Negative binomial maximum likelihood expectation maximization (NB-MLEM) algorithm for reconstruction of pre-corrected PET data

Purpose: Positron emission tomography (PET) image reconstruction is usually performed using maximum like-lihood (ML) iterative reconstruction methods, under the assumption of Poisson distributed data. Pre-correcting raw measured counts, this assumption is no longer realistic. The goal of this work is to develop a reconstruction algorithm based on the Negative Binomial (NB) distribution, which can generalize over the Poisson distribution in case of over-dispersion of raw data, that may occur if sinogram pre-correction is used. Methods: The mathematical derivation of a Negative Binomial Maximum Likelihood Expectation-Maximization (NB-MLEM) algorithm is presented. A simulation study to compare the performance of the proposed NB-MLEM algorithm with respect to a Poisson-based MLEM (P-MLEM) method was performed, in reconstructing PET data. The proposed NB-MLEM reconstruction was tested on a real phantom and human brain data. Results: For the property of NB distribution, it is a generalization of the conventional P-MLEM: for not over dispersed data, the proposed NB-MLEM algorithm behaves like the conventional P-MLEM; for over-dispersed PET data, the additional evaluation of the dispersion parameter after each reconstruction iteration leads to a more accurate final image with respect to P-MLEM. Conclusions: A novel approach for PET image reconstruction from pre-corrected data has been developed, which exhibits a statistical behavior that deviates from the Poisson distribution. Simulation study and preliminary tests on real data showed how the NB-MLEM algorithm, being able to explain the over-dispersion of pre-corrected data, can outperform other algorithms that assume no over-dispersion of pre-corrected data, while still not accounting for the presence of negative data, such as P-MLEM.