Determination of the system matrix used in list-mode EM reconstruction of PET

The system matrix bridges measured lines-of-response (LORs) in PET to the object being imaged. Accurate determination of the system matrix remains challenging since every object is different and pre-computing and storing the entire matrix is not feasible due to the large number of scintillation crystals present in current PET scanners. A ray tracing algorithm in list-mode expectation-maximization (EM) reconstruction is commonly used to simplify and calculate the system matrix on-the-fly. The EM algorithm requires an estimate of the sensitivity image which is the summation of the system matrix over all possible LORs and is sometimes computed through a limited sampling of possible LORs. A problem with this approach is that the variation within the sensitivity image can contribute a large amount to the variance of reconstruction through error propagation. Furthermore, it is not feasible to accurately model the detector response for each reconstruction. In this study, we approximated the sensitivity image to be object-independent, thus making it possible to use an accurate Monte Carlo simulator such as GATE to model all interactions within the detector and to pre-compute the sensitivity image only once. Traditionally, the measured LORs contain paired detector IDs but no 3D position information due to limited spatial resolution. We propose a method to model the 3D positions of LORs including the depth of interaction (DOI) in the determination of the system matrix. We investigated the impact of using different techniques for assigning the end-point positions of the LOR in the projection and backprojection operations. Results showed that assigning the end-points to be located in the center of the crystal surface provided the worst performance. Better performance could be achieved by either randomly perturbing the end-points in 2D (across the crystal face), or 3D (including depth) in a 2D PET. The best performance was achieved by using a Monte Carlo derived position distribution. In conclusion, combining a Monte Carlo simulated sensitivity image for a given PET system with a correct LOR 3D position sampling according to a realistic photon distribution in the corresponding detector crystals can significantly improve list-mode EM reconstruction.