Home A simple cornea deformation model
Article
Licensed
Unlicensed Requires Authentication

A simple cornea deformation model

  • Tobias Kehrer ORCID logo and Samuel Arba Mosquera ORCID logo EMAIL logo
Published/Copyright: November 26, 2021
Become an author with De Gruyter Brill
Author Information
Advanced Optical Technologies
From the journal Advanced Optical Technologies Volume 10 Issue 6

Abstract

In this paper, we present a cornea deformation model based on the idea of extending the ‘neutral axis’ model to two-dimensional deformations. Considering this simple model, assuming the corneal tissue to behave like a continuous, isotropic and non-compressible material, we are able to partially describe, e.g., the observed deviation in refractive power after lenticule extraction treatments. The model provides many input parameters of the patient and the treatment itself, leading to an individual compensation ansatz for different setups. The model is analyzed for a reasonable range of various parameters. A semi-quantitative comparison to real patient data is performed.

Keywords: biomechanics; computer simulation; laser vision correction
Corresponding author: Samuel Arba Mosquera, SCHWIND Eye-Tech-Solutions GmbH, Mainparkstrasse 6-10, D-63801 Kleinostheim, Germany, E-mail:

Acknowledgments

We thank our colleague Pascal Naubereit for a discussion about future improvements of our model.

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article. Both authors were employees of SCHWIND eye-tech-solutions (manufacturer of the SCHWIND ATOS femtosecond laser system for refractive surgery) at the time of this work.

Appendix

A Schematics of deformation

As described in Section 1, our model is inspired by the principle of the ‘neutral axis’, cf. [22]. The schematics of a 1D deformation, e.g., 1D bending of a beam, is presented in Figure 6A. The dashed line represents the ‘neutral axis’ whose length is not changed under the deformation. Other axes that are ‘parallel’ to the ‘neutral axis’ are compressed below and stretched above, see Figure 6A. In contrast, a slice through a cornea that experienced a 2D bending is given in Figure 6B. The schematics of our coordinate system is presented in Figure 6C.

B Residuals

In Figure 7 the residuals of various fitted profiles of Figures 1B, C and 2B are presented.

C Rest of model application

In Figure 8, Figure 9 and Figure 10, the remaining parameter variations in addition to Figure 3 are displayed.

Figure 8: Extension of Figure 3. Various parameter variations for the ratio of D post/D plan, R cap,post/R cap and TZ post/TZ, the deviation in incision angle Δ θ ${\Delta}\theta $ and maximal absolute fit residue maxabs,res. In order to model the ratios and the deviation in incision angle, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 ${y}_{a}\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}$ are fitted to the five scenarios.
Figure 8:

Extension of Figure 3. Various parameter variations for the ratio of D post/D plan, R cap,post/R cap and TZ post/TZ, the deviation in incision angle Δ θ and maximal absolute fit residue maxabs,res. In order to model the ratios and the deviation in incision angle, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 are fitted to the five scenarios.

Figure 9B shows almost no influence on output parameters which is in accordance with the results of [31].

Figure 9: Extension of Figures 3 and 8. Various parameter variations for the ratio of D post/D plan, R cap,post/R cap and TZ post/TZ, the deviation in incision angle Δ θ ${\Delta}\theta $ and maximal absolute fit residue maxabs,res. In order to model the ratios and the deviation in incision angle, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 ${y}_{a}\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}$ are fitted to the five scenarios.
Figure 9:

Extension of Figures 3 and 8. Various parameter variations for the ratio of D post/D plan, R cap,post/R cap and TZ post/TZ, the deviation in incision angle Δ θ and maximal absolute fit residue maxabs,res. In order to model the ratios and the deviation in incision angle, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 are fitted to the five scenarios.

Figure 10: Extension of Figures 3, 8, and 9. Variation of θ for the ratio of D post/D plan, R cap,post/R cap and TZ post/TZ, the deviation in incision angle Δ θ ${\Delta}\theta $ and maximal absolute fit residue maxabs,res. In order to model the ratios and the deviation in incision angle, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 ${y}_{a}\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}$ are fitted to the five scenarios.
Figure 10:

Extension of Figures 3, 8, and 9. Variation of θ for the ratio of D post/D plan, R cap,post/R cap and TZ post/TZ, the deviation in incision angle Δ θ and maximal absolute fit residue maxabs,res. In order to model the ratios and the deviation in incision angle, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 are fitted to the five scenarios.

