A Novel Zernike Application to Differentiate Between Three-dimensional Corneal Thickness of Normal Corneas and Corneas With Keratoconus


To evaluate a novel Zernike algorithm to differentiate 3-dimensional (3-D) corneal thickness distribution of corneas with keratoconus (KC) from normal corneas.


A retrospective development and evaluation of a diagnostic approach.


Corneal tomography with Scheimpflug imaging was performed in normal (43 eyes) and KC (85 eyes) corneas. Axial and tangential cone location magnitude index (axial CLMI and tangential CLMI, respectively) of the anterior and posterior surface were calculated. The aberrations of the anterior corneal surface were analyzed with Zernike polynomials. Pachymetric Zernike analyses (PZA) were used to map the 3-D thickness distribution of the cornea. Logistic regression was performed to develop a diagnostic procedure for KC using CLMI, PZA, and aberrations. A receiver operating characteristic curve was constructed for each regression model. Corneal volume was also compared between normal and KC corneas. Only the central 5 mm zone was used for all analyses.


Among the PZA coefficients, second- and third-order root mean squares of PZA coefficients were the best predictors of KC corneas ( P < .0001). Among the CLMI variables, axial CLMI of anterior and tangential CLMI of posterior surface were the best predictors of KC ( P < .0001). Among the Zernike corneal aberration coefficients, second- and third-order root mean squares of coefficients were the best predictors of KC ( P < .0001). Sensitivity and specificity of Zernike corneal aberrations, CLMI, and PZA logistic regression model were similar ( P > .05).


The entire 3-D corneal thickness was mapped with Zernike. The PZA method was comparable to CLMI and anterior corneal wavefront aberrations in detecting KC.

Corneal tomography and biomechanics play an important role in detection and treatment of keratoconus. A multitude of approaches are used to classify grades of keratoconus, including slit-lamp microscopic features, corneal tomography, corneal aberrations, and corneal biomechanics. Custom indicators beyond maximum keratometry and minimum thickness of the cornea have been developed for improved detection of keratoconus and the more diagnostically challenging subclinical cases. These custom indicators have comparable value in predicting keratoconus among different study groups. Total corneal and epithelial thickness have also been shown to be sensitive indicators of keratoconus. Epithelial thickness may be particularly useful in diagnosis of severity of keratoconus because of its compensatory relationship to the steepness of the stroma.

Spatial variation of corneal thickness may be an important marker of keratoconus as well. Recent studies have shown that corneal thickness distribution and change (percentage progression in thickness) are useful in diagnosis of keratoconus. However, the mathematical formulations used were limited to quadratic or cubic-order polynomials that reduced the complex corneal 3-dimensional (3-D) thickness distribution, which varies as a function of both radius and meridian, to a function of just the radius by averaging the thickness distribution over all the meridians. The objective of this study was to introduce a novel application of the Zernike transform to map the 3-D spatial distribution of corneal thickness measured by Scheimpflug imaging. Further, Zernike coefficients, similar to Zernike transform of corneal aberrations, were defined and analyzed in normal and keratoconus patients using an automated classification system.


This retrospective, observational study was approved by the institutional research and ethics committee of Narayana Nethralaya multi-specialty hospital, India, and conducted in accordance with the tenets of the Declaration of Helsinki. Only the preoperative data of patients who underwent collagen cross-linking and/or topography-guided ablation were included in this retrospective study. The selection criteria of corneas with keratoconus that underwent treatment are detailed below.

Study Population

Patients between 18 and 60 years of age were included in the study. The diagnosis of keratoconus was based on evidence of stromal thinning, focal protrusion, or increase in corneal curvature, Fleischer ring, Vogt striae, scissoring of the red reflex, an abnormal retinoscopy, and curvature asymmetry leading to abnormal corneal astigmatism. The exclusion criteria were ocular hypertension, corneal inflammation, prior eye surgery, and current topical medication use. The severity of keratoconus was assessed using mean keratometry (Kmean). Three keratoconus grades and a normal grade, designated as grade 0, were defined: grade 1, Kmean < 48 diopters (D); grade 2, 48 D ≤ Kmean < 52 D; grade 3, Kmean ≥ 52 D. For “normal” eyes, manifest spherical error and astigmatism were limited to ±2 D.

Study Design

Corneal topography (flat [K1] and steep [K2] axis keratometry, maximum keratometry [Kmax], and Kmean of the anterior surface) and thickness measurements were performed using Scheimpflug imaging (Pentacam; Oculus Optikgeräte GmbH, Wetzlar, Germany). If any normal or keratoconus subject was using contact lenses prior to treatment, tomography was measured after the subject was off contact lenses for at least 2 weeks. Anterior corneal wavefront aberrations were computed by Pentacam using ray tracing (software version 6.07r29). The wavefront aberrations were analyzed using Zernike polynomials up to 12th order and an analysis zone size of 5 mm of the central cornea. A 5 mm analysis zone size was chosen because in most eyes functional vision is limited by the pupil size and the location of the cone was within the central 3 mm cornea in all the patients included in this study. The location of the cone was evaluated as the location of the steepest point on the tangential curvature map. The Zernike polynomials can be expressed as:

W ( ρ , θ ) = ∑ i = 0 n a m i Z m i

In eq. (1) , W = wavefront error, ρ = nondimensional radius (0 ≤ ρ ≤ 1), θ = meridian in radians, a = Zernike coefficients (defocus, coma, etc), Z = Zernike polynomials, i = order of Zernike polynomial, and m = −n, −n+2,…, n−2, n. In this study, eq. (1) was modified to map the 3-D thickness distribution, p, as follows:

p ( ρ , θ ) = ∑ i = 0 n b m i Z m i

In eq. (2) , p was the thickness spatial distribution. Henceforth, the method described by eq. (2) is referred to as “pachymetry Zernike analyses,” or PZA. By converting eq. (2) to the conventional quadratic matrix form (eq. 3 ), all bim
b m i
were computed. Thus, bim
b m i
were the PZA Zernike coefficients.

[ b ] = ( [ Z ] T [ Z ] ) − 1 ( [ Z ] T [ p ] )

Thus, eqs. (2) and (3) were simply a nonlinear regression of the 3-D thickness distribution as a function of Zernike polynomials. To assess the quality of the nonlinear regression, a root mean square (pRMS in eq. 4 ) of the difference between the measured (p) and estimated ( ˆp
p ˆ
) at all the evaluation points (k) on the anterior surface of the cornea in the central 5 mm was computed.

pRMS = ∑ j = 1 k ( p j − p ˆ j ) 2 k

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Jan 6, 2017 | Posted by in OPHTHALMOLOGY | Comments Off on A Novel Zernike Application to Differentiate Between Three-dimensional Corneal Thickness of Normal Corneas and Corneas With Keratoconus
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