Maximum-Likelihood Estimation to Assess the Degree of Reconstruction of Microvasculature from Super-Resolution US Imaging

In clinical applications of super-resolution ultrasound imaging, the complete reconstruction of the microvasculature will not be feasible due to several limiting factors such as measurement times, concentrations of microbubbles, or motion artefacts. Therefore, it is of interest to estimate the degree of reconstruction in order to establish reasonable measurement protocols and to derive meaningful morphological and perfusion parameters. Here, we show that the filling of the voxels with the detected microbubbles (i.e. the reconstruction), can be well modeled with a zero-inflated Poisson (ZIP) process. For the first time - to our knowledge - we derived a closed-form solution for the maximum likelihood estimator (MLE) of the relevant parameters of a ZIP process. From these parameters, the degree of reconstruction can be assessed from the ratio of the number of filled voxels to the number of voxels that are expected to be filled after infinite time. We show, that in preclinical and clinical measurements a degree of reconstruction between 38% and 74% was achieved. We also demonstrate that reliable estimates can be achieved earlier with the MLE than with the least-squares fit to the data as previously proposed. Additionally, the MLE is very easy to implement as the counts observed at the end of the measurement period provide all necessary data for parameter estimation.