  Color Doppler by transthoracic echocardiography creates 2-D fan-shaped maps
of blood velocities in the cardiac cavities. It is a one-component velocimetric
technique since it only returns the velocity components parallel to the
ultrasound beams. Intraventricular vector flow mapping (iVFM) is a method to
recover the blood velocity vectors from the Doppler scalar fields in an
echocardiographic three-chamber view. We improved our iVFM numerical scheme by
imposing physical constraints. The iVFM consisted in minimizing regularized
Doppler residuals subject to the condition that two fluid-dynamics constraints
were satisfied, namely planar mass conservation, and free-slip boundary
conditions. The optimization problem was solved by using the Lagrange
multiplier method. A finite-difference discretization of the optimization
problem, written in the polar coordinate system centered on the cardiac
ultrasound probe, led to a sparse linear system. The single regularization
parameter was determined automatically for non-supervision considerations. The
physics-constrained method was validated using realistic intracardiac flow data
from a patient-specific CFD model. The numerical evaluations showed that the
iVFM-derived velocity vectors were in very good agreement with the CFD-based
original velocities, with relative errors ranged between 0.3 and 12%. We
calculated two macroscopic measures of flow in the cardiac region of interest,
the mean vorticity and mean stream function, and observed an excellent
concordance between physics-constrained iVFM and CFD. The capability of
physics-constrained iVFM was finally tested with in vivo color Doppler data
acquired in patients routinely examined in the echocardiographic laboratory.
The vortex that forms during the rapid filling was deciphered. The
physics-constrained iVFM algorithm is ready for pilot clinical studies.
