The anterior surface of the human cornea is a major refractive element. A sound understanding of the corneal shape is needed for the design of the rigid gas permeable (RGP) contact lens and corneal laser refractive surgery (LASIK). Several mathematical models have been proposed to describe the anterior corneal shape.12.3.–4 In particular, asphericity is an important parameter of the corneal shape that is widely used by corneal topographers.
Guillon et al.1 and Bennett5 assumed that the human cornea had a conic section describable by Baker’s equation: , where represents the corneal asphericity.6 Bennett and Rabbetts7 derived the conic equation, , to calculate the asphericity by the sagittal radius of curvature () from keratometry. However, although Bennett’s equation is widely used in the studies of corneal shape,89.10.11.–12 the sagittal radius of curvature is spherically biased and is not a true radius of curvature.1314.–15 Other researchers have described the corneal shape by using the least-squares fitting of zernike polynomials to the data from the corneal height (elevation) map.16 Recently, Laliberté et al.17 proposed a method to construct a three-dimensional (3-D) average human corneal model using the data from the elevation map from Orbscan II.
Most previous studies have reported that -values are representative of all or two principal corneal meridians. Dubbelman et al.18 measured the -values (where ) of six semimeridians (0 deg, 30 deg, 60 deg, 90 deg, 120 deg, and 150 deg) using Scheimpflug photography and modeled the meridional variation of the -values using the function. However, the results indicated that the function was not an adequate model to describe the variation.
Corneal topography is commonly presented as an axial (sagittal) power, tangential power, or elevation map. The tangential radius of curvature () is a true radius of curvature that can better represent the corneal shape and local curvature changes.19 However, to the best of our knowledge, no study has calculated corneal asphericity with the tangential radius of curvature from the tangential power map.
In this study, we calculated the -values of the semimeridian based on the tangential radius of curvature using linear regression. For the first time, we elucidated the 360-semimeridional variation rule of the -value using the polynomial fitting and constructed a customized 3-D model of the anterior corneal surface. Our mathematical model could be useful in the back-surface fitting of the RGP lens or -value-guided customized LASIK.
The right eyes of 66 healthy, young adult volunteers (25 females, 41 males; mean age 24 years; range 18 to 36 years) were evaluated in this study. Inclusion criteria were: (1) refractive spherical equivalent to diopter sphere (DS) and DS, (2) corneal astigmatism diopter cylinder, and (3) absence of detectable ocular disease or history of ocular surgery.
The Bausch & Lomb Orbscan II corneal topographer (version 3.00) was used to acquire topographic images of the right eye of each subject. Three images were obtained from each subject. Topographic images, in which of the data were available, were selected for further analysis. The tangential radius of curvature (), the perpendicular distance from the point to the optical axis () of all data points on a semimeridian, and the vertex radius of curvature (-value) were obtained from the raw data of the tangential power map of the anterior corneal surface. The data points were arranged on a semimeridian at 0.1-mm intervals. The interval between two semimeridians was 1 deg.
-Value Calculation by Tangential Radius of Curvature
A 3-D Cartesian coordinate system was set with its origin () at the vertex normal to the corneal intersection of the optic axis of the corneal topographer.20 The -, -, and -axes of the coordinate system represent the optical axis, vertical, and horizontal directions, respectively. was the angle between the corneal meridian section and the plane. Our published paper21 has introduced the derivation of the equation in detail for the -value calculation by tangential radius. The equation of the corneal meridian section of any angle could be expressed as
As was a nonlinear function of in Eq. (1), it was difficult to calculate the -value. To transform the nonlinear problem into a linear one, Eq. (1) was converted into the form , where and were constants. A straight-line graph of (on the ordinate) versus (on the abscissa) was plotted. Using linear regression, we obtained and ; that is, . The straight line gives the coefficient of determination (). Considering the reliability of the linear regression equation, the coefficient of determination () should be . The -value of a given semimeridian was calculated from the first point at 0.1 mm to the peripheral point. The mean of three -values of a given semimeridian was regarded as the final value.
