Skip to main content
Log in

On the Energy of a Hydroelastic System: Blood Flow in an Artery with a Cerebral Aneurysm

  • Published:
Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

The energy approach to the study of a hydroelastic system consisting of an elastic blood vessel, viscous fluid flow, and an aneurysm has been developed to evaluate the various energy components of the system: viscous flow dissipation energy and the stretching and bending energies of the aneurysm wall. To calculate the total energy of the system, we have developed a computing complex including commercial and free software and self-developed modules. The performance of the complex has been tested on model geometric configurations and configurations corresponding to blood vessels with cerebral aneurysms of real patients and reconstructed by angiographic images. The calculated values of the Willmore functional characterizing the shell bending energy are consistent with theoretical data.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Scott, G. G. Ferguson, and M. R. Roach, “Comparison of the Elastic Properties of Human Intracranial Arteries and Aneurysms,” Canad. J. Physiol. Pharmacol. 50 (4), 328–332 (1972).

    Article  Google Scholar 

  2. D. V. Parshin, A. I. Lipovka, A. S. Yunoshev, et al., “On the Optimal Choice of a Hyperelastic Model of Ruptured and Unruptured Cerebral Aneurysm,” Sci. Rep. 9, 15865 (2019); DOI: 10.1038/s41598-019-52229-4.

  3. A. K. Khe, A. P. Chupakhin, A. A. Cherevko, et al., “Viscous Dissipation Energy As a Risk Factor in Multiple Cerebral Aneurysms,” Russ. J. Numer. Anal. Math. Model. 30 (5), 277–287 (2015).

    Article  MathSciNet  Google Scholar 

  4. S. N. Wright, P. Kochunov, F. Mut, et al., “Digital Reconstruction and Morphometric Analysis of Human Brain Arterial Vasculature from Magnetic Resonance Angiography,” Neurolmage 82, 170–181 (2013).

    Article  Google Scholar 

  5. X. Zhao, N. Gold, Y. Fang, et al., “Vertebral Artery Fusiform Aneurysm Geometry in Predicting Rupture Risk,” Roy. Soc. Open Sci. 5 (10), 180780 (2018).

    Google Scholar 

  6. T. J. Willmore, Total Curvature in Riemannian Geometry (John Wiley and Sons, Somerset, 1982).

    MATH  Google Scholar 

  7. W. Helfrich, “Lipid Bilayer Spheres: Deformation and Birefringence in Magnetic Fields,” Phys. Lett. A 43 (5), 409–410 (1973).

    Article  ADS  Google Scholar 

  8. W. Blaschke, Vorlesungen Uber Differentialgeometrie (Springer, Berlin-Heidelberg, 1924).

    Book  Google Scholar 

  9. E. A. Evans, “Bending Resistance and Chemically Induced Moments in Membrane Bilayers,” Biophys. J. 14 (12), 923–931 (1974).

    Article  ADS  Google Scholar 

  10. P. B. Canham, “The Minimum Bending Energy As a Possible Explanation of the Biconcave Shape of the Human Red Blood Cell,” J. Theor. Biol. 26 (1), 61–81 (1970).

    Article  Google Scholar 

  11. R. Capovilla and J. Guven, “Stresses in Lipid Membranes,” J. Phys., A: Math. General. 35 (30), 6233–6247 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  12. G. Landolfi, “New Results on the Canham-Helfrich Membrane Model via the Generalized Weierstrass Representation,” J. Phys., A: Math. General. 36 (48), 11937–11954 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  13. Z. C. Tu and Z. C. Ou-Yang, “A Geometric Theory on the Elasticity of Bio-Membranes,” J. Phys., A: Math. General. 37 (47), 11407–11429 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  14. M. A. Il’gamov, Introduction to Nonlinear Hydroelasticity (Nauka, Moscow, 1991) [in Russian].

