Fourier rebinning of time-of-flight PET data

Modular GAN: positron emission tomography image reconstruction using two generative adversarial networks

Introduction

The reconstruction of PET images involves converting sinograms, which represent the measured counts of radioactive emissions using detector rings encircling the patient, into meaningful images. However, the quality of PET data acquisition is impacted by physical factors, photon count statistics and detector characteristics, which affect the signal-to-noise ratio, resolution and quantitative accuracy of the resulting images. To address these influences, correction methods have been developed to mitigate each of these issues separately. Recently, generative adversarial networks (GANs) based on machine learning have shown promise in learning the complex mapping between acquired PET data and reconstructed tomographic images. This study aims to investigate the properties of training images that contribute to GAN performance when non-clinical images are used for training. Additionally, we describe a method to correct common PET imaging artefacts without relying on patient-specific anatomical images.

Methods

The modular GAN framework includes two GANs. Module 1, resembling Pix2pix architecture, is trained on non-clinical sinogram-image pairs. Training data are optimised by considering image properties defined by metrics. The second module utilises adaptive instance normalisation and style embedding to enhance the quality of images from Module 1. Additional perceptual and patch-based loss functions are employed in training both modules. The performance of the new framework was compared with that of existing methods, (filtered backprojection (FBP) and ordered subset expectation maximisation (OSEM) without and with point spread function (OSEM-PSF)) with respect to correction for attenuation, patient motion and noise in simulated, NEMA phantom and human imaging data. Evaluation metrics included structural similarity (SSIM), peak-signal-to-noise ratio (PSNR), relative root mean squared error (rRMSE) for simulated data, and contrast-to-noise ratio (CNR) for NEMA phantom and human data.

Results

For simulated test data, the performance of the proposed framework was both qualitatively and quantitatively superior to that of FBP and OSEM. In the presence of noise, Module 1 generated images with a SSIM of 0.48 and higher. These images exhibited coarse structures that were subsequently refined by Module 2, yielding images with an SSIM higher than 0.71 (at least 22% higher than OSEM). The proposed method was robust against noise and motion. For NEMA phantoms, it achieved higher CNR values than OSEM. For human images, the CNR in brain regions was significantly higher than that of FBP and OSEM (p < 0.05, paired t-test). The CNR of images reconstructed with OSEM-PSF was similar to those reconstructed using the proposed method.

Conclusion

The proposed image reconstruction method can produce PET images with artefact correction.

Analytic time-of-flight positron emission tomography reconstruction: three-dimensional case

In a positron emission tomography (PET) scanner, the time-of-flight (TOF) information gives us rough event position along the line-of-response (LOR). Using the TOF information for PET image reconstruction is able to reduce image noise. The state-of-the-art TOF PET image reconstruction uses iterative algorithms. This study introduces an analytic TOF PET algorithm that focuses on three-dimensional (3D) reconstruction. The proposed algorithm is in the form of backprojection filtering, in which the backprojection is performed first by using a time-resolution profile function, and then a 3D filter is applied to the backprojected image. For the list-mode data, the backprojection is carried out in the event-by-event fashion, and the timing resolution determined weighting function is used along the projection LOR. Computer simulations are carried out to verify the feasibility of the proposed algorithm.

DirectPET: full-size neural network PET reconstruction from sinogram data

Purpose: Neural network image reconstruction directly from measurement data is a relatively new field of research, which until now has been limited to producing small single-slice images (e.g., 1 × 128 × 128 ). We proposed a more efficient network design for positron emission tomography called DirectPET, which is capable of reconstructing multislice image volumes (i.e., 16 × 400 × 400 ) from sinograms. Approach: Large-scale direct neural network reconstruction is accomplished by addressing the associated memory space challenge through the introduction of a specially designed Radon inversion layer. Using patient data, we compare the proposed method to the benchmark ordered subsets expectation maximization (OSEM) algorithm using signal-to-noise ratio, bias, mean absolute error, and structural similarity measures. In addition, line profiles and full-width half-maximum measurements are provided for a sample of lesions. Results: DirectPET is shown capable of producing images that are quantitatively and qualitatively similar to the OSEM target images in a fraction of the time. We also report on an experiment where DirectPET is trained to map low-count raw data to normal count target images, demonstrating the method's ability to maintain image quality under a low-dose scenario. Conclusion: The ability of DirectPET to quickly reconstruct high-quality, multislice image volumes suggests potential clinical viability of the method. However, design parameters and performance boundaries need to be fully established before adoption can be considered.

