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Stent Design Blood Flow Simulation

Kirill Pichon Gostaf

Eric Bonnetier, Didier Bresch, Vuk Milisic, Amélie Rambaud 15 September 2009

People

This work was done in collaboration with: Eric Bonnetier Prof Didier Bresch DR CNRS Vuk Milisic CR1 CNRS Amélie Rambaud ENS Lyon

Objective: Asymptotic modeling and numerical simulations of blood flows in arteries with wired multi-layer stents. Mathematical modeling of multi-layered stents inserted in the human cardio-vascular network. Investigation of realistic 3D geometries. Collaboration with an industrial partner: Cardiatis, Belgium

Kirill Gostaf

Freefem++ workshop 2009

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Aneurysm

An aneurysm is an abnormal bulge in the wall of an artery. Often, no symptoms... becomes large, begins to leak blood, ruptures.

National Heart, Lung and Blood Institute, USA.

Kirill Gostaf

Freefem++ workshop 2009

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Stenting Saccular Aneurysm

The multilayer stent design reduces flow velocity within the aneurysm, improves laminar flow in the main artery and surrounding branches, pressure and tension reduction within the aneurismal sac, Cardiatis. Frontal flow Tangential flow

Kirill Gostaf

Freefem++ workshop 2009

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Stenting Saccular Aneurysm

The multilayer stent design reduces flow velocity within the aneurysm, improves laminar flow in the main artery and surrounding branches, pressure and tension reduction within the aneurismal sac, Cardiatis. Frontal flow Tangential flow

Kirill Gostaf

Freefem++ workshop 2009

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Direct Simulation 2D: Pressure Driven Flow

Blood flow impinges directly upon the struts Stokes problem

Nnodes = 55k

-u +

p=0 u=0

in in on lf ,rt ,strut on in ,out

div u = 0 u· =0

p = pin on in p = pout on out Pressure imposed = Dirichlet velocity Direct finite element simulation to be a reference solution Same number of elements on a stent strut

Kirill Gostaf Freefem++ workshop 2009 Nnodes = 104k

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Direct Simulation 3D: Two Vessels

Stokes direct simulation mesh Catia v5 Nnodes = 280K, Ntet =1.5M P2/P1 FE discretization solver FreeFem++ CG symmetric matrix 4.8Gb

Kirill Gostaf

Freefem++ workshop 2009

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Direct Simulation 3D: Two Vessels

Stokes direct simulation mesh Catia v5 Nnodes = 280K, Ntet =1.5M P2/P1 FE discretization solver FreeFem++ CG symmetric matrix 4.8Gb

EXTREMELY EXPENSIVE!

Kirill Gostaf

Freefem++ workshop 2009

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Direct Simulation 3D: Two Vessels

Stokes direct simulation mesh Catia v5 Nnodes = 280K, Ntet =1.5M P2/P1 FE discretization solver FreeFem++ CG symmetric matrix 4.8Gb

Parallel solution EXTREMELY EXPENSIVE!

one time step: 120 hours

Homogenization theory

Kirill Gostaf

Freefem++ workshop 2009

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Homogenization

We replace struts by the homogenization wall law Bi-domain formulation

-u

,j

+

p ,j = 0

in j

div u ,j = 0 in j R u ,j on [u ,p ] · n =

Discontinuous pressure Mono-domain numerical treatment with continuous pressure on

velocity Kirill Gostaf

pressure Freefem++ workshop 2009 10

Domain Decomposition

Dirichlet-Neumann method [Quarteroni et.al.'88] o Consider an initial guess u on the interface :

(1)

k k L (u1 , p1 ) = 0 in 1 uk = g1 on 1 \ 1 k k u1 = u on k k L (u2 , p2 ) = 0 in 2 k u2 = g2 on 2 \ k [u ,p ] · n = R u k on 2

(2)

Correct the solution u until convergence, relaxation parameter :

(3)

Kirill Gostaf

k k k u +1 = (1 - ) u + u2

Freefem++ workshop 2009

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Velocity Profile: Flow Rate

0 -0.005 -0.01 Velocity Profile -0.015 -0.02 -0.025 -0.03

solid lines = Homogenized symbols = Direct solution

6 struts 10 struts 16 struts 20 struts 0.4 0.6 Artery Cross-Section 0.8 1

-0.035

0

0.2

Kirill Gostaf

Freefem++ workshop 2009

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Velocity Profile: Near the Homogenized Border

0 Direct -0.005 Homogenized (FEM) Theory Velocity Profile -0.01

-0.015

-0.02

-0.025

-0.03

0

0.2

0.4 0.6 Artery Cross-Section

0.8

1

Kirill Gostaf

Freefem++ workshop 2009

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Homogenization Error

10

0

L2 H1 3/2 10

-1

1

Error

10

-2

10

-3

10

-4

10

-2

10

-1

10

0

Kirill Gostaf

Freefem++ workshop 2009

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Aneurysm Side 3D: FreeFem++ code

Solid geometry + mesh created with CATIA v5 CATIA v5 to FreeFem++ convert tool Load mesh, create finite element spaces

mesh3 Th("aneurysm_side_h30.mesh"); fespace Xh(Th,P23d); Xh u1,v1, u2,v2, u3,v3; fespace Mh(Th,P13d); Mh p,q;

//velocity FE space

//pressure FE space

Kirill Gostaf

Freefem++ workshop 2009

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Aneurysm Side 3D: FreeFem++ code

Variational form: Stokes equation

problem Stokes([u1,u2,u3,p],[v1,v2,v3,q],solver=UMFPACK) = int3d(Th)( dx(u1)*dx(v1) + dy(u1)*dy(v1) + dz(u1)*dz(v1) + dx(u2)*dx(v2) + dy(u2)*dy(v2) + dz(u2)*dz(v2) + dx(u3)*dx(v3) + dy(u3)*dy(v3) + dz(u3)*dz(v3) - p*dx(v1) - p*dy(v2) - p*dz(v3) - dx(u1)*q - dy(u2)*q - dz(u3)*q - epsPenalty*p*q - int2d(Th,1)( pin*v1) + int2d(Th,2)(pout*v2) //p*div(w) //div(u)*q //penalization //penalization

//negative 0X //positive 0Y

+ on(1,u2=0,u3=0) //Gamma_in, u2=u3=0 + on(2,u1=0,u3=0) //Gamma_out, u1=u3=0 + on(20,u1=0,u2=0,u3=0);

Kirill Gostaf

Freefem++ workshop 2009

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Aneurysm Side 3D: Velocity Field

Kirill Gostaf

Freefem++ workshop 2009

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Aneurysm Side 3D: Pulsatile flow

Pressure solution

Kirill Gostaf

Freefem++ workshop 2009

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Aneurysm Side 3D: Pulsatile flow

Velocity magnitude

Kirill Gostaf

Freefem++ workshop 2009

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