Fast 3D-EM reconstruction using Planograms for stationary planar positron emission mammography camera.
At the University of Pisa we are building a PEM prototype, the YAP-PEM camera, consisting of two opposite 6 x 6 x 3 cm3 detector heads of 30 x 30 YAP:Ce finger crystals, 2 x 2 x 30 mm3 each. The camera will be equipped with breast compressors. The acquisition will be stationary. Compared with a whole body PET scanner, a planar Positron Emission Mammography (PEM) camera allows a better, easier and more flexible positioning around the breast in the vicinity of the tumor: this increases the sensitivity and solid angle coverage, and reduces cost. To avoid software rejection of data during the reconstruction, resulting in a reduced sensitivity, we adopted a 3D-EM reconstruction which uses all of the collected Lines Of Response (LORs). This skips the PSF distortion given by data rebinning procedures and/or Fourier methods. The traditional 3D-EM reconstruction requires several times the computation of the LOR-voxel correlation matrix, or probability matrix {p(ij)}; therefore is highly time-consuming. We use the sparse and symmetry properties of the matrix {p(ij)} to perform fast 3D-EM reconstruction. Geometrically, a 3D grid of cubic voxels (FOV) is crossed by several divergent 3D line sets (LORs). The symmetries occur when tracing different LORs produces the same p(ij) value. Parallel LORs of different sets cross the FOV in the same way, and the repetition of p(ij) values depends on the ratio between the tube and voxel sizes. By optimizing this ratio, the occurrence of symmetries is increased. We identify a nucleus of symmetry of LORs: for each set of symmetrical LORs we choose just one LOR to be put in the nucleus, while the others lie outside. All of the possible p(ij) values are obtainable by tracking only the LORs of this nucleus. The coordinates of the voxels of all of the other LORs are given by means of simple translation rules. Before making the reconstruction, we trace the LORs of the nucleus to find the intersecting voxels, whose p(ij) values are computed and stored with their voxel coordinates on a hard disk. Only the non-zero p(ij) are considered and their computation is performed just once. During the reconstruction, the stored values are loaded and are available in the random access memory for all of the operations of normalization, backprojection and projection: these are now performed rapidly, because the application of the translation rules is much faster than the probability computations. We tested the algorithm on Monte Carlo data fully simulating the typical YAP-PEM clinical condition. The adopted algorithm gives an excellent positioning capability for hot spots in the camera FOV. To use all of the possible skew LORs in the FOV avoids the software rejection of collected data. Reconstructed images indicate that a 5mm diameter tumor of 37 kBq/cm3, in an active breast with a 10:1 Tissue to Background ratio (T/B), with a 10 min acquisition, for a head distance of 5 cm, can be detected by the YAP-PEM with a SNR of 8.7+/-1.0. The obtained SNR values depend linearly on the tumor volume. The algorithm allows one to discriminate between two hot sources of 5.0 mm diameter if they do not lie on the same axis. The YAP-PEM is now in the assembly stage.