Performance of a Fast Maximum Likelihood Algorithm for Fully 3D PET Reconstruction

The authors studied certain characteristics of the expectation-maximization (EM) algorithm for image reconstruction in positron emission tomography (PET). They have attempted to speed up the algorithm by performing some time-consuming computation only once, at the initiation stage. They have studied the resulting memory size, disk space, and CPU requirements versus the number of pixels in the reconstructed image and the number of gamma -ray detectors in the tomograph. The convergence and image quality have been studied as a function of the number of iterations performed. Preliminary results are presented. >

Ordered Subsets Expectation Maximization (OSEM) has been introduced to greatly reduce reconstruction time compared to Maximum Likelihood Expectation Maximization (MLEM) in Positron Emission Tomography (PET). In case of noise-free data, OSEM converges to a MLEM image, but in presence of noise this might not be true, and also OSEM might not converge at all but cycle between similar but distinct images. Assessing what are the limits of the OSEM in terms of reliable contrast recovery is particularly interesting in situation of very short acquisitions and very sparse data. In this work, we study the convergence value of the contrast recovery as a function of the count statistics (and therefore the noise level) in the PET data, using a NEMA image quality phantom. A comparison between Time-of-Flight (TOF) and non-TOF implementations is performed. The scope of the work is to assess how the count statistics affects the convergence value and the convergence rate for OSEM reconstruction, and then to compare the corresponding convergence rates in conventional PET and in TOF PET reconstructions, by analyzing both experimental and simulated data.

Image quality of positron emission tomography (PET) reconstructions is degraded by subject motion occurring during the acquisition. Magnetic resonance (MR)-based motion correction approaches have been studied for PET/MR scanners and have been successful at capturing regular motion patterns, when used in conjunction with surrogate signals (e.g. navigators) to detect motion. However, handling irregular respiratory motion and bulk motion remains challenging. In this work, we propose an MR-based motion correction method relying on subspace-based real-time MR imaging to estimate motion fields used to correct PET reconstructions. We take advantage of the low-rank characteristics of dynamic MR images to reconstruct high-resolution MR images at high frame rates from highly undersampled k-space data. Reconstructed dynamic MR images are used to determine motion phases for PET reconstruction and estimate phase-to-phase nonrigid motion fields able to capture complex motion patterns such as irregular respiratory and bulk motion. MR-derived binning and motion fields are used for PET reconstruction to generate motion-corrected PET images. The proposed method was evaluated on in vivo data with irregular motion patterns. MR reconstructions accurately captured motion, outperforming state-of-the-art dynamic MR reconstruction techniques. Evaluation of PET reconstructions demonstrated the benefits of the proposed method in terms of motion artifacts reduction, improving the contrast-to-noise ratio by up to a factor 3 and achieveing a target-to-background ratio up to 90% superior compared to standard/uncorrected methods. The proposed method can improve the image quality of motion-corrected PET reconstructions in clinical applications.

In this article, we evaluate Parallel Level Sets (PLS) and Bowsher's method as segmentation-free anatomical priors for regularized brain positron emission tomography (PET) reconstruction. We derive the proximity operators for two PLS priors and use the EM-TV algorithm in combination with the first order primal-dual algorithm by Chambolle and Pock to solve the non-smooth optimization problem for PET reconstruction with PLS regularization. In addition, we compare the performance of two PLS versions against the symmetric and asymmetric Bowsher priors with quadratic and relative difference penalty function. For this aim, we first evaluate reconstructions of 30 noise realizations of simulated PET data derived from a real hybrid positron emission tomography/magnetic resonance imaging (PET/MR) acquisition in terms of regional bias and noise. Second, we evaluate reconstructions of a real brain PET/MR data set acquired on a GE Signa time-of-flight PET/MR in a similar way. The reconstructions of simulated and real 3D PET/MR data show that all priors were superior to post-smoothed maximum likelihood expectation maximization with ordered subsets (OSEM) in terms of bias-noise characteristics in different regions of interest where the PET uptake follows anatomical boundaries. Our implementation of the asymmetric Bowsher prior showed slightly superior performance compared with the two versions of PLS and the symmetric Bowsher prior. At very high regularization weights, all investigated anatomical priors suffer from the transfer of non-shared gradients.

