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Applied Mathematics and Mechanics

, Volume 34, Issue 5, pp 529–540 | Cite as

Influence of random uncertainties of anisotropic fibrous model parameters on arterial pressure estimation

  • A. Eddhahak-Ouni
  • I. Masson
  • F. Mohand-Kaci
  • M. ZidiEmail author
Article

Abstract

This paper deals with a stochastic approach based on the principle of the maximum entropy to investigate the effect of the parameter random uncertainties on the arterial pressure. Motivated by a hyperelastic, anisotropic, and incompressible constitutive law with fiber families, the uncertain parameters describing the mechanical behavior are considered. Based on the available information, the probability density functions are attributed to every random variable to describe the dispersion of the model parameters. Numerous realizations are carried out, and the corresponding arterial pressure results are compared with the human non-invasive clinical data recorded over a mean cardiac cycle. Furthermore, the Monte Carlo simulations are performed, the convergence of the probabilistic model is proven. The different realizations are useful to define a reliable confidence region, in which the probability to have a realization is equal to 95%. It is shown through the obtained results that the error in the estimation of the arterial pressure can reach 35% when the estimation of the model parameters is subjected to an uncertainty ratio of 5%. Finally, a sensitivity analysis is performed to identify the constitutive law relevant parameters for better understanding and characterization of the arterial wall mechanical behaviors.

Key words

arterial pressure non-invasive clinical data hyperelasticity anisotropy random confidence region 

