Quantitative comparison of corneal surface areas in keratoconus and normal eyes

Keratoconus is a highly prevalent corneal disorder characterized by progressive corneal thinning, steepening and irregular astigmatism. To date, pathophysiology of keratoconus development and progression remains debated. In this study, we retrospectively analysed topographic elevation maps from 3227 eyes of 3227 patients (969 keratoconus and 2258 normal eyes) to calculate anterior and posterior corneal surface area. We compared results from normal eyes and keratoconus eyes using the Mann–Whitney U test. The Kruskal–Wallis test was used to compare keratoconus stages according to the Amsler–Krumeich classification. Keratoconus eyes were shown to have statistically significantly larger corneal surface areas, measured at the central 4.0 mm and 8.0 mm, and total corneal diameter. However, no significant increase in corneal surface area was seen with increasing severity of keratoconus. We suggest that these results indicate redistribution, rather than increase, of the corneal surface area with keratoconus severity.


Results
Demographic data. Our study involved 3227 eyes of 3227 patients, of which 969 eyes had keratoconus and 2258 were normal eyes. We classified keratoconus according to the topographic criteria of the Amsler-Krumeich classification ( Table 4). Characteristics of each group are summarized in Table 1. Sex ratio (65.9% and 65.9%; p = 0.98) and side (50.8% of right eyes and 53.3% of right eyes; p = 0.21) of the selected eye were similar in both groups. However, keratoconus patients were slightly younger than the control group (31.8 and 35.2 years, respectively; p < 0.0001). No difference was observed between keratoconus subgroups.
Intergroup comparison of topographic parameters. Minimal pachymetry was significantly thinner in keratoconus eyes than control (445.4 µm and 548.7 µm; p < 0.0001) and corneal diameter was larger in keratoconus eyes compared to normal eyes (12.01 mm and 11.82 mm; p < 0.0001). Maximal keratometry was statistically significantly steeper in the keratoconus group (49.57 and 43.87; p < 0.0001). Mean anterior keratometry values in the centered 3.0 mm zone, in the 3-5 mm ring were statistically steeper in keratoconus eyes (p < 0.0001). This data is summarized in Table 2 and in Fig. 1A-D.
Anterior chamber depth also increased with keratoconus severity (p < 0.001). Moreover, there was a small, but significant difference in corneal diameter between keratoconus subgroups, without any clear trend.
Corneal surface areas. Anterior and posterior surface areas were calculated at the central 4.0 mm and 8.0 mm diameter from raw elevation data. Anterior and posterior surfaces areas from 8.0 mm to the limbus were extrapolated using a best fitting curve method. Distributions showed wide overlap between normal and keratoconus values. Data is summarized in Table 3. Distribution of corneal surface areas is represented in Fig. 2A Anterior corneal surface area measured at the central 4.0 mm and 8.0 mm diameter area were significantly larger in keratoconus (12.96 mm 2 and 55.31 mm 2 , respectively) compared to normal eyes (12.92 mm 2 and 54.99 mm 2 , respectively). Central surface area increased with keratoconus severity, according to the Amsler-Krumeich classification ( Fig. 2A).
Total anterior corneal surface area calculated up to the measured corneal diameter was significantly larger in keratoconus (134.74 mm 2 ) than in controls (129.3 mm 2 ). However, total anterior corneal surface area showed www.nature.com/scientificreports/ no significant difference between keratoconus subgroups according to the Amsler-Krumeich classification (p = 0.1669) (Fig. 2B).
Posterior corneal surface area measured at the central 4.0 mm and 8.0 mm diameter area were also significantly larger in keratoconus, and increased with keratoconus severity (Fig. 2C).
Total posterior corneal surface area calculated up to corneal diameter was larger in keratoconus compared to normal eyes (143.8mm 2 and 138.1 mm 2 , respectively) but showed no significant difference between groups of the Amsler-Krumeich Classification (p = 0.334) (Fig. 2D).

