Blood-flow models of the circle of Willis from magnetic resonance data


Detailed knowledge of the cerebral hemodynamics is important for a variety of clinical applications. Cerebral perfusion depends not only on the status of the diseased vessels but also on the patency of collateral pathways provided by the circle of Willis. Due to the large anatomical and physiologic variability among individuals, realistic patient-specific models can provide new insights into the cerebral hemodynamics. This paper presents an image-based methodology for constructing patient-specific models of the cerebral circulation. This methodology combines anatomical and physiologic imaging techniques with computer simulation technology. The methodology is illustrated with a finite element model constructed from magnetic resonance image data of a normal volunteer. Several of the remaining challenging problems are identified. This work represents a starting point in the development of realistic models that can be applied to the study of cerebrovascular diseases and their treatment.

This is a preview of subscription content, access via your institution.


  1. 1.

    G. W. Petty, R. D. Brown, J. P. Whisnant, J.D. Sicks, W. M. O'Fallon and D.O. Wiebers, Ischemic stroke subtypes; a population-based study of functional outcome, survival, and recurrence. Stroke 31 (2000) 1062–1068.

    Google Scholar 

  2. 2.

    M. Kluytmans, J. van der Grond, K.J. van Everdingen, C. J. M. Klijn, L.J. Kappelle and M.A. Viergever, Cerebral Hemodynamics in Relation to Patterns of Collateral Flow. Stroke 30 (1999) 1432–1439.

    Google Scholar 

  3. 3.

    R. M. K. W. Lee, Morphology of cerebral arteries. Pharmac. Ther. 66 (1995) 149–173.

    Google Scholar 

  4. 4.

    R. H. Kufahl and M. E. Clark, A Circle of Willis Simulation Using Distensible Vessels and Pulsatile Flow. J. Biomech. Engng. 107 (1985) 112–22.

    Google Scholar 

  5. 5.

    J. R. Cebral, R. Löhner and J. E. Burgess, Computer Simulation of Cerebral Artery Clipping: Relevance to Aneurysm Neuro-Surgery Planning. In: E. Oñate, G. Bugeda and B. Suarez (eds.), Proc. ECCOMAS, Barcelona, Spain, Sept. 11–14 (2000).

  6. 6.

    A. Fernandez, T. David and M. D. Brown, Numerical models of autoregulation and blood flow in the cerebral circulation. Comp. Meth. Biomech. Engng. 5 (2002) 7–20.

    Google Scholar 

  7. 7.

    F. T. Charbel, K.H. Guppy, M. Zhao and M.E. Clark, Computerized hemodynamic evaluation of the cerebral circulation for bypass. Neurosurg. Clin. N. Am. 12 (2001) 499–508.

    Google Scholar 

  8. 8.

    M. L. Barr and J. A. Kierman, The Human Nervous System: an Anatomical Viewpoint. Philadelphia: Lippincott Co. (1993), 451 pp.

    Google Scholar 

  9. 9.

    J. A. Moore, D. A. Steinman, D. WHoldsworth and C. R. Ethier, Accuracy of computational hemodynamics in complex arterial geometries reconstructed from magnetic resonance imaging. Ann. Biomed. Engng. 27 (1999) 32–41.

    Google Scholar 

  10. 10.

    C. A. Taylor, M. T. Draney, J. P. Ku, D. Parker, B. N. Steele, K. Wang and C.K. Zarins, Predictive medicine: vomputational techniques in therapeutic decision-making. Comp. Assisted Surgery 4 (1999) 231–247.

    Google Scholar 

  11. 11.

    S. Z. Zhao, X. Y. Xu, A. D. Hughes, S. A. Thom, A. V. Stanton, B. Ariff and Q. Long, Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation. J. Biomech. 33 (2000) 975–984.

    Google Scholar 

  12. 12.

    J. R. Cebral, P. J. Yim, R. Löhner, O. Soto and P. L. Choyke, Blood flow modeling in carotid arteries using computational fluid dynamics and magnetic resonance imaging. Acad. Radiol. 9 (2002) 1286–1299.

    Google Scholar 

  13. 13.

    P. J. Yim, J. R. Cebral, R. Mullick and P. J. Choyke, Vessel surface reconstruction with a tubular deformable model. IEEE Trans. Med. Imag. 20 (2001) 1411–1421.

    Google Scholar 

  14. 14.

    N. J. Pelc, Flow quantification and analysis methods. MRI Clin. N. Am. 3 (1995) 413–424.

    Google Scholar 

  15. 15.

    C. J. G. Bakker, R.M. Hoogeveen and M. A. Viergever, Construction of a protocol for measuring blood flow by two-dimensional phase-contrast MRA. J. Magn. Reson. Imaging 9 (1999) 119–217.

    Google Scholar 

  16. 16.

    B. R. Mustert, D. M. Williams and M. R. Prince, In vitro model of arterial stenosis: correlation of MR signal dephasing and trans-stenotic pressure gradients. J. Magn. Reson. Imaging 16 (1998) 301–310.

    Google Scholar 

  17. 17.

    S. J. Owen, A survey of unstructured mesh generation technology. Proc. 7th International Meshing Roundtable, Sandia National Lab. (1998) 239–267 (

  18. 18.

    R. Löhner, Automatic unstructured grid generators. Finite Elem. Anal. Design 25 (1997) 111–134.

    Google Scholar 

  19. 19.

    J. R. Cebral and R. Löhner, From medical images to anatomically accurate finite element grids. Int. J. Num. Meth. Engng. 51 (2001) 985–1008.

    Google Scholar 

  20. 20.