Analytic time-of-flight positron emission tomography reconstruction: two-dimensional case

In a positron emission tomography (PET) scanner, the time-of-flight (TOF) information gives us rough event position along the line-of-response (LOR). Using the TOF information for PET image reconstruction is able to reduce image noise. The state-of-the-art TOF PET image reconstruction uses iterative algorithms. Analytical image reconstruction algorithm exits for TOF PET which emulates the iterative Landweber algorithm. This paper introduces such an algorithm, focusing on two-dimensional (2D) reconstruction. The proposed algorithm is in the form of backprojection filtering, in which the backprojection is performed first, and then a 2D filter is applied to the backprojected image. For the list-mode data, the backprojection is carried out in the event-by-event fashion, and a profile function may be used along the projection LOR. The 2D filter depends on the TOF timing resolution as well as the backprojection profile function. In order to emulate the iterative algorithm effects, a Fourier-domain window function is suggested. This window function has a parameter, k, which corresponds to the iteration number in an iterative algorithm.

Quantitative image reconstruction for total-body PET imaging using the 2-meter long EXPLORER scanner

The EXPLORER project aims to build a 2 meter long total-body PET scanner, which will provide extremely high sensitivity for imaging the entire human body. It will possess a range of capabilities currently unavailable to state-of-the-art clinical PET scanners with a limited axial field-of-view. The huge number of lines-of-response (LORs) of the EXPLORER poses a challenge to the data handling and image reconstruction. The objective of this study is to develop a quantitative image reconstruction method for the EXPLORER and compare its performance with current whole-body scanners. Fully 3D image reconstruction was performed using time-of-flight list-mode data with parallel computation. To recover the resolution loss caused by the parallax error between crystal pairs at a large axial ring difference or transaxial radial offset, we applied an image domain resolution model estimated from point source data. To evaluate the image quality, we conducted computer simulations using the SimSET Monte-Carlo toolkit and XCAT 2.0 anthropomorphic phantom to mimic a 20 min whole-body PET scan with an injection of 25 MBq ¹⁸F-FDG. We compare the performance of the EXPLORER with a current clinical scanner that has an axial FOV of 22 cm. The comparison results demonstrated superior image quality from the EXPLORER with a 6.9-fold reduction in noise standard deviation comparing with multi-bed imaging using the clinical scanner.

Recent developments in time-of-flight PET

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.

Fourier rebinning and consistency equations for time-of-flight PET planograms

Due to the unique geometry, dual-panel PET scanners have many advantages in dedicated breast imaging and on-board imaging applications since the compact scanners can be combined with other imaging and treatment modalities. The major challenges of dual-panel PET imaging are the limited-angle problem and data truncation, which can cause artifacts due to incomplete data sampling. The time-of-flight (TOF) information can be a promising solution to reduce these artifacts. The TOF planogram is the native data format for dual-panel TOF PET scanners, and the non-TOF planogram is the 3D extension of linogram. The TOF planograms is five-dimensional while the objects are three-dimensional, and there are two degrees of redundancy. In this paper, we derive consistency equations and Fourier-based rebinning algorithms to provide a complete understanding of the rich structure of the fully 3D TOF planograms. We first derive two consistency equations and John's equation for 3D TOF planograms. By taking the Fourier transforms, we obtain two Fourier consistency equations and the Fourier-John equation, which are the duals of the consistency equations and John's equation, respectively. We then solve the Fourier consistency equations and Fourier-John equation using the method of characteristics. The two degrees of entangled redundancy of the 3D TOF data can be explicitly elicited and exploited by the solutions along the characteristic curves. As the special cases of the general solutions, we obtain Fourier rebinning and consistency equations (FORCEs), and thus we obtain a complete scheme to convert among different types of PET planograms: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF planograms. The FORCEs can be used as Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. As a byproduct, we show the two consistency equations are necessary and sufficient for 3D TOF planograms. Finally, we give numerical examples of implementation of a fast 2D TOF planogram projector and Fourier-based rebinning for a 2D TOF planograms using the FORCEs to show the efficacy of the Fourier-based solutions.