Discussion
In order to calculate corneal surface areas, we used a geometrical method inspired from landscape measurements 11 using raw elevation data, and no geometrical assumption on the global shape or local curvature of the measured corneal surface. Elevation data was obtained from topographic measurements using the Orbscan II, which have been shown to be repeatable 12 . Such a method of dividing each discrete point into eight 3D triangles, is commonly used to calculate surface area 11 . We deduced the surface area from 8.0 mm to the measured corneal diameter for each cornea independently, by plotting the surface area for each ring diameter (from central 1.0 mm to central 8.0 mm diameter) and by fitting a polynomial of degree 3 to the available points. Sex-ratio between each group was adjusted in order to eliminate a possible sex-induced bias. Keratometry of eyes with keratoconus were steeper, but this difference decreased with the diameter of analysis. In our opinion, this could corroborate a curvature redistribution induced by the hyperprolate corneal profile of corneas with keratoconus 13 .
Keratoconus anterior and posterior surface areas were greater than control. However, there were wide overlaps in the distribution of values between normal eyes and keratoconus. This emphasizes that, in terms of corneal surface area measurements, no clear cut-off value can be set to discriminate normal corneas from keratoconus. The differences in surface area (0.4-4.1%) between the two groups however, are notably less than the differences observed for maximal keratometry and minimal pachymetry (13% and 19% respectively) and although statistically significant, in our view, are unlikely to be clinically relevant.
Furthermore, although the central 4.0 mm and 8.0 mm diameter corneal surface area increased with keratoconus severity, total corneal surface area did not. When analysed on the largest available zone, average keratometry and surface area values were relatively conserved suggesting a redistribution rather than a net increase. This observation is consistent with the Gauss's Theorema Egregium (Latin for "Remarkable Theorem") that states that the Gaussian curvature, which at a point is the product of the principal curvatures, does not change if one bends the surface without stretching it. Some authors have proposed that the biomechanical modification encountered in keratoconus corneas is focal in nature, rather than a uniform generalized weakening 14 . They proposed the concept that a focal reduction in the elastic modulus precipitates a cycle of biomechanical decompensation, driven by asymmetry in the biomechanical properties. A repeating cycle of increased strain and stress redistribution would then lead to subsequent focal steepening and thinning 15 .
As keratoconus progresses, the central steepening is associated with peripheral flattening, which is consistent with a central increase in surface area and a peripheral relative reduction. The increasing amount of flattening in thicker peripheral corneal regions should allow higher stresses to be tolerated without requiring any www.nature.com/scientificreports/ accompanying increase in surface area. This surface redistribution matched the curvature redistribution in our study and is reflected in the increased prolateness of keratoconus corneas. The observed difference in surface area between control and keratoconus eyes may be explained by the larger corneal diameter and the deeper anterior chamber in keratoconus. Indeed, the entire corneal surface area is largely influenced by the corneal diameter and possibly by the anterior chamber depth. In fact, if we make an analogy between the cornea and a spherical cap, the corneal diameter and the anterior chamber depth intervene according to the theorical formula: This formula emphases the relative importance of the square of corneal diameter and anterior chamber depth in surface area calculation.
Our study demonstrates an increase in corneal surface area in eyes with keratoconus, although the total corneal surface area remains constant with increasing keratoconus severity. We hypothesize that corneal surface area, as well as curvature, is redistributed without significant corneal stretching. As such, this suggests that keratoconus is a permanent corneal deformation with a stable total surface area, and may be considered as an extreme form of corneal warpage caused by structural damage secondary to stromal degeneration and external forces 8 .
Otherwise, when compared to control eyes, keratoconus eyes showed increased corneal surface area, but also increased corneal diameter and anterior chamber depth. As of yet, it is unclear whether these increased values are a sign of keratoconus or may represent potential risk factors.
Further studies are required to improve our understanding of the role of corneal diameter and corneal surface area in the development of keratoconus..

Patients. This retrospective study included subjects examined at the Department of Anterior Segment and
Refractive surgery at Rothschild Foundation Hospital, Paris, France. The study and data collection were achieved with approval from the Rothschild Foundation Institutional Review Board which followed the tenets of the Declaration of Helsinki. Informed consent was obtained from all subjects prior to their participation in this study.
