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.     

Unmatched projector/backprojector pairs in an iterative reconstruction algorithm

Computational burden is a major concern when an iterative algorithm is used to reconstruct a three-dimensional (3-D) image with attenuation, detector response, and scatter corrections. Most of the computation time is spent executing the projector and backprojector of an iterative algorithm. Usually, the projector and the backprojector are transposed operators of each other. The projector should model the imaging geometry and physics as accurately as possible. Some researchers have used backprojectors that are computationally less expensive than the projectors to reduce computation time. This paper points out that valid backprojectors should satisfy a condition that the projector/backprojector matrix must not contain negative eigen-values. This paper also investigates the effects when unmatched projector/backprojector pairs are used.     

3-D SPECT simulations of a complex 3-D mathematical brain model: effects of 3-D geometric detector response, attenuation, scatter, and statistical noise

The quantitative imaging characteristics of ultrahigh-resolution parallel-hole SPECT, including 3-D geometric detector response, attenuation, scatter, and statistical noise, were investigated by simulations based on a complex digitized 3-D brain model of the gray and white matter distributions. The projection data resulting from a uniform distribution of gray and white matter radioactivity, in a ratio of 5:1, were simulated. The results demonstrate significant qualitative and quantitative artifacts in reconstructed human brain images. In the absence of attenuation, scatter, and noise, artifactual variation caused inaccuracies in regional radioactivity quantification. Inclusion of attenuation scatter, and noise in the simulation caused additional artifacts, and resulted in reconstructed images which qualitatively and quantitatively corresponded very closely to reconstructed images of the actual 3-D brain phantom which was constructed from the same set of data as the mathematical 3-D brain model. It is concluded that the major degrading factor in SPECT neuroimaging is the 3-D geometric detector response function.     

Fast SPECT simulation including object shape dependent scatter

A fast simulator of SPECT projection data taking into account attenuation, distance dependent detector response, and scatter has been developed, based on an analytical point spread function model. The parameters of the scatter response are obtained from a single line source measurement with a triangular phantom. The simulator is able to include effects of object curvature on the scatter response to a high accuracy. The simulator has been evaluated for homogeneous media by measurements of (99m)Tc point sources placed at different locations in a water-filled cylinder at energy windows of 15% and 20%. The asymmetrical shapes of measured projections of point sources are In excellent agreement with simulations for both energy windows. Scatter-to-primary ratio (SPR) calculations of point sources at different positions in a cylindrical phantom differ not more than a few percent from measurements. The simulator uses just a few megabytes of memory for storing the tables representing the forward model; furthermore, simulation of 60 SPECT projections from a three-dimensional digital brain phantom with 6-mm cubic voxels takes only ten minutes on a standard workstation. Therefore, the simulator could serve as a projector in iterative true 3-D SPECT reconstruction.     

Improved SPECT quantitation using fully three-dimensional iterative spatially variant scatter response compensation

The quality and quantitative accuracy of iteratively reconstructed SPECT images improves when better point spread function (PSF) models of the gamma camera are used during reconstruction. Here, inclusion in the PSF model of photon crosstalk between different slices caused by limited gamma camera resolution and scatter is examined. A three-dimensional (3-D) projector back-projector (proback) has been developed which models both the distance dependent detector point spread function and the object shape-dependent scatter point spread function of single photon emission computed tomography (SPECT). A table occupying only a few megabytes of memory is sufficient to represent this scatter model. The contents of this table are obtained by evaluating an analytical expression for object shape-dependent scatter. The proposed approach avoids the huge memory requirements of storing the full transition matrix needed for 3-D reconstruction including object shape-dependent scatter. In addition, the method avoids the need for lengthy Monte Carlo simulations to generate such a matrix. In order to assess the quantitative accuracy of the method, reconstructions of a water filled cylinder containing regions of different activity levels and of simulated 3-D brain projection data have been evaluated for technetium-99m. It is shown that fully 3-D reconstruction including complete detector response and object shape-dependent scatter modeling clearly outperforms simpler methods that lack a complete detector response and/or a complete scatter response model. Fully 3-D scatter correction yields the best quantitation of volumes of interest and the best contrast-to-noise curves.     

