Robust numerical approximation of coupled Stokes' and Darcy's flows applied to vascular hemodynamics and biochemical transport

D'Angelo, Carlo; Zunino, Paolo

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 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.

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[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

[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

[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

[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

[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

[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).

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl