Analytical volume calculation of voxel fragments for iterative, fully 3D PET reconstruction

Using iterative algorithms to reconstruct quantitative 3D PET data requires the computation of the system probability matrix. The pure geometrical contribution to the matrix elements can easily be approximated by the length of intersection between detected lines of response (LOR) and individual voxels. However, the natural observables are volume-based, i.e. the coincidence rates which are usually detected in tubes of response (TOR) as well as the activity concentration in each voxel refer to specific spatial fragments. Therefore, we developed a fast method for the analytical calculation of the 3D shape and volume of intersection polyhedrons to build the system matrix with an improved description of the geometrical detection probability. Additionally, we considered effects of detector blurring within the system matrix. Gaussian resolution functions have been assumed in the radial and axial direction describing migrations between adjacent TORs. Finally, exploiting intrinsic symmetry relations and the sparseness of the system matrix allowed to create an efficiently small matrix representation which could be pre-computed and completely stored in memory. Time-consuming voxel addressing calculations due to necessary symmetry operations have been completely avoided by using an octant-wise symmetrically ordered field of voxels. The above algorithm has been applied for an iterative, fully 3D reconstruction (MLEM/OSEM) of pre-corrected 3D sinograms provided by a Siemens/CTI HR+PET scanner. A comparison about the results with other established reconstruction platforms is given where our analytical volume-based approach yields a superior contrast behaviour at a lower level of noise.