Analytic TOF PET reconstruction algorithm within DIRECT data partitioning framework

Iterative reconstruction algorithms are routinely used for clinical practice; however, analytic algorithms are relevant candidates for quantitative research studies due to their linear behavior. While iterative algorithms also benefit from the inclusion of accurate data and noise models the widespread use of time-of-flight (TOF) scanners with less sensitivity to noise and data imperfections make analytic algorithms even more promising. In our previous work we have developed a novel iterative reconstruction approach (DIRECT: direct image reconstruction for TOF) providing convenient TOF data partitioning framework and leading to very efficient reconstructions. In this work we have expanded DIRECT to include an analytic TOF algorithm with confidence weighting incorporating models of both TOF and spatial resolution kernels. Feasibility studies using simulated and measured data demonstrate that analytic-DIRECT with appropriate resolution and regularization filters is able to provide matched bias versus variance performance to iterative TOF reconstruction with a matched resolution model.

Attenuation correction in emission tomography using the emission data--A review

The problem of attenuation correction (AC) for quantitative positron emission tomography (PET) had been considered solved to a large extent after the commercial availability of devices combining PET with computed tomography (CT) in 2001; single photon emission computed tomography (SPECT) has seen a similar development. However, stimulated in particular by technical advances toward clinical systems combining PET and magnetic resonance imaging (MRI), research interest in alternative approaches for PET AC has grown substantially in the last years. In this comprehensive literature review, the authors first present theoretical results with relevance to simultaneous reconstruction of attenuation and activity. The authors then look back at the early history of this research area especially in PET; since this history is closely interwoven with that of similar approaches in SPECT, these will also be covered. We then review algorithmic advances in PET, including analytic and iterative algorithms. The analytic approaches are either based on the Helgason-Ludwig data consistency conditions of the Radon transform, or generalizations of John's partial differential equation; with respect to iterative methods, we discuss maximum likelihood reconstruction of attenuation and activity (MLAA), the maximum likelihood attenuation correction factors (MLACF) algorithm, and their offspring. The description of methods is followed by a structured account of applications for simultaneous reconstruction techniques: this discussion covers organ-specific applications, applications specific to PET/MRI, applications using supplemental transmission information, and motion-aware applications. After briefly summarizing SPECT applications, we consider recent developments using emission data other than unscattered photons. In summary, developments using time-of-flight (TOF) PET emission data for AC have shown promising advances and open a wide range of applications. These techniques may both remedy deficiencies of purely MRI-based AC approaches in PET/MRI and improve standalone PET imaging.

