A 3D/1D geometrical multiscale model of cerebral vasculature

  • Tiziano Passerini
  • Mariarita de Luca
  • Luca Formaggia
  • Alfio Quarteroni
  • Alessandro Veneziani


Geometrical multiscale modeling is a strategy advocated in computational hemodynamics for representing in a single numerical model dynamics that involve different space scales. This approach is particularly useful to describe complex networks such as the circle of Willis in the cerebral vasculature. A multiscale model of the cerebral circulation is presented where a one-dimensional (1D) description of the circle of Willis, relying on the one-dimensional Euler equations, is coupled to a fully three-dimensional model of a carotid artery, based on the solution of the incompressible Navier–Stokes equations. Even if vascular compliance is often not relevant to the meaningfulness of three-dimensional (3D) results by themselves, it is crucial in the multiscale model, since it is the driving mechanism of pressure-wave propagation. Unfortunately, 3D simulations in compliant domains still demand computational costs significantly higher than in the rigid case. Appropriate matching conditions between the two models have been devised to concentrate the effects of the compliance at the interfaces and to obtain reliable results still solving a 3D problem on rigid vessels.


Circle of Willis Domain splitting Geometrical multiscale modeling Matching conditions 


  1. 1.
    Cebral JR, Castro MA, Soto O, Lohner R, Alperin N (2003) Blood-flow models of the circle of Willis from magnetic resonance data. J Eng Math 47: 369–386MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Balboni G. Anatomia Umana 1. Ediz. ErmesGoogle Scholar
  3. 3.
    Mynard JP, Nithiarasu P (2008) A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronoary flow using the locally conservative Galerkin (LCG) method. Commun Numer Methods Eng 24: 367–417MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Formaggia L, Nobile F, Quarteroni A, Veneziani A (1999) Multiscale modelling of the circulatory system: a preliminary analysis. Comput Visual Sci 2(2/3): 75–83MATHCrossRefGoogle Scholar
  5. 5.
    Quarteroni A, Formaggia L, Veneziani A (eds) (2009) Cardiovascular mathematics Springer.
  6. 6.
    Viedma A, Jimenez Ortiz C, Marco V (1997) Extended Willis circle model to explain clinical observations in periorbital arterial flow. J Biomech 30: 265–272CrossRefGoogle Scholar
  7. 7.
    Alastruey J, Moore SM, Parker KH, David T, Peirò J, Sherwin SJ (2008) Reduced modelling of blood flow in the cerebral circulation: coupling 1D, 0-D and cerebral autoregulation models. Int J Num Methods Fl 56: 1061–1067MATHCrossRefGoogle Scholar
  8. 8.
    Ferrandez A, David T, Brown MD (2002) Numerical models of auto-regulation and blood flow in the cerebral circulation. Comput Methods Biomech Biomed Eng 5(1): 7–19CrossRefGoogle Scholar
  9. 9.
    Vignon I, Figueroa CA, Jansen KC, Taylor CA (2006) Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Comput Methods Appl Mech Eng 195: 3776–3796MATHCrossRefGoogle Scholar
  10. 10.
    Moura A (2007) The geometrical multiscale modelling of the cardiovascular system: coupling 3D FSI and 1D models, Mox PhD Thesis, Department of Mathematics, Politecnico di Milano, ItalyGoogle Scholar
  11. 11.
    Quarteroni A, Veneziani A (2003) Analysis of a geometrical multiscale model based on the coupling of PDE’s and ODE’s for Blood Flow Simulations. Mult Models Sim SIAM 1(2): 173–195MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Formaggia L, Lamponi D, Quarteroni A (2003) One-dimensional models for blood flow in arteries. J Eng Math 47: 251–276MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Stergiopulos N, Westerhof BE, Westerhof N (1999) Total arterial inertance as the fourth element of the Windkessel model. Am J Physiol 276: H81–H88Google Scholar
