Data Driven Estimation of Wall Shear Stress from Magnetic Resonance Imaging

Flows occur in various applications in engineering and medicine. Dynamic quantities, in particular the forces, that the flowing viscous fluid exerts on the neighbouring material, are of special interest. Following the widely accepted flow model, these dynamic quantities are representable by derivatives of the velocity field. A modification of magnetic resonance imaging is capable to measure besides morphological data, also velocity fields in the interior of an object. As a non-invasive method, it is in particular suited for in vivo investigations of the cardiovascular system. This thesis deals with the problem of reconstructing the wall shear stress, the distribution of the shear forces, that the blood flow exerts on the aortic vessel wall. This involves the reconstruction of both the flow geometry and velocity from magnetic resonance data and the evaluation of the normal velocity derivative, the shear rate. At first glance, this problem might seem trivial. However, there are several issues: - Magnetic resonance imaging acquires local means of the flow velocity with comparatively low spatial and temporal resolution. Additionally, the measurements are contaminated by noise. - The blood flow exhibits boundary layers, where the flow field dramatically changes over small distances. This complicates an accurate approximation of the velocity field in the region near to the boundary. - The flow geometry and the flow velocity are structurally connected: Just the shear-rate, that has to be evaluated, exhibits a discontinuity at the boundary. In the first part of this work we present a framework for the purely data driven wall shear stress reconstruction. For this purpose, we approximate the flow geometry first, and then the flow velocity using parametric representations. The reconstruction method allows for a continuous analysis as regularization procedure for two coupled inverse problems. Since the corresponding forward operators satisfy a conditional stability estimate, convergence of the reconstruction method can be ...