Advertisement

Journal of Medical and Biological Engineering

, Volume 37, Issue 6, pp 899–911 | Cite as

Approximate Artery Elasticity Using Linear Springs

  • Jürgen A. Baier-Saip
  • Pablo A. Baier
  • Klaus Schilling
  • Jauvane C. Oliveira
Original Article

Abstract

The mechanical properties of arteries play an essential role in the study of the circulatory system dynamics, which has been becoming increasingly important in the treatment of cardiovascular diseases. Similarly, when building virtual reality simulators, it is crucial to have a tissue model able to respond in real time. The aim of this work is to linearize an artery model to calculate the stiffness of springs. Arteries with three tissue layers (Intima, Media, and Adventitia) are considered and, starting from the stretch-energy density, some of the elasticity tensor components are calculated. The artery is discretized by a two dimensional mesh where the nodes are connected by three kinds of linear springs (one normal and two angular ones). The model linearizes and homogenizes the material response, but it still contemplates the geometric nonlinearity. Comparisons showed a good match with a nonlinear model and with a standard two-dimensional finite element model, when the artery undergoes a stretch in the circumferential and axial directions. The agreement is also good if the arterial tissue undergoes bending. Finally, the Intima layer shows the biggest deviation from linearity when there is a large deformation in the axial direction. If the arterial stretch varies by 1% or less, then the agreement between the linear and nonlinear models is trustworthy.

Keywords

Artery Elasticity tensor Two dimensions Angular springs Stiffnesses 

Notes

Acknowledgements

This work was supported by Programa de Capacitação Institucional/Laboratório Nacional de Computação Científica (PCI/LNCC), Instituto Nacional de Ciência e Tecnologia - Medicina Assistida por Computação Científica (INCT-MACC), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Projects 454815/2015-8, 573710/2008-2, 290011/2008-6) and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ, Project E-26/170.030/2008).

