Skip to main content
SpringerLink
Log in

Models of Solid Mechanics in the Problems of Ophthalmology

  • TO THE 300th ANNIVERSARY OF SPbU
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

This paper presents a very brief review of models constructed in cooperation with ophthalmologists, namely, for the change in the stress-strain state of the eye membrane after vision-correction operations and the change in the intraocular pressure after the injection of drugs into the vitreous body. The mathematical models describing the process of measuring the true intraocular pressure (IOP) by applanation methods are discussed. The models of eye biomechanics provide the possibility to obtain a series of new results in solid mechanics, i.e., to solve the problem of the stability of a spherical shell under a concentrated force and inner pressure and the stability of an axisymmetric equilibrium state of inhomogeneous annular orthotropic plates under a uniformly distributed normal load. The problems of the deformation of transversally isotropic spherical and cylindrical layers under inner and outer pressure are analyzed and the solutions obtained within the framework of the three-dimensional theory are compared with those found by the non-classical theories of shells. This comparison makes it possible to estimate the precision of some theories.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.

REFERENCES

  1. E. N. Iomdina, S. M. Bauer, and K. E. Kotlyar, Eye Biomechanics: Theoretical Aspects and Clinical Applications (Real Taim, Moscow, 2015) [in Russian].

    Google Scholar 

  2. S. M. Bauer, P. E. Tovstik, and A. B. Katchanov, “On the stability of the eye shell under encircling band,” Tech. Mech. 15, 183–190 (1995).

    Google Scholar 

  3. E. N. Mishina, “On the calculation of the stress-strain state of the eye shell at an encircling load,” Vestn. S.‑Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 2, 68–72 (1995).

  4. A. N. Mironov and B. N. Semenov, “Zum Problem der mathematischen Modellierung in der Ophtalmologie,” Tech. Mech. 16, 245–249 (1996).

    Google Scholar 

  5. S. M. Bauer, P. E. Tovstik, and B. A. Zimin, The Simple Models of Theory of Shells and Plates in Ophthalmology (S.-Peterb. Gos. Univ., St. Petersburg, 2000) [in Russian].

    Google Scholar 

  6. S. M. Bauer and A. N. Mironov, “On the mathematical simulation of the stress-strain state of the eye shell undergoing the scleral buckling procedure,” Acta Bioeng. Biomech. 4, 726–727 (2002).

    Google Scholar 

  7. S. M. Bauer and A. N. Mironov, “Contact of a spherical shell with an elastic ring,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 3, 111–114 (2007).

  8. S. M. Bauer and P. E. Tovstik, “Buckling of spherical shells under concentrated load and internal pressure,” Tech. Mech. 18, 135–139 (1998).

    Google Scholar 

  9. V. V. Volkov, Glaucoma under Pseudonormal Pressure. A Guide for Doctors (Meditsina, Moscow, 2001) [in Russian].

    Google Scholar 

  10. S. M. Bauer, “Axisymmetric deformation of nonuniform transversely isotropic circular plates,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 3, 65–68 (2002).

  11. S. M. Bauer, “Mechanical models of the development of glaucoma,” in Advances in Mechanics of Solids (World Scientific, Singapore, 2006), in Ser.: Series on Stability, Vibration and Control of Systems, Series B, Vol. 15, pp. 153–178. https://doi.org/10.1142/9789812773166_0008

  12. S. M. Bauer and E. B. Voronkova, “On the deformation of the Lamina Cribrosa under intraocular pressure,” Russ. J. Biomech. 5 (1), 73–82 (2001). https://vestnik.pstu.ru/biomech. Accessed July 21, 2023.

  13. S. M. Bauer and E. B. Voronkova, “The mechanical response of the Lamina Cribrosa to the elevated intraocular pressure,” Acta Bioeng. Biomech. 4 (NS), 712–713 (2002).

  14. S. A. Ambartsumyan, Theory of Anisotropic Plates: Strength, Stability, and Vibrations (Nauka, Moscow, 1987; Hemisphere, New York, 1991).

  15. O. M. Palii and V. E. Spiro, Anisotropic Shells in Ship Design. Theory and Analysis (Sudostroenie, Leningrad, 1977) [in Russian].

    Google Scholar 

  16. V. A. Rodionova, V. F. Titaev, and K. F. Chernykh, Applied Theory of Anisotropic Plates and Shells (S.-Peterb. Gos. Univ., St. Petersburg, 1996) [in Russian].

    Google Scholar 

  17. S. M. Bauer and E. B. Voronkova, “Nonclassical theories of bending analysis of orthotropic circular plate,” in Shell Structures: Theory and Applications (CRC, Boca Raton, Fla., 2013), pp. 57–60. https://doi.org/10.1201/b15684

  18. E. B. Voronkova, S. M. Bauer, and A. Eriksson, “Nonclassical theories of shells in application to soft biological tissues,” in Shell-Like Structures (Springer-Verlag, 2011), in Ser.: Advanced Structured Materials, Vol. 15, pp. 647–654. https://doi.org/10.1007/978-3-642-21855-2_42

  19. E. B. Voronkova, “ Deformation, stability and vibrations of the Lamina Cribrosa of the eye,” in Ocular Biomechanics: Proc. 3rd Seminar held by Moscow Helmholtz Research Inst. of Eye Diseases, Moscow, Russia, 2002, pp. 105–106.

