Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-16T20:46:12.977Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  11 October 2010

Carlo D'Angelo  and
Paolo Zunino
Carlo D'Angelo
Affiliation:
MOX – Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. carlo.dangelo@polimi.it; paolo.zunino@polimi.it
Paolo Zunino
Affiliation:
MOX – Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. carlo.dangelo@polimi.it; paolo.zunino@polimi.it
Get access

Abstract

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.

Keywords

Coupled Stokes/Darcy's problembiological flows and mass transfercardiovascular applicationsfinite element approximationinterior penalty methoditerative splitting strategyoptimal preconditioning
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 3 , May 2011 , pp. 447 - 476
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Badia, S. and Codina, R., Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47 (2009) 1971–2000. CrossRef
Balakrishnan, B., Tzafriri, A.R., Seifert, P., Groothuis, A., Rogers, C. and Edelman, E.R., Strut position, blood flow, and drug deposition. Implications for single and overlapping drug-eluting stents. Circulation 111 (2005) 2958–2965. CrossRef
Balossino, R., Gervaso, F., Migliavacca, F. and Dubini, G., Effects of different stent designs on local hemodynamics in stented arteries. J. Biom. 41 (2008) 1053–1061. CrossRef
Boland, J.M. and Nicolaides, R.A., Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722–731. CrossRef
Bramble, J.H., Pasciak, J.E. and Steinbach, O., On the stability of the L 2 projection in H 1(Ω). Math. Comp. 71 (2002) 147–156. CrossRef
Burman, E., Pressure projection stabilizations for Galerkin approximations of Stokes' and Darcy's problem. Numer. Meth. Partial Diff. Equ. 24 (2008) 127–143. CrossRef
Burman, E. and Hansbo, P., A unified stabilized method for Stokes' and Darcy's equations. J. Comput. Appl. Math. 198 (2007) 35–51. CrossRef
Burman, E. and Zunino, P., A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 44 (2006) 1612–1638. CrossRef
Burman, E., Fernández, M.A. and Hansbo, P., Continuous interior penalty finite element method for Oseen's equations. SIAM J. Numer. Anal. 44 (2006) 1248–1274. CrossRef
Cahouet, J. and Chabard, J.-P., Some fast 3D finite element solvers for the generalized Stokes problem. Int. J. Numer. Methods Fluids 8 (1988) 869–895. CrossRef
Calgaro, C., Deuring, P. and Jennequin, D., A preconditioner for generalized saddle point problems: application to 3D stationary Navier-Stokes equations. Numer. Methods Partial Diff. Equ. 22 (2006) 1289–1313. CrossRef
D'Angelo, C. and Quarteroni, A., On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci. 18 (2008) 1481–1504. CrossRef
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.
D'Angelo, C. and Zunino, P., A finite element method based on weighted interior penalties for heterogeneous incompressible flows. SIAM J. Numer. Anal. 47 (2009) 3990–4020. CrossRef
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).
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).
Delfour, M.C., Garon, A. and Longo, V., Modeling and design of coated stents to optimize the effect of the dose. SIAM J. Appl. Math. 65 (2005) 858–881. CrossRef
Discacciati, M., Quarteroni, A. and Valli, A., Robin-Robin domain decomposition methods for the Stokes Darcy coupling. SIAM J. Numer. Anal. 45 (2007) 1246–1268. CrossRef
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).
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).
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).
R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press, Cambridge (1990), Corrected reprint of the 1985 original.
Klawonn, A., 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). CrossRef
Klawonn, A. and Starke, G., Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis. Numer. Math. 81 (1999) 577–594. CrossRef
Kolachalama, V.B., Tzafriri, A.R., Arifin, D.Y. and Edelman, E.R., Luminal flow patterns dictate arterial drug deposition in stent-based delivery. J. Control Release 133 (2009) 24–30. CrossRef
Layton, W.J., Schieweck, F. and Yotov, I., Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195–2218. CrossRef
Lee, S.L. and Gear, C.W., Second-order accurate projective integrators for multiscale problems. J. Comput. Appl. Math. 201 (2007) 258–274. CrossRef
Loghin, D. and Wathen, A.J., Analysis of preconditioners for saddle-point problems. SIAM J. Sci. Comput. 25 (2004) 2029–2049. CrossRef
Lovich, M.A. and Edelman, E.R., Mechanisms of transmural heparin transport in the rat abdominal aorta after local vascular delivery. Circ. Res. 77 (1995) 1143–1150. CrossRef
Migliavacca, F., Petrini, L., Colombo, M., Auricchio, F. and Pietrabissa, R., Mechanical behavior of coronary stents investigated through the finite element method. J. Biomech. 35 (2002) 803–811. CrossRef
Petrini, L., Migliavacca, F., Auricchio, F. and Dubini, G., Numerical investigation of the intravascular coronary stent flexibility. J. Biomech. 37 (2004) 495–501. CrossRef
Pontrelli, G. and de Monte, F., Mass diffusion through two-layer porous media: an application to the drug-eluting stent. Int. J. Heat Mass Transfer 50 (2007) 3658–3669. CrossRef
Rajagopal, K.R., On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17 (2007) 215–252. CrossRef
Saffman, P., On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 292–315.
Sakharov, D.V., Kalachev, L.V. and Rijken, D.C., Numerical simulation of local pharmacokinetics of a drug after intravascular delivery with an eluting stent. J. Drug Targ. 10 (2002) 507–513. CrossRef
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).
Silvester, D. and Wathen, A., Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners. SIAM J. Numer. Anal. 31 (1994) 1352–1367. CrossRef
Soares, J.S. and Zunino, P., A mixture model for water uptake, degradation, erosion and drug release from polydisperse polymeric networks. Biomaterials 31 (2010) 3032–3042. CrossRef
Starke, G., Field-of-values analysis of preconditioned iterative methods for nonsymmetric elliptic problems. Numer. Math. 78 (1997) 103–117. CrossRef
Tada, S. and Tarbell, J.M., 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. CrossRef
Tzafriri, A.R. and Edelman, E.R., On the validity of the quasi-steady state approximation of bimolecular reactions in solution. J. Theor. Biol. 233 (2005) 343–350. CrossRef
Vergara, C. and Zunino, P., Multiscale boundary conditions for drug release from cardiovascular stents. Multiscale Model. Simul. 7 (2008) 565–588. CrossRef
Wathen, A. and Silvester, D., Fast iterative solution of stabilised Stokes systems. I. Using simple diagonal preconditioners. SIAM J. Numer. Anal. 30 (1993) 630–649. CrossRef
Zunino, P., D'Angelo, C., Petrini, L., Vergara, C., Capelli, C. and Migliavacca, F., Numerical simulation of drug eluting coronary stents: Mechanics, fluid dynamics and drug release. Comput. Methods Appl. Mech. Eng. 198 (2009) 3633–3644. CrossRef