The maximum likelihood expectation maximization (MLEM) reconstruction method is known to yield noisy images at high iteration numbers because emission tomographic reconstruction is an ill-posed problem. The noise can be suppressed by post-filtering the ML estimate or imposing a priori knowledge as a constraint within a Bayesian reconstruction framework. Most of these filters and priors are based on weighting the intensity differences between neighbouring pixels within a small local neighbourhood. Therefore, they have limited information to distinguish edges from noise. We investigated the use of a non-local means (NLM) filter for post-filtering MLEM reconstructed positron emission tomography (PET) images. We further investigated the effect of incorporating anatomical side information obtained from co-registered computed tomography (CT) images into the NLM, resulting in an adaptive non-local means (A-NLM) filter which takes into account the variance within each anatomical region on the PET image. In simulated and physical phantom experiments, the A-NLM filter demonstrated superior performance tradeoff between lesion contrast and noise than conventional Gaussian post-filtering and NLM without anatomical prior. We conclude that the A-NLM filter has potential for improved lesion detection over Gaussian post-filtered MLEM images.

A Maximum Likelihood Expectation Maximization (MLEM) method is proposed for joint tissue activity distribution and photon attenuation map reconstruction from Positron Emission Tomography (PET) data. It is based on a new set of latent random variables, similar to the ones used by Lange and Carson for CT [JCAT, 1984]. It results in a separable likelihood function, which simplifies maximization. The proposed method is consistent with the methods of Krol et al. for SPECT [TMI, 2001] and Shepp and Vardi for PET [TMI, 1982].

Including accurate modeling of the point spread function (PSF) in positron emission tomography (PET) reconstruction algorithms results in improvements in image spatial resolution and contrast. In this work, we sampled the PSF in our first-generation radio-frequency brain dedicated PET insert for simultaneous PET/MR imaging using a 100 µCi NEMA standard 250 µm diameter Na-22 point source at 13 different positions within a subsection of the system field of view (FoV). The acquired list mode data was converted into the canonical sinogram format from which the spatial positioning of the source and standard deviations were calculated. The subset was then used to extrapolate the PSF for the full system FoV. This model was then fed as an input parameter into a graphical processing unit based ordered subset expectation maximization (OSEM) reconstruction algorithm and used to generate reconstructed images with and without spatially varying PSF modeling for the Na-22 point source and a Hoffman brain phantom. Results indicate that for point source reconstruction, the FWHM of the horizontal profile of the point source is smaller with spatially variant PSF especially closer to the edges. Effect of spatially varying PSF modeling is also presented with Hoffman phantom reconstruction and CNR value has increased with spatially varying PSF.

Objective.This article explores a new graphics processing unit (GPU)-based techniques for efficient image reconstruction in organ-targeted positron emission tomography (PET) scanners with planar detectors.Approach.GPU-based reconstruction is applied to the Radialis low-dose organ-targeted PET technology, developed to overcome the issues of high exposure and limited spatial resolution inherent in traditional whole-body PET/CT (Computed Tomography) scans. The Radialis planar detector technology is based on four-side tileable sensor modules that can be seamlessly combined into a sensing area of the needed size, optimizing the axial field-of-view for specific organs, and maximizing geometric sensitivity. The article explores the transition from central processing unit-based maximum likelihood expectation maximization algorithms to a GPU-based counterpart, demonstrating a tenfold overall speedup in image reconstruction with a hundredfold improvement in iteration speed.Main results.Through standardized PET performance tests and clinical image analysis, this work demonstrates that GPU-based image reconstruction maintains diagnostic image quality while significantly reducing reconstruction times. The application of this technology, particularly in breast imaging using the Radialis low-dose positron emission mammography, significantly reduces exam times thus improving patient comfort and throughput in clinical settings.Significance.This study represents an important advancement in the clinical workflow of PET imaging, providing insights into optimizing reconstruction algorithms to effectively leverage the parallel processing capabilities of GPUs.

