# Fluid Structure Interaction of Arterial Walls and Blood Flow

## Introduction

The simulation of the physiological loading situation of arteries with moderate atherosclerotic plaque may provide additional indicators for medical doctors to estimate if the plaque is likely to rupture and if surgical intervention is required. In particular the transmural stresses are important in this context. In order to account for the correct behavior of the arterial wall to predict these stresses we apply fully coupled fluid-structure-interaction algorithms and nonlinear anisotropic hyperelastic material wall models.

Fine element mesh of an artery consisting of adventitia (yellow), media (green), and plaque (red).

## Fluid-Structure-Interaction

In order to model the interaction of fluid and structure, three problems have to be considered under certain coupling conditions.

**Fluid problem:**Navier-Stokes equations in ALE form,**Structural problem:**( are the first Piola-Kirchhoff stresses)**Geometry problem:****Coupling conditions on :**(, displacement of the fluid-structure interface)

Flow in an idealized artery, cf. Section **Benchmark**.

## Monolithic Coupling Algorithm

We use a monolithic coupling scheme, i.e., we consider the **fluid**, the **structure** and the **geometry** problem at once. Applying the convective explicit (**CE**) time discretization scheme [2] yields the fully coupled system

Lagrange multipliers are introduced to enforce the coupling on the fluid-structure interface .

## Material Model

We consider nonlinear anisotropic hyperelastic material models to model the elastic response of the arterial wall. The strain energy function is given by

where the isotropic and the compression part are

with . The transversely isotropic function for the fibers

cf. [1], where the invariants of the right Cauchy-Green-Tensor , , and invariants of the additional structural tensor incorporating the fiber reinforcement.

## Benchmark

We propose a benchmark problem for the testing of fluid-structure-interaction simulations using nonlinear material models.

Geometry of the fluid-structure-interaction benchmark for nonlinear material models.

A parabolic inflow-velocity profile is prescribed in order to enforce an internal pressure of 80 mmHg. Absorbing boundary conditions (see [4]) for reducing wave reflections at the outlet are applied. Additionally the displacement at inlet and outlet is constrained, such that the geometry is statically determined.

Afterwards we impose an appropriate inflow pressure for a heartbeat.

Characteristic values, i. e. flowrate (top), Lumen area (middle), and average pressure (bottom), at inlet (left) and outlet (right ).

## Software

Our software environment is built on the base of

- The
**F**inite**E**lement**A**nalysis**P**rogram (**FEAP**) version 8.2, cf. [6]. - LifeV software library 3.6.2, cf. [5].
- A portable lightweight coupling library for FEAP and LifeV, cf. [7].

## Parallel Performance

We also work on the development of domain decomposition methods for fluid-structure-interaction systems. In our current simulations we use the Composed Dirichlet-Neumann preconditioner, cf. \cite{Crosetto2011}. Neglecting one term of the Jacobian of the monolithic we factorize the matrix

and apply the AAS preconditioner on each part separately.

Strong scaling simulations run on CHEOPS HPC Cluster at the University of Cologne using the Composed Dirichlet-Neumann preconditioner.

## Links

- Numerical Mathematics and Scientific Computing - Universität zu Köln
- Institut für Numerische Mathematik und Optimierung - Technische Universität Bergakademie Freiberg
- Institute of Mechanics - Universität Duisburg-Essen
- Chair of Modelling and Scientific Computing - CMCS, EPF Lausanne

## References

- [1] D. Balzani, P. Neff, J. Schröder and G.A. Holzapfel. A polyconvex framework for soft biological tissues. adjustment to experimental data. IJSS, 43(20), 6052--6070, 2006.
- [2] P. Crosetto, S. Deparis, G. Fourestey, and A. Quarteroni. Parallel algorithms for fluid-structure interaction problems in haemodynamics. SISC, 33(4), 1598--1622, 2011.
- [3] S. Deparis, M. Discacciati, G. Fourestey and A. Quarteroni. Fluid-structure algorithms based on Steklov-Poincare operators. CMAME, 195, 5797--5812, 2006.
- [4] F. Nobile and C. Vergara. An effective fluid-structure interaction formulation for vascular dynamics by generalized Robin conditions. SISC, 30(2):731--763, 2008.
- [5] LifeV webpage: http://www.lifev.org.
- [6] R. L. Taylor. FEAP webpage: http://www.ce.berkeley.edu/projects/feap.
- [7] A. Fischle, A. Heinlein, A. Klawonn, O. Rheinbach. Coupling library for LifeV and FEAP. Unpublished, 2014.