these curves, kinetic parameters were estimated accurately for noiseless data and with some bias for noisy data. A preliminary study of spatial segmentation errors showed that spatial model mismatch adversely affected quantitative accuracy, but also resulted in structured errors (projected model versus raw data) that were easily detected in our simulations. This suggests iterative refinement of the spatial model to reduce structured errors as an area of future research.     

Use of three-dimensional Gaussian interpolation in the projector/backprojector pair of iterative reconstruction for compensation of known rigid-body motion in SPECT

Due to the extended imaging times employed in single photon emission computed tomography (SPECT) and positron emission tomography (PET), patient motion during imaging is a common clinical occurrence. The fast and accurate correction of the three-dimensional (3-D) translational and rotational patient motion in iterative reconstruction is thus necessary to address this important cause of artifacts. We propose a method of incorporating 3-D Gaussian interpolation in the projector/backprojector pair to facilitate compensation for rigid-body motion in addition to attenuation and distance-dependent blurring. The method works as the interpolation step for moving the current emission voxel estimates and attenuation maps in the global coordinate system to the new patient location in the rotating coordinate system when calculating the expected projection. It also is employed for moving back the backprojection of the ratio of the measured projection to the expected projection and backprojection of the unit value (sensitivity factor) to the original location. MCAT simulations with known six-degree-of-freedom (6DOF) motion were employed to evaluate the accuracy of our method of motion compensation. We also tested the method with acquisitions of the data spectrum anthropomorphic phantom where motion during SPECT acquisition was measured using the Polaris IR motion tracking system. No motion artifacts were seen on the reconstructions with the motion compensation.     

Characterization of pinhole SPECT acquisition geometry

A method is presented to estimate the acquisition geometry of a pinhole single photon emission computed tomography (SPECT) camera with a circular detector orbit. This information is needed for the reconstruction of tomographic images. The calibration uses the point source projection locations of a tomographic acquisition of three point sources located at known distances from each other. It is shown that this simple phantom provides the necessary and sufficient information for the proposed calibration method. The knowledge of two of the distances between the point sources proves to be essential. The geometry is estimated by fitting analytically calculated projections to the measured ones, using a simple least squares Powell algorithm. Some mild a priori knowledge is used to constrain the solutions of the fit. Several of the geometrical parameters are however highly correlated. The effect of these correlations on the reconstructed images is evaluated in simulation studies and related to the estimation accuracy. The highly correlated detector tilt and electrical shift are shown to be the critical parameters for accurate image reconstruction. The performance of the algorithm is finally demonstrated by phantom measurements. The method is based on a single SPECT scan of a simple calibration phantom, executed immediately after the actual SPECT acquisition. The method is also applicable to cone-beam SPECT and X-ray CT.     

Reconstruction of attenuation map using discrete consistency conditions

Methods of quantitative emission computed tomography require compensation for linear photon attenuation. A current trend in single-photon emission computed tomography (SPECT) and positron emission tomography (PET) is to employ transmission scanning to reconstruct the attenuation map. Such an approach, however, considerably complicates both the scanner design and the data acquisition protocol. A dramatic simplification could be made if the attenuation map could be obtained directly from the emission projections, without the use of a transmission scan. This can be done by applying the consistency conditions that enable us to identify the operator of the problem and, thus, to reconstruct the attenuation map. In this paper, we propose a new approach based on the discrete consistency conditions. One of the main advantages of the suggested method over previously used continuous conditions is that it can easily be applied in various scanning configurations, including fully three-dimensional (3-D) data acquisition protocols. Also, it provides a stable numerical implementation, allowing us to avoid the crosstalk between the attenuation map and the source function. A computationally efficient algorithm is implemented by using the QR and Cholesky decompositions. Application of the algorithm to computer-generated and experimentally measured SPECT data is considered.     

