1. Numerical simulation of the Fluid-Structure Interaction in stented aneurysms M.-A. FERNÁNDEZ, J.-F. GERBEAU, J. MURA INRIA / REO Team Paris-Rocquencourt France. EndoCom

2. Outline Motivation Mathematical modeling Robin-Neumann coupling conditions Numerical Examples Conclusions

3. Motivation Abdominal Aortic Aneurysms (AAA) is a bulbous enlargement of the aorta that eventually may burst. A common treatment is the implantation of an Stent-Graft. To improve the follow-up of AAA, a device allowing the remote monitoring of the intra-aneurismal pressure is currently in development at ENDOCOM project.

4. We will denote the contact surface (interface) between the solid and the fluid as. Mathematical Modeling Geometry We consider two cases: Aneurysm with and without stent-graft. stent-graft aneurysm wall Mesh generated from medical images: Laboratoire de Biomécanique et Génie Biomédical, UTC.

5. Mathematical Modeling Geometry Two interfaces: Aneurysm Stent fluid - solid - fluid solid - fluid  Fluid at each side of the stent.  To impose continuity in velocity and jump in pressure across the stent structure we follow [Fernández- Gerbeau-Martin M2AN ‘08], where this interface is unfolded creating two portions of fluids communicated through the stent.

6. Mathematical Modeling Fluid and Structure: Partitioned Scheme Structure: Lagrangian formulation Fluid: ALE formulation Where: solid displacement, fluid velocity and pressure, harmonic extension to fluid of the solid velocity at the interface.

7. Restrictions on the interface Kinematical (Dirichlet) Dynamical (Neumann) Mathematical Modeling Interaction  A special issue is the problem of enclosed fluid between the stent and aneurysm wall.  Moreover, we have to face large added-mass effects, as in the case of physiological flows. The condition must be satisfied for the fluid But it is not necessarily true from the solid part.

8. Robin-Neumann coupling conditions Interaction We use of Robin condition for the fluid on : The parameter plays the role of compliance, relaxing the kinematic condition during the Fluid-Structure iterations. It has been shown that this scheme can successfully tackle problems with a large added-mass effect and it shows good convergence properties [ Badia-Nobile-Vergara. J. Comput. Phys.’08 / Fernández-Maday-Mullaert. Preprint ].

9. Robin-Neumann coupling conditions Interaction The Robin-Neumann coupling conditions on are With this scheme, the Dirichlet condition is relaxed through the Robin condition.

10. Navier-Stokes equation + initial conditions + Robin-Neumann coupling conditions Interaction More precisely, the coupling algorithm consist in iterations between the solid and the fluid solvers by exchange force and velocity. MASTER FSI Elasticity equation + initial conditions +

11. Navier-Stokes equation + initial conditions + Robin-Neumann coupling conditions Interaction MASTER FSI Elasticity equation + initial conditions + More precisely, the coupling algorithm consist in iterations between the solid and the fluid solvers by exchange force and velocity.

12. Robin-Neumann coupling conditions Interaction MASTER FSI Elasticity equation + initial conditions + Navier-Stokes equation + initial conditions + More precisely, the coupling algorithm consist in iterations between the solid and the fluid solvers by exchange force and velocity.

13. Robin-Neumann coupling conditions Interaction MASTER FSI until Elasticity equation + initial conditions + Navier-Stokes equation + initial conditions + More precisely, the coupling algorithm consist in iterations between the solid and the fluid solvers by exchange force and velocity.

14. Numerical Example Test: Blocked aneurysm wall  To asses the preservation of the volume in the intra-aneurysmal sac.

15. Test: Aneurysm wall pressure for different sizes Numerical Example

18. Conclusions The Robin-Neumann coupling algorithm can be successfully applied to the simulation of a stented AAA, involving an enclosed fluid. Convergence rate of the method sensitive to the choice of the Robin parameter. Simulations confirm that in presence of the stent the intrasac pressure is reduced. Maximal intrasac pressure decreases as the aneurysm radius increases, which is in agreement with experimental results. The intrasac pressure is almost constant in space (not in time) with respect to the lumen pressure.

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