Assessing Vessel Reconstruction in Ultrasound Localization Microscopy by Maximum Likelihood Estimation of a Zero-Inflated Poisson Model

In clinical applications of super-resolution ultrasound imaging, it is often not possible to achieve a full reconstruction of the microvasculature within a limited measurement time. This makes the comparison of studies and quantitative parameters of vascular morphology and perfusion difficult. Therefore, saturation models were proposed to predict adequate measurement times and estimate the degree of vessel reconstruction. Here, we derive a statistical model for the microbubble counts in super-resolution voxels by a zero-inflated Poisson (ZIP) process. In this model, voxels either belong to vessels with probability P_V and count events with Poisson rate Λ, or they are empty and remain zero. In this model, P_V represents the vessel voxel density in the super-resolution image after infinite measurement time. For the parameters P_V and Λ, we give Cramér-Rao lower bounds (CRLBs) for the estimation variance and derive maximum likelihood estimators (MLEs) in a novel closed-form solution. These can be calculated with knowledge of only the counts at the end of the acquisition time. The estimators are applied to preclinical and clinical data and the MLE outperforms alternative estimators proposed before. The estimated degree of reconstruction lies between 38% and 74% after less than 90 s. Vessel probability P_V ranged from 4% to 20%. The rate parameter Λ was estimated in the range of 0.5-1.3 microbubbles/voxel. For these parameter ranges, the CRLB gives standard deviations of less than 2%, which supports that the parameters can be estimated with good precision already for limited acquisition times.