A majorize-minimize line search algorithm for barrier function optimization

Computed tomography (CT) can gain a significant increase in the signal-to-noise ratio using constraints to better define regions of known intensity. This is accomplished in positron emission tomography (PET) by reducing the region of unknown activity with time-coincidence circuitry. Mathematically, constraints can be implemented into reconstruction algorithms using a priori information, such as the use of an x-ray CT image to define regions of radionuclide uptake in single photon emission computed tomography (SPECT) or PET imaging. In addition to regional constraints, intensity constraints can also be included into the model equations. This may be especiallly useful where the data has a high degree of contrast as in DSA limited-angle tomography. The maximum entropy reconstruction problem with constraints is reduced to a dual optimization program using duality principles. The solution to the reconstruction problem is determined by solving for the optimum solution to this dual program. Inequality con-straints complicates the problem in that the derivatives of the function to be optimized may not exist at the boundary points. Using penalty function techniques, the problem can be structured as an unconstrained optimization program. This way, solutions can be determined using gradient type of algorithms which require an essentially smooth function as feasible solutions approach boundary points.

The promise of new PET image reconstruction

Standard 3D dynamic positron emission tomographic (PET) imaging consists of independent image reconstructions of individual frames followed by application of appropriate kinetic model to the time activity curves at the voxel or region-of-interest (ROI). The emerging field of 4D PET reconstruction, by contrast, seeks to move beyond this scheme and incorporate information from multiple frames within the image reconstruction task. Here we propose a novel reconstruction framework aiming to enhance quantitative accuracy of parametric images via introduction of priors based on voxel kinetics, as generated via clustering of preliminary reconstructed dynamic images to define clustered neighborhoods of voxels with similar kinetics. This is then followed by straightforward maximum a posteriori (MAP) 3D PET reconstruction as applied to individual frames; and as such the method is labeled ‘3.5D’ image reconstruction. The use of cluster-based priors has the advantage of further enhancing quantitative performance in dynamic PET imaging, because: (a) there are typically more voxels in clusters than in conventional local neighborhoods, and (b) neighboring voxels with distinct kinetics are less likely to be clustered together. Using realistic simulated 11C-raclopride dynamic PET data, the quantitative performance of the proposed method was investigated. Parametric distribution-volume (DV) and DV ratio (DVR) images were estimated from dynamic image reconstructions using (a) maximum-likelihood expectation maximization (MLEM), and MAP reconstructions using (b) the quadratic prior (QP-MAP), (c) the Green prior (GP-MAP) and (d, e) two proposed cluster-based priors (CP-U-MAP and CP-W-MAP), followed by graphical modeling, and were qualitatively and quantitatively compared for 11 ROIs. Overall, the proposed dynamic PET reconstruction methodology resulted in substantial visual as well as quantitative accuracy improvements (in terms of noise versus bias performance) for parametric DV and DVR images. The method was also tested on a 90 min 11C-raclopride patient study performed on the high-resolution research tomography. The proposed method was shown to outperform the conventional method in visual as well as quantitative accuracy improvements (in terms of noise versus regional DVR value performance).

Advanced clinical positron-emission tomography (PET) scanners exploit time-of-flight (TOF) information of annihilation photons for improving PET imaging performance, and image reconstruction from TOF-PET data remains an active research topic. In the study, we investigate optimization-based image reconstruction from list-mode TOF-PET data collected with a clinical PET system employing the digital silicon photomultiplier (SiPM) technology. The reconstruction problem is formulated as an image-TV-constrained data-fidelity-minimization problem, and a primal-dual algorithm is tailored for image reconstruction through solving the optimization problem. In an attempt to evaluate the reconstruction performance, we have carried out studies on images reconstructed from data of patient subjects. A focus of the work centers on investigating image reconstruction from data containing different amounts of photon accounts. Results of the study reveal that the proposed algorithm can yield image reconstruction with suppressed noise level and increased axial coverage over that obtained in conventional reconstructions.

