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The 'simultaneous maximum-likelihood attenuation correction factors' (sMLACF) algorithm presented here, is an iterative algorithm to calculate the maximum-likelihood estimate of the activity λ and the attenuation factors a in time-of-flight positron emission tomography, and this from emission data only. Hence sMLACF is an alternative to the MLACF algorithm. sMLACF is derived using the generalized expectation-maximization principle by introducing an appropriate set of complete data. The resulting iteration step yields a simultaneous update of λ and a which, in addition, enforces in a natural way the constraints [Formula: see text] where [Formula: see text] is a fixed lower bound that ensures the boundedness of the reconstructed activities. Some properties-like the monotonic increase of the likelihood and the asymptotic regularity of the estimated [Formula: see text]-of sMLACF are proven. Comparison of sMLACF with MLACF for two data sets reveals that both algorithms show very similar results, although sMLACF converges slower.

Scatter correction is typically done using a simulation of the single scatter, which is then scaled to account for multiple scatters and other possible model mismatches. This scaling factor is determined by fitting the simulated scatter sinogram to the measured sinogram, using only counts measured along LORs that do not intersect the patient body, i.e. 'scatter-tails'. Extending previous work, we propose to scale the scatter with a plane dependent factor, which is determined as an additional unknown in the maximum likelihood (ML) reconstructions, using counts in the entire sinogram rather than only the 'scatter-tails'. The ML-scaled scatter estimates are validated using a Monte-Carlo simulation of a NEMA-like phantom, a phantom scan with typical contrast ratios of a 68 Ga-PSMA scan, and 23 whole-body 18 F-FDG patient scans. On average, we observe a 12.2% change in the total amount of tracer activity of the MLEM reconstructions of our whole-body patient database when the proposed ML scatter scales are used. Furthermore, reconstructions using the ML-scaled scatter estimates are found to eliminate the typical 'halo' artifacts that are often observed in the vicinity of high focal uptake regions.