Human corneal shape prediction with mathematical tools

Human corneal shape prediction with mathematical tools
Hajar Moghadas,^1,*
1. Department of mechanical engineering, Gas and Petroleum Faculty, Yasouj University, 75918-74831, Iran
Introduction: The eye is one of the most vital organs in the human body. The protective envelope of the eye that covers the iris, pupil, and the anterior chamber is the cornea. It is the most transparent tissue and responsible for the optical system well-functioning [1]. In certain diseases, the corneal geometry may alter so accurate construction of the corneal shape can assist in the early detection of certain conditions. For example, the corneal topography is abnormal in astigmatism, hyperopia, keratoconus, and myopia. Additionally, corneal shape prediction can be helpful for lens design. Recently, a nonlinear two-point boundary value equation was developed for human corneal shape based on physical motivation [2]. The solution of the corneal equation, predict the corneal shape. For some special conditions, the corneal equation has been solved using different methods [2-4]. In this work, we propose a simple approach to solve the human corneal shape equation that includes all real conditions correspond to the normal and abnormal cornea. The accuracy and convergence of the presented solutions are evaluated for various cases.
Methods: A dimensionless nonlinear differential equation proposed for the human corneal shape is [2]: u^'' (x)-a u(x)=-b/√(1+〖u^' (x)〗^2 ). u^' (0)=0. u(1)=0. (1) where a and b are positive constants as a = KR2/T and b = PR/T. K is the elasticity coefficient, R is the radius of the cornea, T is the corneal surface tension, and P is intraocular pressure. The nonlinear term, 1/√(1+〖u^' (x)〗^2 ), is the major challenge in solving Eq. (1). Using different assumptions, three accurate approximate analytical solutions are developed for Eq. (1) that are Approximation1, Approximation 2, and Approximation 3. In the following juts, the method of solving in Approximation1 is presented and the others are addressed in the appendixes. In Approximation1, it is assumed that |u^' (x)|<1, and then the finite Maclaurin expansion is applied for the nonlinear term. u_0^' (x) is approximated by the zero-order solution [3] or linear solution and combining the homogeneous and particular solutions, and incorporating the boundary conditions, an analytical approximation solution of Eq. (1), is developed as follows: u(x)=∑_(m=0)^N▒〖C_2m cosh⁡(2mαx)-(∑_(m=0)^N▒〖C_2m cosh⁡(2mα) 〗) cosh⁡〖(αx)〗/cosh⁡〖(α)〗〗. (2) For N=2 as an example, the solution of the corneal equation is obtained as follow: u(x)=〖C_0+C〗_2 cosh⁡〖(2αx)+C_4 cosh⁡(4αx) 〗-[〖C_0+C〗_2 cosh⁡〖(2α)+C_4 cosh⁡(4α) 〗 ] cosh⁡(αx)/cosh⁡(α) . (3) Huge mathematical computation is applied to solve the equation that is not presented here. There is no need to deal with physicians with mathematical computations. By plotting u(x) the corneal shape is obtained which is helpful for eye researchers.
Results: Having a and b which are measured clinically, and differ from person to person, the corneal shape is plotted by u(x). Three cases are selected to validate the proposed solutions at weak, moderate and strong nonlinear conditions. They are, respectively, Case 1: a=2.0, b=0.5, Case 2: a=1.7, b=1.6, and Case 3: a=0.5, b=2. Case 1 represents low intraocular eye fluid pressure caused by Hypotony, Case 2 represents typical cornea [2, 3], and Case 3 represents high intraocular pressure caused by keratoconus. Fig. 2 compares the curve of u(x), the corneal shape, obtained by the proposed solutions, the numerical solution, and the other closed-form solutions [3, 4] for Case 2 and N=2. The results of the Approximation 1 and 2 are in excellent agreement with the numerical results, and they are more accurate than the other closed-form solutions [3, 4]. However, there are significant discrepancies between the results of Approximation 3 and numerical results. Fig. 3 shows the comparison between the results for Case 1 and 3.
Conclusion: The human corneal shape is obtained by solving the human corneal equation. The obtained results are valuable and practical in diagnostic purposes, early detection, custom lens design, and refractive surgery. The presented solutions are not only simple and easy to use but also more accurate than the other available solution in the literature. In contrast to the numerical solution, our approximate analytical solutions can identify the effect of determinant parameters in the corneal performance, which is helpful in ophthalmology.
Keywords: Human corneal shape, early detection, lens design, Nonlinear ordinary differential equation