zbMATH — the first resource for mathematics

A nonlinear mathematical model of the corneal shape. (English) Zbl 1239.34004
Summary: We consider a nonlinear two-point boundary value problem which is derived as a description of corneal shape. We prove some basic results concerning existence, uniqueness and estimates. We suggest some approximate solution fitting over fifteen thousands real corneal data points with an error of order of 1%.

34A05 Explicit solutions, first integrals of ordinary differential equations
37N25 Dynamical systems in biology
92C05 Biophysics
Full Text: DOI
[1] Meja-Barbosa, Y.; Malacara-Hernndez, D., A review of methods for measuring corneal topography, Optometry and vision science, 78, 240-253, (2001)
[2] Ted Mahavier, W.; Hunt, J., An alternative mathematical algorithm for the photo- and videokeratoscope, Nonlinear analysis: real world applications, 7, 1223-1232, (2006) · Zbl 1112.34320
[3] Kasprzak, H.; Iskander, D.R., Approximating ocular surfaces by generalized conic curves, Opthalmic & physiological optics, 26, 602-609, (2006)
[4] Rosales, M.A.; Jurez-Aubry, M.; Lpez-Olazagasti, E.; Ibarra, J.; Tepichn, E., Applied optics, 48, 6594-6599, (2009)
[5] Anderson, K.; El-Sheikh, A.; Newson, T., Application of structural analysis to the mechanical behaviour of the cornea, Journal of the royal society, interface, 1, 3-15, (2004)
[6] Ahmed, E., Finite element modeling of corneal biomechanical behavior, Journal of refractive surgery, 26, 289-300, (2010)
[7] Iskander, D.R.; Collins, M.J.; Davis, B., Optimal modeling of corneal surfaces by Zernike polynomials, IEEE transactions on biomedical engineering, 48, 1, (2001)
[8] Schneider, M.; Iskander, D.R.; Collins, M.J., Modeling corneal surfaces with rational functions for high-speed videokeratoscopy data compression, IEEE transactions on biomedical engineering, 56, 493-499, (2009), art. no. 4637871
[9] Urs, R.; Ho, A.; Manns, F.; Parel, J.M., Age dependent Fourier model of the shape of the isolated ex vivo human crystalline Lens, Vision research, 50, 1041-1047, (2010)
[10] Bakaraju, R.C.; Ehrmann, K.; Falk, D.; Ho, A.; Papas, E., Physical human model eye and methods of its use to analyse optical performance of soft contact lenses, Optics express, 18, 16868-16882, (2010)
[11] Grytz, R.; Meschke, G., Constitutive modeling of crimped collagen fibrils in soft tissues, Journal of the mechanical behavior of biomedical materials, 2, 522-533, (2009)
[12] Cabada, A.; Cid, J., On a class of singular sturm – liouville periodic boundary value problems, Nonlinear analysis: real world applications, 12, 2378-2384, (2011) · Zbl 1229.34033
[13] Alber, H.-D.; Peicheng, Z., Solutions to a model with Neumann boundary conditions for phase transitions driven by configurational forces, Nonlinear analysis: real world applications, 12, 1797-1809, (2011) · Zbl 1215.35012
[14] Nee, J., Nonlinear integral equation from the BCS gap equations of superconductivity, Nonlinear analysis: real world applications, 190-197, (2010) · Zbl 1183.82097
[15] Smith, G.; Atchison, D.A.; Iskander, D.R.; Jones, C.E.; Pope, J.M., Mathematical models for describing the shape of the in vitro unstretched human crystalline Lens, Vision research, 49, 2442-2452, (2009)
[16] Trattler, W.; Majmudar, P.; Luchs, J.I.; Swartz, T., Cornea handbook, (2010), Slack Incorporated
[17] Kasprzak, H.; Jankowska-Kuchta, E., A new analytical approximation of corneal topography, Journal of modern optics, 43, 6, 1135-1148, (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.