Kyungsang Kim1, Donghwan Kim2, Jaewon Yang3, Georges El Fakhri1, Youngho Seo3, Jeffrey A Fessler2, Quanzheng Li1. 1. Gordon Center for Medical Imaging, Massachusetts General Hospital and Harvard Medical School, 125 Nashua Street 6th floor, Suite 660, Boston, MA, 02114, USA. 2. Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48105, USA. 3. Physics Research Laboratory, Department of Radiology and Biomedical Imaging, University of California, San Francisco, CA, 94143, USA.
Abstract
PURPOSE: Time of flight (TOF) PET reconstruction is well known to statistically improve the image quality compared to non-TOF PET. Although TOF PET can improve the overall signal to noise ratio (SNR) of the image compared to non-TOF PET, the SNR disparity between separate regions in the reconstructed image using TOF data becomes higher than that using non-TOF data. Using the conventional ordered subset expectation maximization (OS-EM) method, the SNR in the low activity regions becomes significantly lower than in the high activity regions due to the different photon statistics of TOF bins. A uniform recovery across different SNR regions is preferred if it can yield an overall good image quality within small number of iterations in practice. To allow more uniform recovery of regions, a spatially variant update is necessary for different SNR regions. METHODS: This paper focuses on designing a spatially variant step size and proposes a TOF-PET reconstruction method that uses a nonuniform separable quadratic surrogates (NUSQS) algorithm, providing a straightforward control of spatially variant step size. To control the noise, a spatially invariant quadratic regularization is incorporated, which by itself does not theoretically affect the recovery uniformity. The Nesterov's momentum method with ordered subsets (OS) is also used to accelerate the reconstruction speed. To evaluate the proposed method, an XCAT simulation phantom and clinical data from a pancreas cancer patient with full (ground truth) and 6× downsampled counts were used, where a Poisson thinning process was employed for downsampling. We selected tumor and cold regions of interest (ROIs) and compared the proposed method with the TOF-based conventional OS-EM and OS-SQS algorithms with an early stopping criterion. RESULTS: In computer simulation, without regularization, hot regions of OS-EM and OS-NUSQS converged similarly, but cold region of OS-EM was noisier than OS-NUSQS after 24 iterations. With regularization, although the overall speeds of OS-EM and OS-NUSQS were similar, recovery ratios of hot and cold regions reconstructed by the OS-NUSQS were more uniform compared to those of the conventional OS-SQS and OS-EM. The OS-NUSQS with Nesterov's momentum converged faster than others while preserving the uniform recovery. In the clinical example, we demonstrated that the OS-NUSQS with Nesterov's momentum provides more uniform recovery ratios of hot and cold ROIs compared to the OS-SQS and OS-EM. Although the cost function of all methods is equivalent, the proposed method has higher structural similarity (SSIM) values of hot and cold regions compared to other methods after 24 iterations. Furthermore, our computing time using graphics processing unit was 80× shorter than the time using quad-core CPUs. CONCLUSION: This paper proposes a TOF PET reconstruction method using the OS-NUSQS with Nesterov's momentum for uniform recovery of different SNR regions. In particular, the spatially nonuniform step size in the proposed method provides uniform recovery ratios of different SNR regions, and the Nesterov's momentum further accelerates overall convergence while preserving uniform recovery. The computer simulation and clinical example demonstrate that the proposed method converges uniformly across ROIs. In addition, tumor contrast and SSIM of the proposed method were higher than those of the conventional OS-EM and OS-SQS in early iterations.
PURPOSE: Time of flight (TOF) PET reconstruction is well known to statistically improve the image quality compared to non-TOF PET. Although TOF PET can improve the overall signal to noise ratio (SNR) of the image compared to non-TOF PET, the SNR disparity between separate regions in the reconstructed image using TOF data becomes higher than that using non-TOF data. Using the conventional ordered subset expectation maximization (OS-EM) method, the SNR in the low activity regions becomes significantly lower than in the high activity regions due to the different photon statistics of TOF bins. A uniform recovery across different SNR regions is preferred if it can yield an overall good image quality within small number of iterations in practice. To allow more uniform recovery of regions, a spatially variant update is necessary for different SNR regions. METHODS: This paper focuses on designing a spatially variant step size and proposes a TOF-PET reconstruction method that uses a nonuniform separable quadratic surrogates (NUSQS) algorithm, providing a straightforward control of spatially variant step size. To control the noise, a spatially invariant quadratic regularization is incorporated, which by itself does not theoretically affect the recovery uniformity. The Nesterov's momentum method with ordered subsets (OS) is also used to accelerate the reconstruction speed. To evaluate the proposed method, an XCAT simulation phantom and clinical data from a pancreas cancerpatient with full (ground truth) and 6× downsampled counts were used, where a Poisson thinning process was employed for downsampling. We selected tumor and cold regions of interest (ROIs) and compared the proposed method with the TOF-based conventional OS-EM and OS-SQS algorithms with an early stopping criterion. RESULTS: In computer simulation, without regularization, hot regions of OS-EM and OS-NUSQS converged similarly, but cold region of OS-EM was noisier than OS-NUSQS after 24 iterations. With regularization, although the overall speeds of OS-EM and OS-NUSQS were similar, recovery ratios of hot and cold regions reconstructed by the OS-NUSQS were more uniform compared to those of the conventional OS-SQS and OS-EM. The OS-NUSQS with Nesterov's momentum converged faster than others while preserving the uniform recovery. In the clinical example, we demonstrated that the OS-NUSQS with Nesterov's momentum provides more uniform recovery ratios of hot and cold ROIs compared to the OS-SQS and OS-EM. Although the cost function of all methods is equivalent, the proposed method has higher structural similarity (SSIM) values of hot and cold regions compared to other methods after 24 iterations. Furthermore, our computing time using graphics processing unit was 80× shorter than the time using quad-core CPUs. CONCLUSION: This paper proposes a TOF PET reconstruction method using the OS-NUSQS with Nesterov's momentum for uniform recovery of different SNR regions. In particular, the spatially nonuniform step size in the proposed method provides uniform recovery ratios of different SNR regions, and the Nesterov's momentum further accelerates overall convergence while preserving uniform recovery. The computer simulation and clinical example demonstrate that the proposed method converges uniformly across ROIs. In addition, tumor contrast and SSIM of the proposed method were higher than those of the conventional OS-EM and OS-SQS in early iterations.