Figure 11: Comparison between pre and post rescaling values of k cp and k cap.
Figure 11:

Comparison between pre and post rescaling values of k cp and k cap.

For the closing of the lenticule, Figure 12 shows further evaluations in addition to Figure 4.

Figure 12: Closing lenticule, Section 3.2. Various parameter variations for the ratio of D post/D plan and maximal absolute fit residue maxabs,res. In order to model the ratio, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 ${y}_{a}\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}$ are fitted to the five scenarios.
Figure 12:

Closing lenticule, Section 3.2. Various parameter variations for the ratio of D post/D plan and maximal absolute fit residue maxabs,res. In order to model the ratio, second order polynomials y a ( x ) = a 0 + a 1 x + a 2 x 2 are fitted to the five scenarios.

References

[1] W. Sekundo, K. S. Kunert, and M. Blum, “Small incision corneal refractive surgery using the small incision lenticule extraction (SMILE) procedure for the correction of myopia and myopic astigmatism: results of a 6 month prospective study,” Br. J. Ophthalmol., vol. 95, no. 3, pp. 335–339, 2010, https://doi.org/10.1136/bjo.2009.174284.Search in Google Scholar PubMed

[2] S. Brar, S. Ganesh, and R. R. Arra, “Refractive lenticule extraction small incision lenticule extraction: a new refractive surgery paradigm,” Indian J. Ophthalmol., vol. 66, no. 1, p. 10, 2018, https://doi.org/10.4103/ijo.ijo_761_17.Search in Google Scholar

[3] R. P. Kishore and S. A. Mosquera, “Initial experience with the SCHWIND ATOS and SmartSight lenticule extraction,” J. Clin. Res. Med., vol. 3, no. 4, 2020, https://doi.org/10.31038/jcrm.2020343.Search in Google Scholar

[4] A.-M. Samuel, P. Naubereit, S. Sobutas, et al., “Analytical optimization of the cutting efficiency for generic cavitation bubbles,” Biomed. Opt. Express, vol. 12, no. 7, p. 3819, 2021. https://doi.org/10.1364/boe.425895.Search in Google Scholar

[5] J. S. Mehta and M. Fuest, “Advances in refractive corneal lenticule extraction,” Taiwan J. Ophthalmol., vol. 11, no. 2, p. 113, 2021, https://doi.org/10.4103/tjo.tjo_12_21.Search in Google Scholar PubMed PubMed Central

[6] R. P. Kishore and S. Arba-Mosquera, “Three-month outcomes of myopic astigmatism correction with small incision guided human cornea treatment,” J. Refract. Surg., vol. 37, no. 5, pp. 304–311, 2021, https://doi.org/10.3928/1081597x-20210210-02.Search in Google Scholar

[7] L. IzquierdoJr., S. Daniel, O. Ben-Shaul, et al., “Corneal lenticule extraction assisted by a low-energy femtosecond laser,” J. Cataract Refract. Surg., vol. 46, no. 9, pp. 1217–1221, 2020. https://doi.org/10.1097/j.jcrs.0000000000000236.Search in Google Scholar PubMed

[8] M. Wang, F. Zhang, C. C. Copruz, et al., “First experience in small incision lenticule extraction with the Femto LDV Z8 and lenticule evaluation using scanning electron microscopy,” J. Ophthalmol., vol. 2020, pp. 1–8, 2020. https://doi.org/10.1155/2020/6751826.Search in Google Scholar PubMed PubMed Central

[9] A. V. Doga, S. V. Kostenev, I. A. Mushkova, et al., “Results of corneal lenticule extraction for correction,” Vestn. Oftalmol., vol. 136, no. 6, p. 214, 2020. https://doi.org/10.17116/oftalma2020136062214.Search in Google Scholar PubMed

[10] S. Arba Mosquera, D. de Ortueta, and S. Verma, “The art of nomograms,” Eye Vis., vol. 5, no. 1, 2018, https://doi.org/10.1186/s40662-018-0096-z.Search in Google Scholar PubMed PubMed Central

[11] S. Arba-Mosquera, D. Y. S. Kang, M. H. A. Luger, et al., “Influence of extrinsic and intrinsic parameters on myopic correction in small incision lenticule extraction,” J. Refract. Surg., vol. 35, no. 11, pp. 712–720, 2019. https://doi.org/10.3928/1081597x-20191003-01.Search in Google Scholar

