Ultrasound has been widely used as a diagnostic tool in medicine for more than 30 years. Recent technological advances have allowed echocardiologists to obtain two-dimensional images of the left ventricle of the heart with a temporal resolutions of 150 frames per second (t = 7 ms). FDA-approved microbubbles have been developed that can be injected into the circulating blood and that, due to their very small density compared to and blood, are easily visualized using ultrasound. Particle image velocimetry (PIV) software can be used to determine the displacement of the bubbles between ultrasound frames and to calculate the two components of the blood velocity vector orthogonal to the image based on the bubble displacement and time step size. In addition to the fluid velocity, ultrasound images of the left ventricle can be used to determine the wall position as a function of time, and the inflow and outflow fluid velocity during the cardiac cycle. Despite the abundance of data, ultrasound and PIV alone are insufficient for calculating the flow properties of interest to clinicians. Specifically, the pressure gradient and total energy loss are of primary importance, but their calculation requires a full three-dimensional velocity field. However, numerous technical hurdles prevent three-dimensional ultrasound from having a sufficiently high frame rate (it is approximately 10 frames per second, currently) for PIV analysis.
One potential approach for obtaining the clinically desirable flow properties is to use computational fluid dynamics to calculate the three-dimensional velocity based on the approximate location of the boundary and the approximate inflow and outflow velocities from ultrasound. The drawback of this approach is that it does not utilize the additional information provided by PIV analysis of the ultrasound, which gives the approximate velocity along a single plane in the three-dimensional domain. The question that we seek to answer is whether a computational fluid dynamics approach can properly incorporate the two-dimensional PIV data along a plane inside the three-dimensional domain. For addressing this problem, we examine the potential of least-squares finite element methods (LSFEM) because of their flexibility in the enforcement of various boundary conditions. Further, by weighting the PIV data and boundary conditions in a manner that properly reflects the accuracy with which the values are known, we develop the weighted LSFEM. The potential of weighted LSFEM is explored for three different test problems: the first uses randomly generated Gaussian noise to create artificial ‘experimental' data in a controlled manner, and the second and third use particle imaging velocimetry data in a fixed domain and in a moving mesh domain. After applying the moving domain, the varying geometries will be introduced in the numerical method. With the complexity introduced by the moving domain, a reasonable numerical method needs to be sought. In all test problems, weighted LSFEM produces accurate results even for cases where there is significant noise in the experimental data.