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.     

Simultaneous MR-Compatible Emission and Transmission Imaging for PET Using Time-of-Flight Information

Quantitative positron emission tomography (PET) imaging relies on accurate attenuation correction. Predicting attenuation values from magnetic resonance (MR) images is difficult because MR signals are related to proton density and relaxation properties of tissues. Here, we propose a method to derive the attenuation map from a transmission scan. An annulus transmission source is positioned inside the field-of-view of the PET scanner. First a blank scan is acquired. The patient is injected with FDG and placed inside the scanner. 511-keV photons coming from the patient and the transmission source are acquired simultaneously. Time-of-flight information is used to extract the coincident photons originating from the annulus. The blank and transmission data are compared in an iterative reconstruction method to derive the attenuation map. Simulations with a digital phantom were performed to validate the method. The reconstructed attenuation coefficients differ less than 5% in volumes of interest inside the lungs, bone, and soft tissue. When applying attenuation correction in the reconstruction of the emission data a standardized uptake value error smaller than 9% was obtained for all tissues. In conclusion, our method can reconstruct the attenuation map and the emission data from a simultaneous scan without prior knowledge about the anatomy or the attenuation coefficients of the tissues.     

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.     

An EM algorithm for estimating SPECT emission and transmission parameters from emissions data only

A maximum-likelihood (ML) expectation-maximization (EM) algorithm (called EM-IntraSPECT) is presented for simultaneously estimating single photon emission computed tomography (SPECT) emission and attenuation parameters from emission data alone. The algorithm uses the activity within the patient as transmission tomography sources, with which attenuation coefficients can be estimated. For this initial study, EM-IntraSPECT was tested on computer-simulated attenuation and emission maps representing a simplified human thorax as well as on SPECT data obtained from a physical phantom. Two evaluations were performed. First, to corroborate the idea of reconstructing attenuation parameters from emission data, attenuation parameters (mu) were estimated with the emission intensities (lambda) fixed at their true values. Accurate reconstructions of attenuation parameters were obtained. Second, emission parameters lambda and attenuation parameters mu were simultaneously estimated from the emission data alone. In this case there was crosstalk between estimates of lambda and mu and final estimates of lambda and mu depended on initial values. Estimates degraded significantly as the support extended out farther from the body, and an explanation for this is proposed. In the EM-IntraSPECT reconstructed attenuation images, the lungs, spine, and soft tissue were readily distinguished and had approximately correct shapes and sizes. As compared with standard EM reconstruction assuming a fix uniform attenuation map, EM-IntraSPECT provided more uniform estimates of cardiac activity in the physical phantom study and in the simulation study with tight support, but less uniform estimates with a broad support. The new EM algorithm derived here has additional applications, including reconstructing emission and transmission projection data under a unified statistical model.     

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.     

Evaluation of an efficient compensation method for quantitative fan-beam brain SPECT reconstruction

Fan-beam collimators are designed to improve the system sensitivity and resolution for imaging small objects such as the human brain and breasts in single photon emission computed tomography (SPECT). Many reconstruction algorithms have been studied and applied to this geometry to deal with every kind of degradation factor. This paper presents a new reconstruction approach for SPECT with circular orbit, which demonstrated good performance in terms of both accuracy and efficiency. The new approach compensates for degradation factors including noise, scatter, attenuation, and spatially variant detector response. Its uniform attenuation approximation strategy avoids the additional transmission scan for the brain attenuation map, hence reducing the patient radiation dose and furthermore simplifying the imaging procedure. We evaluate and compare this new approach with the well-established ordered-subset expectation-maximization iterative method, using Monte Carlo simulations. We perform quantitative analysis with regional bias-variance, receiver operating characteristics, and Hotelling trace studies for both methods. The results demonstrate that our reconstruction strategy has comparable performance with a significant reduction of computing time.     

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.     

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.     

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.     

Experimental Assessment of the Gain Achieved by the Utilization of Time-of-Flight Information in a Positron Emission Tomograph (Super PETT I)

The gain achieved in image quality by utilizing, in the image forming process, the time-of-flight information (TOF) of positron annihilation photons between their inception and detection was measured experimentally by means of a positron emission tomograph (PET)-Super PETT I. The measurements were carried out by imaging a 35 cm cylindrical uniform phantom containing different positron activity concentrations. The gain achieved through the incorporation of TOF information, defined as the ratio of variances in images reconstructed with and without TOF information, was found to be approximately 3 at the lowest activity concentration and 5-8 in the activity concentration range typically encountered in clinical studies especially in fast or dynamic studies. This increase in gain with activity was interpreted as resulting from the reduction of random coincidences when TOF information is used. Further image improvement is yielded by incorporating TOF information into the PET attenuation correction provided by the measurement of transmission of annihilation photons in the object imaged.     