A unified Fourier theory for time-of-flight PET data

Fully 3D time-of-flight (TOF) PET scanners offer the potential of previously unachievable image quality in clinical PET imaging. TOF measurements add another degree of redundancy for cylindrical PET scanners and make photon-limited TOF-PET imaging more robust than non-TOF PET imaging. The data space for 3D TOF-PET data is five-dimensional with two degrees of redundancy. Previously, consistency equations were used to characterize the redundancy of TOF-PET data. In this paper, we first derive two Fourier consistency equations and Fourier-John equation for 3D TOF PET based on the generalized projection-slice theorem; the three partial differential equations (PDEs) are the dual of the sinogram consistency equations and John's equation. We then solve the three PDEs using the method of characteristics. The two degrees of entangled redundancy of the TOF-PET data can be explicitly elicited and exploited by the solutions of the PDEs along the characteristic curves, which gives a complete understanding of the rich structure of the 3D x-ray transform with TOF measurement. Fourier rebinning equations and other mapping equations among different types of PET data are special cases of the general solutions. We also obtain new Fourier rebinning and consistency equations (FORCEs) from other special cases of the general solutions, and thus we obtain a complete scheme to convert among different types of PET data: 3D TOF, 3D non-TOF, 2D TOF and 2D non-TOF data. The new FORCEs can be used as new Fourier-based rebinning algorithms for TOF-PET data reduction, inverse rebinnings for designing fast projectors, or consistency conditions for estimating missing data. Further, we give a geometric interpretation of the general solutions--the two families of characteristic curves can be obtained by respectively changing the azimuthal and co-polar angles of the biorthogonal coordinates in Fourier space. We conclude the unified Fourier theory by showing that the Fourier consistency equations are necessary and sufficient for 3D x-ray transform with TOF measurement. Finally, we give numerical examples of inverse rebinning for a 3D TOF PET and Fourier-based rebinning for a 2D TOF PET using the FORCEs to show the efficacy of the unified Fourier solutions.

Improved intrinsic motion detection using time-of-flight PET

Intrinsic or data-driven respiratory and cardiac motion detection often track the center-of-mass (COM) change in PET data to derive a motion gating signal. The effectiveness of this method depends on the contrast of the moving target to the relatively stationary background. The stationary background leads to a reduced COM displacement in PET data. Further, the COM calculated using axially truncated PET data is biased. To improve intrinsic motion detection for motion compensated image reconstruction, we use the time-of-flight (TOF) PET data of the original object f(x) to calculate the non-TOF PET data of a volume-of-interest (VOI) weighted object f(x)w(x) . The VOI-weighting w(x) can be chosen to reduce contribution from the stationary background. The reduced background in f(x)w(x) leads to an observed increase in the COM displacement. We also derive rebinning equations to obtain the exact axial COM using axially truncated PET data. To assess the quality of the motion gating signal, we analyze the variance property of the COM using different methods, including with(out) VOI weighting and with(out) compensation for axial data truncation. Analytical simulations, phantom and patient data demonstrate the effectiveness of the proposed approach in identifying the motion phase and in deriving a gating signal to be used for motion-compensated image reconstruction.

Singular value decomposition for photon-processing nuclear imaging systems and applications for reconstruction and computing null functions

Recent advances in technology are enabling a new class of nuclear imaging systems consisting of detectors that use real-time maximum-likelihood (ML) methods to estimate the interaction position, deposited energy, and other attributes of each photon-interaction event and store these attributes in a list format. This class of systems, which we refer to as photon-processing (PP) nuclear imaging systems, can be described by a fundamentally different mathematical imaging operator that allows processing of the continuous-valued photon attributes on a per-photon basis. Unlike conventional photon-counting (PC) systems that bin the data into images, PP systems do not have any binning-related information loss. Mathematically, while PC systems have an infinite-dimensional null space due to dimensionality considerations, PP systems do not necessarily suffer from this issue. Therefore, PP systems have the potential to provide improved performance in comparison to PC systems. To study these advantages, we propose a framework to perform the singular-value decomposition (SVD) of the PP imaging operator. We use this framework to perform the SVD of operators that describe a general two-dimensional (2D) planar linear shift-invariant (LSIV) PP system and a hypothetical continuously rotating 2D single-photon emission computed tomography (SPECT) PP system. We then discuss two applications of the SVD framework. The first application is to decompose the object being imaged by the PP imaging system into measurement and null components. We compare these components to the measurement and null components obtained with PC systems. In the process, we also present a procedure to compute the null functions for a PC system. The second application is designing analytical reconstruction algorithms for PP systems. The proposed analytical approach exploits the fact that PP systems acquire data in a continuous domain to estimate a continuous object function. The approach is parallelizable and implemented for graphics processing units (GPUs). Further, this approach leverages another important advantage of PP systems, namely the possibility to perform photon-by-photon real-time reconstruction. We demonstrate the application of the approach to perform reconstruction in a simulated 2D SPECT system. The results help to validate and demonstrate the utility of the proposed method and show that PP systems can help overcome the aliasing artifacts that are otherwise intrinsically present in PC systems.