Inclusion criteria for the control group were healthy eyes that did not meet any exclusion criteria and were not diagnosed with keratoconus.
Keratoconus was diagnosed mainly on the basis of associated characteristic topographic patterns and keratometry. Topographic characteristics included an asymmetric bowtie pattern in corneal topography, a central keratometry superior to 47.2 dioptres, pathological anterior or posterior corneal elevation. Eyes with forme fruste keratoconus were included.
Exclusion criteria in both groups were any previous ocular surgery (e.g. penetrating keratoplasty, lamellar keratoplasty, corneal rings, corneal collagen cross-linking) and any other ocular disease. Patients using rigid gas permeable contact lenses were also excluded. Patients using soft contact lenses were asked to stop wearing them two weeks before the examination. Topographies with missing data within the central 8 mm were excluded.
First, elevation maps obtained by Orbscan II at Rothschild Foundation Hospital were classified using a machine learning algorithm 16 as keratoconus or normal. We obtained 12 587 elevation maps from 8162 healthy eyes and 7003 elevation maps from 2041 keratoconus. Topographies were manually checked for inclusion and exclusion criteria.
If more than one topography was available for the same eye, only one map was randomly kept to obtain 6863 elevation maps from 3730 normal patients and 2236 maps from 969 keratoconus patients.
A single elevation map from one eye of each patient was then randomly selected, leaving 3730 maps from 3730 patients and 969 maps from 969 patients.
1472 normal elevation maps from female patients were then randomly removed in order to equalize the sexratio in both groups, and eliminate a potential bias. Figure 3 summarizes the process to select the elevation maps studied. Keratoconus was classified following the Amsler-Krumeich classification 17 . Only topographic criteria were used. Grades are summarized in Table 4.
Corneal topography method. Topography was obtained using the Orbscan II corneal topographer (Bausch & Lomb). This device uses a Placido disk with 40 rings combined with 40 scans of slit lamp acquisition. Scans were obtained by qualified technicians using standard protocol for Orbscan topography acquisition. Poor quality exams were repeated. The Orbscan's software provided anterior and posterior elevation maps including 10,000 measurements for a corneal zone of 10 by 10 mm. The elevation maps were calculated from a reference float best fit sphere 18 .
Orbscan II also provided biometric and keratometry values: corneal diameter (horizontal white to white), minimal pachymetry, maximal keratometry, and cornea irregularities in the central 3.0 mm diameter and the 3-5 mm ring.
All topographies were acquired at Rothschild Foundation Hospital. Data was exported from Orbscan and analysed and classified using NumPy v1. 18 (Fig. 4). The surface area of each point is equal to the sum of the 8 triangle areas.  www.nature.com/scientificreports/ Using this method, the corneal area is gradually calculated for concentric areas ranging from the central 1.0 mm to the central 8.0 mm diameter.
Missing elevation data from 8.0 mm diameter to the limbus did not allow corneal surface area calculation. Thus we extrapolated corneal surface area from the central 8.0 mm to the measured total corneal diameter by plotting the surface area for each ring diameter (from central 1.0 mm to central 8.0 mm diameter) and using a least square polynomial fit of degree 3. These calculations were achieved using Python 3.7 and the NumPy v1.18.0 package. Data analysis. Data was compiled in Microsoft Excel files. Variance analysis and descriptive statistics were performed with Microsoft Excel and Prism GraphPad software. Normality tests were performed for each parameter using the D'agostino-Pearson normality test. Since the data was not normally distributed, we used nonparametric Mann-Whitney U test to compare parameters of normal eyes and keratoconus. To compare different stages of keracotonus, we used the Krukal-Wallis test.
A P value of less than 0.05 was considered statistically significant. The lengths of the sides of each triangles are calculated using the Pythagorean theorem. The distance between two elevation points is calculated using the formula DG = DX 2 + XG 2 where DX is the difference of elevation values and XG the distance between two measured points. Then using Euclid's theorem of similar triangles, the length values are divided by two to fit the area surrounding the central point (E). The surface area of each triangle is calculated using those half lengths. Finally, the sum of the 8 triangle area values corresponds to the area of the cell surrounding E.