A methodology for testing for statistically significant differences between fully 3D PET reconstruction algorithms

We present a practical methodology for evaluating 3D PET reconstruction methods. It includes generation of random samples from a statistically described ensemble of 3D images resembling those to which PET would be applied in a medical situation, generation of corresponding projection data with noise and detector point spread function simulating those of a 3D PET scanner, assignment of figures of merit appropriate for the intended medical applications, optimization of the reconstruction algorithms on a training set of data, and statistical testing of the validity of hypotheses that say that two reconstruction algorithms perform equally well (from the point of view of a particular figure of merit) as compared to the alternative hypotheses that say that one of the algorithms outperforms the other. Although the methodology was developed with the 3D PET in mind, it can be used, with minor changes, for other 3D data collection methods, such as fully 3D cr or SPECT.     

ROC analysis for assessment of lesion detection performance in 3D PET: influence of reconstruction algorithms

Image quality in positron emission tomography (PET) can be assessed with physical parameters, as spatial resolution and signal-to-noise ratio, or using psychophysical approaches, which include the observer performance and the considered task (ROC analysis). For PET in oncology, such a task is the detection of hot lesions. The aim of the present study was to assess the lesion detection performance due to adequate modeling of the scanner and the measurement process in the image reconstruction process. We compared the standard OSEM software of the manufacturer with a sophisticated fully 3D iterative reconstruction technique (USC MAP). A rectangular phantom with 6 oblique line sources in a homogeneous background (2.6 kBq/ml 18F) was imaged dynamically with an ECAT EXACT HR+ scanner in 3D mode. Reconstructed activity contrasts varied between 15 and 0, as the line sources were filled with 11C (3.2 MBq/ml). Measured attenuation and standard randoms, dead time, and scatter corrections of the manufacturer were employed. For the ROC analysis, a software tool presented a cut-out of the phantom (15 x 15 pixels) to two observers. These cut-outs were rated (5 classes) and the area Az under the ROC curve was determined as a measure of detection performance. The improvement for Az with USC MAP compared to the OSEM reconstructions ranged between 0.02 and 0.23 for signal-to-noise ratios of the background between 2.8 and 3.1 and lesion contrast between 2.1 and 4.2. This study demonstrates that adequate modeling of the measurement process in the reconstruction algorithm improves the detection of small hot lesions markedly.     

Pragmatic fully 3D image reconstruction for the MiCES mouse imaging PET scanner

We present a pragmatic approach to image reconstruction for data from the micro crystal elements system (MiCES) fully 3D mouse imaging positron emission tomography (PET) scanner under construction at the University of Washington. Our approach is modelled on fully 3D image reconstruction used in clinical PET scanners, which is based on Fourier rebinning (FORE) followed by 2D iterative image reconstruction using ordered-subsets expectation-maximization (OSEM). The use of iterative methods allows modelling of physical effects (e.g., statistical noise, detector blurring, attenuation, etc), while FORE accelerates the reconstruction process by reducing the fully 3D data to a stacked set of independent 2D sinograms. Previous investigations have indicated that non-stationary detector point-spread response effects, which are typically ignored for clinical imaging, significantly impact image quality for the MiCES scanner geometry. To model the effect of non-stationary detector blurring (DB) in the FORE+OSEM(DB) algorithm, we have added a factorized system matrix to the ASPIRE reconstruction library. Initial results indicate that the proposed approach produces an improvement in resolution without an undue increase in noise and without a significant increase in the computational burden. The impact on task performance, however, remains to be evaluated.     

A performance study of 3D reconstruction algorithms for positron emission tomography

This paper investigates the statistical and systematic accuracy of five three-dimensional reconstruction algorithms for multi-ring PET scanners operated without septa: the reprojection method, the direct Fourier reconstruction, the FAVOR algorithm, and the single-slice and multi-slice rebinning algorithms. Simulated data of a uniform cylinder, of Gaussian sources, and of spherical sources are used to compare respectively the noise properties, the modulation transfer function, and the recovery coefficients of the algorithms. Brain scans reconstructed with the different algorithms are compared by calculating the linear regression of the mean values within regions of interest. The most significant observations are a slight loss of transaxial resolution with the reprojection algorithm in the external slices of the scanner, and increased noise in the images reconstructed using multi-slice rebinning.     

A fully 3D small PET scanner.

A fully 3D small PET scanner based on a novel detection principle for gamma rays is described. It uses BaF2 scintillator and photosensitive wire chambers. Extensive tests with technical prototypes have shown that such a system will have a detection efficiency for gamma rays comparable with what can be obtained with the more traditional approach, and a spatial resolution determined by the size of the crystals. The expected performances of the scanner, based on our measurements and on simulations, are given.     

