Temporally invariant uniform spatial resolution in dynamic PET

We investigate methods for obtaining uniform spatial resolution over the entire scan duration for penalized ML dynamic PET image reconstructions. We approximate the true continuous time log-likelihood function by a bin-mode log-likelihood, which is obtained by dividing the scan duration into multiple constant-width bins and expressing the likelihood of the total number of counts in each spatiotemporal bin. We then exploit the space-time separability of the bin-mode imaging equation using Kronecker products and derive approximate expressions for two types of local impulse response (LIRs). The first type of LIR is defined as the derivative of the mean reconstructed dynamic PET image to a perturbation to the coefficients of the temporal basis functions. The second LIR definition involves the derivative of the mean reconstructed image to a spatiotemporal impulse which falls outside the temporal model. Temporally invariant uniform spatial resolution can be achieved under the first LIR definition by scaling both spatial and temporal hyperparameters by the diagonal entries of the dynamic Fisher information matrix (FIM). Under the second definition this scaling ameliorates the non-uniform resolution, however further modifications are necessary. We conclude with simulation results.