title: ODE/PDE analysis of corneal curvature.
abstract: The starting point for this paper is a nonlinear, two-point boundary value ordinary differential equation (BVODE) that defines corneal curvature according to a static force balance. A numerical solution to the BVODE is computed by first converting the BVODE to a parabolic partial differential equation (PDE) by adding an initial value (t, pseudo-time) derivative to the BVODE. A numerical solution to the PDE is then computed by the method of lines (MOL) with the calculation proceeding to a sufficiently large value of t such that the derivative in t reduces to essentially zero. The PDE solution at this point is also the solution for the BVODE. This procedure is implemented in R (an open source scientific programming system) and the programming is discussed in some detail. A series approximation to the solution is derived from which an estimate for the rate of convergence is obtained. This is compared to a fitted exponential model. Also, two linear approximations are derived, one of which leads to a closed form solution. Both provide solutions very close to that obtained from the full nonlinear model. An estimate for the cornea radius of curvature is also derived. The paper concludes with a discussion of the features of the solution to the ODE/PDE system.
authors.affiliation: Institute of Mathematics and Computer Science, Wroclaw University of Technology, ul. Janiszewskiego 14a, 50-372 Wroclaw, Poland. Electronic address: lukasz.plociniczak@pwr.wroc.pl.
authors.forename: Lukasz
authors.initials: L
authors.lastname: Płociniczak
authors.affiliation: City University, Northampton Square, London EC1V 0HB, UK. Electronic address: graham@griffiths1.com.
authors.forename: Graham W
authors.initials: GW
authors.lastname: Griffiths
authors.affiliation: Lehigh University, D118 Iacocca, 111 Research Drive, Bethlehem, PA 18015, USA. Electronic address: wes1@lehigh.edu.
authors.forename: William E
authors.initials: WE
authors.lastname: Schiesser
mesh_terms: D000465:Algorithms; D001696:Biomechanical Phenomena / Q000502:physiology; D003198:Computer Simulation; D003315:Cornea / Q000033:anatomy & histology*; D006801:Humans; D007429:Intraocular Pressure / Q000502:physiology; D008954:Models, Biological
keywords: Biomechanical shell; Boundary value ordinary differential equation (BVODE); Corneal curvature; Method of Lines (MOL); Ordinary differential equation (ODE); Parabolic PDE; Partial differential equation (PDE); R language