Transmission-less attenuation estimation from time-of-flight PET histo-images using consistency equations

In positron emission tomography (PET) imaging, attenuation correction with accurate attenuation estimation is crucial for quantitative patient studies. Recent research showed that the attenuation sinogram can be determined up to a scaling constant utilizing the time-of-flight information. The TOF-PET data can be naturally and efficiently stored in a histo-image without information loss, and the radioactive tracer distribution can be efficiently reconstructed using the DIRECT approaches. In this paper, we explore transmission-less attenuation estimation from TOF-PET histo-images. We first present the TOF-PET histo-image formation and the consistency equations in the histo-image parameterization, then we derive a least-squares solution for estimating the directional derivatives of the attenuation factors from the measured emission histo-images. Finally, we present a fast solver to estimate the attenuation factors from their directional derivatives using the discrete sine transform and fast Fourier transform while considering the boundary conditions. We find that the attenuation histo-images can be uniquely determined from the TOF-PET histo-images by considering boundary conditions. Since the estimate of the attenuation directional derivatives can be inaccurate for LORs tangent to the patient boundary, external sources, e.g. a ring or annulus source, might be needed to give an accurate estimate of the attenuation gradient for such LORs. The attenuation estimation from TOF-PET emission histo-images is demonstrated using simulated 2D TOF-PET data.

Evaluation of image reconstruction algorithms encompassing Time-Of-Flight and Point Spread Function modelling for quantitative cardiac PET: phantom studies

BACKGROUND

To perform kinetic modelling quantification, PET dynamic data must be acquired in short frames, where different critical conditions are met. The accuracy of reconstructed images influences quantification. The added value of Time-Of-Flight (TOF) and Point Spread Function (PSF) in cardiac image reconstruction was assessed.

METHODS

A static phantom was used to simulate two extreme conditions: (i) the bolus passage and (ii) the steady uptake. Various count statistics and independent noise realisations were considered. A moving phantom filled with two different radionuclides was used to simulate: (i) a great range of contrasts and (ii) the cardio/respiratory motion. Analytical and iterative reconstruction (IR) algorithms also encompassing TOF and PSF modelling were evaluated.

RESULTS

Both analytic and IR algorithms provided good results in all the evaluated conditions. The amount of bias introduced by IR was found to be limited. TOF allowed faster convergence and lower noise levels. PSF achieved near full myocardial activity recovery in static conditions. Motion degraded performances, but the addition of both TOF and PSF maintained the best overall behaviour.

CONCLUSIONS

IR accounting for TOF and PSF can be recommended for the quantification of dynamic cardiac PET studies as they improve the results compared to analytic and standard IR.

MAP reconstruction for Fourier rebinned TOF-PET data

Time-of-flight (TOF) information improves the signal-to-noise ratio in positron emission tomography (PET). The computation cost in processing TOF-PET sinograms is substantially higher than for nonTOF data because the data in each line of response is divided among multiple TOF bins. This additional cost has motivated research into methods for rebinning TOF data into lower dimensional representations that exploit redundancies inherent in TOF data. We have previously developed approximate Fourier methods that rebin TOF data into either three-dimensional (3D) nonTOF or 2D nonTOF formats. We refer to these methods respectively as FORET-3D and FORET-2D. Here we describe maximum a posteriori (MAP) estimators for use with FORET rebinned data. We first derive approximate expressions for the variance of the rebinned data. We then use these results to rescale the data so that the variance and mean are approximately equal allowing us to use the Poisson likelihood model for MAP reconstruction. MAP reconstruction from these rebinned data uses a system matrix in which the detector response model accounts for the effects of rebinning. Using these methods we compare the performance of FORET-2D and 3D with TOF and nonTOF reconstructions using phantom and clinical data. Our phantom results show a small loss in contrast recovery at matched noise levels using FORET compared to reconstruction from the original TOF data. Clinical examples show FORET images that are qualitatively similar to those obtained from the original TOF-PET data but with a small increase in variance at matched resolution. Reconstruction time is reduced by a factor of 5 and 30 using FORET3D+MAP and FORET2D+MAP respectively compared to 3D TOF MAP, which makes these methods attractive for clinical applications.

