Optimal rebinning of time-of-flight PET data

Retrospective data-driven respiratory gating for PET using TOF information

Traditional data-driven respiratory gating method is capable of detecting breathing cycles directly from positron emission tomography (PET) data, but usually fails at low SNR, particularly at low dose PET/CT study. Time-of-flight (TOF) PET has the potential to improve the SNR. In order for TOF information to reduce the statistical noise and boost the performance of respiratory gating, we present a robust data-driven respiratory gating method using TOF information, which retrospectively derived the respiratory signal from the acquired TOF-PET data. The PET data was acquired in list mode format and analyzed in sinogram space. The method was demonstrated with patient datasets acquired on a TOF PET/CT system. Data-driven gating methods by center of mass (COM) and principle component analysis (PCA) algorithm were successfully performed on nonTOF PET and TOF PET dataset. To assess the accuracy of the data-driven respiratory signal, a hardware-based signal was acquired for comparison. The study showed that retrospectively respiratory gating using TOF sinograms has improved the SNR, and outperforms the non-TOF gating under both COM and PCA algorithms.

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.

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.

Reconstruction for 3D PET Based on Total Variation Constrained Direct Fourier Method

This paper presents a total variation (TV) regularized reconstruction algorithm for 3D positron emission tomography (PET). The proposed method first employs the Fourier rebinning algorithm (FORE), rebinning the 3D data into a stack of ordinary 2D data sets as sinogram data. Then, the resulted 2D sinogram are ready to be reconstructed by conventional 2D reconstruction algorithms. Given the locally piece-wise constant nature of PET images, we introduce the total variation (TV) based reconstruction schemes. More specifically, we formulate the 2D PET reconstruction problem as an optimization problem, whose objective function consists of TV norm of the reconstructed image and the data fidelity term measuring the consistency between the reconstructed image and sinogram. To solve the resulting minimization problem, we apply an efficient methods called the Bregman operator splitting algorithm with variable step size (BOSVS). Experiments based on Monte Carlo simulated data and real data are conducted as validations. The experiment results show that the proposed method produces higher accuracy than conventional direct Fourier (DF) (bias in BOSVS is 70% of ones in DF, variance of BOSVS is 80% of ones in DF).

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.