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.     

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.     

3-D maximum a posteriori estimation for single photon emission computed tomography on massively-parallel computers

A fully three-dimensional (3-D) implementation of the maximum a posteriori (MAP) method for single photon emission computed tomography (SPECT) is demonstrated. The 3-D reconstruction exhibits a major increase in resolution when compared to the generation of the series of separate 2-D slice reconstructions. As has been noted, the iterative EM algorithm for 2-D reconstruction is highly computational; the 3-D algorithm is far worse. To accommodate the computational complexity, previous work in the 2-D arena is extended, and an implementation on the class of massively parallel processors of the 3-D algorithm is demonstrated. Using a 16000- (4000-) processor MasPar/DECmpp-Sx machine, the algorithm is demonstrated to execute at 2.5 (7.8) s/EM-iteration for the entire 64x64x64 cube of 96 planar measurements obtained from the Siemens Orbiter rotating camera operating in the high-resolution mode.     

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.     

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.     

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.     

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     

Performance characteristics of the 3-D OSEM algorithm in the reconstruction of small animal PET images. Ordered-subsets expectation-maximixation

Rat brain images acquired with a small animal positron emission tomography (PET) camera and reconstructed with the three-dimensional (3-D) ordered-subsets expectation-maximization (OSEM) algorithm with resolution recovery have better quality when the brain is imaged by itself than when inside the head with surrounding background activity. The purpose of this study was to characterize the dependence of this effect on the level of background activity, attenuation, and scatter. Monte Carlo simulations of the imaging system were performed. The coefficient of variation from replicate images, full-width at half-maximum (FWHM) from point sources and image profile fitting, and image contrast and uniformity were used to evaluate algorithm performance. A rat head with the typical levels of five and ten times the brain activity in the surrounding background requires additional iterations to achieve the same resolution as the brain-only case at a cost of 24% and 64% additional noise, respectively. For the same phantoms, object scatter reduced contrast by 3%-5%. However, attenuation degraded resolution by 0.2 mm and was responsible for up to 12% nonuniformity in the brain images suggesting that attenuation correction is useful. Given the effects of emission and attenuation distribution on both resolution and noise, simulations or phantom studies should be used for each imaging situation to select the appropriate number of OSEM iterations to achieve the desired resolution-noise levels.     

Simultaneously constraining SPECT activity estimates with primary and secondary energy window projection data

Single photon emission computed tomography (SPECT) projection data acquired in primary (photopeak) and secondary energy windows provide independent constraints on the source activity distribution. These constraints are used simultaneously and a single matrix equation is solved to estimate source activity. This simultaneous constraint reconstruction method (I) is evaluated in a Monte Carlo experiment and compared with activity estimation using only primary energy window projection data with (m) and without (II) modeling the detection of scattered photons in the system matrix, and with dual energy window scatter subtraction (TV). The transport and detection of photons emitted by Tc-99m within a water-filled cylinder is simulated. Images are reconstructed using generalized matrix inverses, which are designed so that the estimated variance of the reconstructed source voxel activities is approximately constant. For 40 projection data sets with about one million counts/slice in the primary energy window, the mean activity in regions of interest is within 2.2%, 28.2%, 2.2% and 3.5% of the source model activity for reconstruction methods I-IV, respectively. The full-widths at half-maximum (FWHM) of resolution kernels from methods I and III differ on average by less than 1%; the FWHM of kernels from methods II and IV average 8% larger. In this experiment, the simultaneous constraint method yields no appreciable improvement in quantitative accuracy or resolution over the use of primary energy window data alone when the detection of scattered photons is modeled. These two methods are only slightly superior to activity estimation using dual energy window scatter subtraction.     

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.     

Resolution and noise properties of MAP reconstruction for fully 3-D PET

We derive approximate analytical expressions for the local impulse response and covariance of images reconstructed from fully three-dimensional (3-D) positron emission tomography (PET) data using maximum a posteriori (MAP) estimation. These expressions explicitly account for the spatially variant detector response and sensitivity of a 3-D tomograph. The resulting spatially variant impulse response and covariance are computed using 3-D Fourier transforms. A truncated Gaussian distribution is used to account for the effect on the variance of the nonnegativity constraint used in MAP reconstruction. Using Monte Carlo simulations and phantom data from the microPET small animal scanner, we show that the approximations provide reasonably accurate estimates of contrast recovery and covariance of MAP reconstruction for priors with quadratic energy functions. We also describe how these analytical results can be used to achieve near-uniform contrast recovery throughout the reconstructed volume.     

