Skip to main content
Log in

A Mathematical Approach in the Design of Arterial Bypass Using Unsteady Stokes Equations

  • In Honor of David Gottlieb’s 60th Birthday
  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

In this paper we present an approach for the study of Aorto-Coronaric bypass anastomoses configurations using unsteady Stokes equations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary according to several optimality criteria.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Agoshkov V.I. (2003). Optimal Control Approaches and Adjoint Equations in the Mathematical Physics Problems. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow

    Google Scholar 

  2. Agoshkov, V. I., Quarteroni, A., and Rozza, G. (2005). Shape design in aorto coronaric bypass using perturbation theory. To appear in SIAM J. Numer. Anal.

  3. Cole J.S., Watterson J.K., O’Reilly M.J.G. (2002). Numerical investigation of the haemodynamics at a patched arterial bypass anastomosis. Med. Eng. Phys 24:393–401

    Article  PubMed  CAS  Google Scholar 

  4. Lions, J. L. (1971). Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag.

  5. Lions, J. L., Magenes, E. (1972). Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag.

  6. Marchuk G.I. (1989). Methods of Numerical Mathematics. Nauka, Moscow

    Google Scholar 

  7. Prud’homme C., Rovas D., Veroy K., Maday Y., Patera A.T., and Turinici G. (2002). Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluid. Eng. 172:70–80

    Article  Google Scholar 

  8. Quarteroni A., and Rozza G. (2003). Optimal control and shape optimization in aorto-coronaric bypass anastomoses. M3AS Math. Mod. Meth. Appl. Sci. 13(12):1801–23

    Article  MATH  MathSciNet  Google Scholar 

  9. Quarteroni A., and Formaggia L. (2004). Mathematical modelling and numerical simulation of the cardiovascular system. In: Ciarlet P.G., and Lions J.L. (eds). Modelling of Living Systems, Handbook of Numerical Analysis Series. Elsevier, Amsterdam

    Google Scholar 

  10. Quarteroni A., Tuveri M., and Veneziani A. (2000). Computational vascular fluid dynamics: problems, models and methods. Comput. Visual Sci 2:163–197

    Article  MATH  Google Scholar 

  11. Quarteroni A., and Valli A. (1994). Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin

    MATH  Google Scholar 

  12. Quarteroni A., Sacco R., and Saleri F. (2000). Numerical Mathematics. Springer, New York

    Book  Google Scholar 

  13. Tikhonov A.N., and Arsenin V.Ya. (1974). Methods for Solving Ill-posed Problems. Nauka, Moscow

    Google Scholar 

  14. Rozza G. (2005). On optimization, control and shape design of an arterial bypass. Int. J. Numer. Meth. Fluid 47(10–11): 1411–1419

    Article  MATH  MathSciNet  Google Scholar 

  15. Rozza, G. (2005). Real time reduced basis techniques for arterial bypass geometries. Bathe, K. J. (ed.), Computational Fluid and Solid Mechanics, Elsevier, pp. 1283–1287.

  16. Vainikko G.M., Veretennikov A.Yu. (1986). Iterative Procedures in Ill-posed problems. Nauka, Moscow

    Google Scholar 

  17. Van Dyke M. (1975) Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford

    MATH  Google Scholar 

  18. Vasiliev F.P. (1981). Methods for Solving the Extremum Problems. Nauka, Moscow

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Valery Agoshkov.

Additional information

Dedicated to David Gottlieb on the occasion of his 60th birthday

Valery Agoshkov: This work has been prepared when the first author was a visiting professor at Institut d’Analyse et Calcul Scientifique of the École Polytechnique Fédérale de Lausanne

Rights and permissions

Reprints and permissions

About this article

Cite this article

Agoshkov, V., Quarteroni, A. & Rozza, G. A Mathematical Approach in the Design of Arterial Bypass Using Unsteady Stokes Equations. J Sci Comput 28, 139–165 (2006). https://doi.org/10.1007/s10915-006-9077-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-006-9077-9

Keywords

Navigation