Purpose: Positron emission tomography (PET) using open dual ring geometry was investigated as an on‐board system for functional imaging and PET marker tracking, specifically with tomotherapy. The open dual ring PET would allow measurement of both inter and intra‐fractional variation, improving the delineation of tumor volume at any stage in the radiation treatment delivery process. We compare results from data obtained with ray tracing to data from Monte Carlo simulations. This study demonstrates that depth of information (DOI) along the z axis of the crystal is required to reduce artifact from DOI blurring due to long axial PET geometry. Methods: PET design was accomplished via Monte Carlo simulations with GATE, the Geant4 Application for Emission Tomography. Images were reconstructed using Software for Image Reconstruction (STIR) with a fully 3D OSEM. Investigation of whether the image blurring is the result of DOI was carried out by scaling axially the PET design to one half and one forth the original proposed open PET. Image reconstruction with varying the ring difference or level of obliqueness of the accepted lines of response (LORs) for reconstruction also indicates the role of the DOI blurring. Results: Monte Carlo simulations of the open PET geometry indicate that the prominent physical factor that degrades image quality is the DOI. When comparing the Monte Carlo simulation for a phantom with four hot sources and cold background in air (reducing the effect of attenuation, randoms, and scatter) to that of the ray tracing simulations, we can visualize the degradation in the image quality. The reason is that GATE simulates the depth information but does not correct for DOI blurring, but STIR generated data assumes all LORs interact at the crystal surface. Conclusions: Onboard open PET with Tomotherapy must include DOI detectors to correct for the observed blurring.

High spatial resolution Positron Emission Tomography (PET) imaging provides functional data about a patient's body and provides us with a range of new possibilities, including medicine development, new diagnostic methods, and others. The use of Depth of Interaction (DOI) detectors is one effective technique for the improvement of whole-body PET imaging, as is evident in the fact that most small animal PET images are taken using a DOI system. In this study, we developed a prototype whole-body DOI- equipped PET system and evaluated its performance. This scanner's DOI detectors consist of dual-layer (front and rear) 9 x 10 array of GSO/GSO crystals, a light guide and two rectangular double-anode photo multiplier tubes (PMTs). The DOI layers are discriminated according to decay time differences, which are controlled by Ce concentrations. The DOI detectors are arranged in a circular detector ring, with a diameter of 664 mm. In order to avoid parallax errors, coincidental events, including rear layer detection, were addressed to the nearest neighbor LOR front-layer coincidence pair. To evaluate the spatial resolution of this PET system at various radial positions, 22Na point sources at 1, 10, 20, and 25 cm offset from the center were reconstructed, both with and without DOI information. Results showed that spatial resolution degradation at 25 cm improved from 195 % to 154 % as a result of using DOI data. During a visual evaluation using a Derenzo phantom, image reconstruction using DOI data clarified the image and corrected hot rod shapes. The results presented here show that DOI detectors were useful in creating an effective whole-body PET system.

This paper presents a logarithmic barrier method without line search for solving linear programming problem. The descent direction is the classical Newton's one. However, the displacement step is determined by a simple and efficient technique based on the notion of the minorant function approximating the barrier function. Numerical tests show the efficiency of this approach versus classical line search methods.