  20. D. Yu. Panov and V. I. Feodos’ev, “On the equilibrium and loss of stability of shallow shells in the case of large displacement,” Prikl. Math. Mekh. 12, 389–406 (1948).

    Google Scholar 

  21. L. S. Cheo and E. L. Reiss, “Unsymmetric wrinkling of circular plates,” Q. Appl. Math. 31, 75–91 (1973).

    Article  Google Scholar 

  22. S. M. Bauer and E. B. Voronkova, “On the unsymmetrical buckling of the nonuniform orthotropic circular plates,” in Numerical Analysis and Its Applications (NAA 2012): 5th Int. Conf., Lozenetz, Bulgaria, June 15–20, 2012 (Springer-Verlag, Berlin, 2013), in Ser.: Lecture Notes in Computer Science, Vol. 8236, pp. 198–205. https://doi.org/10.1007/978-3-642-41515-9_20

  23. S. M. Bauer and E. B. Voronkova, “Influence of boundary constraints on the appearance of asymmetrical equilibrium states in circular plates under normal pressure,” Zh. Bel. Gos. Univ., Mat. Inf., No. 1, 38–46 (2020). https://doi.org/10.33581/2520-6508-2020-1-38-46

  24. S. M. Bauer and E. B. Voronkova, “On non-axisymmetric buckling modes of inhomogeneous circular plates,” Vestn. St. Petersburg Univ.: Math. 54, 113–118 (2021). https://doi.org/10.1134/S1063454121020023

    Article  MathSciNet  Google Scholar 

  25. S. M. Bauer, E. B. Voronkova, and B. N. Semenov, “On the nonsymmetric equilibrium forms of circular plates under normal pressure,” Vestn. St. Petersburg Univ.: Math. 55, 275–280 (2022). https://doi.org/10.1134/S1063454122030050

    Article  MathSciNet  Google Scholar 

  26. S. M. Bauer, D. A. Indeitsev, B. N. Semenov, and E. B. Voronkova, “Asymmetric buckling of orthortropic plates under normal pressure,” in Advances in Solid and Fracture Mechanics (Springer-Verlag, Cham, 2022), in Ser.: Advanced Structured Materials, Vol. 180, pp. 13–22 . https://doi.org/10.1007/978-3-031-18393-5_2

  27. S. M. Bauer and E. B. Voronkova, “Asymmetric buckling of heterogeneous annular plates,” in Recent Approaches in the Theory of Plates and Plate-Like Structures, Ed. by S. Bauer, V. A. Eremeyev, G. I. Mikhasev, N. F. Morozov, and H. Altenbach (Springer-Verlag, Cham, 2022), in Ser.: Advanced Structured Materials, Vol. 151, pp. 17–26. https://doi.org/10.1007/978-3-030-87185-7_2

  28. S. M. Bauer and E. B. Voronkova, “On buckling behavior of inhomogeneous shallow spherical caps with elastically restrained edge,” in Analysis of Shells, Plates, and Beams (Springer-Verlag, Cham, 2020), in Ser.: Advanced Structured Materials, Vol. 134, pp. 65–74. https://doi.org/10.1007/978-3-030-47491-1_4

  29. S. M. Bauer and E. B. Voronkova, “Unsymmetrical wrinkling of nonuniform annular plates and spherical caps under internal pressure,” in Recent Developments in the Theory of Shells (Springer-Verlag, Cham, 2019), in Ser.: Advanced Structured Materials, Vol. 110, pp. 79–89. https://doi.org/10.1007/978-3-030-17747-8_6

  30. V. V. Strakhov, “Obvious anatomy and incredible physiology of accommodation.” https://www.detskoezrenie.ru/uploads/lectur/ochevidnaya-anatomiya-i-neveroyatnaya-fiziologiya-akkomodatsii-strahov-vladimir-vitalevich.pdf. Accessed July 21, 2023.

  31. D. Y. Ljubimova, A. Eriksson, and S. M. Bauer, “Numerical study of effect of vitreous support on eye accommodation,” Acta Bioeng. Biomech. 7 (2), 2–15 (2005). https://www.actabio.pwr.wroc.pl/Vol7No2/1.pdf. Accessed: July 21, 2023.

  32. D. Y. Ljubimova, A. Eriksson, and S. M. Bauer, “Aspects of eye accommodation evaluated by finite elements,” Biomech. Model. Mechanobiol. 7, 139–150 (2008). https://doi.org/10.1007/s10237-007-0081-2

    Article  Google Scholar 

  33. S. Yu. Kal’fa, “Eye elastometry,” Russ. Oftal’mol. Zh. 8 (2), 250–262 (1928).

    Google Scholar 

  34. S. M. Bauer, G. A. Lyubimov, and P. E. Tovstik, “On the mathematical simulation of the measuring of the intraocular pressure by Maklakov method,” Tech. Mech. 24, 231–235 (2004). https://journals.ub.ovgu.de/index.php/techmech/article/view/924. Accessed July 21, 2023.