Optimization of a fully 3D single scatter simulation algorithm for 3D PET

We describe a new implementation of a single scatter simulation (SSS) algorithm for the prediction and correction of scatter in 3D PET. In this implementation, out of field of view (FoV) scatter and activity, side shields and oblique tilts are explicitly modelled. Comparison of SSS predictions with Monte Carlo simulations and experimental data from uniform, line and cold-bar phantoms showed that the code is accurate for uniform as well as asymmetric objects and can model different energy resolution crystals and low level discriminator (LLD) settings. Absolute quantitation studies show that for most applications, the code provides a better scatter estimate than the tail-fitting scatter correction method currently in use at our institution. Several parameters such as the density of scatter points, the number of scatter distribution sampling points and the axial extent of the FoV were optimized to minimize execution time, with particular emphasis on patient studies. Development and optimization were carried out in the case of GSO-based scanners, which enjoy relatively good energy resolution. SSS estimates for scanners with lower energy resolution may result in different agreement, especially because of a higher fraction of multiple scatter events. The algorithm was applied to a brain phantom as well as to clinical whole-body studies. It proved robust in the case of large patients, where the scatter fraction increases. The execution time, inclusive of interpolation, is typically under 5 min for a whole-body study (axial FoV: 81 cm) of a 100 kg patient.     

Fast and efficient fully 3D PET image reconstruction using sparse system matrix factorization with GPU acceleration

Statistically based iterative image reconstruction has been widely used in positron emission tomography (PET) imaging. The quality of reconstructed images depends on the accuracy of the system matrix that defines the mapping from the image space to the data space. However, an accurate system matrix is often associated with high computation cost and huge storage requirement. In this paper, we present a method to address this problem using sparse matrix factorization and graphics processor unit (GPU) acceleration. We factor the accurate system matrix into three highly sparse matrices: a sinogram blurring matrix, a geometric projection matrix and an image blurring matrix. The geometrical projection matrix is precomputed based on a simple line integral model, while the sinogram and image blurring matrices are estimated from point-source measurements. The resulting factored system matrix has far less nonzero elements than the original system matrix, which substantially reduces the storage and computation cost. The smaller matrix size also allows an efficient implementation of the forward and backward projectors on a GPU, which often has a limited memory space. Our experimental studies show that the proposed method can dramatically reduce the computation cost of high-resolution iterative image reconstruction, while achieving better performance than existing factorization methods.     

Model-based normalization for iterative 3D PET image reconstruction

We describe a method for normalization in 3D PET for use with maximum a posteriori (MAP) or other iterative model-based image reconstruction methods. This approach is an extension of previous factored normalization methods in which we include separate factors for detector sensitivity, geometric response, block effects and deadtime. Since our MAP reconstruction approach already models some of the geometric factors in the forward projection, the normalization factors must be modified to account only for effects not already included in the model. We describe a maximum likelihood approach to joint estimation of the count-rate independent normalization factors, which we apply to data from a uniform cylindrical source. We then compute block-wise and block-profile deadtime correction factors using singles and coincidence data, respectively, from a multiframe cylindrical source. We have applied this method for reconstruction of data from the Concorde microPET P4 scanner. Quantitative evaluation of this method using well-counter measurements of activity in a multicompartment phantom compares favourably with normalization based directly on cylindrical source measurements.     

Three penalized EM-type algorithms for PET image reconstruction

Based on Bayes theory, Green introduced the maximum a posteriori (MAP) algorithm to obtain a smoothing reconstruction for positron emission tomography. This algorithm is flexible and convenient for most of the penalties, but it is hard to guarantee convergence. For a common goal, Fessler penalized a weighted least squares (WLS) estimator by a quadratic penalty and then solved it with the successive over-relaxation (SOR) algorithm, however, the algorithm was time-consuming and difficultly parallelized. Anderson proposed another WLS estimator for faster convergence, on which there were few regularization methods studied. For three regularized estimators above, we develop three new expectation maximization (EM) type algorithms to solve them. Unlike MAP and SOR, the proposed algorithms yield update rules by minimizing the auxiliary functions constructed on the previous iterations, which ensure the cost functions monotonically decreasing. Experimental results demonstrated the robustness and effectiveness of the proposed algorithms.     

