Hayes, Sally

Hwang, Yanhui

Ma, Yanhui

Meek, Keith M.

Pinsky, Peter M.

Online Research @ Cardiff

© Journal for Modeling in Ophthalmology 2018;1:7-11Meeting highlights article: ARVO 2017Correspondence: PeterM.Pinsky, DepartmentofMechanical Engineering, StanfordUniversity,Stanford, CA 94305-4040, USA.E-mail: pinsky@stanford.eduA simplemathematical model forcollagen fibril organization innormal and keratoconic corneasPeter M. Pinsky1, Yanhui Ma2, Yunjae Hwang1, Sally Hayes2, Keith M. Meek21Department of Mechanical Engineering, Stanford University, Stanford, California, USA,2School of Optometry and Vision Sciences, Cardi University, Cardi, UKKey words: anisotropy, collagen fibril organization, cornea, keratoconus, x-ray scat-tering1. Background and purposeWide-angle x-ray scattering techniques have been used to provide quantitative in-formation about the angular distribution and preferred orientation of collagen fibrilsthroughout the corneaMeek, Boote [1]. Using this technique, it has been shown thatcollagen fibrils in the human cornea exhibit a preferential alignment in the inferior-superior and nasal-temporal directionsAghamohammadzadeh et al. [2]. However,this arrangement is oenseen tobeabsentordisturbed in casesof keratoconusHayeset al. [3] and Meek et al. [4], a dystrophy in which the cornea becomes progressivelythin and weak, resulting in the appearance of a cone-shaped cornea.The elasticity of stromal tissue depends on the distribution of collagen fibrilsthroughout the cornea, and preferential fibril alignments imply that the tissue is me-chanically anisotropic. Finite element models for corneal biomechanics can accountfor this mechanical anisotropy by employing a fibril angular orientation probabilityfunction based on x-ray scattering patternsCheng et al. [5]. However, the scatteringpatterns are obtained only at a discrete set of collection points, whereas finite el-ement models need a continuous representation over the cornea. To address thisneed, amathematical modeling approach is proposed that converts the x-ray data atthe collection points into a single continuous function over the corneaMa et al. [6].This representation of the x-ray data is necessary for finite element analysis, as wellas valuable for quantitatively analyzing and comparing fibril organization in normaland keratoconic corneas.8 P.M. Pinsky et al.2.MethodsThemodeling approach uses two fitting steps, both based on least squares. First, thex-ray scattering data at each collection point is modeled as a continuous function ofthe x-ray scan angle, θ, using the following representation:s1 (θ; p) = a1 cosn1(θ − β1) + a2 sinn2(θ − β2) + c, (1)where p = {a1, a2, β1, β2, c} are five parameters determined from a non-linearleast squares problem (trials indicated that n1 = n2 = 6 in all cases). In this way,x-ray scattering patterns at each collection point are reduced to five parameter val-ues. In the second fitting step, the five parameters at each collection point are givena continuous representation over the corneal unit disk using Zernike basis functions,such that:s2 (r′, ϕ; θ) =(pa1∑i=1κ̂ia1Zi(r′, ϕ))cos6(θ −pβ1∑i=1κ̂iβ1Zi(r′, ϕ))+(pa2∑i=1κ̂ia2Zi(r′, ϕ))sin6(θ −pβ2∑i=1κ̂iβ2Zi(r′, ϕ))+pc∑i=1κ̂icZi(r′, ϕ),(2)where (r′, ϕ) are polar coordinates on the unit disk and the five summations are ex-pansions for the five parameters in p using Zernike basis functionsZi(r′, ϕ). The co-eicients κ̂ip, for p ∈ p, are obtained by solving a linear least squares problem. Toprevent over-fitting and tomaximize interpolation accuracy, multiple applications often-fold cross-validationGeisser [7] were used to establish the optimal radial order ofthe Zernike expansions in Equation (2).3.ResultsData from four normal post-mortem corneas and four severe keratoconus cornealdisks (removed during routine transplant surgery), were analyzed. X-ray scatteringdata were collected over the entire cornea in the case of the healthy samples andover a central 8 mm corneal disk in the case of the keratoconic samples. The accu-racy of Equation (1) in reproducing the raw x-ray scattering plots for normal and ker-atoconic corneas at three typical collection points is illustrated in Figure 1. Fittingsbased on Equation (1) for both healthy and keratoconic corneas were obtained withhigh precision in all cases. In the second fitting step based, on Equation (2), ten-foldcross-validation studies determined that the maximum Zernike radial order is case-dependent; for normal corneas the optimal radial order was near 10, while for kera-toconic corneas the optimal orderwas near 15. Figure 2 (row 1) illustrates the result ofModel for collagen organization in the human cornea 9the second fitting, showing the contours of the β1 and β2 coeicients (phase-shi ofpreferred orientations) for a normal and a keratoconic cornea. By integrating Equa-tion (2) over θ ∈ [0, 2π] ,maps of total collagen mass and aligned collagen mass canbe obtained. This is illustrated in Figure 2 (row 2), which contrasts mass contours fora normal and a keratoconic cornea.Fig. 1. Comparison of x-ray scattering data (red circles) and fitting by Equation (1) (blue curve).