  35. S. M. Bauer, G. A. Lyubimov, and P. E. Tovstik, “Mathematical modeling of Maklakoff’s method for measuring the intraocular pressure,” Fluid Dyn. 40, 20–23 (2005).

    Article  MathSciNet  Google Scholar 

  36. S. M. Bauer and A. S. Tipyasev, “On the mathematical model of the measuring of the intraocular pressure by Maklakov method,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 4, 98–101 (2008).

  37. L. I. Balashevich, A. B. Kachanov, S. A. Nikulin, S. P. Golovatenko, S. M. Bauer, and B. A. Zimin, “Influence of the corneal thickness on pneumotonometric parameters of intraocular pressure,” Oftal’mokhirurgiya, No. 1, 31–33 (2005).

    Google Scholar 

  38. S. M. Bauer, “On applanational tonometry methods of intraocular pressure measurement,” in Computer Methods in Continuum Mechanics: Proc. Seminar, 2006–2007 (S.-Peterb. Gos. Univ., SanktPeterburg, 2007), pp. 84–99.

  39. L. A. Karamshina, “Mechanical models of applanation tonometry taking into account the cornea multilayer structure,” Ross. Zh. Biomekh. 15 (3), 37–44 (2011). http://vestnik.pstu.ru/biomech/. Accessed July 21, 2023.

  40. S. M. Bauer and A. M. Ermakov, “Buckling of a spherical segment under the flat base load,” in Proc. 2nd Int. Conf. on Optimization and Analysis of Structures (2013), pp. 24–27. http://oas2015.ut.ee/OAS2013book.pdf.

    Google Scholar 

  41. S. M. Bauer, L. A. Venatovskaya, A. B. Kachanov, and V. V. Kornikov, “Mathematical models of laser correction of myopia by LASIK, SMILE and PRK methods,” Russ. J. Biomech. 25, 317–322 (2021). https://doi.org/10.15593/RZhBiomeh/2021.4.02

    Article  Google Scholar 

  42. A. Daxer, “Corneal intrastromal implantation surgery for the treatment of moderate and high myopia,” J. Cataract Refractive Surg. 34, 194–198 (2008). https://doi.org/10.1016/j.jcrs.2007.10.011

    Article  Google Scholar 

  43. K. Kotliar, M. Maier, S. Bauer, N. Feucht, C. Lohmann, and I. Lanzl, “Effect of intravitreal injections and volume changes on intraocular pressure: Clinical results and biomechanical model,” Acta Ophthalmol. Scand. 85, 777–781 (2007). https://doi.org/10.1111/j.1600-0420.2007.00939.x

    Article  Google Scholar 

  44. S. M. Bauer, E. B. Voronkova, and A. S. Tipyasev, “On pressure-volume relationship for a human eye shell,” Vestn. S.-Peterb. Univ., Ser. 1: Mat., Mekh., Astron., No. 4, 86–93 (2010).

  45. S. M. Bauer and E. B. Voronkova, “Nonclassical shell theories in ocular biomechanics,” in Shell and Membrane Theories in Mechanics and Biology (Cham, Springer-Verlag, 2015), in Ser.: Advanced Structured Materials, Vol. 45, pp. 81–97. https://doi.org/10.1007/978-3-319-02535-3_4

  46. A. P. Nesterov, A. Ya. Bunin, and L. A. Katsnel’son, Intraocular Pressure. Physiology and Pathology (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  47. K. A. Rezaei and J. C. Wen, “Intravitreal injection technique,” MedEdPORTAL: J. Teach. Learn. Resour. 12, 10502 (2016). https://doi.org/10.15766/mep_2374-8265.10502

    Article  Google Scholar 

  48. S. M. Bauer, L. A. Venatovskaya, E. B. Voronkova, and A. B. Kachanov, “Modeling approaches for an eyeball deformation after intravitreal injection,” in Advanced Materials Modelling for Mechanical, Medical and Biological Applications (Springer-Verlag, Cham, 2022), in Ser.: Advanced Structured Materials, Vol. 155, pp. 77–85. https://doi.org/10.1007/978-3-030-81705-3_6

  49. G. K. Lang, Ophthalmology (Thieme, Stuttgart, 2000).

    Google Scholar 

Download references

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

Author information

Authors and Affiliations

  1. St. Petersburg State University, 199034, St. Petersburg, Russia

    S. M. Bauer, L. A. Venatovskaya & E. B. Voronkova

Authors
  1. S. M. Bauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. Venatovskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. B. Voronkova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to S. M. Bauer, L. A. Venatovskaya or E. B. Voronkova.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Translated by E. Glushachenkova

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bauer, S.M., Venatovskaya, L.A. & Voronkova, E.B. Models of Solid Mechanics in the Problems of Ophthalmology. Vestnik St.Petersb. Univ.Math. 56, 493–511 (2023). https://doi.org/10.1134/S1063454123040039

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454123040039

Keywords:

Search

Navigation