Efficient fully 3-D iterative SPECT reconstruction with Monte Carlo-based scatter compensation

Quantitative accuracy of single photon emission computed tomography (SPECT) images is highly dependent on the photon scatter model used for image reconstruction. Monte Carlo simulation (MCS) is the most general method for detailed modeling of scatter, but to date, fully three-dimensional (3-D) MCS-based statistical SPECT reconstruction approaches have not been realized, due to prohibitively long computation times and excessive computer memory requirements. MCS-based reconstruction has previously been restricted to two-dimensional approaches that are vastly inferior to fully 3-D reconstruction. Instead of MCS, scatter calculations based on simplified but less accurate models are sometimes incorporated in fully 3-D SPECT reconstruction algorithms. We developed a computationally efficient fully 3-D MCS-based reconstruction architecture by combining the following methods: 1) a dual matrix ordered subset (DM-OS) reconstruction algorithm to accelerate the reconstruction and avoid massive transition matrix precalculation and storage; 2) a stochastic photon transport calculation in MCS is combined with an analytic detector modeling step to reduce noise in the Monte Carlo (MC)-based reprojection after only a small number of photon histories have been tracked; and 3) the number of photon histories simulated is reduced by an order of magnitude in early iterations, or photon histories calculated in an early iteration are reused. For a 64 x 64 x 64 image array, the reconstruction time required for ten DM-OS iterations is approximately 30 min on a dual processor (AMD 1.4 GHz) PC, in which case the stochastic nature of MCS modeling is found to have a negligible effect on noise in reconstructions. Since MCS can calculate photon transport for any clinically used photon energy and patient attenuation distribution, the proposed methodology is expected to be useful for obtaining highly accurate quantitative SPECT images within clinically acceptable computation times.     

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.     

Implementation of a model-based nonuniform scatter correctionscheme for SPECT

Four scatter-compensation schemes are considered. The 4 schemes are all based on a previously developed two-dimensional (2-D) scatter model. Reconstruction is achieved using the iterative expectation-maximization maximum-likelihood (EM-ML) algorithm. The schemes consist of: (1) including the model in both the forward and back projector; (2) just including the model in the forward projector; (3) and (4) implementing the model in a subtraction and addition scheme, respectively. Monte Carlo simulated projection data are used to test the accuracy, convergence properties, and noise properties of the 4 scatter-compensation schemes. Data are simulated for both uniformly and nonuniformly attenuating objects. The results show that all 4 correction schemes yield images which are similar in terms of accuracy to that obtained from reconstructing scatter-free data. The subtraction scheme is shown to converge faster than the other compensation schemes, both in terms of iterations and actual time required for reconstruction. The scheme in which the model is only used in the forward-projector and the scatter-addition scheme both performs slightly better, in terms of signal-to-noise ratio (SNR), than the subtraction scheme. However, the forward projector scheme requires significantly more CPU time for reconstruction. The correction scheme in which the scatter model was included in both the forward and backprojectors is shown to produce accurate images with SNR's higher than even a perfect scatter rejection scheme. While the scatter correction scheme with the model in both the forward projector and backprojector has superior noise properties to the other algorithms, the results suggest that the faster subtraction/addition schemes will probably prove most useful for routine clinical scatter compensation     

Scatter correction in 3-D PET

Modern multiring positron emission tomographs allow the acquisition of 3-D data sets to increase their sensitivity. A substantial part of this data is due to scattered radiation. The authors describe the experimental dependence of point source scatter distributions on energy window setting, source location, and scatter volume in geometries relevant for brain studies. The point source scatter distribution was parametrized accurately by a broad, 2-D Gaussian, which included a shift parameter to account for asymmetry of the scatter medium relative to the source. This parametrization was used to formulate two fast scatter correction algorithms suitable for brain scans. In both algorithms, a 2-D subset of the measured projections was transformed into a scatter projection. An image of the 3-D scatter distribution was reconstructed using 2-D algorithms. It was then subtracted from the total (true+scattered) 3-D image. Both algorithms were implemented in different combinations with the additional attenuation correction and were tested on point source and phantom measurements. It was shown that, for the situation typical for brain scans, reconstructed scatter fractions could be reduced to 5% or less.     