The promise of new PET image reconstruction

Several algorithms for image reconstruction in positron emission tomography (PET) have been described in the medical and statistical literature. We study a continuous idealization of the PET reconstruction problem, considered as an example of bivariate density estimation based on indirect observations. Given a large sample of indirect observations, we consider the size of the equivalent sample of observations, whose original exact positions would allow equally accurate estimation of the image of interest. Both for indirect and for direct observations, we establish exact minimax rates of convergence of estimation, for all possible estimators, over suitable smoothness classes of functions. A key technical device is a modulus of continuity appropriate to global function estimation. For indirect data and (in practice unobservable) direct data, the rates for mean integrated square error are $n^{-p/(p + 2)}$ and $(n/\\log n)^{-p/(p + 1)}$, respectively, for densities in a class corresponding to bounded square-integrable $p$th derivatives. We obtain numerical values for equivalent sample sizes for minimax linear estimators using a slightly modified error criterion. Modifications of the model to incorporate attenuation and the third dimension effect do not affect the minimax rates. The approach of the paper is applicable to a wide class of linear inverse problems.

In recent years, various approaches to overcome the problems associated with iterative statistical image reconstruction methods have been investigated, including issues such as slow convergence rate, costly computational time and nonuniform correction efficiency. A multiscale Bayesian image reconstruction algorithm based on a 1-D wavelet technique is proposed. In this approach, the authors first apply a wavelet transform to the sinogram measurements, which yields a multiresolution object representation. The image is reconstructed at the corresponding grid by the Bayesian image reconstruction algorithm for each scale level in the frequency space; the backprojector is also modified to match each corresponding resolution scale. After the low-frequency components of the image have been recovered sufficiently on the coarser scale level, the resulted image is used as the starting point for the finer level at a new iteration. An important feature in the authors' new approach is that the wavelet decomposition is carried out in the sinogram space, while multiple grids are also used in the reconstructed image space.

An algorithm is described for three-dimensional whole-body (3DWB) image reconstruction in positron emission tomography. For whole-body applications, improvements to the popular fixed-axial-acceptance-angle technique are achieved by combining axially adjacent projection data available with a long-axis data set. Time-consuming reprojection of unmeasured oblique lines of response is reduced or eliminated by axial overlap of bed positions, while pixel variance and reconstructed axial resolution are made more uniform by the overlap. Improvements in noise and resolution uniformity are accompanied by gains in reconstruction efficiency, and may be optimized against increased acquisition time due to overlapping acquisition segments and reduced axial coverage. An 11-detector-ring overlap improves axial uniformity in coronal images of a long, uniform cylinder from 23% to 8% with uniform axial resolution. Associated with a 37% improvement in reconstruction time is a 34% reduction in axial coverage for four bed positions. A smaller degree of overlap is found to provide the best trade-off between image uniformity, total scan duration, and reconstruction time because of a proportionally greater reduction in reprojected lines of response. Using a sample optimization scheme, we find a three-ring overlap is best for a 60 cm axial field of view and a five-ring overlap for an 80 cm axial field of view.

