2.1.

Subjects

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 $>-0.25$ diopter sphere (DS) and $<+0.50$ DS, (2) corneal astigmatism $<1.00$ diopter cylinder, and (3) absence of detectable ocular disease or history of ocular surgery.

2.2.

Data Acquisition

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 $\ge 75\%$ of the data were available, were selected for further analysis. The tangential radius of curvature ($r_t$), the perpendicular distance from the point to the optical axis ($y$) of all data points on a semimeridian, and the vertex radius of curvature ($r_0$-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.

2.3.

$Q$-Value Calculation by Tangential Radius of Curvature

A 3-D Cartesian coordinate system was set with its origin ($O$) at the vertex normal to the corneal intersection of the optic axis of the corneal topographer.²⁰ The $Z$-, $Y$-, and $X$-axes of the coordinate system represent the optical axis, vertical, and horizontal directions, respectively. $\theta$ was the angle between the corneal meridian section and the $XOZ$ plane. Our published paper²¹ has introduced the derivation of the equation in detail for the $Q$-value calculation by tangential radius. The equation of the corneal meridian section of any angle $\theta$ could be expressed as

As $r_t$ was a nonlinear function of $y$ in Eq. (1), it was difficult to calculate the $Q$-value. To transform the nonlinear problem into a linear one, Eq. (1) was converted into the form $y^2=b+c{r}_t^{2/3}$, where $b$ and $c$ were constants. A straight-line graph of $y^2$ (on the ordinate) versus $r_t^{2/3}$ (on the abscissa) was plotted. Using linear regression, we obtained $b=(r_0^2/Q)$ and $c=-(r_0^{4/3}/Q)$; that is, $Q=-(b^2/c^3)$. The straight line gives the coefficient of determination ($R^2$). Considering the reliability of the linear regression equation, the coefficient of determination ($R^2$) should be $>0.5$. The $Q$-value of a given semimeridian was calculated from the first point at 0.1 mm to the peripheral point. The mean of three $Q$-values of a given semimeridian was regarded as the final value.

2.4.

Modeling the 360-Semimeridional Variation Rule of the $Q$-Value

We previously found that the near-horizontal region showed a good coefficient of determination ($R^2$), 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 $Q$-values of the near-vertical region.

In the present study, according to the $Q$-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 $Q$-values for each subject was modeled by polynomial fitting with MATLAB (MathWorks, Inc., Natick, Massachusetts). Then, we fitted the $Q$-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 $f(x)=p_0+p_{1}x+p_2{x}^2+p_3{x}^3+p_4{x}^4+\dots$, where $x$ is the semimeridian angle $\theta$ (deg) and $f(x)$ is the corresponding $Q$-value. The degree was converted to a radian when we performed the polynomial fitting.

3.1.

Rotation of the Coordinate System

A new coordinate system ($\overline{X}O\overline{Y}$) was obtained by rotating the original coordinate system ($XOY$) counter-clockwise by $\theta$ deg (Fig. 1). $P$ was an arbitrary point in the coordinate system, with $P(x,y)$ in the original and $P(\overline{x},\overline{y})$ in the new coordinate systems, respectively. We obtained the following coordinate rotation formula:

Fig. 1

Schematic diagram of the rotation of the coordinate system.

3.2.

Generation of a 3-D Corneal Model

We set the angle between a given corneal meridian section and the $XOZ$ plane to be $\theta$ deg. A new coordinate system ($\overline{X}O\overline{Y}$) was obtained by rotating the original coordinate system ($XOY$) around the $Z$-axis counterclockwise by $-(90\mathrm{deg}-\theta )\mathrm{deg}$. Thus, the $\overline{Y}OZ$ plane coincided with the corneal meridian section in the new coordinate system ($\overline{X}O\overline{Y}$). The equations of the corneal meridian section in the new coordinate system ($\overline{X}O\overline{Y}$) were as follows:

We substituted $\overline{x},\overline{y}$ in Eq. (3) to Eq. (4). The equations of the corneal meridian section in the original coordinate system ($XOY$) were as follows:

Finally, we transformed Eq. (5) into the following form:

A total of 360-semimeridians were chosen, and each point had $(x,y,z)$ coordinates. The $z$-value of each semimeridian was selected from 0 to 3.5 mm at 0.1-mm intervals. The $x$ and $y$ coordinate value of every point was calculated by substituting the corresponding $z$-value into Eq. (6). Finally, a 3-D corneal surface plot was generated with the Visual C++ 6.0 program.²²

4.1.

Function Relationship

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 ($y^2$) versus the tangential radius of curvature to the two-thirds power ($r_t^{2/3}$) on the nasal horizontal principal semimeridian of the right eye for subject number 1. The coefficient of determination ($R^2$) in the near-horizontal region of most of the right eyes was $>0.85$.

Fig. 2

Scatterplot of perpendicular distance squared ($y^2$) versus tangential radius of curvature to the two-thirds power ($r_t^{\frac{2}{3}}$) on the nasal horizontal principal semimeridian of the right eye for subject number 1.

4.2.

360-Semimeridional Variation Rule of the $Q$-Value

To determine which degree of polynomial would provide an optimal fit to the 360-semimeridional variation of the $Q$-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 $Q$-value was well fitted with a seventh-degree polynomial function for all subjects.

Figure 3 shows an example of the variation of the asphericity ($Q$) as a function of the semimeridian for subject number 22 with the following seventh-degree polynomial function:

Fig. 3

Typical example of the variation of the asphericity ($Q$) as a function of semimeridian for subject number 22. Blue: $Q$-value of each semimeridian in the near-horizontal region. Red: Fitted curve of 360-semimeridional variation of the $Q$-value. $r^2$: Goodness of fit of the polynomial function. RMSE: RMS fit error.

Most right eyes displayed a good fit $(r^2)>0.9$ for the asphericity of all subjects (Fig. 4). The median value was 0.94, and the mean RMSE was $0.02\pm 0.008$.

Fig. 4

Box and whisker plot for the goodness of fit ($r^2$) of the polynomial function for the right eyes of all subjects for the asphericity ($Q$).

Table 1 shows the mean values of $Q$ at different semimeridian regions in the four quadrants of the anterior corneal surface. The $Q$-values for the sample analyzed in our study displayed negative values ($-1<Q<0$), 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 $Q$-value distribution of the anterior corneal surface presented bimodal variation with the two peak values representing the least-negative $Q$-values. The 360-semimeridional variation of $Q$-values in subjects in Fig. 3 were similar to that of the subjects in Fig. 5.

Table 1

Values for asphericity (Q) at different semimeridian regions in four quadrants of the anterior corneal surface.

Fig. 5

Variation in asphericity as a function of the semimeridian region of the anterior corneal surface for all subjects. Bars denote 95% confidence interval (CI).

4.3.

3-D Corneal Model

Figure 6 shows a colorized 3-D surface plot of the anterior corneal surface described by asphericity ($Q$) from the different perspectives of the same subject in Fig. 3. The color variation reflects the semimeridional variation of the $Q$-value with 0.02 color steps. The $Q$-value gradually becomes more negative from the top to the bottom of the color scale. Because the $Q$-value of each semimeridian was a negative value ($-1<Q<0$) corresponding to the most common corneal shape,²³ the 3-D surface plot of the anterior corneal surface approximated a prolate ellipsoid.

Fig. 6

Three-dimensional (3-D) surface plot of anterior corneal surface described by asphericity ($Q$) for the same subject as in Fig. 2. Color scale (left) represents variation of the $Q$-value.