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 quantitatively accurate activity (tracer distribution) in the body. Generally, this is carried out based on the regulated or synthetic attenuation map obtained by computed tomography or magnetic resonance imaging respectively. However, except for errors in the attenuation correction factors obtained, the additional scan not only brings in more radiation dose and/or increases the scanning time but also leads to severe misalignments induced by the body motion 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, referred to as MLAAS, from time-of-flight (TOF)-PET emission data only. Particularly, in the proposed model, we make full use of the exclusively exponential form of the attenuation correction factors, and consider the constraint on a total amount of the activity in some mask region. Furthermore, we analyze the well-posedness of the model, including the existence, uniqueness and stability of its solution. We propose an alternating update algorithm to solve the model, and also analyze its convergence. Finally, numerical experiments with various emission data demonstrate that the proposed method MLAAS is of numerical convergence and robust to noise, and outperforms some state-of-the-art methods in terms of accuracy and efficiency, and enables TOF-PET to have the capability of autonomous attenuation correction.