Magnetic resonance-guided positron emission tomography image reconstruction

The resolution of positron emission tomography (PET) images is limited by the physics of positron-electron annihilation and instrumentation for photon coincidence detection. Model-based methods that incorporate accurate physical and statistical models have produced significant improvements in reconstructed image quality when compared with filtered backprojection reconstruction methods. However, it has often been suggested that by incorporating anatomical information, the resolution and noise properties of PET images could be further improved, leading to better quantitation or lesion detection. With the recent development of combined MR-PET scanners, we can now collect intrinsically coregistered magnetic resonance images. It is therefore possible to routinely make use of anatomical information in PET reconstruction, provided appropriate methods are available. In this article, we review research efforts over the past 20 years to develop these methods. We discuss approaches based on the use of both Markov random field priors and joint information or entropy measures. The general framework for these methods is described, and their performance and longer-term potential and limitations are discussed.

Comparison of list-mode and DIRECT approaches for time-of-flight PET reconstruction

Early clinical results with time-of-flight (TOF) positron emission tomography (PET) systems have demonstrated the advantages of TOF information in PET reconstruction. Reconstruction approaches in TOF-PET systems include list-mode and binned iterative algorithms as well as confidence-weighted analytic methods. List-mode iterative TOF reconstruction retains the resolutions of the data in the spatial and temporal domains without any binning approximations but is computationally intensive. We have developed an approach [DIRECT (direct image reconstruction for TOF)] to speed up TOF-PET reconstruction that takes advantage of the reduced angular sampling requirement of TOF data by grouping list-mode data into a small number of azimuthal views and co-polar tilts and depositing the grouped events into histo-images, arrays with the sampling and geometry of the final image. All physical effects are included in the system model and deposited in the same histo-image structure. Using histo-images allows efficient computation during reconstruction without ray-tracing or interpolation operations. The DIRECT approach was compared with 3-D list-mode TOF ordered subsets expectation maximization (OSEM) reconstruction for phantom and patient data taken on the University of Pennsylvania research LaBr (3) TOF-PET scanner. The total processing and reconstruction time for these studies with DIRECT without attention to code optimization is approximately 25%-30% that of list-mode TOF-OSEM to achieve comparable image quality. Furthermore, the reconstruction time for DIRECT is independent of the number of events and/or sizes of the spatial and TOF kernels, while the time for list-mode TOF-OSEM increases with more events or larger kernels. The DIRECT approach is able to reproduce the image quality of list-mode iterative TOF reconstruction both qualitatively and quantitatively in measured data with a reduced time.

Optimal rebinning of time-of-flight PET data

Time-of-flight (TOF) positron emission tomography (PET) scanners offer the potential for significantly improved signal-to-noise ratio (SNR) and lesion detectability in clinical PET. However, fully 3D TOF PET image reconstruction is a challenging task due to the huge data size. One solution to this problem is to rebin TOF data into a lower dimensional format. We have recently developed Fourier rebinning methods for mapping TOF data into non-TOF formats that retain substantial SNR advantages relative to sinograms acquired without TOF information. However, mappings for rebinning into non-TOF formats are not unique and optimization of rebinning methods has not been widely investigated. In this paper we address the question of optimal rebinning in order to make full use of TOF information. We focus on FORET-3D, which approximately rebins 3D TOF data into 3D non-TOF sinogram formats without requiring a Fourier transform in the axial direction. We optimize the weighting for FORET-3D to minimize the variance, resulting in H(2)-weighted FORET-3D, which turns out to be the best linear unbiased estimator (BLUE) under reasonable approximations and furthermore the uniformly minimum variance unbiased (UMVU) estimator under Gaussian noise assumptions. This implies that any information loss due to optimal rebinning is as a result only of the approximations used in deriving the rebinning equation and developing the optimal weighting. We demonstrate using simulated and real phantom TOF data that the optimal rebinning method achieves variance reduction and contrast recovery improvement compared to nonoptimized rebinning weightings. In our preliminary study using a simplified simulation setup, the performance of the optimal rebinning method was comparable to that of fully 3D TOF MAP.

