For citation:
Muslov S. A., Sukhochev P. Y. Comparison of hyperelastic and formally defined deformation models of the stapedius tendon of the middle ear. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2026, vol. 26, iss. 2, pp. 251-264. DOI: 10.18500/1816-9791-2026-26-2-251-264, EDN: PRWOZZ
Comparison of hyperelastic and formally defined deformation models of the stapedius tendon of the middle ear
This article examines some unresolved issues concerning the mechanical properties of the middle ear organs — its tendons, in particular the stapedius tendon. The mechanical properties of biological tissues are a central topic in biomechanics and bioengineering. Mechanical characteristics are important parameters in computer modeling of organs and tissues during their functioning or under external influences. Mechanical properties of the stapedius tendon of the human middle ear are examined within the framework of the most commonly used hyperelastic models in the literature, as well as formally defined deformation models that allow for the description of the experimental curve with minimal error. The calculations were performed in the Mathcad 15.0 computer algebra system using specially developed functionality. The agreement between mechanical test data and model data was assessed using descriptive statistics. The results showed that the polynomial, Veronda-Westmann, and exponential models were the most accurate in terms of fitting the experimental data. The Hill – Drucker criterion $E > 0$ and the condition $\partial E / \partial \lambda > 0$ are satisfied by the Ogden, Yeoh, Veronda – Westmann, Fung, and Gent models, as well as one formally defined model (the exponential model). It is not recommended to use the 2-parameter Mooney – Rivlin model in the undeformed state and under small deformations due to the loss of mechanical stability of the model in this range $\lambda$. The results obtained in the work can be used for practical purposes in the creation of a physical model and finite element modeling of the middle ear, as well as in reconstructive surgery in the selection of artificial replacement materials for prosthetics and plastic surgery (stapedoplasty).
- Batuev A. S. Fiziologiya vysshej nervnoj deyatel’nosti i sensornykh sistem [Physiology of higher nervous activity and sensory systems]. Moscow, Piter, 2012. 316 p. (in Russian). EDN: QKUHPV
- Prendergast E. J., Ferris E., Rice H. J., Blayney A. W. Vibro-acoustic modeling of the outer and middle ear using the finite element method. Audiol Neurootol, 1999, vol. 4, pp. 185–191. DOI: https://doi.org/10.1159/000013839
- Selyaninov A. A., Elovikov A. M., Charntseva O. V., Elovikov V. A. Biomechanical modeling of the functioning of the stapes of the human middle ear. Russian Journal of Biomechanics, 2016, vol. 20, iss. 4, pp. 358–367 (in Russian). DOI: https://doi.org/10.15593/RZhBiomeh/2016.4.08, EDN: XXMNZN
- Howard J., Roberts W. M., Hudspeth A. J. Mechanicoelectrical transduction by hair cells. Annual Review of Biophysics and Biophysical Chemistry, 1988, vol. 17, pp. 99–124 (in Russian). DOI: https://doi.org/10.1146/annurev.bb.17.060188.000531
- Funnell W. R. J., Maftoon N., Decraemer W. F. Modeling of middle ear mechanics. In: Puria S., Fay R., Popper A. (eds.) The Middle Ear. New York, Springer Handbook of Auditory Research, 2013, pp. 171–210. DOI: https://doi.org/10.1007/978-1-4614-6591-1_7
- Lobato L. C., Paul S., Cordioli J. A. Statistical analysis of the human middle ear mechanical properties. The Journal of the Acoustical Society of America, 2022, vol. 151, iss. 3, art. 2043. DOI: https://doi.org/10.1121/10.0009890
- Ebrahimian A., Mohammadi H., Maftoon N. Material characterization of human middle ear using machine-learning-based surrogate models. Journal of the Mechanical Behavior of Biomedical Materials, 2024, vol. 153, art. 106478. DOI: https://doi.org/10.1016/j.jmbbm.2024.106478
- Gentil F., Natal Jorge R., Ferreira A. J. M., Parente M. P. L., Martins P. A. L. S., Almeida E. Biomechanical simulation of middle ear using hyperelastic models. Journal of Biomechanics, 2006, vol. 39, iss. 1, pp. S388–S389. DOI: https://doi.org/10.1016/S0021-9290(06)84569-0
- Martins E. A. L. S., Jorge R. M. N., Ferreira A. J. M., Figueiredo M., Fernandes R. A. A., Figueiredo M., Silva R. Modelling the mechanical behavior of soft tissues using hyperelastic constitutive models. In: Rodrigues H. et al. (eds.) International Conference on Computational Bioengineering (ICCB2005). Lisbon, Portugal, 2005, pp. 403–410.
