Christian Vergara

Fluid-structure partitioned procedures based on Robin transmission conditions

In this article we design new partitioned procedures for fluid-structure interaction problems, based on Robin-type transmission conditions. The choice of the coefficient in the Robin conditions is justified via simplified models. The strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane, as in hemodynamics applications. We analyze theoretically the new iterative procedures on a model problem, which represents a simplified blood-vessel system. In particular, the Robin-Neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures. The theoretical results are checked using numerical experimentation.

An Effective Fluid-Structure Interaction Formulation for Vascular Dynamics by Generalized Robin Conditions

In this work we focus on the modeling and numerical simulation of the fluid-structure interaction mechanism in vascular dynamics. We first propose a simple membrane model to describe the deformation of the arterial wall, which is derived from the Koiter shell equations and is applicable to an arbitrary geometry. Secondly, we consider a reformulation of the fluid-structure problem, in which the newly derived membrane model, thanks to its simplicity, is embedded into the fluid equations and will appear as a generalized Robin boundary condition. The original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only. We also derive a stability estimate for the resulting numerical scheme. Finally, we propose new outflow absorbing boundary conditions, which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains. We present several numerical results showing the effectiveness of the proposed approaches.

Comparative finite element model analysis of ascending aortic flow in bicuspid and tricuspid aortic valve.

In bicuspid aortic valve (BAV) disease, the role of genetic and hemodynamic factors influencing ascending aortic pathology is controversial. To test the effect of BAV geometry on ascending aortic flow, a finite element analysis was undertaken. A surface model of aortic root and ascending aorta was obtained from magnetic resonance images of patients with BAV and tricuspid aortic valve using segmentation facilities of the image processing code Vascular Modeling Toolkit (developed at the Mario Negri Institute). Analytical models of bicuspid (antero-posterior [AP], type 1 and latero-lateral, type 2 commissures) and tricuspid orifices were mathematically defined and turned into a volumetric mesh of linear tetrahedra for computational fluid dynamics simulations. Numerical simulations were performed with the finite element code LifeV. Flow velocity fields were assessed for four levels: aortic annulus, sinus of Valsalva, sinotubular junction, and ascending aorta. Comparison of finite element analysis of bicuspid and tricuspid aortic valve showed different blood flow velocity pattern. Flow in bicuspid configurations showed asymmetrical distribution of velocity field toward the convexity of mid-ascending aorta returning symmetrical in distal ascending aorta. On the contrary, tricuspid flow was symmetrical in each aortic segment. Comparing type 1 BAV with type 2 BAV, more pronounced recirculation zones were noticed in the latter. Finally, we found that in both BAV configurations, maximum wall shear stress is highly localized at the convex portion of the mid-ascending aorta level. Comparison between models showed asymmetrical and higher flow velocity in bicuspid models, in particular in the AP configuration. Asymmetry was more pronounced at the aortic level known to be more exposed to aneurysm formation in bicuspid patients. This supports the hypothesis that hemodynamic factors may contribute to ascending aortic pathology in this subset of patients.

A Variational Approach for Estimating the Compliance of the Cardiovascular Tissue: An Inverse Fluid-Structure Interaction Problem

Estimation of the stiffness of a biological soft tissue is useful for the detection of pathologies such as tumors or atherosclerotic plaques. Elastography is a method based on the comparison between two images before and after a forced deformation of the tissue of interest. An inverse elasticity problem is then solved for Youngs modulus estimation. In the case of arteries, no forced deformation is required, since vessels naturally move under the action of blood. Youngs modulus can therefore be estimated by solving a coupled inverse fluid-structure interaction problem. In this paper we focus on the mathematical properties of this problem and its numerical solution. We give some well posedness analysis and some preliminary results based on a synthetic data set, i.e., test cases where the exact Youngs modulus is known and the displacement dataset is numerically generated by solving a forward fluid-structure interaction problem. We address the problem of the presence of the noise in the measured displacement and of the proper sampling frequency for obtaining reliable estimates.

A New Approach to Numerical Solution of Defective Boundary Value Problems in Incompressible Fluid Dynamics

We consider the incompressible Navier-Stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed. There are basically two strategies for solving these defective boundary problems: the variational approach (see J. Heywood, R. Rannacher, and S. Turek, Internat. J. Numer. Methods Fluids, 22 (1996), pp. 325-352) and the augmented formulation (see L. Formaggia, J. F. Gerbeau, F. Nobile, and A. Quarteroni, SIAM J. Numer. Anal., 40 (2002), pp. 376-401, and A. Veneziani and C. Vergara, Internat. J. Numer. Methods Fluids, 47 (2005), pp. 803-816). However, both of these approaches present some drawbacks. For the flow rate problem, the former resorts to nonstandard functional spaces, which are quite difficult to discretize. On the other hand, for the mean pressure problem, it yields exact solutions only in very special cases. The latter is applicable only to the flow rate problem, since for the mean pressure problem it provides unfeasible boundary conditions. In this paper, we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional. This approach allows us to treat the two kinds of problems (flow rate and mean pressure) successfully within the same framework, which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries. Moreover, it is more versatile, being prone to be extended to other kinds of defective conditions. We analyze the problems obtained with this approach and propose some algorithms for their numerical solution. Several numerical results are presented supporting the effectiveness of our approach.