Efficient 3-D TOF PET reconstruction using view-grouped histo-images: DIRECT-direct image reconstruction for TOF

For modern time-of-flight (TOF) positron emission tomography (PET) systems, in which the number of possible lines of response and TOF bins is much larger than the number of acquired events, the most appropriate reconstruction approaches are considered to be list-mode methods. However, their shortcomings are relatively high computational costs for reconstruction and for sensitivity matrix calculation. Efficient treatment of TOF data within the proposed DIRECT approach is obtained by 1) angular (azimuthal and co-polar) grouping of TOF events to a set of views as given by the angular sampling requirements for the TOF resolution, and 2) deposition (weighted-histogramming) of these grouped events, and correction data, into a set of "histo-images," one histo-image per view. The histo-images have the same geometry (voxel grid, size and orientation) as the reconstructed image. The concept is similar to the approach involving binning of the TOF data into angularly subsampled histo-projections-projections expanded in the TOF directions. However, unlike binning into histo-projections, the deposition of TOF events directly into the image voxels eliminates the need for tracing and/or interpolation operations during the reconstruction. Together with the performance of reconstruction operations directly in image space, this leads to a very efficient implementation of TOF reconstruction algorithms. Furthermore, the resolution properties are not compromised either, since events are placed into the image elements of the desired size from the beginning. Concepts and efficiency of the proposed data partitioning scheme are demonstrated in this work by using the DIRECT approach in conjunction with the row-action maximum-likelihood (RAMLA) algorithm.

Exact and approximate Fourier rebinning of PET data from time-of-flight to non time-of-flight

The image reconstruction problem for fully 3D TOF PET is challenging because of the large data sizes involved. One approach to this problem is to first rebin the data into one of the following lower dimensional formats: 2D TOF, 3D non TOF or 2D non TOF. Here we present a unified framework based on a generalized projection slice theorem for TOF data that can be used to compute each of these mappings. We use this framework to develop approaches for rebinning into non TOF formats without significant loss of information. We first derive the exact mappings and then describe approximations which address the missing data problem for oblique sinograms. We evaluate the performance of approximate rebinning using Monte Carlo simulations. Our results show that rebinning into non TOF sinograms retains significant SNR advantages over sinograms collected without TOF information.

State of the art and challenges of time-of-flight PET

After a brief review of the history of time-of-flight (TOF) positron emission tomography (PET) instrumentation from the 1980s to present, the principles of TOF PET are introduced, the concept of time resolution and its effect on TOF gain in signal-to-noise ratio (SNR) are discussed. The factors influencing the time resolution of a TOF PET scanner are presented, with focus on the intrinsic properties of scintillators of particular interest for TOF PET. Finally, some open issues, challenges and achievements of today's TOF PET reconstruction are reviewed: the structure of the data organization, the choice of analytical or iterative method, the recent experimental assessment of TOF image quality, and the most promising applications of TOF PET.

Continuous and discrete data rebinning in time-of-flight PET

This paper investigates data compression methods for time-of-flight (TOF) positron emission tomography (PET), which rebin the 3-D TOF measurements into a set of 2-D TOF data for a stack of transaxial slices. The goal of this work is to develop rebinning algorithms that are more accurate than the TOF single-slice-rebinning (TOF-SSRB) method proposed by Mullani in 1982. Two approaches are explored. The first one is based on a partial differential equation, which expresses a consistency condition for TOF-PET data with a Gaussian TOF profile. From this equation we derive an analytical rebinning algorithm, which is unbiased in the limit of continuous sampling. The second approach is discrete: each 2-D rebinned data sample is calculated as a linear combination of the 3-D TOF samples in the same axial plane parallel to the axis of the scanner. The coefficients of the linear combination are precomputed by optimizing a cost function which enforces both accuracy and good variance reduction, models the TOF profile, the axial PSF of the LORs, and the specific sampling scheme of the scanner. Measurements of a thorax phantom on a prototype TOF-PET scanner with a resolution of 550 ps show that the proposed discrete method improves the bias-variance trade-off and is a promising alternative to TOF-SSRB when data compression is required to achieve clinically acceptable reconstruction time.

