Data Assimilation in Computational Hemodynamics

Abstract

Analysis of hemodynamics shows great potential to provide indications for the risk of cardiac malformations and is essential for diagnostic purposes in clinical applications. Although medical imaging techniques such as phase-contrast magnetic resonance imaging (also known as 4D flow MRI) deliver useful information about the flow patterns in the lumen of large arteries, they cannot provide sufficient information at near-wall regions especially due to the noise in the observed signals, coarse resolution and partial volume effects. As an alternative providing noise-free solution, computational fluid dynamics (CFD) has been established as a valuable tool for the detailed characterization of volumetric blood flow and its effects on the arterial wall. However, CFD requires awareness of boundary conditions and initial flow, which is usually not known beforehand. Besides, the flow is heavily influenced by the dynamic nature of the heart beat, which results in unsteady and periodic flow phenomena. This work aims to combine the superiority of CFD with the advantages of 4D flow MRI by introducing a novel approach for variational data assimilation and at the same time taking into account the dynamic nature of the heart beat. Phase-contrast MRI is utilized for the prescription of the initial flow and boundary conditions. Due to the noisy nature of these observations, the velocity components are controlled at the boundaries through a mathematical optimization of flow patterns at the inlets. The adjustment is supported by the more reliable flow measurements in the middle of the lumen, where a least-squares flow-matching is considered. The norm of the control and the control surface gradient are augmented by Tikhonov regularization terms, which result (along with the flow-matching term) in the final objective function. The minimization is performed under the constraint that the Navier-Stokes equations are satisfied. In addition, the time-periodic heart beat is captured by a set of harmonically balanced equations. The latter is achieved by a temporal discretization using a Fourier-spectral collocation approach, where the collocation points are aligned with 4D flow MRI measurements. Compared to the raw measurements, the proposed approach significantly improves the reconstructed flow field at the aortic root, which is one of the most important clinically relevant locations where flow disturbances can easily lead to pathological modifications of the arterial wall. Thus the new method has a great potential for revealing clinically relevant hemodynamic phenomena.