Using linear time-invariant system theory to estimate kinetic parameters directly from projection measurements

In a dynamic SPECT (single photon emission computed tomography) study, if an image is reconstructed for each detector rotation, the image will contain errors as the data for each projection angle are acquired at different times in the dynamic study. These errors will affect the kinetic parameter estimation. We hypothesize that more accurate estimates can be obtained if the parameters are estimated directly from the projections than from the reconstructions. It is more difficult to estimate the parameters from the projections than from the reconstructions. In the reconstructed images, all the regions are well separated, while in projections the regions are mixed. Estimation from the projections requires all projections of the kinetic models for each region. This results in a relatively complicated objective function, which is usually not well behaved and has many local minima due to the non-linear exponential terms. We propose a technique to estimate the exponential parameters separately using a linear model, which has a closed-form solution for the one parameter case and has a unique solution via the singular value decomposition (SVD) for the multiple parameter case. The other multiplicative parameters also have a linear model and are estimated once the exponential parameters have been fixed. The objective function with fixed exponential parameters is quadratic and well-behaved. No local minima are observed in computer simulations, which enabled us to obtain accurate estimation. Our computer simulations demonstrate that estimation directly from the projections are more accurate than the estimation from the reconstructed images