Physics-Informed Tensor Velocity Reconstruction From 2-D Velocity Estimates Using Row-Column Probes

A method is presented for estimating 3-D velocity fields over space-time (tensor velocity fields) by combining 2-D velocity estimates acquired in multiple imaging planes with the continuity equation. The proposed tensor velocity estimator accurately reconstructs flow within the aliasing limit and mitigates motion-related artifacts observed in synthetic aperture (SA) and plane wave imaging (PWI) techniques, which typically reduce estimation accuracy at higher velocities. The estimator is demonstrated using both a short imaging sequence, which estimates the tensor velocity field in a thin volume (effectively a 2-D slice), and a longer sequence for reconstructing flow over a larger volume. A singular value decomposition (SVD) procedure for phase shift noise reduction is also proposed. In addition to improving the estimation precision, the method significantly reduces the required storage space and input size for the reconstruction algorithm: from approximately 20 GB-110 MB in one measurement. Furthermore, the procedure enables quick analysis of the SVD components' contributions to the flow. Flow-rig measurements of parabolic flow were acquired at various combinations of beam-to-flow angles (60°, 75°, and 90°) and flow-rotation angles (50° and 90°), with peak velocities ranging from approximately 10%-95% of the velocity aliasing limit. The average bias for the velocity magnitude was found to be -0.71% $ \pm ~2.96$ %, and the standard deviation (SD) was 0.57% $ \pm ~ 0.19\%$ . From Field II simulations of pulsatile carotid artery flows, the estimator reconstructs the velocity magnitude over a 3.5 s period with a normalized root-mean-square error (nRMSE) of 3.57%. In measurements of flow in a carotid artery phantom, the average velocity magnitude pulse profiles of the mean and peak velocities were estimated with SDs of 0.75% and 2.63%. Similarly, in vivo measurements of the carotid artery bifurcation yielded mean and peak velocity magnitude profiles with estimated SDs of 1.06% and 2.51%. These results demonstrate that the proposed estimator can accurately and precisely reconstruct tensor velocity fields in clinically relevant and high-velocity flow scenarios.