While the first time-of-flight (TOF)-positron emission tomography (PET) systems were already built in the early 1980s, limited clinical studies were acquired on these scanners. PET was still a research tool, and the available TOF-PET systems were experimental. Due to a combination of low stopping power and limited spatial resolution (caused by limited light output of the scintillators), these systems could not compete with bismuth germanate (BGO)-based PET scanners. Developments on TOF system were limited for about a decade but started again around 2000. The combination of fast photomultipliers, scintillators with high density, modern electronics, and faster computing power for image reconstruction have made it possible to introduce this principle in clinical TOF-PET systems. This paper reviews recent developments in system design, image reconstruction, corrections, and the potential in new applications for TOF-PET. After explaining the basic principles of time-of-flight, the difficulties in detector technology and electronics to obtain a good and stable timing resolution are shortly explained. The available clinical systems and prototypes under development are described in detail. The development of this type of PET scanner also requires modified image reconstruction with accurate modeling and correction methods. The additional dimension introduced by the time difference motivates a shift from sinogram- to listmode-based reconstruction. This reconstruction is however rather slow and therefore rebinning techniques specific for TOF data have been proposed. The main motivation for TOF-PET remains the large potential for image quality improvement and more accurate quantification for a given number of counts. The gain is related to the ratio of object size and spatial extent of the TOF kernel and is therefore particularly relevant for heavy patients, where image quality degrades significantly due to increased attenuation (low counts) and high scatter fractions. The original calculations for the gain were based on analytical methods. Recent publications for iterative reconstruction have shown that it is difficult to quantify TOF gain into one factor. The gain depends on the measured distribution, the location within the object, and the count rate. In a clinical situation, the gain can be used to either increase the standardized uptake value (SUV) or reduce the image acquisition time or administered dose. The localized nature of the TOF kernel makes it possible to utilize local tomography reconstruction or to separate emission from transmission data. The introduction of TOF also improves the joint estimation of transmission and emission images from emission data only. TOF is also interesting for new applications of PET-like isotopes with low branching ratio for positron fraction. The local nature also reduces the need for fine angular sampling, which makes TOF interesting for limited angle situations like breast PET and online dose imaging in proton or hadron therapy. The aim of this review is to introduce the reader in an educational way into the topic of TOF-PET and to give an overview of the benefits and new opportunities in using this additional information.

This work explores the feasibility and impact of including both the motion correction and the image registration transformation parameters from positron emission tomography (PET) image space to magnetic resonance (MR), or stereotaxic, image space within the system matrix of PET image reconstruction. This approach is motivated by the fields of neuroscience and psychiatry, where PET is used to investigate differences in activation patterns between different groups of participants, requiring all images to be registered to a common spatial atlas. Currently, image registration is performed after image reconstruction which introduces interpolation effects into the final image. Furthermore, motion correction (also requiring registration) introduces a further level of interpolation, and the overall result of these operations can lead to resolution degradation and possibly artifacts. It is important to note that performing such operations on a post-reconstruction basis means, strictly speaking, that the final images are not ones which maximize the desired objective function (e.g. maximum likelihood (ML), or maximum a posteriori reconstruction (MAP)). To correctly seek parameter estimates in the desired spatial atlas which are in accordance with the chosen reconstruction objective function, it is necessary to include the transformation parameters for both motion correction and registration within the system modeling stage of image reconstruction. Such an approach not only respects the statistically chosen objective function (e.g. ML or MAP), but furthermore should serve to reduce the interpolation effects. To evaluate the proposed method, this work investigates registration (including motion correction) using 2D and 3D simulations based on the high resolution research tomograph (HRRT) PET scanner geometry, with and without resolution modeling, using the ML expectation maximization (MLEM) reconstruction algorithm. The quality of reconstruction was assessed using bias-variance and root mean squared error analyses, comparing the proposed method to conventional post-reconstruction registration methods. An overall reduction in bias (for a cold region: from 41% down to 31% (2D) and 97% down to 65% (3D), and for a hot region: from 11% down to 8% (2D) and from 16% down to 14% (3D)) and in root mean squared error analyses (for a cold region: from 43% to 37% (2D) and from 97% to 65% (3D), and for a hot region: from 11% to 9% (2D) and from 16% down to 14% (3D)) in reconstructed regional mean activities (full regions of interest; all with statistical significance: p < 5 × 10−10) is found when including the motion correction and registration in the system matrix of the MLEM reconstruction, with resolution modeling. However, this improvement in performance comes with an extra computational cost of about 40 min. In this context, this work constitutes an important step toward the goal of estimating parameters of interest directly from the raw Poisson-distributed PET data, and hence toward the complete elimination of post-processing steps.