  14. 14.
    Alastruey J, Parker KH, Peirò J, Sherwin SJ (2008) Lumped parameter outflow models for 1D blood flow simulations: effects on pulse waves and parameter estimation. Commun Comp Phys 4: 317Google Scholar
  15. 15.
    Alastruey J, Parker KH, Peirò J, Byrd SM, Sherwin SJ (2007) Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J Biomech 40: 1794–1805CrossRefGoogle Scholar
  16. 16.
    Lippert H, Pabst R (1985) Arterial variations in man: classification and frequency. JF Bergmann, MunichGoogle Scholar
  17. 17.
    Hetzel A, von Reuternc G-M, Wernza MG, Drostea DW, Schumacher M (2000) The carotid compression test for therapeutic occlusion of the internal carotid artery. Cerebrovasc Dis 10: 194–199CrossRefGoogle Scholar
  18. 18.
    Rogers L (1947) The function of the circulus arteriosus of Willis. Brain 70: 171–178CrossRefGoogle Scholar
  19. 19.
    Avman N, Bering EA (1961) A plastic model for the study of pressure changes in the circle of Willis and major cerebral arteries following arterial occlusion. J Neurosurg 18: 361–365CrossRefGoogle Scholar
  20. 20.
    Murray KD (1964) Dimensions of circle of Willis and dynamic studies using electrical analogy. J Neurosurg 21: 26CrossRefGoogle Scholar
  21. 21.
    Fasano VA, Broggi G (1966) Discussion sur le polygone de Willis. Neurochirurgie 12: 761Google Scholar
  22. 22.
    Clark ME, Himwich WA, Martin JD (1967) Simulation studies of factors influencing cerebral circulation. Acta Neurol Scand 43: 189CrossRefGoogle Scholar
  23. 23.
    Hillen B (1986) On the meaning of the variability of the circle of Willis. Acta Morphol Neerlando-Scand 24: 79–80Google Scholar
  24. 24.
    Hillen B, Drinkenburg BAH, Hoogstraten HW, Post L (1988) Analysis of flow and vascular resistance in a model of the circle of Willis. J Biomech 21: 807–814CrossRefGoogle Scholar
  25. 25.
    Lyden P, Nelson T (1997) Visualization of the cerebral circulation using three-dimensional transcranial power Doppler ultrasound imaging. J Neuroimag 7: 35–39Google Scholar
  26. 26.
    Macchi C, Catini C, Federico C et al (1996) Magnetic resonance angiographic evaluation of circulus arteriosus cerebri (circle of Willis): a morphology study in 100 human healthy subjects. Ital J Anat Embryol 101: 115–123Google Scholar
  27. 27.
    David T, Brown M, Ferrandez A (2003) Auto-regulation and blood flow in the cerebral circulation. Int J Numer Methods Fluids 43: 701–713MATHCrossRefGoogle Scholar
  28. 28.
    Matthys KS, Alastruey J, Peirò J, Khir AW, Segers P, Verdonck PR, Parker KH, Sherwin SJ (2007) Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements. J Biomech 40: 3476–3486CrossRefGoogle Scholar
  29. 29.
    Prosi M, Zunino P, Perktold K, Quarteroni A (2005) Mathematical and numerical models for transfer of low-density lipoproteins through the arterial walls: a new methodology for the model set up with applications to the study of disturbed lumenal flow. J Biomech 38(4): 903–17CrossRefGoogle Scholar
  30. 30.
    Martin V, Clement F, Decoene A, Gerbeau JF (2005) Parameter identification for a one-dimensional blood flow model. Proceedings Cemracs 2004, Esaim, September 2005Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Tiziano Passerini
    • 1
  • Mariarita de Luca
    • 1
    • 2
  • Luca Formaggia
    • 1
  • Alfio Quarteroni
    • 1
    • 3
  • Alessandro Veneziani
    • 4
  1. 1.MOX, Department of MathematicsPolitecnico di MilanoMilanoItaly
  2. 2.Department of Mechanical EngineeringUniversity of PittsburghPittsburghUSA
  3. 3.CMCS, EPFLLausanneSwitzerland
  4. 4.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

Personalised recommendations