Fourier rebinning applied to multiplanar circular-orbit cone-beam SPECT

We study the application of Fourier rebinning methods to dual-planar cone-beam SPECT. Dual-planar cone-beam SPECT involves the use of a pair of dissimilar cone-beam collimators on a dual-camera SPECT system. Each collimator has its focus in a different axial plane. While dual-planar data is best reconstructed with fully three-dimensional (3-D) iterative methods, these methods are slow and have prompted a search for faster reconstruction techniques. Fourier rebinning was developed to estimate equivalent parallel projections from 3-D PET data, but it simply expresses a relationship between oblique projections taken in planes not perpendicular to the axis of rotation and direct projections taken in those that are. We find that it is possible to put cone-beam data in this context as well. The rebinned data can then be reconstructed using either filtered backprojection (FBP) or parallel iterative algorithms such as OS-EM. We compare the Feldkamp algorithm and fully 3-D OSEM reconstruction with Fourier-rebinned reconstructions on realistically-simulated Tc-99m HMPAO brain SPECT data. We find that the Fourier-rebinned reconstructions exhibit much less image noise and lower variance in region-of-interest (ROI) estimates than Feldkamp. Also, Fourier-rebinning followed by OSEM with nonuniform attenuation correction exhibits less bias in ROI estimates than Feldkamp with Chang attenuation correction. The Fourier-rebinned ROI estimates exhibit bias and variance comparable to those from fully 3-D OSEM and require considerably less processing time. However, in areas off the axis of rotation, the axial-direction resolution of FORE-reconstructed images is poorer than that of images reconstructed with 3-D OSEM. We conclude that Fourier rebinning is a practical and potentially useful approach to reconstructing data from dual-planar circular-orbit cone-beam systems.     

A fully automated optimization algorithm for determining the 3-D patient contour from photo-peak projection data in SPECT

The class of noninverse problems versus inverse problems is discussed. A general optimization approach to solving certain constrained noninverse problems is presented. This approach is applied in the area of single photon emission computed tomography to estimate the patient outer contour directly from the photo-peak projection data. The resulting algorithm is fully automated, fast, and the determined 3-D patient contour satisfies smoothness constraints. The accuracy and reliability of the algorithm is evaluated through Monte Carlo simulations, patient studies, and an experimental validation study.     

Aperture collimation correction and maximum-likelihood image reconstruction for near-field coded aperture imaging of single photon emission computerized tomography

Coded aperture (CA) imaging originally developed in X-ray astronomy has not been widely used in nuclear medicine due to the decoding complexity of near-field CA images. In this paper, we present a near-field CA imaging technique and image reconstruction method for high sensitivity and high resolution single photon emission computerized tomography (SPECT). Our approach makes three contributions. First, a correction method for the aperture collimation effect is used to eliminate the near-field artifacts without dual acquisitions of mask and anti-mask images. Second, a maximum-likelihood expectation-maximization (MLEM) deconvolution method is used to restore CA images. Finally, a new MLEM-based algorithm is used to partially reconstruct three-dimensional (3-D) objects from a single projection of CA images. Experiments conducted using a dual-head SPECT system equipped with a parallel-hole collimator and a CA module show a tenfold increase in count sensitivity and significant improvement in image resolution with CA collimation as compared to parallel-hole collimation. Experiments conducted using the same dual-head SPECT system equipped with a pinhole collimator show that when the object is closer to the pinhole collimator the CA image resolution is only slightly inferior to the pinhole collimated image. We found that the MLEM deconvolution method provides an inherent nonnegativity constraint on pixel values and remarkably reduces background activities of CA images. The MLEM reconstruction algorithm for CA images is capable of reconstructing 3-D objects from a single projection and can be potentially extended to full 3-D SPECT reconstruction for CA images.     

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.      

Optimized acquisition time and image sampling for dynamic SPECT ofTl-201

With the recent development in scatter and attenuation correction algorithms, dynamic single photon emission computerized tomography (SPECT) can potentially yield physiological parameters, with tracers exhibiting suitable kinetics such as thallium-201 (Tl-201). A systematic way is proposed to investigate the minimum data acquisition times and sampling requirements for estimating physiological parameters with quantitative dynamic SPECT. Two different sampling schemes were investigated with Monte Carlo simulations: (1) Continuous data collection for total study duration ranging from 30-240 min. (2) Continuous data collection for first 10-45 min followed by a delayed study at approximately 3 h. Tissue time activity curves with realistic noise were generated from a mean plasma time activity curve and rate constants (K₁-k₄) derived from Tl-201 kinetic studies in 16 dogs. Full dynamic sampling schedules (DynSS) were compared to optimum sampling schedules (OSS). The authors found that OSS can reliably estimate the blood flow related K₁ and V_d comparable to DynSS. A 30-min continuous collection was sufficient if only K₁ was of interest. A split session schedule of a 30-min dynamic followed by a static study at 3 h allowed reliable estimation of both K₁ and V_d avoiding the need for a prolonged (>60-min) continuous dynamic acquisition. The methodology developed should also be applicable to optimizing sampling schedules for other SPECT tracers     

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.     

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.     

Fully three-dimensional reconstruction for a PET camera with retractable septa

A fully 3-D reconstruction algorithm has been developed to reconstruct data from a 16 ring PET camera (a Siemens/CTI 953B) with automatically retractable septa. The tomograph is able to acquire coincidences between any pair of detector rings and septa retraction increases the total system count rate by a factor of 7.8 (including scatter) and 4.7 (scatter subtracted) for a uniform, 20 cm diameter cylinder. The reconstruction algorithm is based on 3-D filtered backprojection, expressed in a form suitable for the multi-angle sinogram data. Sinograms which are not measured due to the truncated cylindrical geometry of the tomograph, but which are required for a spatially invariant response function, are obtained by forward projection. After filtering, the complete set of sinograms is backprojected into a 3-D volume of 128x128x31 voxels using a voxel-driven procedure. The algorithm has been validated with simulation, and tested with both phantom and clinical data from the 953B.