[12] I. Jun, D. S. Y. Kang, S. Arba-Mosquera, et al., “Comparison of clinical outcomes between vector planning and manifest refraction planning in SMILE for myopic astigmatism,” J. Cataract Refract. Surg., vol. 46, no. 8, pp. 1149–1158, 2020. https://doi.org/10.1097/j.jcrs.0000000000000100.Search in Google Scholar PubMed

[13] H. P. Studer, K. R. Pradhan, D. Z. Reinstein, et al., “Biomechanical modeling of femtosecond laser keyhole endokeratophakia surgery,” J. Refract. Surg., vol. 31, no. 7, pp. 480–486, 2015. https://doi.org/10.3928/1081597x-20150623-07.Search in Google Scholar

[14] V. Vavourakis, J. H. Hipwell, and D. J. Hawkes, “An inverse finite element u/p-Formulation to predict the unloaded state of in vivo biological soft tissues,” Ann. Biomed. Eng., vol. 44, no. 1, pp. 187–201, 2015, https://doi.org/10.1007/s10439-015-1405-5.Search in Google Scholar PubMed

[15] P.-J. Shih, I. J. Wang, W. F. Cai, et al.., “Biomechanical simulation of stress concentration and intraocular pressure in corneas subjected to myopic refractive surgical procedures,” Sci. Rep., vol. 7, no. 1, 2017. https://doi.org/10.1038/s41598-017-14293-0.Search in Google Scholar PubMed PubMed Central

[16] B. Pajic, D. M. Aebersold, A. Eggspuehler, et al., “Biomechanical modeling of pterygium radiation surgery: a retrospective case study,” Sensors, vol. 17, no. 6, p. 1200, 2017. https://doi.org/10.3390/s17061200.Search in Google Scholar PubMed PubMed Central

[17] M. Francis, P. Khamar, R. Shetty, et al., “In vivo prediction of air-puff induced corneal deformation using LASIK, SMILE, and PRK finite element simulations,” Investig. Opthalmol. Vis. Sci., vol. 59, no. 13, p. 5320, 2018. https://doi.org/10.1167/iovs.18-2470.Search in Google Scholar PubMed

[18] D. Zhou, A. Ahmed, E. Ashkan, et al., “Microstructure-based numerical simulation of the mechanical behaviour of ocular tissue,” J. R. Soc. Interface, vol. 16, no. 154, p. 20180685, 2019. https://doi.org/10.1098/rsif.2018.0685.Search in Google Scholar PubMed PubMed Central

[19] L. Fang, W. Ma, Y. Wang, et al., “Theoretical analysis of wave-front aberrations induced from conventional laser refractive surgery in a biomechanical finite element model,” Investig. Opthalmol. Vis. Sci., vol. 61, no. 5, p. 34, 2020. https://doi.org/10.1167/iovs.61.5.34.Search in Google Scholar PubMed PubMed Central

[20] W. J. Dupps and C. J. Roberts, Eds. Journal of Cataract and Refractive Surgery, vol. 40, no. 6, in Corneal Biomechanics, 2014. Available at: https://journals.lww.com/jcrs/toc/2014/06000.10.1016/j.jcrs.2014.04.012Search in Google Scholar PubMed

[21] S. Arba-Mosquera, S. Verma, and S. T. Awwad, “Theoretical effect of coma and spherical aberrations translation on refractive error and higher order aberrations,” Photonics, vol. 7, no. 4, p. 116, 2020, https://doi.org/10.3390/photonics7040116.Search in Google Scholar

[22] S. P. Timoshenko, Course of Elasticity Theory. Part 2—Columns and Plates, St. Petersburg, AE Collins Publishers, 1916.Search in Google Scholar

[23] L. N. Thibos, A. Bradley, and X. Hong, “A statistical model of the aberration structure of normal, wellcorrected eyes,” Ophthalmic Physiol. Opt., vol. 22, no. 5, pp. 427–433, 2002, https://doi.org/10.1046/j.1475-1313.2002.00059.x.Search in Google Scholar PubMed

[24] J. J. Rozema, D. A. Atchison, and M.-J. Tassignon, “Statistical eye model for normal eyes,” Investig. Opthalmol. Vis. Sci., vol. 52, no. 7, p. 4525, 2011, https://doi.org/10.1167/iovs.10–6705.10.1167/iovs.10-6705Search in Google Scholar PubMed

[25] A. Elsheikh, C. Whitford, R. Hamarashid, et al., “Stress free configuration of the human eye,” Med. Eng. Phys., vol. 35, no. 2, pp. 211–216, 2013. https://doi.org/10.1016/j.medengphy.2012.09.006.Search in Google Scholar PubMed

[26] P. Rodríguez, R. Navarro, and J. J. Rozema, “Eigencorneas: application of principal component analysis to corneal topography,” Ophthalmic Physiol. Opt., vol. 34, no. 6, pp. 667–677, 2014, https://doi.org/10.1111/opo.12155.Search in Google Scholar PubMed

[27] J. J. Rozema, P. Rodriguez, R. Navarro, et al., “SyntEyes: a higher-order statistical eye model for healthy eyes,” Investig. Opthalmol. Vis. Sci., vol. 57, no. 2, p. 683, 2016. https://doi.org/10.1167/iovs.15-18067.Search in Google Scholar PubMed

[28] J. J. Esteve-Taboada, R. Montés-Micó, and T. Ferrer-Blasco, “Schematic eye models to mimic the behavior of the accommodating human eye,” J. Cataract Refract. Surg., vol. 44, no. 5, pp. 627–641, 2018, https://doi.org/10.1016/j.jcrs.2018.02.024.Search in Google Scholar PubMed

[29] R. Navarro, J. J. Rozema, M. H. Emamian, et al., “Average biometry of the cornea in a large population of Iranian school children,” J. Opt. Soc. Am. A, vol. 36, no. 4, p. B85, 2019. https://doi.org/10.1364/josaa.36.000b85.Search in Google Scholar

[30] J. H. Talamo, P. Gooding, D. Angeley, et al., “Optical patient interface in femtosecond laser–assisted cataract surgery: contact corneal applanation versus liquid immersion,” J. Cataract Refract. Surg., vol. 39, no. 4, pp. 501–510, 2013. https://doi.org/10.1016/j.jcrs.2013.01.021.Search in Google Scholar PubMed

[31] S. Taneri, S. Arba-Mosquera, A. Rost, et al., “Results of thin-cap small-incision lenticule extraction,” J. Cataract Refract. Surg., vol. 47, no. 4, pp. 439–444, 2021. https://doi.org/10.1097/j.jcrs.0000000000000470.Search in Google Scholar PubMed

Received: 2021-08-18
Accepted: 2021-10-29
Published Online: 2021-11-26
Published in Print: 2021-12-20

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Community
  3. Preview: ICO World Congress of optics and photonics 2022
  4. Topical Issue: Lasers in Ophthalmology; Guest Editor: Holger Lubatschowski
  5. Editorial
  6. Lasers in ophthalmology
  7. Views
  8. Analytical optimization of the laser induced refractive index change (LIRIC) process: maximizing LIRIC without reaching the damage threshold
  9. Tutorial
  10. Probing biomechanical properties of the cornea with air-puff-based techniques – an overview
  11. Review Article
  12. Femtosecond lasers for eye surgery applications: historical overview and modern low pulse energy concepts
  13. Research Articles
  14. Effect of laser beam truncation (pinhole), (ordered) dithering, and jitter on residual smoothness after poly(methyl methacrylate) ablations, using a close-to-Gaussian beam profile
  15. Towards temperature controlled retinal laser treatment with a single laser at 10 kHz repetition rate
  16. A simple cornea deformation model
https://doi.org/10.1515/aot-2021-0039
Keywords for this article
biomechanics; computer simulation; laser vision correction

Articles in the same Issue

  1. Frontmatter
  2. Community
  3. Preview: ICO World Congress of optics and photonics 2022
  4. Topical Issue: Lasers in Ophthalmology; Guest Editor: Holger Lubatschowski
  5. Editorial
  6. Lasers in ophthalmology
  7. Views
  8. Analytical optimization of the laser induced refractive index change (LIRIC) process: maximizing LIRIC without reaching the damage threshold
  9. Tutorial
  10. Probing biomechanical properties of the cornea with air-puff-based techniques – an overview
  11. Review Article
  12. Femtosecond lasers for eye surgery applications: historical overview and modern low pulse energy concepts
  13. Research Articles
  14. Effect of laser beam truncation (pinhole), (ordered) dithering, and jitter on residual smoothness after poly(methyl methacrylate) ablations, using a close-to-Gaussian beam profile
  15. Towards temperature controlled retinal laser treatment with a single laser at 10 kHz repetition rate
  16. A simple cornea deformation model
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 7.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/aot-2021-0039/html
Scroll to top button