    MATH  Google Scholar 

  15. A. V. Pogorelov, Geometric Methods in the Nonlinear Theory of Elastic Shells (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  16. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 7: Theory of Elasticity (Fizmatlit, Moscow, 2007; 1987; Pergamon Press, 1975).

    Google Scholar 

  17. S. S. Antman, Nonlinear Problems of Elasticity (Springer, New York, 2005).

    MATH  Google Scholar 

  18. V. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry: Methods and Applications (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

  19. L. S. Velimirovic, M. S. Ciric, and N. M. Velimirovic, “On the Willmore Energy of Shells under Infinitesimal Deformations,” Comput. Math. Appl. 61 (11), 3181–3190 (2011).

    Article  MathSciNet  Google Scholar 

  20. R. V. Garimella and B. K. Swartz, “Curvature Estimation for Unstructured Triangulations of Surfaces,” Tech. Report No. LA-UR-03-8240. (Los Alamos Nat. Lab., Los Alamos, 2003).

    Google Scholar 

  21. C. Geuzaine and J. F. Remacle, “Gmsh: A 3-D Finite Element Mesh Generator with Built-in Pre- and Post-Processing Facilities,” Int. J. Numer. Meth. Eng. 79 (11), 1309–1331 (2009).

    Article  MathSciNet  Google Scholar 

  22. S. N. Krivoshapko and V. N. Ivanov, Encyclopedia of Analytical Surfaces (Librokom, Moscow, 2010) [in Russian].

    MATH  Google Scholar 

  23. P. A. Yushkevich, J. Piven, H. C. Hazlett, et al., “User-Guided 3D Active Contour Segmentation of Anatomical Structures: Significantly Improved Efficiency and Reliability,” Neurolmage 31 (3), 1116–1128 (2006).

    Article  Google Scholar 

  24. J. Wu, Q. Hu, and X. Ma, “Comparative Study of Surface Modeling Methods for Vascular Structures,” Comput. Med. Imag. Graph. 37 (1), 4–14 (2013).

    Article  Google Scholar 

  25. A. P. Chupakhin, A. A. Cherevko, A. K. Khe, et al., “Measurements and Analysis of Local Cerebral Hemodynamics in Patients with Vascular Brain Malformations,” Pat. Krovoobr. Kardiokhir. 16 (4), 27–31 (2012).

    Google Scholar 

  26. A. K. Khe, A. A. Cherevko, A. P. Chupakhin, et al., “Monitoring of Hemodynamics of Brain Vessels,” Prikl. Mekh. Tekh. Fiz. 58 (5), 7–16 (2017) [J. Appl. Mech. Phys. 58 (5), 763–770 (2017)].

    MathSciNet  Google Scholar 

  27. A. L. Krivoshapkin, V. A. Panarin, K. Yu. Orlov, et al., “Algorithm for Preventing Hemodynamic Hemorrhage during Embolization of Cerebral Arteriovenous Malformations,” Byull. SO RAMN 33 (6), 65–73 (2013).

    Google Scholar 

  28. A. A. Yanchenko, A. A. Cherevko, A. P. Chupakhin, et al., “Nonstationary Hemodynamics Modelling in a Cerebral Aneurysm of a Blood Vessel,” Russ. J. Numer. Anal. Math. Model. 29 (5), 307–317 (2014).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to D. V. Parshin or A. P. Chupakhin.

Additional information

Original Russian Text © M.Yu. Mamatyukov, A.K. Khe, D.V. Parshin, P.I. Plotnikov, A.P. Chupakhin.

__________

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 60, No. 6, pp. 3–16, November-December, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mamatyukov, M.Y., Khe, A.K., Parshin, D.V. et al. On the Energy of a Hydroelastic System: Blood Flow in an Artery with a Cerebral Aneurysm. J Appl Mech Tech Phy 60, 977–988 (2019). https://doi.org/10.1134/S0021894419060014

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0021894419060014

Keywords

Navigation