Abstract
We propose a regularized least-squares method for reconstructing 2D velocity vector ields within the left ventricular cavity from single-view color Doppler echocardiographic images. Vector low mapping is formulated as a quadratic optimization problem based on an ℓ2-norm minimization of a cost function composed of a Doppler data-idelity term and a regularizer. The latter contains three physically interpretable expressions related to 2D mass conservation, Dirichlet boundary conditions, and smoothness. A inite difference discretization of the continuous problem was adopted in a polar coordinate system, leading to a sparse symmetric positive-deinite system. The three regularization parameters were determined automatically by analyzing the L-hypersurface, a generalization of the L-curve. The performance of the proposed method was numerically evaluated using (1) a synthetic low composed of a mixture of divergence-free and curl-free low ields and (2) simulated low data from a patient-speciic CFD (computational luid dynamics) model of a human left heart. The numerical evaluations showed that the vector low ieldsreconstructed from the Doppler components were in good agreement with the original velocities, with a relative error less than 20%. It was also demonstrated that a perturbation of the domain contour has little effect on the rebuilt velocity ields. The capability of our intraventricular vector low mapping (iVFM) algorithm was inally illustrated on in vivo echocardiographic color Doppler data acquired in patients. The vortex that forms during the rapid illing was clearly deciphered. This improved iVFM algorithm is expected to have a signiicant clinical impact in the assessment of diastolic function.