Direct 4D reconstruction of parametric images incorporating anato-functional joint entropy

We developed a closed-form 4D algorithm to directly reconstruct parametric images as obtained using the Patlak graphical method for (nearly) irreversible tracers. Conventional methods consist of individually reconstructing 2D/3D PET data, followed by graphical analysis on the sequence of reconstructed images. The proposed approach maintains the simplicity and accuracy of the EM algorithm by extending the system matrix to include the relation between the parametric images and the measured data. The proposed technique achieves a closed-form solution by utilizing a different hidden complete-data formulation within the EM framework. Additionally, the method is extended to maximum a posterior (MAP) reconstruction via incorporating MR image information, with the joint entropy between the MR and parametric PET features. A Parzen window method was used to estimate the joint probability density of the MR and parametric PET images. Using realistic simulated [¹¹C]-Naltrindole PET and MR brain images/data, the quantitative performance of the proposed methods was investigated. Significant improvements in terms of noise vs. bias performance have been achieved, when performing direct parametric reconstruction, and additionally when extending the algorithm to its Bayesian counter-part using MR-PET join entropy.