European Numerical Mathematics and
 08:30 25 mins Flow field mapping and vasculature resistance estimation for CFD in human arteries Bruno Frackowiak, Ralph Maessen, Marco Baragona, Sergei Shulepov, Kevin Lau, Juliana Franz, Frans van de Vosse Abstract: 1. Introduction Computational Fluid Dynamics (CFD) is a promising non-invasive method to predict heart failure and understand better the blood flow physiology in arteries. However, the correct flow distribution relies on appropriate boundary conditions (BCs) setting, e.g. outlet flow resistance. For models derived only from images, the flow distribution is unknown. Thus appropriate BCs are usually derived from generic correlations from literature [1] or from machine learning algorithm trained on a limited set of ground truth data [2], leading to some uncertainties in the CFD outcome. Possible ways to reduce these uncertainties lie in using more elaborated sub-models for the boundary conditions estimation, or in mapping the CFD flow field with experimental data in a specific area where measurements are possible. In the present work, we will present a Structural Fractal Tree Model (SFTM) based on Olufsen scaling laws [3] to provide an absolute value of outlet flow resistance. Moreover we will also describe mapping methods using 3D Doppler velocity field in carotids arteries for improved estimation of the upstream flow rate. Next to these two main topics, we will briefly touch upon other related developments such as pressure reconstruction from MR velocity data and KF based data assimilation in the context of 0D/1D modelling. 2. Materials and Methods Typically the geometry of the arterial tree is reconstructed from CT images, over a few branches down to the resolution limit, using segmentation methods, as shown in Figure 1. At each outlet of the reconstructed tree, the full distal vasculature is reconstructed recursively by the SFTM, assuming its self-similar nature. Hence flow resistance absolute value Routlet is calculated analytically, assuming viscosity driven flow in vessel segments network Using CFD, the blood flow can be simulated in carotid arteries too, where the velocity field can be measured externally with Ultrasound Doppler technique. The exact probe position and orientation are not known. Hence they are used as fitting parameters to map the CFD field with experimental data. This yields a more accurate estimation of the inlet flow rate used as boundary condition for the CFD model. 3. Results & Discussion Figure 2 exhibits results for different branches of the reconstructed coronary tree and for the set of patients used in [2]. The agreement between the SFTM and the values derived from machine learning is quite good, whereas the power law trend agrees with Kassab scaling law [1]. For estimating the outlet flow resistances as boundary conditions of the CFD model, the SFTM exhibits several advantages: - The absolute value of flow resistance Routlet arises from the model itself, using the outlet radius routlet as input - Parameters of the SFTM are estimated from the reconstructed coronary tree, making the SFTM image dependent only and patient specific. - Those parameters can be assigned within a statistical margin, yielding a confidence interval for flow resistance. Figure 3 exhibits reconstructed 3D geometry of the carotid artery and mapped velocity field with experimental data. It clearly shows a tilt of the Doppler measurement plane with respect to the main vessel. Hence the determination of this tilt angle and of the plane position is essential to estimate accurately the inlet flow rate. The CFD model allows some simulation of the blood flow in a larger area than the experimental one, especially in the post-stenotic branch, providing relevant insights on blood flow physiology of such complex recirculating flow patterns. 5. References 1. Frackowiak B et al., VPH conference 2016, poster session 2. Freiman M et al., Med Phys 2018 (publication in progress) 3. Olufsen M S et al., Am. J. of Physiol. heart and circ. Physiol., 276(1):H257-H268 (1999). 08:55 25 mins A priori numerical analysis of relative pressure estimators from full-field velocity measurements Rodolfo Araya, Cristobal Bertoglio, Cristian Carcamo Abstract: The purpose of this article is to compare two methods for estimate the relative pressure diference which have a unknown $q$. The first method is a Poisson problem and the second method is a Stokes problem. Both methods are obtained from the Navier - Stokes Equation and then the errors to compare are the form $\Vert p_{NS}- q \Vert_{*} \leq C^{meth}(h),$ where $p_{NS}$ is the continous pressure of Navier - Stokes equation, $h$ is the mesh size, $C^{meth} \in \left\lbrace C^{ppe}, C^{ste} \right\rbrace$ with $C^{ppe}$ and $C^{ste}$ are the error constant for the Poisson Pressure Estimator and Stokes Estimator methods described in \cite{BerNuGaNoS} and $\Vert \cdot \Vert_{*}$ is a convenient norm. This way, our goal is to compare these constants to establish a order relationship of the form $C^{ppe} \leq C^{ste} \quad\text{or}\quad C^{ste} \leq C^{ppe}.