Optimization of 3-D Divergence-Free Flow Field Reconstruction Using 2-D Ultrasound Vector Flow Imaging

3-D blood vector flow imaging is of great value in understanding and detecting cardiovascular diseases. Currently, 3-D ultrasound vector flow imaging requires 2-D matrix probes, which are expensive and suffer from suboptimal image quality. Our recent study proposed an interpolation algorithm to obtain a divergence-free reconstruction of the 3-D flow field from 2-D velocities obtained by high-frame-rate ultrasound particle imaging velocimetry (High Frame Rate echo-Particle Imaging Velocimetry, also known as HFR Ultrasound Imaging Velocimetry (UIV)), using a 1-D array transducer. The aim of this work was to significantly improve the accuracy and reduce the time-to-solution of our previous approach, thereby paving the way for clinical translation. More specifically, accuracy was improved by optimising the divergence-free basis to reduce Runge phenomena near domain boundaries, and time-to-solution was reduced by demonstrating that under certain conditions, the resulting system could be solved using widely available and highly optimised generalised minimum residual algorithms. To initially illustrate the utility of the approach, coarse 2-D subsamplings of an analytical unsteady Womersely flow solution and a steady helical flow solution obtained using computational fluid dynamics were used successfully to reconstruct full flow solutions, with 0.82% and 4.8% average relative errors in the velocity field, respectively. Subsequently, multiplane 2-D velocity fields were obtained through HFR UIV for a straight-tube phantom and a carotid bifurcation phantom, from which full 3-D flow fields were reconstructed. These were then compared with flow fields obtained via computational fluid dynamics in each of the two configurations, and average relative errors of 6.01% and 12.8% in the velocity field were obtained. These results reflect 15%-75% improvements in accuracy and 53- to 874-fold acceleration of reconstruction speeds for the four cases, compared with the previous divergence-free flow reconstruction method. In conclusion, the proposed method provides an effective and fast method to reconstruct 3-D flow in arteries using a 1-D array transducer.