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Bessel function model of corneal topography. (English) Zbl 1329.92058
Summary: In this paper we propose a new nonlinear mathematical model of corneal topography. This model is stated as a two-point boundary value problem. We derive the governing equation from the first physical principles and provide a mathematical analysis concerning its solution. The existence and uniqueness theorems are proved and various estimates are shown. At the end, we fit the simplified model based on a modified Bessel function of the first kind with the corneal data. The fitting error is of order of $$1\%$$, which is sufficiently accurate for this type of data and biomedical applications.

##### MSC:
 92C50 Medical applications (general)
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##### References:
 [1] H. von Helmholtz, H. Southall, James P.C. (Ed), Helmholtz’s treatise on physiological optics, Optical Society of America, New York, 1924. [2] Canning, C. R.; Dewynne, J. N.; Fitt, A. D.; Greaney, M. J., Fluid flow in the anterior chamber of a human eye, IMA Journal of Mathematics Applied in Medicine and Biology, 19, 31-60, (2002) · Zbl 1013.92014 [3] Braun, R. J.; Usha, R.; McFadden, G. B.; Driscoll, T. A.; Cook, L. P.; King-Smith, P. E., Thin film dynamics on a prolate spheroid with application to the cornea, Journal of Engineering Mathematics, 73, 121-138, (2012) [4] Braun, R. J.; Fitt, A. D., Modelling drainage of the precorneal tear film after a blink, Mathematical Medicine and Biology, 20, 1-28, (2003) · Zbl 1042.92004 [5] Meja-Barbosa, Y.; Malacara-Hernndez, D., A review of methods for measuring corneal topography, Optometry and Vision Science, 78, 240-253, (2001) [6] Fowler, C. W.; Dave, T. N., Review of past and present techniques of measuring corneal topography, Ophthalmic and Physiological Optics, 14, 1, 49-58, (1994) [7] 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 [8] Kasprzak, H.; Iskander, D. R., Approximating ocular surfaces by generalized conic curves, Ophthalmic and Physiological Optics, 26, 602-609, (2006) [9] Rosales, M. A.; Jurez-Aubry, M.; Lpez-Olazagasti, E.; Ibarra, J.; Tepichn, E., Applied Optics, 48, 6594-6599, (2009) [10] 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) [11] Ahmed, E., Finite element modeling of corneal biomechanical behavior, Journal of Refractive Surgery, 26, 289-300, (2010) [12] Iskander, D. R.; Collins, M. J.; Davis, B., Optimal modeling of corneal surfaces by Zernike polynomials, IEEE Transactions on Biomedical Engineering, 48, 1, (2001) [13] 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) [14] 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) [15] Okrasiński, W.; Płociniczak, Ł., A nonlinear mathematical model of the corneal shape, Nonlinear Analysis: Real World Applications, 13, 1498-1505, (2012) · Zbl 1239.34004 [16] He, J.-H., A remark on a nonlinear mathematical model of the corneal shape, Nonlinear Analysis: Real World Applications, 13, 6, 2863-2865, (2012) · Zbl 1257.34010 [17] Trattler, W.; Majmudar, P.; Luchs, J. I.; Swartz, T., Cornea handbook, (2010), Slack Incorporated [18] Tikhonov, A. N.; Samarskii, A. A., Equations of mathematical physics, (1963), Dover Publications · Zbl 0111.29008 [19] Płociniczak, Ł.; Okrasiński, W., Regularization of an ill-posed problem in corneal topography, Inverse Problems in Science and Engineering, (2013) · Zbl 1308.35326 [20] Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions: with formulas, graphs, and mathematical tables, (1965), Dover Publications · Zbl 0515.33001 [21] Klein, S. A., Axial curvature and the skew ray error in corneal topography, Optometry and Vision Science, 74, 11, 931-944, (1997)
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