Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 3, p. 447-476

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations, we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for the approximation of incompressible flow coupled problems. We exploit stabilized mixed finite elements together with Nitsche's type matching conditions that automatically adapt to the coupling of different combinations of coefficients. We study in details the stability of the method using weighted norms, emphasizing the robustness of the stability estimate with respect to the coefficients. We also consider an iterative method to split the coupled heterogeneous problem in possibly homogeneous local problems, and we investigate the spectral properties of suitable preconditioners for the solution of the global as well as local problems. Finally, we present the simulation of the fully coupled blood flow and plasma filtration problems on a realistic geometry of a cardiovascular artery after the implantation of a drug eluting stent (DES). A similar finite element method for mass transport is then employed to study the evolution of the drug released by the DES in the blood stream and in the arterial walls, and the role of plasma filtration on the drug deposition is investigated.

DOI : https://doi.org/10.1051/m2an/2010062
Classification:  65M60,  76D05,  76Z05,  92C50
Keywords: coupled Stokes/Darcy's problem, biological flows and mass transfer, cardiovascular applications, finite element approximation, interior penalty method, iterative splitting strategy, optimal preconditioning
@article{M2AN_2011__45_3_447_0,
     author = {D'Angelo, Carlo and Zunino, Paolo},
     title = {Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {3},
     year = {2011},
     pages = {447-476},
     doi = {10.1051/m2an/2010062},
     zbl = {1274.92010},
     mrnumber = {2804646},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_3_447_0}
}
D'Angelo, Carlo; Zunino, Paolo. Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 3, pp. 447-476. doi : 10.1051/m2an/2010062. http://www.numdam.org/item/M2AN_2011__45_3_447_0/

[1] S. Badia and R. Codina, Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47 (2009) 1971-2000. | MR 2519591 | Zbl pre05736082

[2] B. Balakrishnan, A.R. Tzafriri, P. Seifert, A. Groothuis, C. Rogers and E.R. Edelman, Strut position, blood flow, and drug deposition. Implications for single and overlapping drug-eluting stents. Circulation 111 (2005) 2958-2965.

[3] R. Balossino, F. Gervaso, F. Migliavacca and G. Dubini, Effects of different stent designs on local hemodynamics in stented arteries. J. Biom. 41 (2008) 1053-1061.

[4] J.M. Boland and R.A. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. | MR 708453 | Zbl 0521.76027

[5] J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the L2 projection in H1(Ω). Math. Comp. 71 (2002) 147-156. | MR 1862992 | Zbl 0989.65122

[6] E. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem. Numer. Meth. Partial Diff. Equ. 24 (2008) 127-143. | MR 2371351 | Zbl 1139.76029

[7] E. Burman and P. Hansbo, A unified stabilized method for Stokes' and Darcy's equations. J. Comput. Appl. Math. 198 (2007) 35-51. | MR 2250387 | Zbl 1101.76032

[8] E. Burman and P. Zunino, A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 44 (2006) 1612-1638. | MR 2257119 | Zbl 1125.65113

[9] E. Burman, M.A. Fernández and P. Hansbo, Continuous interior penalty finite element method for Oseen's equations. SIAM J. Numer. Anal. 44 (2006) 1248-1274. | MR 2231863 | Zbl pre05167773

[10] J. Cahouet and J.-P. Chabard, Some fast 3D finite element solvers for the generalized Stokes problem. Int. J. Numer. Methods Fluids 8 (1988) 869-895. | MR 953141 | Zbl 0665.76038

[11] C. Calgaro, P. Deuring and D. Jennequin, A preconditioner for generalized saddle point problems: application to 3D stationary Navier-Stokes equations. Numer. Methods Partial Diff. Equ. 22 (2006) 1289-1313. | MR 2257634 | Zbl pre05123804

[12] C. D'Angelo and A. Quarteroni, On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18 (2008) 1481-1504. | MR 2439847 | Zbl pre05360522

[13] C. D'Angelo and P. Zunino, A numerical study of the interaction of blood flow and drug release from cardiovascular stents, in Numerical Mathematics and Advanced Applications - Proceedings of ENUMATH 2007, Springer, Berlin (2008) 75-82. | Zbl 1153.92016

[14] C. D'Angelo and P. Zunino, A finite element method based on weighted interior penalties for heterogeneous incompressible flows. SIAM J. Numer. Anal. 47 (2009) 3990-4020. | MR 2576529 | Zbl pre05815164

[15] C. D'Angelo and P. Zunino, Multiscale models of drug delivery by thin implantable devices, in Applied and industrial mathematics in Italy III, Ser. Adv. Math. Appl. Sci. 82, World Sci. Publ. (2009). | MR 2856873 | Zbl pre05802062

[16] C. D'Angelo and P. Zunino, Numerical approximation with Nitsche's coupling of transient Stokes'/Darcy's flow problems applied to hemodynamics. Technical report, MOX, Department of Mathematics, Politecnico di Milano (submitted). | Zbl pre06030367

[17] M.C. Delfour, A. Garon and V. Longo, Modeling and design of coated stents to optimize the effect of the dose. SIAM J. Appl. Math. 65 (2005) 858-881. | MR 2136035 | Zbl 1075.92033