Segmentation of the body and lungs from Compton scatter and photopeak window data in SPECT: a Monte-Carlo investigation

In SPECT imaging of the chest, nonuniform attenuation correction requires use of a patient specific attenuation (mu) map. Such a map can be obtained by estimating the regions of (1) the lungs and (2) the soft tissues and bones, and then assigning an appropriate value of attenuation coefficient (mu) to each region. The authors proposed a method to segment such regions from the Compton scatter and photopeak window SPECT slices of Tc-99m Sestamibi studies. The Compton scatter slices are used to segment the body outline and to estimate the regions of the lungs. Locations of the back bone and sternum are estimated from the photopeak window slices to assist in the segmentation. To investigate the accuracy of using Compton scatter slices in estimating the regions of the body and the lungs, a Monte-Carlo SPECT simulation of an anthropomorphic phantom with an activity distribution and noise characteristics similar to patient data was conducted. Energy windows of various widths were simulated for use in locating a suitable Compton scatter window for imaging, The effects of attenuation correction using a mu map based on segmentation were also studied. The results demonstrated for the activity and mu maps studied herein that: (1) reasonable contrast could be obtained from Compton scatter data for the segmentation of the lung regions, (2) true positive rates of 99% and 89% for determining the body and lung regions, respectively, with total error rates of 4% and 29%, could be achieved, (3) usage of a mu map based on segmentation for attenuation correction improved relative quantification over filtered backprojection, (4) variations in the assigned mu value of 40% smaller or 40% larger in the lung regions had an insignificant impact on the results of relative quantification, (5) a wide energy window away from the photopeak window for recording scattered events could benefit both the segmentation of the lung regions and the attenuation correction of the activity in the myocardium region, and (6) usage of a smaller than true mu value in the lung regions of an assigned mu map might benefit attenuation correction for absolute quantification.     

Improved regional activity quantitation in nuclear medicine using a new approach to correct for tissue partial volume and spillover effects

We have developed a new method of compensating for effects of partial volume and spillover in dual-modality imaging. The approach requires segmentation of just a few tissue types within a small volume-of-interest (VOI) surrounding a lesion; the algorithm estimates simultaneously, from projection data, the activity concentration within each segmented tissue inside the VOI. Measured emission projections were fitted to the sum of resolution-blurred projections of each such tissue, scaled by its unknown activity concentration, plus a global background contribution obtained by reprojection through the reconstructed image volume outside the VOI. The method was evaluated using multiple-pinhole μSPECT data simulated for the MOBY mouse phantom containing two spherical lung tumors and one liver tumor, as well as using multiple-bead phantom data acquired on μSPECT and μCT scanners. Each VOI in the simulation study was 4.8 mm (12 voxels) cubed and, depending on location, contained up to four tissues (tumor, liver, heart, lung) with different values of relative (99m)Tc concentration. All tumor activity estimates achieved bias after ～ 15 ordered-subsets expectation maximization (OSEM) iterations (×10 subsets) , with better than 8% precision ( ≤ 25% greater than the Cramer-Rao lower bound). The projection-based fitting approach also outperformed three standardized uptake value (SUV)-like metrics, one of which was corrected for count spillover. In the bead phantom experiment, the mean ± standard deviation of the bias of VOI estimates of bead concentration were 0.9±9.5%, comparable to those of a perturbation geometric transfer matrix (pGTM) approach (-5.4±8.6%); however, VOI estimates were more stable with increasing iteration number than pGTM estimates, even in the presence of substantial axial misalignment between μCT and μSPECT image volumes.     

Accelerated image reconstruction using ordered subsets of projection data

The authors define ordered subset processing for standard algorithms (such as expectation maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is defined as a single pass through all the subsets, in each subset using the current estimate to initialize application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al. (1981) for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positivity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT, the OS-EM algorithm provides an order-of-magnitude acceleration over EM, with restoration quality maintained.     

Quantitative rotating multisegment slant-hole SPECT mammography with attenuation and collimator-detector response compensation

Rotating multisegment slant-hole (RMSSH) single photon emission computed tomography (SPECT) is suitable for detecting small and low-contrast breast lesions since it has much higher detection efficiency than conventional SPECT with a parallel-hole collimator and can image the breast at a closer distance. Our RMSSH SPECT reconstruction extends a previous rotation-shear transformation-based method to include nonuniform attenuation and collimator-detector response (CDR) compensation. To evaluate our reconstruction method, we performed two phantom simulation studies with 1) an isolated breast and 2) a breast phantom attached to the body torso. The reconstructed RMSSH SPECT images with attenuation and CDR compensation showed improved quantitative accuracy and less image artifacts than without. To evaluate the clinical efficacy of RMSSH SPECT mammography, we used a simulation study to compare with planar scintimammography in terms of the signal-to-noise ratio (SNR) value of a breast lesion. The RMSSH SPECT reconstruction images showed higher SNR value than the planar scintimammography images and even more so as we applied compensation for attenuation and collimator detector response. We conclude that attenuation and CDR compensation provide RMSSH SPECT mammography images with improved quality and quantitative accuracy.     

