Convergence properties of algorithms for direct parametric estimation of linear models in dynamic PET

In dynamic PET studies, the time changing activity of the radiotracer is measured through multiple consecutive frames. Subsequently, voxel-wise application of the kinetic model is expected to estimate parametric images. In this work we investigate the convergence properties of direct reconstruction algorithms of parametric images in 3D PET for the case where the kinetic model is linear in its parameters. As direct reconstruction algorithms we use a modification of the PIR algorithm, corresponding to the MLEM formula for parametric images, and a transformed version of the Separable Paraboloid Surrogate (SPS) algorithm formula. The directly reconstructed images are compared with indirectly generated parametric maps using filtered back projection where the kinetic parameters are estimated using the Patlak plot, a standard linear regression method for the estimation of irreversibly bound tracers. Results show that direct MLEM and SPS parametric reconstruction algorithms have remarkably slow convergence. This is explained by the high correlation of the kinetic parameters. The method has been implemented in STIR library (Software for Tomographic Image Reconstruction).