Alternating strategies and ordered subset acceleration schemes for maximum likelihood activity and attenuation reconstruction in time-of-flight PET

Background: Simultaneous reconstruction of activity and attenuation with MLAA offers an interesting option to solve the problem of attenuation correction in hybrid PET/MR scanners. The algorithm maximizes the joint likelihood by alternating MLEM and MLTR steps. Any scheme where a number of MLTR iterations follow any number of MLEM ones converges to the true solution, but convergence speed and computation time might differ significantly. We investigate which alternating strategy is expected to give optimal results as a function of TOF coincidence timing resolution (CTR), given the different computation time and convergence speed between TOF-MLEM and non-TOF MLTR. Following, acceleration strategies exploiting ordered subset (OS) techniques are proposed and tested. Methods: A slice of a 2D digital body phantom was simulated and forward projected with CTR of 200, 400, 600 and 800 ps. Reconstructions were performed using the corresponding CTR using 1, 2, 4 and 8 MLTR iterations for each MLEM iteration, using native-geometry, ray-driven forward and back-projectors. Convergence curves were plotted as a function of computation time. Different strategies of OS acceleration were then tested and compared using simulations both noiseless and featuring Poisson noise, using 32 subsets. Results: Convergence was achieved with the least computation time when 4 or 8 MLTR iterations were performed for each MLEM one. All proposed OS strategies had results similar to reference non-accelerated versions and performed equivalently also in presence of noise, despite the high subsets number. Conclusions: Performing 4 or 8 MLTR iterations for each MLEM one appears to be the optimal solution for current implementations of the MLAA algorithm. OS techniques are effective in reducing computation time and perform comparably to non-accelerated versions.