Towards a Geometrical Multiscale Approach to Non-Newtonian Blood Flow Simulations

  • João Janela
  • Alexandra Moura
  • Adélia SequeiraEmail author


In this paper we address some problems that arise when modelling the human cardiovascular system. On one hand, blood is a complex fluid and in many situations Newtonian models may not be capable of capturing important aspects of blood rheology, for example its shear-thinning viscosity, viscoelasticity or yield stress. On the other hand, the geometric complexity of the cardiovascular system does not permit the use of full three-dimensional (3D) models in large regions. We deal with these problems by using a relatively simple non-Newtonian model capturing the shear-thinning behaviour of blood in a confined region of interest, and coupling it with a zero dimensional (0D) model (also called lumped parameters model) accounting for the remaining circulatory system. More specifically, the 0D system emulates the global circulation, providing proper boundary conditions to the 3D model.


Blood rheology Geometrical multiscale approach Lumped parameters models Numerical coupling strategies 


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  1. 1.
    Burman, E., Fernández, M.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numerische Mathematik 107(1), 39–77 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Fernández, M., Moura, A., Vergara, C.: Defective boundary conditions applied to multiscale analysis of blood flow. In: E. Cancès, J.F. Gerbeau (eds.) CEMRACS 2004 – Mathematics and applications to biology and medicine, Marseille, France, July 26 – September 3, 2004, vol. 14, pp. 89–100. ESAIM: Proceedings (2005)Google Scholar
  3. 3.
    Formaggia, L., Veneziani, A.: Reduced and multiscale models for the human cardiovascular system. Lecture notes VKI Lecture Series 2003–07, Brussels (2003)Google Scholar
  4. 4.
    Heywood, J., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Num. Meth. Fluids 22, 325–352 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Janela, J.: Mathematical and Numerical Modelling in Hemodynamics and Hemorheology. Ph.D. thesis, Instituto Superior Técnico (2008)Google Scholar
  6. 6.
    Janela, J., Sequeira, A.: High accuracy semi-analytical solutions for generalized Newtonian flows. In: Proc. of the Conf. on Topical problems in Fluid Mech., Inst. Thermomechanics, Prague (2007)Google Scholar
  7. 7.
    Kim, S., Cho, Y.I., Jeon, A.H., Hogenauer, B., Kensey, K.R.: A new method for blood viscosity measurements. J. non-Newtonian Fluid Mech. 94, 47–56 (2000)zbMATHCrossRefGoogle Scholar
  8. 8.
    Laganà, K., Dubini, G., Migliavacca, F., Pietrabissa, R., Pennati, G., Veneziani, A., Quarteroni, A.: Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures. Biorheology 39, 359–364 (2002)Google Scholar
  9. 9.
    Moura, A.: The geometrical multiscale modelling of the cardiovascular system: coupling 3D and 1D models. Ph.D. thesis, Politecnico di Milano (2007)Google Scholar
  10. 10.
    Quarteroni, A., Formaggia., L.: Handbook of Numerical Analysis, vol. XII, chap. Mathematical modelling and numerical simulation of the cardiovascular system. Elsevier, Amsterdam (2002)Google Scholar
  11. 11.
    Quarteroni, A., Ragni, S., Veneziani, A.: Coupling between lumped and distributed models for blood flow problems. Comput. Vis. Sci. 4, 111–124 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Robertson, A., Sequeira, A., Kameneva, M.: Hemorheology. In: G.P. Galdi, R. Rannacher, A. Robertson, S. Turek (eds.) Haemodynamical Flows: Modelling Analysis and Simulation, Oberwolfach Seminars, vol. 37, pp. 63–120. Birkhauser Basel, Switzerland (2008)Google Scholar
  13. 13.
    Robertson, A., Sequeira, A., Owens, R.G.: Rheological models for blood. In: A. Quarteroni, L. Formaggia, A. Veneziani (eds.) Cardiovascular Mathematics: Modelling and simulation of the cardiovascular system, pp. 211–241. Springer-Verlag, Italia (2009)Google Scholar
  14. 14.
    Steinman, D., Vorp, D., Ethier, C.: Computational modeling of the arterial biomechanics: insights into pathogenesis and treatment of vascular disease. J. Vasc. Surg. 37, 1118–1128 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • João Janela
    • 1
  • Alexandra Moura
    • 2
  • Adélia Sequeira
    • 3
    Email author
  1. 1.Department of Mathematics/ISEG and CEMAT/ISTLisboaPortugal
  2. 2.CEMAT/ISTLisboaPortugal
  3. 3.Department of Mathematics and CEMAT/ISTLisboaPortugal

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