Analytical properties of time-of-flight PET data

We investigate the analytical properties of time-of-flight (TOF) positron emission tomography (PET) sinograms, where the data are modeled as line integrals weighted by a spatially invariant TOF kernel. First, we investigate the Fourier transform properties of 2D TOF data and extend the 'bow-tie' property of the 2D Radon transform to the time-of-flight case. Second, we describe a new exact Fourier rebinning method, TOF-FOREX, based on the Fourier transform in the time-of-flight variable. We then combine TOF-FOREX rebinning with a direct extension of the projection slice theorem to TOF data, to perform fast 3D TOF PET image reconstruction. Finally, we illustrate these properties using simulated data.

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.

From 2D PET to 3D PET: Issues of Data Representation and Image Reconstruction

Positron emission tomography (PET), intrinsically a 3D imaging technique, was for a long time exclusively operated in 2D mode, using septa to shield the detectors from photons emitted obliquely to the detector planes. However, the use of septa results in a considerable loss of sensitivity. From the late 1980s, significant efforts have been made to develop a methodology for the acquisition and reconstruction of 3D PET data. This paper focuses on the differences between data acquisition in 2D and 3D mode, especially in terms of data set sizes and representation. Although the real time data acquisition aspect in 3D has been mostly solved in modern PET scanner systems, there still remain questions on how to represent and how to make best use of the information contained in the acquired data sets. Data representation methods, such as list-mode and matrix-based methods, possibly with additional compression, will be discussed. Moving from 2D to 3D PET has major implications on the way these data are reconstructed to images. Two fundamentally different approaches exist, the analytical one and the iterative one. Both, at different expenses, can be extended to directly handle 3D data sets. Either way the computational burden increases heavily compared to 2D reconstruction. One possibility to benefit from the increased sensitivity in 3D PET while sticking to high-performance 2D reconstruction algorithms is to rebin 3D into 2D data sets. The value of data rebinning will be explored. An ever increasing computing power and the concept of distributed or parallel computing have made direct 3D reconstruction feasible. Following a short review of reconstruction methods and their extensions to 3D, we focus on numerical aspects that improve reconstruction performance, which is especially important in solving large equation systems in 3D iterative reconstruction. Finally exemplary results are shown to review the properties of the discussed algorithms. This paper concludes with an overview on future trends in data representation and reconstruction.

First experimental results of time-of-flight reconstruction on an LSO PET scanner

Time-of-flight (TOF) positron emission tomography (PET) was studied and preliminarily developed in the 1980s, but the lack of a scintillator able to deliver at the same time proper time resolution and stopping power has prevented this technique from becoming widespread and commercially available. With the introduction of LSO in PET, TOF is now a feasible option. TOF reconstruction has been implemented in the CPS Hi-Rez PET scanner, both with 2D filtered-back-projection (FBP2D) and 3D ordered subset expectation maximization (OSEM3D). A new procedure has been introduced in the time alignment to compensate for the limited digital time resolution of the present electronics. A preliminary version of scatter correction for TOF has been devised and is presented. The measured time resolution of 1.2 ns (FWHM) allowed for a signal-to-noise ratio increase of about 50% in phantoms of about 40 cm transaxial size, or a gain larger than 2 in noise equivalent counts (NEC). TOF reconstruction has shown the expected improvement in SNR, both in simulation and experimental data. First experimental results show two improvements of TOF reconstruction over conventional (non-TOF) reconstruction: a lower noise level and a better capability to resolve structures deep inside large objects.