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.     

Data consistency based translational motion artifact reduction in fan-beam CT

A basic assumption in the classic computed tomography (CT) theory is that an object remains stationary in an entire scan. In biomedical CT/micro-CT, this assumption is often violated. To produce high-resolution images, such as for our recently proposed clinical micro-CT (CMCT) prototype, it is desirable to develop a precise motion estimation and image reconstruction scheme. In this paper, we first extend the Helgason-Ludwig consistency condition (HLCC) from parallel-beam to fan-beam geometry when an object is subject to a translation. Then, we propose a novel method to estimate the motion parameters only from sinograms based on the HLCC. To reconstruct the moving object, we formulate two generalized fan-beam reconstruction methods, which are in filtered backprojection and backprojection filtering formats, respectively. Furthermore, we present numerical simulation results to show that our approach is accurate and robust.     

Respiratory compensation in projection imaging using a magnification and displacement model

Respiratory motion during the collection of computed tomography (CT) projections generates structured artifacts and a loss of resolution that can render the scans unusable. This motion is problematic in scans of those patients who cannot suspend respiration, such as the very young or intubated patients. Here, the authors present an algorithm that can be used to reduce motion artifacts in CT scans caused by respiration. An approximate model for the effect of respiration is that the object cross section under interrogation experiences time-varying magnification and displacement along two axes. Using this model an exact filtered backprojection algorithm is derived for the case of parallel projections. The result is extended to generate an approximate reconstruction formula for fan-beam projections. Computer simulations and scans of phantoms on a commercial CT scanner validate the new reconstruction algorithms for parallel and fan-beam projections. Significant reduction in respiratory artifacts is demonstrated clinically when the motion model is satisfied. The method can be applied to projection data used in CT, single photon emission computed tomography (SPECT), positron emission tomography (PET), and magnetic resonance imaging (MRI).     

Signal-to-noise ratios for attenuation correction in PET imaging

Attenuation correction is an important part of accurate image reconstruction in positron tomography. The usual correction method involves direct measurement of attenuation correction factors (ACFs). A reconstruct-reproject method, which has been suggested as providing superior noise properties, is sometimes employed; an attenuation image is first reconstructed from the measurement and then ACFs are obtained by reprojection through this image. Here the authors present a model which follows the signal-to-noise ratio (SNR) through the attenuation correction by both the direct and reconstruct-reproject methods. This model is applicable to both 2D and 3D imaging geometry, but applies to the central elements of emission and transmission objects with circular symmetry and constant amplitude. For this simplified geometry, the model predicts that the SNR of the emission image following attenuation correction is the same for both direct and reconstruct-reproject methods, although the SNRs of the ACFs are themselves substantially different. The authors also present the measured SNR at the various steps of attenuation correction for both the direct and reconstruct-reproject methods using simulated transmission and emission data. The measured SNRs agree with the model; no significant difference between the direct and reconstruct-reproject SNRs was observed.     

Three-dimensional PET emission scan registration and transmissionscan synthesis

The duration of a positron emission tomography (PET) imaging scan can be reduced if the transmission scan of one patient which is used for emission correction can be synthesized by using the reference transmission scan of another patient. In this paper, we propose a new intersubjects PET emission scan registration method and PET transmission synthesis method by using the boundary information of the body or brain scan of the PET emission scans. The PET emission scans have poor image quality and different intensity statistics so that we preprocess the emission scans to have a similar histogram and then apply the point distribution model (PDM) to extract the contours of the emission scan. The extracted boundary contour of every slice is used to reconstruct the three-dimensional (3-D) surface of the reference set and the target set. Our registration is 3-D surface-based which uses the normal flow method to find the correspondence vector field between two 3-D reconstructed surfaces. Since it is difficult to analyze internal organs using PET emission scan imaging without correction, we assume that the deformation of internal organ is homogeneous. With the corresponding vector field between the two emission scans and the transmission scan of the reference set, we can synthesize the transmission scan of the target set     

The circular harmonic transform for SPECT reconstruction and boundary conditions on the Fourier transform of the sinogram

The circular harmonic transform (CHT) solution of the exponential Randon transform (ERT) is applied to single-photon emission computed tomography (SPECT) for uniform attenuation within a convex boundary. An important special case also considered is the linear (unattenuated) Radon transform (LRT). The solution is on the form of an orthogonal function expansion matched to projections that are in parallel-ray geometry. This property allows for efficient and accurate processing of the projections with fast Fourier transform (FFT) without interpolation or beam matching. The algorithm is optimized by the use of boundary conditions on the 2-D Fourier transform of the sinogram. These boundary conditions imply that the signal energy of the sinogram is concentrated in well-defined sectors in transform space. The angle defining the sectors depends in a direct way on the radius of the field view. These results are also obtained for fan-beam geometry and the linear Radon transform (the Fourier-Chebyshev transform of the sinogram) to demonstrate that the boundary conditions are a more general property of the Radon transform and a not a property unique to rectangular coordinates.     

