title: Data Driven Estimation of Wall Shear Stress from Magnetic Resonance Imaging
Creators: name: Teschner, Gabriel
Publisher: name: UNSPECIFIED
Year: 2023

description: 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 established under reasonable smoothness assumptions on the geometry and the flow velocity. These results widely carry over to the discrete setting, where we introduce discrete versions of the forward operators to minimize the data error. In the second part of this work we use methods of data assimilation, to enhance the purely data driven reconstruction using a fluid dynamical model. In a first study, we utilize a variational approach for the enhancement of the velocity reconstruction under known geometry, that minimizes a functional consisting of a data error and a model error and was formerly developed and analyzed in our research group. The variational approach is formally equivalent to an optimal control problem. Secondly, we demonstrate the basic possibility to enhance the geometry identification based on a fixed fluid dynamical model. For both methods, we utilize the widely accurate purely data driven reconstructions for linearization and localization of the applied fluid dynamic model. For the assessment of the developed methods we have conducted a comprehensive validation in collaboration with experts of fluid dynamics and radiology from the Institute for Fluid Mechanics and Aerodynamics, Technical University Darmstadt, and the Department of Radiology – Medical Physics, University Hospital Freiburg, respectively. A big issue is the lack of ground truth. The wall shear stress is highly sensitve to perturbations of the environmental conditions. Hence, the experiments have to be performed with meticulous diligence, to ensure reproducibility and hence validity of the reference values, that are obtained from high-resolution laser Doppler anemometry or computational fluid dynamics. Additionally, we have developed highly robust and accurate but to specific flows limited reconstruction methods, to estimate the wall shear stress directly from the magnetic resonance data. Furthermore we fall back on virtual in silico data in some cases. Already the purely data driven reconstruction method provides largely convincing results. However, the reconstruction is highly sensitve to perturbations of the geometry identification and reveals sometimes systematic errors due to the coarse resolution. Our analysis identifies the error sources and their contribution to the overall error. This offers a guideline for a suitable choice of several parameters in the measurement setup. Furthermore, the described shortcomings of the purely data driven reconstruction are essentially corrected by the provided data assimilation techniques. The specialization to the application in the aorta leads to a fully integrated reconstruction method with low computational effort, typical running times for all provided methods are in the range of several minutes using common hardware. Therefore, a valid estimation of wall shear stress in the aorta is feasible, even under the limitations of clinical routine. This thesis was funded by the DFG via grant EG-331/1-1. The collaboration with the project partners resulted in the following papers: - H. Egger and G. Teschner. On the Stable Estimation of Flow Geometry and Wall Shear Stress from Magnetic Resonance Images. Inverse Problems, 35:095001, 2019. - A. Bauer, S. Wegt, M. Bopp, S. Jakirlic, C. Tropea, A. J. Krafft, N. Shokina, J. Hennig, G. Teschner and H. Egger. Comparison of Wall Shear Stress Estimates Obtained by Laser Doppler Velocimetry, Magnetic Resonance Imaging and Numerical Simulations. Experiments in Fluids, 60:1–16, 2019. - N. Shokina, A. Bauer, G. Teschner, W. B. Buchenberg, C. Tropea, H. Egger, J. Hennig and A. J. Krafft. MR-based Wall Shear Stress Measurements in Fully Developed Turbulent Flow using the Clauser Plot Method. Journal of Magnetic Resonance, 305:16–21, 2019. - N. Shokina, G. Teschner, A. Bauer, C. Tropea, H. Egger, J. Hennig and A. J. Krafft. Quantification of Wall Shear Stress in Large Blood Vessels using Magnetic Resonance Imaging. Computational Technologies, 24:4–27, 2019. - N. Shokina, G. Teschner, A. Bauer, C. Tropea, H. Egger, J. Hennig and A. J. Krafft. Parametric Sequential Method for MRI-based Wall Shear Stress Quantification. IEEE Transactions on Medical Imaging, 40:1105–1112, 2020. In this thesis, we will summarize the findings of the papers mentioned above and appropriatly extend them to an entire analysis of the wall shear stress reconstruction.