Journal of Engineering Mathematics

, Volume 47, Issue 3–4, pp 251–276 | Cite as

One-dimensional models for blood flow in arteries

  • Luca Formaggia
  • Daniele Lamponi
  • Alfio Quarteroni


In this paper a family of one-dimensional nonlinear systems which model the blood pulse propagation in compliant arteries is presented and investigated. They are obtained by averaging the Navier-Stokes equation on each section of an arterial vessel and using simplified models for the vessel compliance. Different differential operators arise depending on the simplifications made on the structural model. Starting from the most basic assumption of pure elastic instantaneous equilibrium, which provides a well-known algebraic relation between intramural pressure and vessel section area, we analyse in turn the effects of terms accounting for inertia, longitudinal pre-stress and viscoelasticity. The problem of how to account for branching and possible discontinuous wall properties is addressed, the latter aspect being relevant in the presence of prosthesis and stents. To this purpose a domain decomposition approach is adopted and the conditions which ensure the stability of the coupling are provided. The numerical method here used in order to carry out several test cases for the assessment of the proposed models is based on a finite element Taylor-Galerkin scheme combined with operator splitting techniques.

blood-flow models cardiovascular system finite elements simulation 


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  1. 1.
    F.C. Hoppensteadt and C.S. Peskin, Modeling and Simulation in Medicine and the Life Sciences. New York: Springer Verlag (2001) 376 pp.Google Scholar
  2. 2.
    T.J. Pedley, The Fluid Mechanics of Large Blood Vessels. Cambridge, G.B.: Cambridge University Press (1980) 461 pp.Google Scholar
  3. 3.
    A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system. In: N. Ayache (ed.), Modelling of Living Systems. Amsterdam: Elsevier (2003) to appear.Google Scholar
  4. 4.
    L. Formaggia, F. Nobile, A. Quarteroni, and A. Veneziani, Multiscale modelling of the circulatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75–83.Google Scholar
  5. 5.
    V.L. Streeter, W.F. Keitzer and D.F. Bohr, Pulsatile pressure and flow through distensible vessels. Circulation Res. 13 (1963) 3–20.Google Scholar
  6. 6.
    F. Phythoud, N. Stergiopulos and J.-J. Meister, Forward and backward waves in the arterial system: nonlinear separation using Riemann invariants. Technol. Health Care 3 (1995) 201–207.Google Scholar
  7. 7.
    L. Formaggia, F. Nobile and A. Quarteroni, A one-dimensional model for blood flow: application to vascular prosthesis. In: I. Babuska, T. Miyoshi and P.G. Ciarlet (eds), Mathematical Modeling and Numerical Simulation in Continuum Mechanics. Volume 19 of Lecture Notes in Computational Science and Engineering. Berlin: Springer-Verlag (2002) pp. 137–153.Google Scholar
  8. 8.
    S. Čani?, Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties. Comput. Visual. Sci. 4 (2002) 147–155.Google Scholar
  9. 9.
    G. Pontrelli, A mathematical model of flow through a viscoelastic tube. Med. Biol. Eng. Comput. 40 (2002) 550–556.Google Scholar
  10. 10.
    G. Pontrelli. Nonlinear pulse propagation in blood flow problems. In: M. Anile, V. Capasso and A. Greco (eds.), Progress in Industrial Mathematics at ECMI 2000, Berlin: Springer-Verlag (2002) pp. 201–207.Google Scholar
  11. 11.
    M. Olufsen, Modeling the Arterial System with Reference to an Anestesia Simulator. PhD thesis, Roskilde Univ. (1998) Tekst 345.Google Scholar
  12. 12.
    S.J. Sherwin, L. Formaggia, J. Peiró and V. Franke, Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Int. J. Numer. Meth. Fluids (2002) to appear.Google Scholar