Enhanced 3D PET OSEM reconstruction using inter-update Metz filtering

We present an enhancement of the OSEM (ordered set expectation maximization) algorithm for 3D PET reconstruction, which we call the inter-update Metz filtered OSEM (IMF-OSEM). The IMF-OSEM algorithm incorporates filtering action into the image updating process in order to improve the quality of the reconstruction. With this technique, the multiplicative correction image--ordinarily used to update image estimates in plain OSEM--is applied to a Metz-filtered version of the image estimate at certain intervals. In addition, we present a software implementation that employs several high-speed features to accelerate reconstruction. These features include, firstly, forward and back projection functions which make full use of symmetry as well as a fast incremental computation technique. Secondly, the software has the capability of running in parallel mode on several processors. The parallelization approach employed yields a significant speed-up, which is nearly independent of the amount of data. Together, these features lead to reasonable reconstruction times even when using large image arrays and non-axially compressed projection data. The performance of IMF-OSEM was tested on phantom data acquired on the GE Advance scanner. Our results demonstrate that an appropriate choice of Metz filter parameters can improve the contrast-noise balance of certain regions of interest relative to both plain and post-filtered OSEM, and to the GE commercial reprojection algorithm software.     

The performance of monotonic and new non-monotonic gradient ascent reconstruction algorithms for high-resolution neuroreceptor PET imaging

Iterative expectation maximization (EM) techniques have been extensively used to solve maximum likelihood (ML) problems in positron emission tomography (PET) image reconstruction. Although EM methods offer a robust approach to solving ML problems, they usually suffer from slow convergence rates. The ordered subsets EM (OSEM) algorithm provides significant improvements in the convergence rate, but it can cycle between estimates converging towards the ML solution of each subset. In contrast, gradient-based methods, such as the recently proposed non-monotonic maximum likelihood (NMML) and the more established preconditioned conjugate gradient (PCG), offer a globally convergent, yet equally fast, alternative to OSEM. Reported results showed that NMML provides faster convergence compared to OSEM; however, it has never been compared to other fast gradient-based methods, like PCG. Therefore, in this work we evaluate the performance of two gradient-based methods (NMML and PCG) and investigate their potential as an alternative to the fast and widely used OSEM. All algorithms were evaluated using 2D simulations, as well as a single [(11)C]DASB clinical brain dataset. Results on simulated 2D data show that both PCG and NMML achieve orders of magnitude faster convergence to the ML solution compared to MLEM and exhibit comparable performance to OSEM. Equally fast performance is observed between OSEM and PCG for clinical 3D data, but NMML seems to perform poorly. However, with the addition of a preconditioner term to the gradient direction, the convergence behaviour of NMML can be substantially improved. Although PCG is a fast convergent algorithm, the use of a (bent) line search increases the complexity of the implementation, as well as the computational time involved per iteration. Contrary to previous reports, NMML offers no clear advantage over OSEM or PCG, for noisy PET data. Therefore, we conclude that there is little evidence to replace OSEM as the algorithm of choice for many applications, especially given that in practice convergence is often not desired for algorithms seeking ML estimates.     

Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography

We investigate fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography for applications in small animal imaging. Our forward model uses a diffusion approximation for optically inhomogeneous tissue, which we solve using a finite element method (FEM). We examine two approaches to incorporating the forward model into the solution of the inverse problem. In a conventional direct calculation approach one computes the full forward model by repeated solution of the FEM problem, once for each potential source location. We describe an alternative on-the-fly approach where one does not explicitly solve for the full forward model. Instead, the solution to the forward problem is included implicitly in the formulation of the inverse problem, and the FEM problem is solved at each iteration for the current image estimate. We evaluate the convergence speeds of several representative iterative algorithms. We compare the computation cost of those two approaches, concluding that the on-the-fly approach can lead to substantial reductions in total cost when combined with a rapidly converging iterative algorithm.     

Quantitative analysis of a reconstruction method for fully three-dimensional PET.

The major advantage of positron emission tomography (PET) using large area planar detectors over scintillator-based commercial ring systems is the potentially larger (by a factor of two or three) axial field-of-view (FOV). However, to achieve the space invariance of the point spread function necessary for Fourier filtering a polar angle rejection criterion is applied to the data during backprojection resulting in a trade-off between FOV size and sensitivity. A new algorithm due to Defrise and co-workers developed for list-mode data overcomes this problem with a solution involving the division of the image into several subregions. A comparison between the existing backprojection-then-filter algorithm and the new method (with three subregions) has been made using both simulated and real data collected from the MUP-PET positron camera. Signal-to-noise analysis reveals that improvements of up to a factor of 1.4 are possible resulting from an increased data usage of up to a factor of 2.5 depending on the axial extent of the imaged object. Quantitation is also improved.     