Chinese Library Classification

O343.5 

2010 Mathematics Subject Classification

74B20 

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References

  1. [1]
    Fung, Y. C. Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., Springer, New York (1993)Google Scholar
  2. [2]
    Hayashi, K. Experimental approaches on measuring the mechanical properties and constitutive laws of arterial walls. Journal of Biomechanical Engineering, 115, 481–488 (1993)CrossRefGoogle Scholar
  3. [3]
    Holzapfel, G. A., Gasser, T. C., and Ogden, R. W. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity, 61, 1–48 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Humphrey, J. D. Cardi Ovascular Solid Mechanics: Cells, Tissues, and Organs, Springer-Verlag, New York (2002)Google Scholar
  5. [5]
    Zidi, M. and Cheref, M. Finite deformations of a hyperelastic, compressible and fiber reinforced tube. European Journal of Mechanics, A/Solids, 21, 971–980 (2002)zbMATHCrossRefGoogle Scholar
  6. [6]
    Gasser, T. C., Ogden, R. W., and Holzapfel, G. A. Hyperelastic modelling of arterial layers with distributed collagen fiber orientations. Journal of the Royal Society of Interface, 3, 15–35 (2006)CrossRefGoogle Scholar
  7. [7]
    Pena, E., Martinez, A., Calvo, B., and Doblaré, M. On the numerical treatment of initial strains in biological soft tissues. International Journal for Numerical Methods in Engineering, 68, 836–860 (2006)zbMATHCrossRefGoogle Scholar
  8. [8]
    Delfino, A., Stergiopulos, N., Moore, J. E., and Meister, J. J. Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation. Journal of Biomechanics, 30, 777–786 (1997)CrossRefGoogle Scholar
  9. [9]
    Chuong, C. J. and Fung, Y. C. On residual stresses in arteries. Journal of Biomechanical Engineering, 108, 189–192 (1986)CrossRefGoogle Scholar
  10. [10]
    Zeinali-Davarani, S., Choi, J., and Baek, S. On parameter estimation for biaxial mechanical behavior of arteries. Journal of Biomechanics, 42, 524–530 (2009)CrossRefGoogle Scholar
  11. [11]
    Eddhahak-Ouni, A., Masson, I., Allaire, E., and Zidi, M. Stochastic approach to estimate the arterial pressure. European Journal of Mechanics, A/Solids, 28, 712–719 (2009)zbMATHCrossRefGoogle Scholar
  12. [12]
    Tang, D., Yang, C., Zheng, J., Woodard, P. K., Sivard, G. A., Saffitz, J. E., and Yuan, C. 3D MRI-based multicomponent FSI models for atherosclerosis plaques. Annals of Biomedical Engineering, 32, 947–960 (2004)CrossRefGoogle Scholar
  13. [13]
    Holzapfel, G. A. and Gasser, T. C. A viscoelastic model for fiber-reinforced composites at finite strains: continum basis, computational aspects and applications. Computer Methods in Applied Mechanics and Engineering, 190, 4379–4403 (2001)CrossRefGoogle Scholar
  14. [14]
    Baek, S., Gleason, R. L., Rajagopal, K. R., and Humphrey, J. D. Theory of small on large: potential utility in computations of fluid-solid interactions in arteries. Computer Methods in Applied Mechanics and Engineering, 196, 3070–3078 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Hu, J. J., Baeck, S., and Humphrey, J. D. Stress-strain behavior of the passive basilar artery in normotension and hypertension. Journal of Biomechanics, 40, 2559–2563 (2007)CrossRefGoogle Scholar
  16. [16]
    Masson, I., Boutouyrie, P., Laurent, S., Humphrey, J. D., and Zidi, M. Characterization of arterial wall mechanical behavior and stresses from human clinical data. Journal of Biomechanics, 41, 2618–2627 (2008)CrossRefGoogle Scholar
  17. [17]
    Humphrey, J. D. and Na, S. Elastodynamics and arterial wall stress. Annals of Biomedical Engineering, 30, 509–523 (2002)CrossRefGoogle Scholar
  18. [18]
    Beaussier, H., Masson, I., Collin, C., Bozec, E., Laloux, B., Calvet, D., Zidi, M., Boutouyrie, P., and Laurent, S. Carotid plaque, arterial stiffness gradient, and remodeling in hypertension. Hypertension, 52, 729–736 (2008)CrossRefGoogle Scholar
  19. [19]
    Ren, J. S. and Yuan, X. G. Mechanics of formation and rupture of human aneurysm. Applied Mathematics and Mechanics (English Edition), 31(5), 593–604 (2010) DOI 10.1007/s10483-010-0507-9MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Rachev, A. and Hayashi, K. Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries. Annals of Biomedical Engineering, 27, 459–468 (1999)CrossRefGoogle Scholar
  21. [21]
    Jaynes, E. T. Information theory and statistical mechanics. Physical Review, 106, 620–630 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Luenberger, D. G. Optimization by Vector Space Methods, Wiley, New York (1969)zbMATHGoogle Scholar
  23. [23]
    Soize, C. Maximum entropy approach for modeling random uncertainties in transient elastodynamics. Journal of the Acoustical Society of America, 109, 1979–1996 (2001)CrossRefGoogle Scholar
  24. [24]
    Feller, W. An Introduction to Probability Theory and Its Applications, John Wiley and Sons, New York, 243–245 (1968)zbMATHGoogle Scholar
  25. [25]
    Rubinstein, R. Y. and Kroese, D. P. Simulation and the Monte Carlo Method, John Wiley and Sons, New York (1981)zbMATHCrossRefGoogle Scholar
  26. [26]
    Chobanian, A. V., Bakris, G. L., Black, H. R., Cushman, W. C., Green, L. A., Izzo, J. L., Jones, D. W., Materson, B. J., Oparil, S., Wright, J. T., Roccella, E. J., and the National High Blood Pressure Education Program Coordinating Committee. Seventh report of the Joint National Committee on prevention, detection, evaluation, and treatment of high blood pressure. Hypertension, 42, 1206–1252 (2003)CrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • A. Eddhahak-Ouni
    • 1
  • I. Masson
    • 2
  • F. Mohand-Kaci
    • 2
  • M. Zidi
    • 2
    Email author
  1. 1.Arts et Métiers ParisTech (ENSAM-ESTP/IRC), Institut de Recherche en ConstructibilitéUniversité Paris-Est CréteilCachanFrance
  2. 2.Centre de Recherches Chirurgicales, Faculté de MédecineUniversité Paris-Est CréteilCréteilFrance

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