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Peter M Pinsky, Yanhui Ma, Yunjae Hwang, Sally Hayes, Keith M Meek; A simple mathematical model for the collagen architecture of normal and keratoconic human corneas. Invest. Ophthalmol. Vis. Sci. 2017;58(8):4312.
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© ARVO (1962-2015); The Authors (2016-present)
Wide-angle X-ray scattering patterns, collected at discrete points across the cornea, provide information about the collagen fibril angular orientation probability at those points. A mathematical modeling approach is proposed that converts the X-ray data into a continuous function over the cornea. The function is easy to evaluate and directly characterizes fibril orientation everywhere over the cornea. This representation of the X-ray data is important for finite element analysis of the cornea and valuable for analyzing and comparing fibril organization in normal and keratoconic corneas.
The modeling approach uses two fitting steps, both based on least squares. First, the scattering data at each collection point is projected onto a set of special Fourier-type basis functions, allowing the data to be represented with only five coefficients. The coefficients at each collection point are then projected into continuous functions on a circular disk using Zernike polynomials. The functions are finally combined to give the fibril angular probability function. To prevent overfitting and maximize interpolation accuracy, 10-fold cross-validation was used to establish the optimal radial order of the Zernike expansion. The collagen architecture of an entire cornea is reduced to 5×Nz coefficients, where Nz is the number of Zernike coefficients.
Four normal and four severe keratoconus corneas were analyzed. The 10-fold cross-validation studies determined that Zernike radial order should be limited to 10 and 15 for normal and keratoconic corneas, respectively. The first (Fourier) fitting for normal and keratoconic corneas was able to reproduce scattering intensity at all points and for all corneas with high precision. The second (Zernike) fitting was assessed by comparing the model prediction and X-ray data for total and aligned collagen mass. In all cases, the agreement was good.
Completed tests confirm that the approach can accurately describe collagen organization in normal and keratoconic corneas. Preferred fibril directions can be precisely defined everywhere on the cornea, including keratoconic corneas which display great irregularity. The model is suitable for use with finite element analysis and provides the true anisotropic collagen organization. Comparative biomechanical finite element analysis of specific fibril architectures is currently underway.
This is an abstract that was submitted for the 2017 ARVO Annual Meeting, held in Baltimore, MD, May 7-11, 2017.
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