    J. R. Cebral, Realistic modeling of arterial hemodynamics from anatomic and physiologic image data. In: R. C. Batra and E.G. Henneke (eds.), Proc. 14th U.S. National Congress in Theoretical and Applied Mechanics, Blacksburg, Virginia, June 23–28 (2002) p. 79.

  21. 21.

    J. R. Cebral, R. Löhner, P. L. Choyke and P. J. Yim, Merging of intersecting triangulations for finite element modeling. J. Biomech. 34 (2001) 815–819.

    Google Scholar 

  22. 22.

    P. J. Yim, J. R. Cebral, R. Löhner, V. B. Ho and P. L. Choyke, Estimation of mechanical stress on the carotid artery. Biomed. Engng Annual Conf., Durham (N.C.), Oct 4–7 (2001).

  23. 23.

    G. Taubin, A signal processing approach to fair surface design. Comp. Graphics Proc. (1995) 351–358.

  24. 24.

    R. Löhner, Regridding surface triangulations. J. Comp. Phys. 126 (1996) 1–10.

    Google Scholar 

  25. 25.

    J. Mazumdar. Biofluid Mechanics. Singapore: World Scientific (1992) 191 pp.

    Google Scholar 

  26. 26.

    F. G. Basombrio, E. A. Dari, G. C. Buscaglia and R. A. Feijoo, Numerical experiments in complex haemodynamics flows. Non-Newtonian effects. Proc. XI Congress on Numerical Methods and Their Applications ENIEF, San Carlos de Bariloche, Argentina, Nov. 24–29 (2000).

  27. 27.

    K. Perktold. and G. Rappitsch, Mathematical modeling of local arterial flow and vessel mechanics. In: J. Crolet and R. Ohayon (eds.), Computational Methods for Fluid-Structure Interaction. Pitman Research Notes in Mathematics 306 (1994) pp. 230–245.

  28. 28.

    P. Neofitou and D. Drikakis, Non-Newtonian modeling effects on stenotic channel flows. In: E. Oñate, G. Bugeda and B. Suarez (eds.), Proc. ECCOMAS CFD, Swansea, Wales, U.K., Sept. 4–7 (2001).

  29. 29.

    O. Soto, R. Löhner, J. R. Cebral and R. Codina, A time-accurate implicit monolithic finite element scheme for incompressible flow problems. In: E. Oñate, G. Bugeda and B. Suarez (eds.), Proc. ECCOMAS CFD, Swansea, UK, Sept. 4–7 (2001).

  30. 30.

    Y. Saad, Iterative Methods for Sparse Linear Systems. Boston: PWS Pub. Co. (1996) 447 pp.

    Google Scholar 

  31. 31.

    J. R. Cebral, R. Löhner, O. Soto, P. L. Choyke and P. J. Yim, Image-based finite element modeling of hemodynamics in stenosed carotid artery. Proc. SPIE Medical Imaging 4683 (2002) 297–304.

    Google Scholar 

  32. 32.

    J. R. Cebral, P. J. Yim, R. Löhner, O. Soto, H. Marcos and P. L. Choyke, New methods for computational fluid dynamics of carotid artery from magnetic resonance angiography. Proc. SPIE Medical Imaging 4321 (2001) 177–187.

    Google Scholar 

  33. 33.

    J. S. Stroud, S. A. Berger and D. Saloner, Numerical analysis of flow through a severely stenotic artery bifurcation. J. Biomech. Engng. 124 (2002) 9–20.

    Google Scholar 

  34. 34.

    N. Alperin, S. Lee, PUBS: Pulsatile-based segmentation of lumens conductiong non-steady flow. Magn. Reson. Med. 49 (2003) 934–944.

    Google Scholar 

  35. 35.

    J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127 (1955) 553–563.

    Google Scholar 

  36. 36.

    C. A. Taylor, T. J. R. Hughes and C. K. Zarins, Finite element modeling of blood flow in arteries. Comput. Methods Appl. Mech. Engng. 158 (1998) 155–196.

    Google Scholar 

  37. 37.

    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. Ann. Biomed. Engng. 28 (2002) 1281–1299.

    Google Scholar 

  38. 38.

    L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modeling of the circulatory system: a preliminary analysis. Visual. Sci. 2 (1999) 75–83.

    Google Scholar 

  39. 39.

    R. Karch, F. Neumann, M. Neumann and W. Schreiner, Staged growth of optimized arterial tree models. Ann. Biomed. Engng. 28 (2000) 495–511.

    Google Scholar 

  40. 40.

    M. Zamir, On fractal properties of arterial trees. J. Theor. Biol. 197 (1999) 517–526.

    Google Scholar 

  41. 41.

    F. Calamante, P. J. Yim and J. R. Cebral, Estimation of bolus dispersion effects in perfusion MRI using image-based computational fluid dynamics. NeuroImage 19 (2003) 342–353.

    Google Scholar 

  42. 42.

    L. R. Caplan. Stroke: a clinical approach. Boston: Butterworths (1986) 343 pp..

    Google Scholar 

  43. 43.

    J. S. Milner, J. A. Moore, B. K. Rutt and D. A. Steinman, Hemodynamics of human carotid artery bifurcations: computational studies in models reconstructed from magnetic resonance imaging of normal subjects. J. Vasc. Surg. 28 (1998) 143–156.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cebral, J.R., Castro, M.A., Soto, O. et al. Blood-flow models of the circle of Willis from magnetic resonance data. Journal of Engineering Mathematics 47, 369–386 (2003).

Download citation

  • circle of Willis
  • computational fluid dynamics
  • hemodynamics
  • magnetic resonance