Assessment of SPM in perfusion brain SPECT studies. A numerical simulation study using bootstrap resampling methods

Statistical parametric mapping (SPM) has become the technique of choice to statistically evaluate positron emission tomography (PET), functional magnetic resonance imaging (fMRI), and single photon emission computed tomography (SPECT) functional brain studies. Nevertheless, only a few methodological studies have been carried out to assess the performance of SPM in SPECT. The aim of this paper was to study the performance of SPM in detecting changes in regional cerebral blood flow (rCBF) in hypo- and hyperperfused areas in brain SPECT studies. The paper seeks to determine the relationship between the group size and the rCBF changes, and the influence of the correction for degradations. The assessment was carried out using simulated brain SPECT studies. Projections were obtained with Monte Carlo techniques, and a fan-beam collimator was considered in the simulation process. Reconstruction was performed by using the ordered subsets expectation maximization (OSEM) algorithm with and without compensation for attenuation, scattering, and spatial variant collimator response. Significance probability maps were obtained with SPM2 by using a one-tailed two-sample f-test. A bootstrap resampling approach was used to determine the sample size for SPM to detect the between-group differences. Our findings show that the correction for degradations results in a diminution of the sample size, which is more significant for small regions and low-activation factors. Differences in sample size were found between hypo- and hyperperfusion. These differences were larger for small regions and low-activation factors, and when no corrections were included in the reconstruction algorithm.     

An analytical scatter correction for singles-mode transmission data in PET

We present an analytical scatter correction, based upon the Klein-Nishina formula, for singles-mode transmission data in positron emission tomography (PET) and its implementation as part of an iterative image reconstruction algorithm. We compared our analytically-calculated scatter sinogram data with previously validated simulation data for a small animal PET scanner with 68 Ge (a positron emitter) and 57 Co (approximately 122-keV photon emitter) transmission sources using four different phantom configurations (three uniform water cylinders with radii of 25, 30, and 45 mm and a nonuniform phantom consisting of water, Teflon, and air). Our scatter calculation correctly predicts the contribution from single-scattered (one incoherent scatter interaction) photons to the simulated sinogram data and provides good agreement for the percent scatter fraction (SF) per sinogram for all phantoms and both transmission sources. We then applied our scatter correction as part of an iterative reconstruction algorithm for PET transmission data for simulated and experimental data using uniform and nonuniform phantoms. For both simulated and experimental data, the reconstructed linear attenuation coefficients (mu-values-values) agreed with expected values to within 4% when scatter corrections were applied, for both the 68 Ge and 57 Co transmission sources. We also tested our reconstruction and scatter correction procedure for two experimental rodent studies (a mouse and rat). For the rodent studies, we found that the average mu-values for soft-tissue regions of interest agreed with expected values to within 4%. Using a 2.2-GHz processor, each scatter correction iteration required between 6-27 min of CPU time (without any code optimization) depending on the phantom size and source used. This extra calculation time does not seem unreasonable considering that, without scatter corrections, errors in the reconstructed mu-values were between 18%-45% depending on the phantom size and transmission source used.     

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.     

Incorporation of system resolution compensation (RC) in the ordered-subset transmission (OSTR) algorithm for transmission imaging in SPECT

In order to reconstruct attenuation maps with improved spatial resolution and quantitative accuracy, we developed an approximate method of incorporating system resolution compensation (RC) in the ordered-subset transmission (OSTR) algorithm for transmission reconstruction. Our method approximately models the blur caused by the finite intrinsic detector resolution, the nonideal source collimation and detector collimation. We derived the formulation using the optimization transfer principle as in the derivation of the OSTR algorithm. The formulation includes one forward-blur step and one back-blur step, which do not severely slow down reconstruction. The formulation could be applicable to various transmission geometries, such as point-source, line-source, and sheet-source systems. Through computer simulations of the MCAT phantom and transmission measurements of the air-filled Data Spectrum Deluxe single photo emission computed tomography (SPECT) Phantom on a system which employed a cone-beam geometry and a system which employed a scanning-line-source geometry, we showed that incorporation of RC increased spatial resolution and improved the quantitative accuracy of reconstruction. In simulation studies, attenuation maps reconstructed with RC correction improved the quantitative accuracy of emission reconstruction.