Algorithms for joint activity-attenuation estimation from positron emission tomography scatter

Background: Attenuation correction in positron emission tomography remains challenging in the absence of measured transmission data. Scattered emission data may contribute missing information, but quantitative scatter-to-attenuation (S2A) reconstruction needs to input the reconstructed activity image. Here, we study S2A reconstruction as a building block for joint estimation of activity and attenuation. Methods: We study two S2A reconstruction algorithms, maximum-likelihood expectation maximization (MLEM) with one-step-late attenuation (MLEM-OSL) and a maximum-likelihood gradient ascent (MLGA). We study theoretical properties of these algorithms with a focus on convergence and convergence speed and compare convergence speeds and the impact of object size in simulations using different spatial scale factors. Then, we propose joint estimation of activity and attenuation from scattered and nonscattered (true) emission data, combining MLEM-OSL or MLGA with scatter-MLEM as well as trues-MLEM and the maximum-likelihood transmission (MLTR) algorithm. Results: Shortcomings of MLEM-OSL inhibit convergence to the true solution with high attenuation; these shortcomings are related to the linearization of a nonlinear measurement equation and can be linked to a new numerical criterion allowing geometrical interpretations in terms of low and high attenuation. Comparisons using simulated data confirm that while MLGA converges largely independent of the attenuation scale, MLEM-OSL converges if low-attenuation data dominate, but not with high attenuation. Convergence of MLEM-OSL can be improved by isolating data satisfying the aforementioned low-attenuation criterion. In joint estimation of activity and attenuation, scattered data helps avoid local minima that nonscattered data alone cannot. Combining MLEM-OSL with trues-MLEM may be sufficient for low-attenuation objects, while MLGA, scatter-MLEM, and MLTR may additionally be needed with higher attenuation. Conclusions: The performance of S2A algorithms depends on spatial scales. MLGA provides lower computational complexity and convergence in more diverse setups than MLEM-OSL. Finally, scattered data may provide additional information to joint estimation of activity and attenuation through S2A reconstruction.