Parameter estimation in fluid flow models from undersampled frequency space data

4D Flow MRI is the state of the art technique for measuring blood flow, and it provides valuable information for inverse problems in the cardiovascular system. However, 4D Flow MRI has a very long acquisition time, straining healthcare resources and inconveniencing patients. Due to this, usually only a part of the frequency space is acquired, where then further assumptions need to be made in order to obtain an image. Inverse problems from 4D Flow MRI data have the potential to compute clinically relevant quantities without the need for invasive procedures, and/or expanding the set of biomarkers for a more accurate diagnosis. However, reconstructing MRI measurements with Compressed Sensing techniques introduces artifacts and inaccuracies, which can compromise the results of the inverse problems. Additionally, there is a high number of different sampling patterns available, and it is often unclear which of them is preferable. Here, we present a parameter estimation problem directly using highly undersampled frequency space measurements. This problem is numerically solved by a Reduced-Order Unscented Kalman Filter (ROUKF). We show that this results in more accurate parameter estimation for boundary conditions in a synthetic aortic blood flow than using measurements reconstructed with Compressed Sensing. We also compare different sampling patterns, demonstrating how the quality of the parameter estimation depends on the choice of the sampling pattern. The results show a considerably higher accuracy than an inverse problem using velocity measurements reconstructed via compressed sensing. Finally, we confirm these findings on real MRI data from a mechanical phantom.