Modeling the 360-Semimeridional Variation Rule of the -Value
We previously found that the near-horizontal region showed a good coefficient of determination (), whereas the coefficient of determination for the near-vertical region was relatively poor. This difference between the coefficients of determination may have been due to problems associated with acquiring a good image or because the eyelids induced a nonconic form on the corneal meridian section. Thus, in our previous studies, we were unable to obtain the -values of the near-vertical region.
In the present study, according to the -value of each semimeridian in the near-horizontal regions, including 0 deg to 50 deg, 130 deg to 180 deg, 181 deg to 230 deg, and 310 deg to 359 deg, the 360-semimeridional variation of the -values for each subject was modeled by polynomial fitting with MATLAB (MathWorks, Inc., Natick, Massachusetts). Then, we fitted the -value of each semimeridian in the near-vertical regions, including 51 deg to 129 deg and 231 deg to 309 deg. The polynomial function took the form , where is the semimeridian angle (deg) and is the corresponding -value. The degree was converted to a radian when we performed the polynomial fitting.
Construction of a 3-D Model of Corneal Shape
Rotation of the Coordinate System
A new coordinate system () was obtained by rotating the original coordinate system () counter-clockwise by deg (Fig. 1). was an arbitrary point in the coordinate system, with in the original and in the new coordinate systems, respectively. We obtained the following coordinate rotation formula:
Generation of a 3-D Corneal Model
We set the angle between a given corneal meridian section and the plane to be deg. A new coordinate system () was obtained by rotating the original coordinate system () around the -axis counterclockwise by . Thus, the plane coincided with the corneal meridian section in the new coordinate system (). The equations of the corneal meridian section in the new coordinate system () were as follows:2), we obtained the following coordinate rotation equations of our corneal model:
Finally, we transformed Eq. (5) into the following form:
A total of 360-semimeridians were chosen, and each point had coordinates. The -value of each semimeridian was selected from 0 to 3.5 mm at 0.1-mm intervals. The and coordinate value of every point was calculated by substituting the corresponding -value into Eq. (6). Finally, a 3-D corneal surface plot was generated with the Visual C++ 6.0 program.22
The Kolmogorov–Smirnov test showed that the parameter distributions were not significantly different from normal, except for the goodness of fit () of the polynomial function.
The peripheral points of the semimeridians in the near-horizontal region deviated from the corneal center by up to 4 mm. Figure 2 shows the function scatterplot of the perpendicular distance squared () versus the tangential radius of curvature to the two-thirds power () on the nasal horizontal principal semimeridian of the right eye for subject number 1. The coefficient of determination () in the near-horizontal region of most of the right eyes was .
360-Semimeridional Variation Rule of the -Value
To determine which degree of polynomial would provide an optimal fit to the 360-semimeridional variation of the -value, we calculated the root mean square error (RMSE) of the fit of the polynomial function from the fifth to ninth degrees. The RMSE was relatively stable at approximately 0.02 for fits higher than the sixth degree. The 360-semimeridional variation of the -value was well fitted with a seventh-degree polynomial function for all subjects.
Figure 3 shows an example of the variation of the asphericity () as a function of the semimeridian for subject number 22 with the following seventh-degree polynomial function:
Most right eyes displayed a good fit for the asphericity of all subjects (Fig. 4). The median value was 0.94, and the mean RMSE was .
Table 1 shows the mean values of at different semimeridian regions in the four quadrants of the anterior corneal surface. The -values for the sample analyzed in our study displayed negative values (), which gradually became less negative as one moved from the horizontal to the vertical semimeridian regions in each quadrant. Figure 5 shows the variation in asphericity with the semimeridian region of the anterior corneal surface for all subjects. The -value distribution of the anterior corneal surface presented bimodal variation with the two peak values representing the least-negative -values. The 360-semimeridional variation of -values in subjects in Fig. 3 were similar to that of the subjects in Fig. 5.