Simultaneous maximum a posteriori reconstruction of attenuation andactivity distributions from emission sinograms

In order to perform attenuation correction in emission tomography an attenuation map is required. We propose a new method to compute this map directly from the emission sinogram, eliminating the transmission scan from the acquisition protocol. The problem is formulated as an optimization task where the objective function is a combination of the likelihood and an a priori probability. The latter uses a Gibbs prior distribution to encourage local smoothness and a multimodal distribution for the attenuation coefficients. Since the attenuation process is different in positron emission tomography (PET) and single photon emission tomography (SPECT), a separate algorithm for each case is derived. The method has been tested on mathematical phantoms and on a few clinical studies. For PET, good agreement was found between the images obtained with transmission measurements and those produced by the new algorithm in an abdominal study. For SPECT, promising simulation results have been obtained for nonhomogeneous attenuation due to the presence of the lungs.     

Effects of myocardial wall thickness on SPECT quantification

The effects of changing myocardial wall thickness in single photon emission computed tomography (SPECT) imaging are characterized, and a method which may be used to compensate for these effects is presented. The underlying principle is that the phenomena of attenuation, Compton scatter, and finite resolution can be separated and treated independently. Only finite resolution and its effects, along with a proposed method for correcting these effects, are addressed. A cardiac phantom with varying wall thickness (9-23 mm) was developed to characterize the dependence effects on (201)Tl myocardial SPECT images. Correction factors in the form of recovery coefficients have been developed with the use of a convolution simulation, and are shown to improve substantially the agreement of counts extracted from SPECT images of the phantom with the actual (201)Tl concentration. The degree of improvement, however, is markedly affected by external attenuation. Clinical application of this method will require corrections for attenuation and scatter or the development of regional recovery coefficients which include these effects.     

A theoretical study of some maximum likelihood algorithms for emission and transmission tomography

This paper has the dual purpose of introducing some new algorithms for emission and transmission tomography and proving mathematically that these algorithms and related antecedent algorithms converge. Like the EM algorithms for positron, single-photon, and transmission tomography, the algorithms provide maximum likelihood estimates of pixel concentration or linear attenuation parameters. One particular innovation we discuss is a computationally practical scheme for modifying the EM algorithms to include a Bayesian prior. The Bayesian versions of the EM algorithms are shown to have superior convergence properties in a vicinity of the maximum. We anticipate that some of the other algorithms will also converge faster than the EM algorithms.     

Practical aspects of a data-driven motion correction approach for brain SPECT

Patient motion can cause image artifacts in single photon emission computed tomography despite restraining measures. Data-driven detection and correction of motion can be achieved by comparison of acquired data with the forward projections. This enables the brain locations to be estimated and data to be correctly incorporated in a three-dimensional (3-D) reconstruction algorithm. Digital and physical phantom experiments were performed to explore practical aspects of this approach. METHODS: Noisy simulation data modeling multiple 3-D patient head movements were constructed by projecting the digital Hoffman brain phantom at various orientations. Hoffman physical phantom data incorporating deliberate movements were also gathered. Motion correction was applied to these data using various regimes to determine the importance of attenuation and successive iterations. Studies were assessed visually for artifact reduction, and analyzed quantitatively via a mean registration error (MRE) and mean square difference measure (MSD). RESULTS: Artifacts and distortion in the motion corrupted data were reduced to a large extent by application of this algorithm. MRE values were mostly well within 1 pixel (4.4 mm) for the simulated data. Significant MSD improvements (>2) were common. Inclusion of attenuation was unnecessary to accurately estimate motion, doubling the efficiency and simplifying implementation. Moreover, most motion-related errors were removed using a single iteration. The improvement for the physical phantom data was smaller, though this may be due to object symmetry. CONCLUSION: These results provide the basis of an implementation protocol for clinical validation of the technique.