(Top row, le to right) Collection points and fit at three selected points for a healthy cornea.(Bottom row, le to right) Collection points and fit at three selected points for a keratoconiccornea.Fig. 2. Selected results from the second fitting based on Equation (2). (Top row, le to right)Contours of β1 and β2 for a normal cornea and for a keratoconic cornea. (Bottom row, le toright) Contours of predicted total collagen mass and aligned (anisotropic) collagen mass for anormal cornea and for a keratoconic cornea.10 P.M. Pinsky et al.4. Conclusions and future perspectivesThe proposed mathematical approach is based on two sequential fittings. The firstfitting, based on the Fourier-type form given by Equation (1), proved to be accuratefor every cornea analyzed. The Zernike expansion of the coeicients, expressed byEquation (2), also proved to have high predictive accuracy, but requires strict adher-ence to the optimal radial order as determined by the cross-validation studies.Total and aligned collagen mass contours of the normal corneal pair have beenpreviously published and agree with contours found from the model, providing ameasure of the overall accuracy of the model. While keratoconic corneas may ap-pear to exhibit great irregularity in collagen angular orientation, it is significant thatEquation (1) models the x-ray data of both healthy and keratoconic corneas with highprecision, suggesting that keratoconic collagen organization, even in severe cases,may result from a redistribution of the normal arrangement, possibly by lamellarslippageMeek et al. [4]. This conclusion is further reinforced by considering, for ex-ample, the contours of β1 and β2 shown in Figure 2. The normal cornea shows valuesthat agree with preferred orientations in the inferior-superior and nasal-temporaldirections; likewise, the keratoconic cornea has an essentially similar pattern exceptthat it has become profoundly diuse.All completed tests have confirmed that the proposed modeling approach canaccurately describe collagen organization in normal and keratoconic corneas. Nextsteps include implementation in a finite element analysis code and further geomet-ric analysis tomap themodel onto the corneal sphere in order to reveal an improvedinterpretation of collagen organization.AcknowledgementsThis work has been supported by a Medical Research Council UK Programme GrantMR/K000837/1 awarded to Keith M. Meek. We thank Diamond Light Source (UK) foraccess to beamline IO2 for the collection of x-ray scattering data, Mr. Stephen Tufor the provision of keratoconus corneas, and the UK National Health Service Bloodand Transplant for the provision of healthy donor corneas. A travel grant awardedto Yanhui Ma from the PGR International Experience Fund, Cardi University (UK), isacknowledged.References1. Meek K, Boote C. The use of X-ray scattering techniques to quantify the orientation and distribution ofcollagen in the corneal stroma. Prog Retin Eye Res. 2009;28(5): 369–392.2. Aghamohammadzadeh H, Newton R, Meek K. X-ray scattering used to map the preferred collagen ori-entation in the human cornea and limbus. Prog Retin Eye Res. 2004;12(2): 249–256.3. Hayes S, Boote C, Tu S, Quantock A, Meek K. A study of corneal thickness, shape and collagen or-ganisation in keratoconus using videokeratography and X-ray scattering techniques. Exp Eye Res.2007;84(3): 423–434.Model for collagen organization in the human cornea 114. Meek K, Tu S, Huang Y, Gill P, Hayes S, Newton R. Changes in collagen orientation and distribution inkeratoconus corneas. Invest Ophthalmol Vis Sci. 2005;46(6): 1948–1956.5. Cheng X, Petsche S, Pinsky P. A structural model for the in vivo human cornea including collagen-swelling interaction. JR Soc Interface. 2015;12(109): doi: 10.1098/rsif.2015.0241.6. MaY, Hwang Y, Hayes S,MeekK, Pinsky P. A simplemathematicalmodel for collagen fibril organizationin normal and keratoconic corneas. Proceedings, ARVO Annual Meeting, Baltimore, MD, 2017.7. Geisser S. Predictive inference. Chapman and Hall, 1993.

A simple mathematical model for collagen fibril organization in normal and keratoconic corneas

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A simple mathematical model for collagen fibril organization in normal and keratoconic corneas

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