Numerical simulation of blood flows through a porous interface
ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 6, pp. 961-990.

We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.

DOI: 10.1051/m2an:2008031
Classification: 65M60, 74K25, 76D05, 76Z05
Keywords: stabilized finite element, sieve problem, blood flow, terminal aneurysm, stent, fluid-structure interaction
Fernández, Miguel A. ; Gerbeau, Jean-Frédéric ; Martin, Vincent 1

1 University of Technology of Compiègne, LMAC, GI, Royallieu, BP 20529, 60205 Compiègne, France.
@article{M2AN_2008__42_6_961_0,
     author = {Fern\'andez, Miguel A. and Gerbeau, Jean-Fr\'ed\'eric and Martin, Vincent},
     title = {Numerical simulation of blood flows through a porous interface},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {961--990},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {6},
     year = {2008},
     doi = {10.1051/m2an:2008031},
     mrnumber = {2473316},
     zbl = {1148.92017},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2008031/}
}
TY  - JOUR
AU  - Fernández, Miguel A.
AU  - Gerbeau, Jean-Frédéric
AU  - Martin, Vincent
TI  - Numerical simulation of blood flows through a porous interface
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 961
EP  - 990
VL  - 42
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2008031/
DO  - 10.1051/m2an:2008031
LA  - en
ID  - M2AN_2008__42_6_961_0
ER  - 
%0 Journal Article
%A Fernández, Miguel A.
%A Gerbeau, Jean-Frédéric
%A Martin, Vincent
%T Numerical simulation of blood flows through a porous interface
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 961-990
%V 42
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2008031/
%R 10.1051/m2an:2008031
%G en
%F M2AN_2008__42_6_961_0
Fernández, Miguel A.; Gerbeau, Jean-Frédéric; Martin, Vincent. Numerical simulation of blood flows through a porous interface. ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 6, pp. 961-990. doi : 10.1051/m2an:2008031. http://www.numdam.org/articles/10.1051/m2an:2008031/

[1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Rational Mech. Anal. 113 (1991) 261-298. | MR | Zbl

[2] J.L. Berry, A. Santamarina, J.E. Jr. Moore, S. Roychowdhury and W.D. Routh, Experimental and computational flow evaluation of coronary stents. Ann. Biomed. Eng. 28 (2000) 386-398.

[3] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin (1991). | MR | Zbl

[4] A. Brillard, Asymptotic flow of a viscous and incompressible fluid through a plane sieve, in Progress in partial differential equations: calculus of variations, applications (Pont-à-Mousson, 1991), Pitman Res. Notes Math. Ser. 267, Longman Sci. Tech., Harlow (1992) 158-172. | MR | Zbl

[5] E. Burman and P. Hansbo, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg. 195 (2006) 2393-2410. | MR | Zbl

[6] D. Chapelle and K.J. Bathe, The finite element analysis of shell - Fundamentals. Springer-Verlag (2004). | MR | Zbl

[7] P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. | MR | Zbl

[8] P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9(R-2) (1975) 77-84. | Numdam | MR | Zbl

[9] R. Codina and J. Blasco, A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput. Methods Appl. Mech. Engrg. 143 (1997) 373-391. | MR | Zbl

[10] C. Conca, Étude d'un fluide traversant une paroi perforée, I. Comportement limite près de la paroi. J. Math. Pures Appl. 66 (1987) 1-43. | MR | Zbl

[11] C. Conca, Étude d'un fluide traversant une paroi perforée, II. Comportement limite loin de la paroi. J. Math. Pures Appl. 66 (1987) 45-70. | MR | Zbl

[12] C. Conca and M. Sepúlveda, Numerical results in the Stokes sieve problem. Rev. Internac. Métod. Numér. Cálc. Diseñ. Ingr. 5 (1989) 435-452. | MR

[13] A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York (2004). | MR | Zbl

[14] L. Formaggia, J.-F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Engrg. 191 (2001) 561-582. | MR | Zbl

[15] P. Frey, Yams: A fully automatic adaptive isotropic surface remeshing procedure. Technical report 0252, INRIA, Rocquencourt, France, Nov. (2001).

[16] P. Frey, Medit: An interactive mesh visualisation software. Technical report 0253, INRIA, Rocquencourt, France, Dec. (2001).

[17] J.-F. Gerbeau and M. Vidrascu, A quasi-Newton algorithm based on a reduced model for fluid structure problems in blood flows. ESAIM: M2AN 37 (2003) 631-647. | Numdam | MR | Zbl

[18] J.-F. Gerbeau, M. Vidrascu and P. Frey, Fluid-structure interaction in blood flows on geometries coming from medical imaging. Comput. Struct. 83 (2005) 155-165.

[19] V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations - Theory and algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986). | MR | Zbl

[20] T.J.R. Hughes, L.P. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comp. Meth. App. Mech. Eng. 59 (1986) 85-99. | MR | Zbl

[21] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, in Handbook of Numerical Analysis XII, North-Holland, Amsterdam (2004) 3-127. | MR

[22] S. Salmon, M. Thiriet and J.-F. Gerbeau, Medical image-based computational model of pulsatile flow in saccular aneurisms. ESAIM: M2AN 37 (2003) 663-679. | Numdam | MR | Zbl

[23] E. Sánchez-Palencia, Problèmes mathématiques liés à l'écoulement d'un fluide visqueux à travers une grille, in Ennio De Giorgi colloquium (Paris, 1983), Res. Notes in Math. 125, Pitman, Boston, USA (1985) 126-138. | Zbl

[24] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190) (1990) 483-493. | MR | Zbl

[25] D.A. Steinman, J.S. Milner, C.J. Norley, S.P. Lownie and D.W. Holdsworth, Image-based computational simulation of flow dynamics int a giant intracranial aneurysms. Am. J. Neuroradiol. 24 (2003) 559-566.

[26] G.R. Stuhne and D.A. Steinman, Finite-element modeling of the hemodynamics of stented aneurysms. J. Biomech. Eng. 126 (2004) 382-387.

[27] V. Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics 25. Springer-Verlag, Berlin, second edition (2006). | MR | Zbl

[28] L. Tobiska and V. Verfurth, Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J. Numer. Anal. 33 (1996) 107-127. | MR | Zbl

[29] I.E. Vignon-Clementel, C.A. Figueroa, K.E. Jansen and C.A. Taylor, Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3776-3796. | MR

[30] N.T. Wang and A.L. Fogelson, Computational methods for continuum models of platelet aggregation. J. Comput. Phys. 151 (1999) 649-675. | MR | Zbl

Cited by Sources: