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

Abstract- 3D blood Vector Flow Imaging (VFI) is of great value for understanding and detecting cardiovascular diseases. Currently 3D Ultrasound (US) VFI requires 2D matrix probes which are expensive and suffer from sub-optimal image quality. Our recent study proposed an interpolation algorithm to obtain a divergence free reconstruction of the 3D flow field from 2D velocities obtained by High Frame Rate US Particle Imaging Velocimetry (HFR echo-PIV, also known as HFR UIV), using a 1D array transducer. This work aims 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 optimized Generalized Minimum Residual (GMRES) algorithms. To initially demonstrate the utility of the approach, coarse 2D sub-samplings of an analytical unsteady Womersely flow solution and a steady helical flow solution obtained using Computational Fluid Dynamics (CFD) were used to successfully reconstruct full flow solutions, with 0.82% and 4.8% average relative errors in the velocity field respectively. Subsequently, multi-plane 2D velocity fields were obtained through HFR UIV for a straight tube phantom and a carotid bifurcation phantom, from which full 3D flow fields were reconstructed. These were then compared with flow fields obtained via CFD 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 53~874 fold acceleration of reconstruction speeds for the four cases, when compared with the previous divergence free flow reconstruction method. In conclusion the proposed method provides an effective and fast method to reconstruct 3D flow in arteries using a 1D array transducer