[18] M. Discacciati, A. Quarteroni and A. Valli, Robin-Robin domain decomposition methods for the Stokes Darcy coupling. SIAM J. Numer. Anal. 45 (2007) 1246-1268. | MR 2318811 | Zbl 1139.76030

[19] H.C. Elman, D.J. Silvester and A.J. Wathen, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation, Oxford University Press, New York (2005). | MR 2155549 | Zbl 1083.76001

[20] L. Formaggia, A. Quarteroni and A. Veneziani Eds., Cardiovascular mathematics - Modeling and simulation of the circulatory system, MS&A Modeling, Simulation and Applications 1. Springer-Verlag Italia, Milan (2009). | MR 2524089 | Zbl 1300.92005

[21] L. Formaggia, S. Minisini and P. Zunino, Modeling erosion controlled drug release and transport phenomena in the arterial tissue. Math. Models Methods Appl. Sci. (to appear). | MR 2735913 | Zbl 1201.92037

[22] R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press, Cambridge (1990), Corrected reprint of the 1985 original. | MR 1084815 | Zbl 0576.15001

[23] A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term. SIAM J. Sci. Comput. 19 (1998) 172-184. Special issue on iterative methods (Copper Mountain, CO, 1996). | MR 1616885 | Zbl 0917.73069

[24] A. Klawonn and G. Starke, Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis. Numer. Math. 81 (1999) 577-594. | MR 1675216 | Zbl 0922.65021

[25] V.B. Kolachalama, A.R. Tzafriri, D.Y. Arifin and E.R. Edelman, Luminal flow patterns dictate arterial drug deposition in stent-based delivery. J. Control Release 133 (2009) 24-30.

[26] W.J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195-2218. | MR 1974181 | Zbl 1037.76014

[27] S.L. Lee and C.W. Gear, Second-order accurate projective integrators for multiscale problems. J. Comput. Appl. Math. 201 (2007) 258-274. | MR 2293553 | Zbl 1110.65063

[28] D. Loghin and A.J. Wathen, Analysis of preconditioners for saddle-point problems. SIAM J. Sci. Comput. 25 (2004) 2029-2049. | MR 2086829 | Zbl 1067.65048

[29] M.A. Lovich and E.R. Edelman, Mechanisms of transmural heparin transport in the rat abdominal aorta after local vascular delivery. Circ. Res. 77 (1995) 1143-1150.

[30] F. Migliavacca, L. Petrini, M. Colombo, F. Auricchio and R. Pietrabissa, Mechanical behavior of coronary stents investigated through the finite element method. J. Biomech. 35 (2002) 803-811.

[31] L. Petrini, F. Migliavacca, F. Auricchio and G. Dubini, Numerical investigation of the intravascular coronary stent flexibility. J. Biomech. 37 (2004) 495-501.

[32] G. Pontrelli and F. De Monte, Mass diffusion through two-layer porous media: an application to the drug-eluting stent. Int. J. Heat Mass Transfer 50 (2007) 3658-3669. | Zbl 1113.80013

[33] K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17 (2007) 215-252. | MR 2292356 | Zbl 1123.76066

[34] P. Saffman, On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 292-315. | Zbl 0271.76080

[35] D.V. Sakharov, L.V. Kalachev and D.C. Rijken, Numerical simulation of local pharmacokinetics of a drug after intravascular delivery with an eluting stent. J. Drug Targ. 10 (2002) 507-513.

[36] Ch. Schwab, p- and hp-Finite Element Methods - Theory and applications in solid and fluid mechanics. Numerical Mathematics and Scientific Computation, Oxford University Press, New York (1998). | Zbl 0910.73003

[37] D. Silvester and A. Wathen, Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners. SIAM J. Numer. Anal. 31 (1994) 1352-1367. | MR 1293519 | Zbl 0810.76044

[38] J.S. Soares and P. Zunino, A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks. Biomaterials 31 (2010) 3032-3042.

[39] G. Starke, Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems. Numer. Math. 78 (1997) 103-117. | MR 1483571 | Zbl 0888.65037

[40] S. Tada and J.M. Tarbell, Internal elastic lamina affects the distribution of macromolecules in the arterial wall: a computational study. Am. J. Physiol. Heart Circ. Physiol. 287 (2004) H905-H913.

[41] A.R. Tzafriri and E.R. Edelman, On the validity of the quasi-steady state approximation of bimolecular reactions in solution. J. Theor. Biol. 233 (2005) 343-350.

[42] C. Vergara and P. Zunino, Multiscale boundary conditions for drug release from cardiovascular stents. Multiscale Model. Simul. 7 (2008) 565-588. | MR 2443003 | Zbl 1183.93040

[43] A. Wathen and D. Silvester, Fast iterative solution of stabilised Stokes systems. I. Using simple diagonal preconditioners. SIAM J. Numer. Anal. 30 (1993) 630-649. | MR 1220644 | Zbl 0776.76024

[44] P. Zunino, C. D'Angelo, L. Petrini, C. Vergara, C. Capelli and F. Migliavacca, Numerical simulation of drug eluting coronary stents: Mechanics, fluid dynamics and drug release. Comput. Methods Appl. Mech. Eng. 198 (2009) 3633-3644. | MR 2571827 | Zbl 1229.76122