Recent advancements in the field of echocardiography have introduced various methods to image blood flow in the heart . Our particular interest is in the left ventricle (LV) 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 micro-bubbles and the frame rate of the ultrasound scan (i.e., using Particle Imagining Velocimetry or echo-PIV), 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 noisy. Our goal is to assimilate two-dimensional velocity data from micro-bubble 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 equation and assimilate an arbitrary number of noisy experimental data points at arbitrary locations within the domain of interests to provide a ‘most probable’ approximate solution that is properly influenced by the experimental data .
We propose a new numerical method for combing two-dimensional noisy echo-PIV data, which is more computationally efficient than previous approaches . The approximate solution is calculated using continuous interior penalty finite element method (IPFEM) coupled with a least-squares finite element method (LSFEM) for integrating the noisy echo-PIV data. This framework allows the flexibility of using any existing numerical approaches for the differential equations that model the original problem, and then complement the solution using a weighted LSFEM framework for data assimilation. The choice of LSFEM approach is due the flexibility when assimilating noisy echo-PIV data since the method can strongly weight more accurate echo-PIV data and use a lower weight for less accurate data. The new numerical approach have been used to predict the 3-dimensional LV blood flow. Results from the current method show the impact of assimilating echo-PIV data combining the data with a full 3-dimensional simulation result.
1. Borazjani, I., et al., Left ventricular flow analysis: recent advances in numerical methods and applications in cardiac ultrasound. Computational and mathematical methods in medicine, 2013. 2013.
2. Heys, J.J., et al., Weighted least-squares finite elements based on particle imaging velocimetry data. Journal of Computational Physics, 2010. 229(1): p. 107-118.
3. Rajaraman, P.K., et al., Echocardiographic particle imaging velocimetry data assimilation with least square finite element methods. Computers & Mathematics with Applications, 2014. 68(11): p. 1569-1580.
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