- Zhang J., Jiao C., Zou D., Ta N., Rao Z. Assigning viscoelastic and hyperelastic properties to the middle-ear soft tissues for sound transmission. Biomechanics and Modeling in Mechanobiology, 2020, vol. 19, iss. 3, pp. 957–970. DOI: https://doi.org/10.1007/s10237-019-01263-w
- Cheng T., Gan R. Z. Mechanical properties of stapedial tendon in human middle ear. Journal of Biomechanical Engineering, 2007, vol. 129, iss. 6, pp. 913–918. DOI: https://doi.org/10.1115/1.2800837
- Wang B., Lu H., Kim G. A damage model for the fatigue life of elastomeric materials. Mechanics of Materials, 2002, vol. 34, iss. 8, pp. 475–483. DOI: https://doi.org/10.1016/S0167-6636(02)00175-8
- Shmurak M. I., Kuchumov A. G., Voronova N. O. Hyperelastic models analysis for description of soft human tissues behavior. Master‘s Journal, 2017, iss. 1, pp. 230–243 (in Russian). EDN: YUOPFB
- Ivanov D. V., Fomkina O. A. Determination of constants for the neo-Hookean and Mooney – Rivlin models based on the results of experiments on uniaxial tension. Mathematics. Mechanics. Saratov, Saratov State University Publ., 2008, iss. 10, pp. 114–117 (in Russian). EDN: UIRZIV
- Ogden R. W., Saccomandi G., Sgura I. Fitting hyperelastic models to experimental data. Computational Mechanics, 2004, vol. 34, iss. 6, pp. 484–502. DOI: https://doi.org/10.1007/s00466-004-0593-y
- Rackl M. Material testing and hyperelastic material model curve fitting for Ogden, polynomial and Yeoh models. Proceedings of the ScilabTEC, 7th International Scilab Users Conference. At: Paris, France, May 2015. DOI: https://doi.org/10.13140/RG.2.2.29552.25600/1
- Yeoh O. H. Some forms of the strain energy function for rubber. Rubber Chemistry and Technology, 1993, vol. 66, iss. 5, pp. 754–771. DOI: https://doi.org/10.5254/1.3538343
- Veronda D., Westmann R. Mechanical characterizations of skin-finite deformations. Journal of Biomechanics, 1970, vol. 3, iss. 1, pp. 111–124. DOI: https://doi.org/10.1016/0021-9290(70)90055-2
- Bone A., Kaoye M. B.-A. L., Baidi B. B., Samon J.-B. Comparison of hyperelastic models for analysis of human and pig skins behavior. Journal of Applied Mathematics and Physics, 2025, vol. 13, iss. 6, pp. 2045–2062. DOI: https://doi.org/10.4236/jamp.2025.136114
- Arruda E. M., Boyce M. C. A three-dimensional model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 1993, vol. 41, iss. 2, pp. 389–412. DOI: https://doi.org/10.1016/0022-5096(93)90013-6
- Gent A. N. A new constitutive relation for rubber. Rubber Chemistry and Technology, 1996, vol. 69, iss. 1, pp. 59–61. DOI: https://doi.org/10.5254/1.3538357
- Fung Y.-C. Biomechanics: Mechanical properties of living tissues. New York, Springer, 1993. 586 p. DOI: https://doi.org/10.1007/978-1-4757-2257-4
- Muslov S. A., Pertsov S. S., Arutyunov S. D. Fiziko-mekhanicheskie svojstva biologicheskikh tkanej [Physico-mechanical properties of biological tissues]. Moscow, Practical Medicine, 2023. 456 p. (in Russian). DOI: https://doi.org/10.17513/np.594, EDN: MNOSIQ
- Wang X., Cheng T., Gan R. Z. Finite-element analysis of middle-ear pressure effects on static and dynamic behavior of human ear. The Journal of the Acoustical Society of America, 2007, vol. 122, iss. 2, pp. 906–917. DOI: https://doi.org/10.1121/1.2749417
- Muslov S. A., Nikishenko A. N., Pertsov S. S. Calculator of parameters of hyperelastic models of biological tissues. Patent RU 2025613849 (in Russian). EDN: SJTYQQ
- Wex C., Arndt S., Stoll A., Bruns C., Kupriyanova Yu. Isotropic incompressible hyperelastic models for modelling the mechanical behaviour of biological tissues: A review. Biomedical Engineering / Biomedizinische Technik, 2015, vol. 60, iss. 6, pp. 577–592. DOI: https://doi.org/10.1515/bmt-2014-0146
- Muslov S. A., Arutyunov S. D., Maev I. V., Zolotnitsky I. V., Solodov A. A., Rasner P. I. Giperuprugie svojstva biologicheskikh tkanej [Hyperelastic properties of biological tissues]. Moscow, Practical Medicine, 2025. 232 p. (in Russian). EDN: BPALJZ
- Golubinsky A. N. Methods of approximating experimental data and constructing models. Vestnik of Voronezh Institute of the Ministry of Interior of Russia, 2007, iss. 2, pp. 138–143 (in Russian). EDN: JXUUHF
- Muslov S. A., Gvetadze R. Sh., Arutyunov S. D., Korneev A. A., Chistyakov M. V., Zaitseva N. V., Sukhochev P. Yu. On the bilinear model, deformation mechanisms and parameters of the elastincollagen transition in biological tissues. Molecular Medicine, 2025, vol. 23, iss. 2, pp. 48—58 (in Russian). DOI: https://doi.org/10.29296/24999490-2025-02-08, EDN: UAOALC
- Hill R. A general theory of uniqueness and stability in elastic-plastic solids. Journal of the Mechanics and Physics of Solids, 1958, vol. 6, iss. 3, pp. 236–249. DOI: https://doi.org/10.1016/0022-5096(58)90029-2
- Drucker D. C. A definition of a stable inelastic material. Journal of Applied Mechanics, 1959, vol. 26, iss. 1, pp. 101–195. DOI: https://doi.org/10.1115/1.4011929
- 19 reads