$ In other words, we want to prove a inequality of the form $\Vert p_{NS} - q^{ste} \Vert_{*} \preceq \Vert p_{NS} - q^{ppe} \Vert_{*} .$ 09:20 25 mins Newton-Multigrid Solver for Optimal Control of Fluid-Structure Interaction Lukas Failer, Thomas Richter Abstract: We present a monolithic Newton multigrid solver for linear-systems occurring in optimal control with BFGS of three dimensional fluid-structure interaction. To compute sensitivity information, an adjoint equation is solved. The key idea of the algorithm is to neglect all derivatives with respect to mesh deformation in the Jacobian in every Newton step. To compute the adjoint, corresponding parts are neglected in the Jacobian. This step enables to rewrite the system in three smaller systems and has a positive effect on the conditioning such that a geometric multigrid solver can be applied. Thereby state and sensitivity information of fluid-structure interaction problems with a large number of degrees of freedom, as in 3D configurations, can be computed. The new linear solver enables parameter estimation and optimal control for various applications. For example material parameters in the outflow condition can be determined to model blood flow in a vein or artery segment. 09:45 25 mins An effective numerical model for fluid-structure interaction in carotid arteries based on CINE MRI images Silvia Pozzi, Maurizio Domanin, Laura Forzenigo, Alberto Redaelli, Christian Vergara, Emiliano Votta, Paolo Zunino Abstract: Hemodynamics play an important role in the development, growth and risk of rupture of atherosclerotic plaques. In this work, we present appropriate numerical methods to study the fluid-dynamics in carotid arteries, which represent a preferential site of development of this pathology. In particular, we propose an effective numerical model for fluid-structure interaction (FSI) simulations in carotids in presence of plaque, performed using subject-specific geometries and boundary conditions based on clinical imaging. Due to difficulties in 3D plaque reconstruction and modelling from the available radiological data, a reduced model was introduced to surrogate the presence of the atherosclerotic plaque. In an FSI framework, the support of surrounding tissues can be modelled by enforcing Robin boundary conditions on the external wall of the artery. In this work, the parameter describing the elastic response alpha was spatially differentiated to take into account the different external tissues. The presence of the atherosclerotic plaque was also surrogated through the differentiation of parameter alpha in the stenosed area. As a first application of the method, we considered three subjects with a degree of stenosis greater than 70%. For each subject, velocity signals were acquired from Echo-Color Doppler (ECD) measurements using a 8 MHz Philips Ultrasound probe and were used as inflow Dirichlet boundary conditions. Radiological acquisitions were performed with a Siemens 1.5T Avanto MR scanner. Subject-specific geometries were obtained by segmenting the interface between the lumen and the arterial wall from MRI images (Gradient Echo T1-weighted 3D Fat Sat sequence) acquired after the injection of a paramagnetic contrast medium. The surface model was used to obtain the volumetric tetrahedral mesh of the fluid domain. The volumetric mesh of the structure domain was obtained by extruding the surface model as to have a realistic thickness of the arterial wall. The resulting FSI simulations were quantitatively assessed by comparing the results with dynamic image sequences. ECD measurements taken at the internal branch (2 cm downstream from the site of maximum stenosis) were used as reference data to assess the flow division. Reference displacements on the lumen-wall interface were obtained throughout the cardiac cycle by segmenting CINE MRI images (True-FISP 2D sequence) acquired in the axial plane at several positions along the vessel. The comparison between computed and imaging wall displacements shows a very good agreement, confirming the validity of the proposed approach.