Positron emission tomography (PET) is an important imaging technique to visualize a number of functions in the brain or human body. For reconstructing PET images from the sinogram data, an inverse problem has to be solved using numerical optimizations such as expectation-maximization (EM)-based methods. However, the standard EM method suffers from measurement noise added in the sinogram data. In this paper, we propose a new simultaneous PET image reconstruction and parts extraction method using constrained non-negative matrix factorization. In contrast that the many existing methods reconstruct a single PET image independently, we reconstruct the time-series of PET images simultaneously from the time-series of sinograms using non-negative matrix factorization. Furthermore, we impose the smoothness constraint for the temporal feature, and the exclusive LASSO-based sparseness constraint for the spatial feature for robust image reconstruction and physically meaningful feature extraction.

This paper develops an iterative optimization technique that can be applied to mode scheduling. The algorithm provides both a mode schedule and timing of that mode schedule with convergence guarantees. Moreover, the algorithm takes advantage of a line search, and the number of iterations in the line search is bounded. There are two key ingredients in the algorithm. First, a projection operation is used that takes arbitrary curves and maps them to feasible switching controls. Second, a descent direction that incorporates the projection is calculated using the mode insertion gradient. Similar to derivative-based finite dimensional optimization, the convergence guarantees and sufficient decrease criteria follow from a local approximation of the cost in the direction of the search direction, but this local approximation is not the standard quadratic approximation. An example demonstrates the steps to implement the optimization algorithm and illustrates convergence.

Molecular imaging using dynamic positron emission tomography (PET) can provide in vivo images of physiologically or biochemically important parameters. Direct reconstruction of parametric images from dynamic PET sinograms is statistically more efficient than the conventional indirect methods, which perform image reconstruction and kinetic modeling in two separate steps. Most existing direct reconstruction methods are derived based on a known blood input function. This paper presents a direct reconstruction algorithm using a simplified reference tissue model, which does not require a blood input function. We have derived a minorization-maximization algorithm to find the penalized maximum likelihood solution. Computer simulations show that the proposed method has better bias-variance tradeoff than the conventional indirect method for estimating parametric images of receptor binding potential using dynamic PET.

Recent advances in the understanding of brain function are opening new frontiers in the investigation of movement disorders and neurodegeneration. The importance of the brain network-like characteristics is rapidly emerging together with increasing evidence that brain diseases imprint specific alterations on such networks. There is a strong need to determine molecular correlates associated with the network-type alterations to enable understanding of disease origin and mapping between clinical disease manifestations, genetic predispositions, and disease-triggering mechanisms. These considerations justify and highlight the importance of recent technological developments in positron emission tomography (PET) and integration of PET and magnetic resonance imaging (MRI), where the high neurochemical sensitivity of PET is complemented by MRI-derived measures of structural and functional connectivity. Ongoing developments of PET tracers suitable to image novel molecular targets and improvements in image reconstruction and analysis methods are further enhancing the relevance of imaging in addressing the complexity of brain function and disease-induced multidimensional alterations. This paper describes a conceptual justifications for the synergy between PET and MRI as related to neurodegeneration and movement disorders, discusses some predominantly PET-related developments relevant to and catalyzed by such synergy, and describes some novel multimodal metrics relevant to fundamental aspects of brain function altered early by disease.

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.

Despite its success in many biomedical applications, Positron Emission Tomography (PET) has the drawback of typically having lower spatial resolution and higher noise respect to other medical imaging techniques. The best achievable spatial resolution in PET scanners is limited by factors such as the positron range, non-collinearity and the size of the detector crystals. In this work, we present a novel method that uses series of image reconstructions (super-iterations) to go beyond the expected resolution-noise limits for a given PET acquisition. The image quality improvement is achieved using the projections of the previous image reconstruction to redistribute the measured counts of each line-of-response (LOR) into several subLORs, from which a new activity distribution with better quality is reconstructed. The method was evaluated with data from the preclinical scanner 4R-SuperArgus PET/CT, using the NEMA NU4-2008 image quality phantom, a cold Derenzo phantom, and an in-vivo FDG cardiac study on a rat. Resolution and recovery coefficient (RC) improvement of ~10% was achieved while keeping the same noise level. Qualitative results from the in-vivo study also confirm this improvement in image quality. The proposed method is able to achieve significantly better images at the expense of a modest increase of the computational time, and it could be also applied to other modalities, such as SPECT and Compton Cameras.