Modeling Elastic Vessels with the LBGK Method in Three Dimensions

  • Daniel Leitner
  • Siegfried Wassertheurer
  • Michael Hessinger
  • Andreas Holzinger
  • Felix Breitenecker
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4799)


The Lattice Bhatnagar Gross and Krook (LBGK) method is widely used to solve fluid mechanical problems in engineering applications. In this work a brief introduction of the LBGK method is given and a new boundary condition is proposed for the cardiovascular domain. This enables the method to support elastic walls in two and three spatial dimensions for simulating blood flow in elastic vessels. The method is designed to be used on geometric data obtained from magnetic resonance angiography without the need of generating parameterized surfaces. The flow field is calculated in an arbitrary geometry revealing characteristic flow patterns and geometrical changes of the arterial walls for different time dependent input contours of pressure and flow. For steady flow the results are compared to the predictions of the model proposed by Y. C. Fung which is an extension of Poiseuille’s theory. The results are very promising for relevant Reynolds and Womersley numbers, consequently very useful in medical simulation applications.


Simulation Lattice Boltzmann Model Haemodynamics Elasticity Computer Fluid Dynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Suo, J., Ferrara, D.E., Sorescu, D., Guldberg, R.E., Taylor, W.R., Giddens, D.P.: Hemodynamic shear stresses in mouse aortas: Implications for atherogenesis. Thromb. Vasc. Biol. 27, 346–351 (2007)CrossRefGoogle Scholar
  2. 2.
    Wolf-Gladrow, D.A.: Lattice-Gas Cellular Automata and Lattice Boltzmann Models- An Introduction. Lecture Notes in Mathematics. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  3. 3.
    Artoli, A.M.M., Hoekstra, A.G., Sloot, P.M.A.: Mesoscopic simulations of systolic flow in the human abdominal aorta. Journal of Biomechanics 39(5), 873–884 (2006)CrossRefGoogle Scholar
  4. 4.
    Fang, H., Wang, Z., Lin, Z., Liu, M.: Lattice boltzmann method for simulating he viscous flow in large distensible blood vessels. Phys. Rev. E. (2001)Google Scholar
  5. 5.
    Leitner, S., Wasssertheurer, M., Hessinger, M., Holzinger, A.: Lattice Boltzmann model for pulsative blood flow in elastic vessels. Springer Elektronik und Informationstechnik e&i 4, 152–155 (2006)Google Scholar
  6. 6.
    Leitner, D., Kropf, J., Wassertheurer, S., Breitenecker, F.: ASIM 2005. In: F, H. (ed.) 18thSymposium on Simulationtechnique (2005)Google Scholar
  7. 7.
    Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, Oxford (2001)zbMATHGoogle Scholar
  8. 8.
    Artoli, A.M.M., Kandhai, B.D., Hoefsloot, H.C.J., Hoekstra, A.G., Sloot, P.M.A.: Lattice bgk simulations of flow in a symmetric bifurcation. Future Generation Computer Systems 20(6), 909–916 (2004)CrossRefGoogle Scholar
  9. 9.
    Kowarschik, M.: Data locality optimizations for iterative numerical algorithms and cellular automata on hierarchical memory architectures. SCS Publishing House (2004)Google Scholar
  10. 10.
    Wilke, J., Pohl, T., Kowarschik, M.: Cache performance optimizations for parallel lattice boltzmann codes. In: Kosch, H., Böszörményi, L., Hellwagner, H. (eds.) Euro-Par 2003. LNCS, vol. 2790, pp. 441–450. Springer, Heidelberg (2003)Google Scholar
  11. 11.
    Hoeksta, A.G., van Hoff, J., Artoli, A.M.M., Sloot, P.M.A.: Unsteady flow in a 2d elastic tube with the lbgk method. Future Generation Computer Systems 20(6), 917–924 (2004)CrossRefGoogle Scholar
  12. 12.
    Fung, Y.C.: Biomechanics, Mechanical Properties of Living Tissues, 2nd edn. Springer, Heidelberg (1993)Google Scholar
  13. 13.
    Wolfram, S.: Cellular Automata and Complexity. Westview, Boulder (1994)zbMATHGoogle Scholar
  14. 14.
    Fung, Y.C.: Biodynamics. In: Circulation, Springer, Heidelberg (1984)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Daniel Leitner
    • 1
    • 2
  • Siegfried Wassertheurer
    • 1
  • Michael Hessinger
    • 3
  • Andreas Holzinger
    • 4
  • Felix Breitenecker
    • 2
  1. 1.Austrian Research Centers GmbH - ARC, Donau-City-Straße 1, A-1220 ViennaAustria
  2. 2.Vienna University of Technology, Institute for Analysis and Scientific Computing, Wiedner Hauptstraße 8-10, A-1040 ViennaAustria
  3. 3.Medical University of Graz, Clinical Department for vascular surgery, Auenbruggerplatz 29, 8036 Graz 
  4. 4.Medical University of Graz, Institute for Medical Informatics, Statistics and, Documentation, Auenbruggerplatz 2/V, A-8036 GrazAustria

Personalised recommendations