Internet2-based 3D PET image reconstruction using a PC cluster

We describe an approach to fast iterative reconstruction from fully three-dimensional (3D) PET data using a network of PentiumIII PCs configured as a Beowulf cluster. To facilitate the use of this system, we have developed a browser-based interface using Java. The system compresses PET data on the user's machine, sends these data over a network, and instructs the PC cluster to reconstruct the image. The cluster implements a parallelized version of our preconditioned conjugate gradient method for fully 3D MAP image reconstruction. We report on the speed-up factors using the Beowulf approach and the impacts of communication latencies in the local cluster network and the network connection between the user's machine and our PC cluster.     

Analytical study of performance in a 3D PET scanner.

The use of arrays of small discrete detectors and thin slices to achieve a high isotropic spatial resolution in positron emission tomography (PET) results in systems with a low ring sensitivity. In a multi-ring system, the overall sensitivity can be considerably improved by removing the interslice collimators to make full use of all cross slices coincidences, but this is achieved at the expense of increased scatter and accidental rates in the image. For imaging of small laboratory animals (diameter of 10 cm or less), the relieved burden of scatters, and to some extent accidentals, suggests that volumetric imaging may be of particular value. In order to evaluate the performance to be expected from a small animal PET scanner (10 cm diameter field) with and without the interplane collimators, the incident event rates for singles (unscattered and single scattered) and true, scatter and accidental coincidences were evaluated analytically. The performance was evaluated for various source sizes and activities in terms of five criteria: true/single ratio, scatter fraction, accidental fraction, image contrast and noise effective sensitivity. As a result of a 3-fold increase in scatter fraction and of a significant increase in accidental fraction for larger sources, the image contrast (true/(scatter+accidental)) is observed to always be inferior with the collimators removed. However, a significant improvement in noise effective sensitivity is obtained with the volumetric configuration, especially for small size sources placed at the centre of the field of view. It is concluded that the volumetric configuration is more advantageous than the multislice configuration to image small animals because the gain in sensitivity overcomes the loss of accuracy due to higher scatter and accidental rates.     

Figures of merit for comparing reconstruction algorithms with a volume-imaging PET scanner

For volume-imaging PET scanners, no septa are used to maximize the sensitivity by collecting events oblique to the scanner axis. We answer two questions: (i) how does the performance of an image reconstruction algorithm for a volume-imaging PET scanner depend on its general dimensions? and (ii) at what point is a three-dimensional (3D) reconstruction algorithm needed for a volume-imaging scanner, as the axial extent is increased? A 3D reconstruction algorithm will accurately incorporate the oblique events in a reconstruction of the original source distribution. From simulations of an existing volume PET scanner with a maximum axial acceptance angle (+/-alpha) of alpha = 9 degrees, however, we show that the single-slice rebinning algorithm is a good compromise between sensitivity, speed, and accuracy when compared to standard two-dimensional reconstruction (alpha = 1 degrees), and a 3D reconstruction with alpha = 9 degrees. We also show with simulations that a new scanner with alpha = 27 degrees requires 3D reconstruction in order to achieve maximum sensitivity without unacceptable losses in accuracy. Measurements of scanner performance are based on a series of figures of merit that characterize image quality and quantitative accuracy measured from a set of simulated test phantoms.     

Fast reconstruction of 3D time-of-flight PET data by axial rebinning and transverse mashing

Faster scintillators like LaBr(3) and LSO have sparked renewed interest in PET scanners with time-of-flight (TOF) information. The TOF information adds another dimension to the data set compared to conventional three-dimensional (3D) PET with the size of the projection data being multiplied by the number of TOF bins. Here we show by simulations and analytical reconstruction that angular sampling for two-dimensional (2D) TOF PET can be reduced significantly compared to what is required for conventional 2D PET. Fully 3D TOF PET data, however, have a wide range of oblique and transverse angles. To make use of the smaller necessary angular sampling we reduce the 3D data to a set of 2D histoprojections. This is done by rebinning the 3D data to 2D data and by mashing these 2D data into a limited number of angles. Both methods are based on the most likely point given by the TOF measurement. It is shown that the axial resolution loss associated with rebinning reduces with improved timing resolution and becomes less than 1 mm for a TOF resolution below 300 ps. The amount of angular mashing that can be applied without tangential resolution loss increases with improved TOF resolution. Even quite coarse angular mashing (18 angles out of 324 measured angles for 424 ps) does not significantly reduce image quality in terms of the contrast or noise. The advantages of the proposed methods are threefold. Data storage is reduced to a limited number of 2D histoprojections with TOF information. Compared to listmode format we have the advantage of a predetermined storage space and faster reconstruction. The method does not require the normalization of projections prior to rebinning and can be applied directly to measured listmode data.