Novel scatter compensation of list-mode PET data using spatial and energy dependent corrections

With the widespread use of positron emission tomography (PET) crystals with greatly improved energy resolution (e.g., 11.5% with LYSO as compared to 20% with BGO) and of list-mode acquisitions, the use of the energy of individual events in scatter correction schemes becomes feasible. We propose a novel scatter approach that incorporates the energy of individual photons in the scatter correction and reconstruction of list-mode PET data in addition to the spatial information presently used in clinical scanners. First, we rewrite the Poisson likelihood function of list-mode PET data including the energy distributions of primary and scatter coincidences and show that this expression yields an MLEM reconstruction algorithm containing both energy and spatial dependent corrections. To estimate the spatial distribution of scatter coincidences we use the single scatter simulation (SSS). Next, we derive two new formulae which allow estimation of the 2-D (coincidences) energy probability density functions (E-PDF) of primary and scatter coincidences from the 1-D (photons) E-PDFs associated with each photon. We also describe an accurate and robust object-specific method for estimating these 1-D E-PDFs based on a decomposition of the total energy spectra detected across the scanner into primary and scattered components. Finally, we show that the energy information can be used to accurately normalize the scatter sinogram to the data. We compared the performance of this novel scatter correction incorporating both the position and energy of detected coincidences to that of the traditional approach modeling only the spatial distribution of scatter coincidences in 3-D Monte Carlo simulations of a medium cylindrical phantom and a large, nonuniform NCAT phantom. Incorporating the energy information in the scatter correction decreased bias in the activity distribution estimation by ~20% and ~40% in the cold regions of the large NCAT phantom at energy resolutions 11.5% and 20% at 511 keV, respectively, compared to when using the spatial information alone.     

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.     

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.     

Effect of Voxel Size and Computation Method on Tc-99m MAA SPECT/CT-Based Dose Estimation for Y-90 Microsphere Therapy

The use of selective internal radiation therapy for treatment of hepatocellular carcinoma and liver metastases using Y-90 labeled microspheres has become an effective and widely used treatment regimen. However, dosimetric evaluations of this treatment are still primitive as uniform distribution models based only on injected activity are often used. This investigation attempts to quantify the effectiveness of several sophisticated patient-specific techniques which utilize the source distribution of Tc-99m MAA simulation studies to perform voxelized dosimetric computations. Among these techniques are complete Monte-Carlo radiation transport computation in patient-specific CT-based voxel phantoms, local energy deposition in patient specific phantoms and kernel transport techniques in water. Each technique was evaluated using three different phantom voxel dimensions and SPECT reconstruction matrix sizes. Dose evaluation results using all methods were compared to the exact solution, obtained using fully 3-D Monte-Carlo simulations with source distribution based not on SPECT data, but on the injected activity and exact boundaries of the anthropomorphic phantom used in the study. The results of this study show that at large voxel sizes and using SPECT reconstructions with a small matrix size (64 $,times,$64), Monte-Carlo and local deposition methods are nearly equivalent. However, using a large SPECT reconstruction matrix (256$,times,$ 256) the local deposition method is significantly more accurate than full 3-D Monte-Carlo transport, and with a negligible computational burden.      

A comparison of deconvolution and windowed subtraction techniques for scatter compensation in SPECT

Three procedures for the removal of Compton-scattered data in SPECT by constrained deconvolution are presented. The first is a deconvolution of a 2-D measured PSRF containing scatter from a single reconstructed transaxial image; the second is a deconvolution of a 2-D measured point-source response function (PSRF) from each frame of projection data prior to reconstruction; the third involves deconvolution of a 3-D measured PSRF from a stack of reconstructed slices. Results of applying these procedures to data obtained from a phantom containing cold cylinders and to data from a cold spot-resolution phantom are presented and are shown to be superior to the results of correcting for scatter by scatter-window substraction. Both 3-D deconvolution from reconstructed images and 2-D deconvolution from projection data show major improvements in image contrast, resolution, and quantitation. Improvements are especially marked for small (1.0-3.0 cm) cold sources.     

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.     

A deconvolution scatter correction for a 3-D PET system

A method to remove the scattered background from a reconstructed image by deconvolution with a point response function which includes the scatter contribution is presented. The amplitude of the scattered response function is obtained by constraining a region of the corrected image to zero average amplitude. This method assumes that the shape of scatter distribution is shift invariant and independent of the shape of the scattering object and the distribution of the positron activity. The validity of these approximations for the QPET geometry was tested using simulations. An average scatter response function for the system was obtained from these simulations and compared with results from measurements. The method was tested using experimental data from an irregularly shaped acrylic phantom. It was simple to implement and resulted in a satisfactory correction of the scattered background for a small-volume system.     