Values for asphericity (Q) at different semimeridian regions in four quadrants of the anterior corneal surface.
|Corneal semimeridian region (deg)|
|n=66||0 to 30||31 to 50||51 to 70||71 to 90||91 to 110||111 to 129||130 to 150||151 to 180|
|Range||−0.72 to −0.10||−0.61 to −0.07||−0.42 to −0.06||−0.29 to −0.01||−0.31 to 0.01||−0.44 to −0.01||−0.55 to −0.04||−0.54 to −0.06|
|n=66||181 to 210||211 to 230||231 to 250||251 to 270||271 to 290||291 to 309||310 to 330||331 to 359|
|Range||−0.56 to −0.05||−0.47 to −0.03||−0.36 to −0.02||−0.30 to −0.02||−0.27 to −0.02||−0.29 to −0.03||−0.45 to −0.05||−0.70 to −0.07|
Note: n=number of eyes; SD=standard deviation.
3-D Corneal Model
Figure 6 shows a colorized 3-D surface plot of the anterior corneal surface described by asphericity () from the different perspectives of the same subject in Fig. 3. The color variation reflects the semimeridional variation of the -value with 0.02 color steps. The -value gradually becomes more negative from the top to the bottom of the color scale. Because the -value of each semimeridian was a negative value () corresponding to the most common corneal shape,23 the 3-D surface plot of the anterior corneal surface approximated a prolate ellipsoid.
Douthwaite et al.24 used linear regression to calculate -values with the sagittal radius of curvature () according to Bennett’s equation . There are two differences between -values calculated using the sagittal or tangential radius of curvature. First, because the sagittal radius of curvature () is spherically biased, it is not a true radius of curvature. The tangential radius of curvature () is a true radius of curvature that better represents corneal shape and local curvature changes. Second, Bennett’s equation does not include rotation of the coordinate system. Therefore, it can only be used to calculate -values of the principal semimeridians or principal semiaxes. Although the -value calculation of our method was more complex, it could calculate -values of the four principal semiaxes as well as those of other semimeridians. Current corneal topographers provide -values that are representative of all or the two principal corneal meridians. We modeled the 360-semimeridional variation of the -value, which was well fitted with a seventh-degree polynomial function for all subjects.
Laliberte et al.17 proposed a method for constructing an average 3-D human corneal model based on elevation data from Orbscan II. They used the best-fit sphere (BFS) as a reference surface. The colorized anterior elevation map showed that the elevation with respect to the BFS (green) was slightly positive (yellow-red) in the central region. Additionally, a mid-peripheral yellow-green pattern was located above the BFS.
In our study, we proposed a method for constructing a customized 3-D human corneal model by asphericity based on the tangential radius of curvature. Figure 6 shows a typical 3-D surface plot of the anterior corneal surface, which was approximated as a prolate ellipsoid. The corneal curvature gradually became flatter from the center toward the periphery. -values in the near-vertical region were less negative (warmer colors) than those in the near-horizontal region, and those in the nasal cornea were more negative (colder colors) than those in the temporal cornea. A corneal model constructed from elevation data does not have a specific mathematical equation to describe the corneal shape and cannot describe the corneal asphericity (). Instead, such a model shows the elevation difference at each point on the corneal surface from the BFS. In contrast, we constructed a corneal model that was quantitatively described by asphericity.
Anterior corneal asphericity is an important parameter for back-surface fitting of the RGP lens and for -value-guided customized LASIK. The goal of the aspheric back-surface RGP lens design is to optimize alignment with the anterior corneal surface. The goal of the -value-guided customized LASIK is to maintain the physiological asphericity of the anterior corneal surface. At present, -values of the two principal meridians or the four principal semimeridians are used to guide RGP lens fitting and LASIK design. Our mathematical model improves on previous methods by providing 360 -values of the semimeridians of the whole anterior corneal surface.
In conclusion, we have proposed a method to calculate the anterior corneal asphericity () of the semimeridian using the tangential radius of curvature (). Furthermore, our study provided 360 -values of semimeridians of the whole anterior corneal surface, and we constructed a customized 3-D corneal model that was completely described by asphericity. Our mathematical model could help to optimize the back-surface fitting of the RGP lens or -value-guided customized LASIK, thereby improving a patient’s visual quality.
This study was supported by Grant No. 30872816 from the National Natural Science Foundation of China.