The objective of this research is to investigate the potential of total variation (TV) nullspace smoothing to mitigate the edge artifacts produced in expectation maximization (EM) reconstruction. Its success in reducing noise and preserving edges in image restoration, suggests that TV norm minimization could be explored in the reconstruction of positron emission tomography (PET) images. For example, the following criteria could be used for reconstruction: Find the image that has the minimum TV norm such that the measurements produced by the image are sufficiently close to the actual measurements. It is argued here, however, that this criterion may cause distortion of edges because of the Poisson nature of PET measurements. Moreover, the above criterion does not exploit the Poisson model of the data as the EM algorithm does. As a potentially better alternative, TV nullspace smoothing of EM reconstructions is proposed here. Rather than minimizing the whole EM reconstructed image with respect to the TV norm, in nullspace smoothing, only a portion of the reconstructed image is minimized; namely, the nullspace component of the reconstructed image. This may be advantageous because PET data contains no information about the nullspace component of the object being scanned. A second potential advantage stems from the fact that EM algorithm attempts to find an image that maximizes the likelihood function associated with a Poisson model of the data. The nullspace smoothed image is as optimal, in the maximum likelihood sense, as the original unsmoothed image. Methodology. To perform this investigation, noise-less projections of a mathematical phantom were simulated. Results. It was seen that the edge artifacts of the TV nullspace smoothed image were uniformly better than those in the original unsmoothed EM reconstruction. Conclusion. Although the results presented here are encouraging, more work needs to be done in the future to demonstrate the usefulness of nullspace smoothing with real data.

MRI-guided (MRIg) positron emission tomography (PET) reconstruction can potentially reduce noise and increase spatial resolution compared to standard clinical ordered-subsets expectation-maximization (OSEM) image quality. However, to adjust for the desired image quality, the balance between measured data and prior information usually requires manual tuning. This work presents an adaptive method to automatically control the influence of the magnetic resonance imaging (MRI) information on the PET emission data using maximum a posteriori (MAP) image reconstruction, robust against a wide range of counts. The method was evaluated on different static brain PET datasets using [18F]-FDG, [18F]-Florbetapir and [11C]-PiB, acquired in a simultaneous PET/MRI scanner and a PET/CT scanner, followed by an MRI scan. Noise in gray and white matter was measured for a wide range of statistics. Furthermore, noise and quantification accuracy were analyzed in different cortical and subcortical brain regions with different levels of tracer uptake, and at different levels of counts. Results demonstrated consistent improved image quality in terms of noise and spatial resolution with MRI-guided MAP PET (MRIg-MAP) reconstruction compared to OSEM. Additionally, it was shown that the number of collected counts could be reduced by ~1.6– <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.3\times $ </tex-math></inline-formula> using MRIg-MAP reconstruction compared to OSEM, without increasing the noise significantly, either by reducing the scan time or injected activity. In conclusion, we presented a novel method to adaptively balance the influence of the anatomical information on the emission data for MRIg-MAP reconstruction, which showed image quality improvements compared to OSEM for different radiotracers, at different levels of counts, and applicable to simultaneous and sequential PET-MRI scans.

List-mode positron emission tomography (PET) image reconstruction is an important tool for PET scanners with many lines-of-response and additional information such as time-of-flight and depth-of-interaction. Deep learning is one possible solution to enhance the quality of PET image reconstruction. However, the application of deep learning techniques to list-mode PET image reconstruction has not been progressed because list data is a sequence of bit codes and unsuitable for processing by convolutional neural networks (CNN). In this study, we propose a novel list-mode PET image reconstruction method using an unsupervised CNN called deep image prior (DIP) which is the first trial to integrate list-mode PET image reconstruction and CNN. The proposed list-mode DIP reconstruction (LM-DIPRecon) method alternatively iterates the regularized list-mode dynamic row action maximum likelihood algorithm (LM-DRAMA) and magnetic resonance imaging conditioned DIP (MR-DIP) using an alternating direction method of multipliers. We evaluated LM-DIPRecon using both simulation and clinical data, and it achieved sharper images and better tradeoff curves between contrast and noise than the LM-DRAMA, MR-DIP and sinogram-based DIPRecon methods. These results indicated that the LM-DIPRecon is useful for quantitative PET imaging with limited events while keeping accurate raw data information. In addition, as list data has finer temporal information than dynamic sinograms, list-mode deep image prior reconstruction is expected to be useful for 4D PET imaging and motion correction.