  13. 13.
    M.S. Olufsen, C.S. Peskin, W.Y. Kim, E.M. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Annals Biomed. Engng. 28 (2000) 1281–1299.Google Scholar
  14. 14.
    A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: Problems, models and methods. Comput. Visual. Sci. 2 (2000) 163–197.Google Scholar
  15. 15.
    V. Rideout and D. Dick, Difference-differential equations for fluid flow in distensible tubes. IEEE Trans. Biomed. Engng. 14 (1967) 171–177.Google Scholar
  16. 16.
    N. Westerhof, F. Bosman, C. Vries and A. Noordergraaf, Analog studies of the human systemic arterial tree. J. Biomech. 2 (1969) 121–143.Google Scholar
  17. 17.
    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. Comp. Methods Appl. Mech. Engng. 191 (2001) 561–582.Google Scholar
  18. 18.
    L. Formaggia, F. Nobile, J.-F. Gerbeau and A. Quarteroni, Numerical treatment of defective boundary conditions for the Navier-Stokes equations. SIAM J. Num. Anal. 40 (2002) 376–401.Google Scholar
  19. 19.
    N.P. Smith, A.J. Pullan and P.J. Hunter, An anatomically based model of coronary blood flow and myocardial mechanics. SIAM J. Appl. Math. 62 (2002) 990–1018.Google Scholar
  20. 20.
    J. Donea, S. Giuliani, H. Laval and L. Quartapelle, Time-accurate solutions of advection-diffusion problems by finite elements. Comp. Meth. Appl. Mech. Engng. 45 (1984) 123–145.Google Scholar
  21. 21.
    D. Ambrosi and L. Quartapelle, A Taylor-Galerkin method for simulating nonlinear dispersive water waves. J. Comp. Phys. 146 (1998) 546–569.Google Scholar
  22. 22.
    L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations. Basel: Birkhäuser Verlag (1993) 191 pp.Google Scholar
  23. 23.
    A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Berlin: Springer-Verlag (1994) 544 pp.Google Scholar
  24. 24.
    F. Dubois and P. Le Floch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Diff. Eq. 71 (1988) 93–122.Google Scholar
  25. 25.
    E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Volume 118 of Applied Mathematical Sciences. New York: Springer (1996) 509 pp.Google Scholar
  26. 26.
    K.W. Thompson, Time dependent boundary conditions for hyperbolic systems. J. Comp. Phys. 68 (1987) 1–24.Google Scholar
  27. 27.
    K. Boukir, Y. Maday and B. Métivet, A high order characteristics method for the incompressible Navier-Stokes equations. Comp. Methods Appl. Mech. Engng. 116 (1994) 211–218.Google Scholar
  28. 28.
    A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford/New York: The Clarendon Press (1999) 360 pp. Oxford Science Publications.Google Scholar
  29. 29.
    F.T. Smith, N.C. Ovenden, P.T. Franke and D.J. Doorly, What happens to pressure when a flow enters a side branch? J. Fluid Mech. 479 (2003) 231–258.Google Scholar
  30. 30.
    J.C. Stettler, P. Niederer and M. Anliker, Theoretical analysis of arterial hemodynamics including the influence of bifurcations, part i: Mathematical model and prediction of normal pulse patterns. Annals Biomed. Engng. 9 (1981) 145–164.Google Scholar
  31. 31.
    H. Holden and N.H. Risebro, Riemann problems with a kink. SIAM J. Math. Anal. 30 (1999) 497–515.Google Scholar
  32. 32.
    G.A. Holzapfel, T.C. Gasser and R.W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity 61 (2000) 1–48.Google Scholar
  33. 33.
    Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues. New York: Springer-Verlag (1993) 568 pp.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Luca Formaggia
    • 1
  • Daniele Lamponi
    • 2
  • Alfio Quarteroni
    • 2
    • 1
  1. 1.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Institut de Mathématiques (FSB/IMA)Ecole Polytecnique Fédérale de LausanneLausanneSwitzerland

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