Recent advancements in the field of echocardiography have introduced various methods to image blood flow in the heart. Of particular interest is the left ventricle of the heart, which pumps oxygenated blood from the lungs out through the aorta. One method for imaging blood flow is injecting FDA-approved micro-bubbles into the left ventricle, and then, using the motion of the microbubbles and the frame rate of the ultrasound scan, the blood velocity can be calculated. In addition to blood velocity, echocardiologists are also interested in calculating pressure gradients and other flow properties, but this is not currently possible because the velocity data obtained is two-dimensional and contains noise. In order to realize the full potential of microbubbles as a tool for determining the pumping efficiency and health of the LV, three-dimensional velocity data is required. Our goal is to assimilate two-dimensional velocity data from ultrasound experiments into a three-dimensional computer model. In order to achieve this objective, a numerical method is needed that can approximate the solution of a system of differential equations and assimilate an arbitrary number of noisy experimental values at arbitrary points within the domain of interests to provide a "most probable" approximate solution that is accordingly influenced by the experimental data. In this thesis we present two different approaches for data assimilation, the first approach is more computationally expensive, but requires only a single step. The second approach uses a two stage data assimilation technique but is computationally less expensive. The motivation for using the least-squares finite element method approach is that it provides many advantages such as the ability to match the numerical solution more closely to more accurate data and less closely to the less accurate data.
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