An efficient approach with theoretical guarantees to simultaneously reconstruct activity and attenuation sinogram for TOF-PET

In positron emission tomography (PET), it is indispensable to perform attenuation correction in order to obtain the quantitatively accurate activity map (tracer distribution) in the body. Generally, this is carried out based on the estimated attenuation map obtained from computed tomography or magnetic resonance imaging. However, except for errors in the attenuation correction factors obtained, the additional scan not only brings in new radiation doses and/or increases the scanning time but also leads to severe misalignment induced by various motions during and between the two sequential scans. To address these issues, based on maximum likelihood estimation, we propose a new mathematical model for simultaneously reconstructing the activity and attenuation sinogram from the time-of-flight (TOF)-PET emission data only. Particularly, we make full use of the exclusively exponential form for the attenuation correction factors, and consider the constraint of a total amount of the activity in some mask region in the proposed model. Furthermore, we prove its well-posedness, including the existence, uniqueness and stability of the solution. We propose an alternating update algorithm to solve the model, and also analyze its convergence. Finally, numerical experiments with various TOF-PET emission data demonstrate that the proposed method is of numerical convergence and robust to noise, and outperforms some state-of-the-art methods in terms of accuracy and efficiency, and has the capability of autonomous attenuation correction.