Application of task-based measures of image quality to optimization and evaluation of three-dimensional reconstruction-based compensation methods in myocardial perfusion SPECT

In this paper, we apply the channelized Hotelling observer (CHO) using a defect detection task to the optimization and evaluation of three-dimensional iterative reconstruction-based compensation methods for myocardial perfusion single-photon emission computed tomography (SPECT). We used a population of 24 mathematical cardiac-torso phantoms that realistically model the activity and attenuation distribution in three classes of patients: females, and males with flat diaphragms and raised diaphragms. Projection data were generated and subsequently reconstructed using methods based on the ordered subsets-expectation maximization (OSEM) algorithm. The methods evaluated included compensation for attenuation, detector response blurring, and scatter in various combinations. We applied the CHO to optimize the number of iterations for OSEM and the cutoff frequency and order of a three-dimensional postreconstruction Butterworth filter. Using the optimal parameters, we then compared the compensation methods. The index of comparison in these studies was the area under the receiver operating characteristics curve (AUC) for the CHO. We found that attenuation compensation with either detector response or scatter compensation gave statistically significant increases in the AUC compared to attenuation compensation alone. The greatest increase in the AUC occurred when all three compensations were applied. These results indicate that compensation for detector response and scatter, in addition to attenuation compensation, will improve defect detectability in myocardial SPECT images.     

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.     

Comparison of residence time estimation methods for radioimmunotherapy dosimetry and treatment planning--Monte Carlo simulation studies

Estimating the residence times in tumor and normal organs is an essential part of treatment planning for radioimmunotherapy (RIT). This estimation is usually done using a conjugate view whole body scan time series and planar processing. This method has logistical and cost advantages compared to 3-D imaging methods such as Single photon emission computed tomography (SPECT), but, because it does not provide information about the 3-D distribution of activity, it is difficult to fully compensate for effects such as attenuation and background and overlapping activity. Incomplete compensation for these effects reduces the accuracy of the residence time estimates. In this work we compare residence times estimates obtained using planar methods to those from methods based on quantitative SPECT (QSPECT) reconstructions. We have previously developed QSPECT methods that provide compensation for attenuation, scatter, collimator-detector response, and partial volume effects. In this study we compared the use of residence time estimation methods using QSPECT to planar methods. The evaluation was done using the realistic NCAT phantom with organ time activities that model (111)In ibritumomab tiuxetan. Projection data were obtained using Monte Carlo simulations (MCS) that realistically model the image formation process including penetration and scatter in the collimator-detector system. These projection data were used to evaluate the accuracy of residence time estimation using a time series of QSPECT studies, a single QSPECT study combined with planar scans and the planar scans alone. The errors in the residence time estimates were 3.8%, 15%, and 2%-107% for the QSPECT, hybrid planar/QSPECT, and planar methods, respectively. The quantitative accuracy was worst for pure planar processing and best for pure QSPECT processing. Hybrid planar/QSPECT methods, where a single QSPECT study was combined with a series of planar scans, provided a large and statistically significant improvement in quantitative accuracy for most organs compared to the planar scans alone, even without sophisticated attention to background subtraction or thickness corrections in planar processing. These results indicate that hybrid planar/QSPECT methods are generally superior to pure planar methods and may be an acceptable alternative to performing a time series of QSPECT studies.     

Efficient Monte Carlo based scatter artifact reduction in cone-beam micro-CT

Cupping and streak artifacts caused by the detection of scattered photons may severely degrade the quantitative accuracy of cone-beam X-ray computed tomography (CT) images. In order to overcome this problem, we propose and validate the following iterative scatter artifact reduction scheme. First, an initial image is reconstructed from the scatter-contaminated projections. Next, the scatter component of the projections is estimated from the initial reconstruction by a Monte Carlo (MC) simulation. The estimate obtained is then utilized during the reconstruction of a scatter-corrected image. The last two steps are repeated until an adequate correction is obtained. The estimation of the noise-free scatter projections in this scheme is accelerated in the following way: first, a rapid (i.e., based on a low number of simulated photon tracks) MC simulation is executed. The noisy result of this simulation is de-noised by a three-dimensional fitting of Gaussian basis functions. We demonstrate that, compared to plain MC, this method shortens the required simulation time by three to four orders of magnitude. Using simulated projections of a small animal phantom, we show that one cycle of the scatter correction scheme is sufficient to produce reconstructed images that barely differ from the reconstructions of scatter-free projections. The reconstructions of data acquired with a charge-coupled device based micro-CT scanner demonstrate a nearly complete removal of the scatter-induced cupping artifact. Quantitative errors in a water phantom are reduced from around 12% for reconstructions without the scatter correction to 1% after the proposed scatter correction has been applied. In conclusion, a general, accurate, and efficient scatter correction algorithm is developed that requires no mechanical modifications of the scanning equipment and results in only a moderate increase in the total reconstruction time.