References

  1. 1.
    Pereira, T., Correia, C., & Cardoso, J. (2015). Novel methods for pulse wave velocity measurement. Journal of Medical and Biological Engineering, 35, 555–565.CrossRefGoogle Scholar
  2. 2.
    Holzapfel, G. A., Gasser, T. C., & Stadler, M. (2002). A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis. European Journal of Mechanics, A/Solids, 21, 441–463.CrossRefGoogle Scholar
  3. 3.
    Alderliesten, T., Konings, M. K., & Niessen, W. J. (2007). Modeling friction, intrinsic curvature, and rotation of guide wires for simulation of minimally invasive vascular interventions. IEEE Transactions on Biomedical Engineering, 54, 29–38.CrossRefGoogle Scholar
  4. 4.
    Wang, Y., Guo, S., Tamiya, T., Hirata, H., & Ishihara, H. (2014). A blood vessel deformation model based virtual-reality simulator for the robotic catheter operating system. Neuroscience and Biomedical Engineering, 2, 126–131.CrossRefGoogle Scholar
  5. 5.
    Baier, P. A., Baier-Saip, J. A., Schilling, K., & Oliveira, J. C. (2016). Simulator for minimally invasive vascular interventions: Hardware and software. Presence, 25, 108–128.CrossRefGoogle Scholar
  6. 6.
    Wells, P. N. T., & Liang, H. D. (2011). Medical ultrasound: Imaging of soft tissue strain and elasticity. Journal of the Royal Society Interface, 8, 1521–1549.CrossRefGoogle Scholar
  7. 7.
    Li, W. (2016). Damage models for soft tissues: A survey. Journal of Medical and Biological Engineering, 36, 285–307.CrossRefGoogle Scholar
  8. 8.
    Pierce, D. M., Fastl, T. E., Rodriguez-Vila, B., Verbrugghe, P., Fourneau, I., Maleux, G., et al. (2015). A method for incorporating three-dimensional residual stretches/stresses into patient-specific finite element simulations of arteries. Journal of the Mechanical Behavior of Biomedical Materials, 47, 147–164.CrossRefGoogle Scholar
  9. 9.
    Vito, R. P., & Dixon, S. A. (2003). Blood vessel constitutive models: 1995–2002. Annual Review of Biomedical Engineering, 5, 413–439.CrossRefGoogle Scholar
  10. 10.
    Fung, Y. C. (1993). Biomechanics: Mechanical properties of living tissues. New York: Springer.Google Scholar
  11. 11.
    Humphrey, J. D. (2002). Cardiovascular solid mechanics: Cells, tissues, and organs. New York: Springer.CrossRefGoogle Scholar
  12. 12.
    Sokolis, D. P. (2008). Passive mechanical properties and constitutive modeling of blood vessels in relation to microstructure. Medical and Biological Engineering and Computing, 46, 1187–1199.CrossRefGoogle Scholar
  13. 13.
    Driessen, N. J., Bouten, C. V., & Baaijens, F. P. (2005). A structural constitutive model for collagenous cardiovascular tissues incorporating the angular fiber distribution. Biomechanical Engineering, 127, 494–503.CrossRefGoogle Scholar
  14. 14.
    Carboni, M., Desch, G. W., & Weizsacker, H. W. (2007). Passive mechanical properties of porcine left circumflex artery and its mathematical description. Medical Engineering and Physics, 29, 8–16.CrossRefGoogle Scholar
  15. 15.
    Holzapfel, G. A., Niestrawska, J. A., Ogdeon, R. W., Reinisch, A. J., & Schrief, A. J. (2015). Modelling non-symmetric collagen fibre dispersion in arterial walls. Journal of the Royal Society Interface, 12, 20150188.CrossRefGoogle Scholar
  16. 16.
    Garcia, M., Mendoza, C., Pastor, L., & Rodriguez, A. (2006). Optimized linear FEM for modeling deformable objects. Computer Animation and Virtual Worlds, 17, 393–402.CrossRefGoogle Scholar
  17. 17.
    Misra, S., Ramesh, K. T., & Okamura, A. M. (2008). Modeling of tool-tissue interactions for computer-based surgical simulation: A literature review. Presence (Camb), 17, 463.CrossRefGoogle Scholar
  18. 18.
    Bro-Nielsen, M. (1998). Finite element modeling in surgery simulation. Proceedings of the IEEE, 86, 490–503.CrossRefGoogle Scholar
  19. 19.
    Zhang, D., Wang, T., Liu, D., & Lin, G. (2010). Vascular deformation for vascular interventional surgery simulation. International Journal of Medical Robotics and Computer Assisted Surgery, 6, 170–177.CrossRefGoogle Scholar
  20. 20.
    Johnson, E., Zhang, Y., & Shimada, K. (2011). Estimating an equivalent wall-thickness of a cerebral aneurysm through surface parameterization and a non-linear spring system. International Journal for Numerical Methods in Biomedical Engineering, 27, 1054–1072.CrossRefGoogle Scholar
  21. 21.
    Holzapfel, G. A., Sommer, G., Gasser, C. T., & Regitnig, P. (2005). Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling. American Journal of Physiology Heart and Circulatory Physiology, 289, H2048–H2058.CrossRefGoogle Scholar
  22. 22.
    Holzapfel, G. A., Gasser, T. C., & Ogden, R. W. (2000). A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity, 61, 1–48.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ogden, R. W. (1997). Non-linear elastic deformations. New York: Dover Publications.Google Scholar
  24. 24.
    Garcia-Herrera, C. M., Celentano, D. J., Cruchaga, M. A., Rojo, F. J., Atienza, J. M., Guinea, G. V., et al. (2012). Mechanical characterization of the human thoracic descending aorta: Experiments and modelling. Computer Methods in Biomechanics and Biomedical Engineering, 15, 185–193.CrossRefGoogle Scholar
  25. 25.
    Baier, P. A., Srinivasan, L., Baier-Saip, J. A., Voelker, W., & Schilling, K. (2015). Surfaces for modeling arteries in virtual reality simulators. IFAC-PapersOnLine, 48, 031–036.CrossRefGoogle Scholar
  26. 26.
    Saez, P., Pena, E., & Martinez, M. A. (2014). A structural approach including the behavior of collagen cross-links to model patient-specific human carotid arteries. Annals of Biomedical Engineering, 42, 1158–1169.CrossRefGoogle Scholar
  27. 27.
    Chen, P., Barner, K. E., & Steiner, K. V. (2006). A displacement driven real-time deformable model for haptic surgery simulation. In Proceedings of 2006 14th symposium on haptic interfaces for virtual environment and teleoperator systems (pp. 499–505).Google Scholar
  28. 28.
    Schmid, H., Grytsan, A., Poshtan, E., Watton, P. N., & Itskov, M. (2013). Influence of differing material properties in media and adventitia on arterial adaptation application to aneurysm formation and rupture. Computer Methods in Biomechanics and Biomedical Engineering, 16, 33–53.CrossRefGoogle Scholar
  29. 29.
    Schulze-Bauer, C. A. J., Regitnig, P., & Holzapfel, G. A. (2002). Mechanics of the human femoral adventitia including the high-pressure response. American Journal of Physiology and Heart and Circulatory Physiology, 282, H2427–H2440.CrossRefGoogle Scholar
  30. 30.
    Kerdok, A. E., Cotin, S. M., Ottensmeyer, M. P., Galea, A. M., Hove, R. D., & Dawson, S. L. (2003). Truth cube: Establishing physical standards for soft tissue simulation. Medical Image Analysis, 7, 283–291.CrossRefGoogle Scholar
  31. 31.
    Hemmasizadeh, A., Autieri, M., & Darvish, K. (2012). Multilayer material properties of aorta determined from nanoindentation tests. Journal of the Mechanical Behavior of Biomedical Materials, 15, 199–207.CrossRefGoogle Scholar

Copyright information

© Taiwanese Society of Biomedical Engineering 2017

Authors and Affiliations

  • Jürgen A. Baier-Saip
    • 1
  • Pablo A. Baier
    • 2
  • Klaus Schilling
    • 3
  • Jauvane C. Oliveira
    • 4
  1. 1.Facultad de Ciencias BásicasUniversidad Católica del MauleTalcaChile
  2. 2.Instituto Federal de Educação, Ciência e Tecnologia do CearáLimoeiro do NorteBrazil
  3. 3.Lehrstuhl für Informatik VII, Am HublandWürzburgGermany
  4. 4.Laboratório Nacional de Computação CientíficaPetrópolisBrazil

Personalised recommendations