Theoretical and numerical study of MLEM and OSEM reconstruction algorithms for motion correction in emission tomography

Patient body-motion and respiratory-motion impacts the image quality of cardiac PET or SPECT perfusion images. Several algorithms exist in the literature to correct for motion within the iterative maximum-likelihood reconstruction framework. In this work, three algorithms are derived using Poisson statistics to correct for patient motion. The first one is a motion compensated MLEM algorithm (MC-MLEM). The next two algorithms called MGEM-1 and MGEM-2 (short for Motion Gated EM, 1 and 2) use the motion states as subsets, in two different ways. Experiments were performed with NCAT phantoms with exactly known motion as the source and attenuation distributions. The SIMIND Monte Carlo simulation software was used to create SPECT projection images of the NCAT phantoms. The projection images were then modified to have Poisson noise levels equivalent to that of clinical acquisition. We investigated application of these algorithms to correction of (1) a large body-motion of 2 cm in Superior-Inferior (SI) and Anterior-Posterior (AP) directions each and (2) respiratory motion of 2 cm in SI and 0.6 cm in AP. We obtained the bias with respect to the NCAT phantom activity for noiseless reconstructions as well as the bias-variance for noisy reconstructions. The MGEM-1 advanced along the biasvariance curve faster than the MC-MLEM with iterations. The MGEM-1 also lowered the noiseless bias (with respect to NCAT truth) faster with iterations, compared to the MC-MLEM algorithms, as expected with subset algorithms. For the body motion correction with two motion states, after the 9^th iteration the bias was close to that of MC-MLEM at iteration 17, reducing the number of iterations by a factor of 1.89. For the respiratory motion correction with 9 motion states, based on the noiseless bias, the iteration reduction factor was approximately 7. For the MGEM-2, however, bias-plot or the bias-variance-plot saturates with iteration because of successive interpolation error.