Influence of Bicuspid Valve Geometry on Ascending Aortic Fluid Dynamics: A Parametric Study

Bicuspid aortic valve (BAV) predisposes to aortic aneurysms with a high prevalence. A first hypothesis for this phenomenon is related to fibrillin deficiency (genetic hypothesis). The present article focused on a complementary, hemodynamic hypothesis stating that it is the peculiar fluid dynamics of blood in the ascending aorta of patients with BAV configurations that leads to aneurysm formation. To corroborate this hypothesis, a parametric study was performed based on numerical simulations of ascending aorta hemodynamics with different configurations of orifice area and valve orientation. The resulting wall shear stress (WSS) distributions and degree of asymmetry of the blood jet were investigated, and surrogate indices introduced. The results showed that WSS was more pronounced in subjects with BAV morphologies, also in the nonstenotic case. In particular, a maximum WSS of 3Pa was found (vs. 1.5Pa in subjects with a tricuspid configuration). It is localized at the mid-ascending aorta, the segment more prone to dilate as shown by the index related to maximum WSS (0.869 in BAV vs. 0.322 in tricuspid). Moreover, the asymmetry of the blood flow was found larger for decreasing valve area, the related index at mid-ascending aorta being more than three times higher than that found for tricuspid configuration (0.70 vs. 0.20). Further, WSS and flow asymmetry were higher also at the sinus of Valsalva and sinotubolar junction for a latero-lateral (LL) BAV configuration in keeping with the clinical observation on association between BAV configurations and different aortic aneurysm morphology. These findings may help explain the higher risk of aneurysm formation in BAV patients. The proposed indices will require validation prior to application in clinical settings.

Analysis and Optimization of Robin-Robin Partitioned Procedures in Fluid-Structure Interaction Problems

In the solution of fluid-structure interaction (FSI) problems, partitioned procedures are modular algorithms that involve separate fluid and structure solvers that interact in an iterative framework through the exchange of suitable transmission conditions at the FS interface. In this work we study, using Fourier analysis, the convergence of partitioned algorithms based on Robin transmission conditions. We derive, for different models of the fluid and the structure, a frequency-dependent reduction factor at each iteration of the partitioned algorithm, which is minimized by choosing optimal values of the coefficients in the Robin transmission conditions. Two-dimensional numerical results are also reported, which highlight the effectiveness of the optimization procedure.

On the physical consistency between three-dimensional and one-dimensional models in haemodynamics

In this work we discuss the reliability of the coupling among three-dimensional (3D) and one-dimensional (1D) models that describe blood flowing into the circulatory tree. In particular, we study the physical consistency of the 1D model with respect to the 3D one. To this aim, we introduce a general criterion based on energy balance for the proper choice of coupling conditions between models. We also propose a way to include in the 1D model the effect of the external tissue surrounding the vessel and we discuss its importance whenever this effect is considered in the 3D model. Finally, we propose several numerical results in real human carotids, studying different configurations for the 1D model and highlighting the best one in view of the physical consistency.

Multiscale Boundary Conditions for Drug Release from Cardiovascular Stents

In this study, we focus on a specific application, the modeling and simulation of drug release from cardiovascular drug eluting stents. In particular, we analyze the drug release dynamics from the stent coating, where the drug is initially stored, to the surrounding arterial wall. The main challenge consists of accounting for multiple space scales. To this purpose, we derive suitable boundary conditions accounting for the smaller scales on the macroscopic model. This approach, applied to drug delivery, significantly cuts down the computational cost of the numerical simulations and allows us to consider a realistic problem setting.

Helical flows and asymmetry of blood jet in dilated ascending aorta with normally functioning bicuspid valve

Bicuspid aortic valve (BAV) is associated with aortic dilatation and aneurysm. Several studies evidenced an eccentric systolic flow in ascending aorta associated with increased wall shear stresses (WSS) and the occurrence of an helical systolic flow. This study seeks to elucidate the connections between jet asymmetry and helical flow in patients with normally functioning BAV and dilated ascending aorta. We performed a computational parametric study by varying, for a patient-specific geometry, the valve area and the flow rate entering the aorta and drawing also a tricuspid valve (TAV). We considered also phase-contrast magnetic resonance imaging of four BAV and TAV patients. Measurement of normalized flow asymmetry index, systolic WSS and of a new index (positive helix fraction, PHF) quantifying the presence of a single a single helical flow were performed. In our computation, BAV cases featured higher values of all indices with respect to TAV in both numerical and imaged-based results. Moreover, all indices increased with decreasing valve area and/or with increasing flow rate. This allowed to separate the BAV and TAV cases with respect to the jet asymmetry, WSS localization and helical flow. Interestingly, these results were obtained without modeling the leaflets.