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(Investigative Ophthalmology and Visual Science. 2001;42:1736-1742.)
© 2001 by The Association for Research in Vision and Ophthalmology, Inc.

Determination of Corneal Asphericity after Myopia Surgery with the Excimer Laser: A Mathematical Model

Damien Gatinel1, Thanh Hoang-Xuan1 and Dimitri T. Azar1,2,3

1 From the Rothschild Foundation, Paris, France; and the 2 Massachusetts Eye and Ear Infirmary and 3 Schepens Eye Research Institute, Harvard Medical School, Boston, Massachusetts.


   Abstract
Top
 Abstract
Introduction
 Materials and Methods
 Results
 Discussion
 References
 
PURPOSE. To determine the theoretical change of corneal asphericity within the zone of laser ablation after a conventional myopia treatment, which conforms to Munnerlyn’s paraxial formula and in which the initial corneal asphericity is not taken into consideration.

METHODS. The preoperative corneal shape in cross section was modeled as a conic section of apical radius R1 and shape factor p1. A myopia treatment was simulated, and the equation of the postoperative corneal section within the optical zone was calculated by subtracting the ablation profile conforming to a general equation published by Munnerlyn et al. The apical radius of curvature r2 of the postoperative profile was calculated analytically. The postoperative corneal shape was fitted by a conic section, with an apical radius equal to r2 and a shape factor p2 equal to the value that induced the lowest sum of horizontal residuals and the lowest sum of squared residuals. These calculations were repeated for a range of different dioptric treatments, initial shape factor values, and radii of curvature to determine the change of corneal asphericity within the optical zone of treatment.

RESULTS. Analytical calculation of r2 showed it to be independent of the initial preoperative shape factor p1. The determination of p2 was unambiguous, because the same value induced both the lowest sum of residuals and the lowest sum of the squared residuals. For corneas initially prolate (p1 < 1), prolateness increased (p2 < p1 < 1), whereas for oblate corneas (p1 > 1), oblateness increased (p2 > p1 > 1) within the treated zone after myopia treatment. This trend increased with the increasing magnitude of treatment and decreased with increasing initial apical radius of curvature R1.

CONCLUSIONS. After conventional myopic excimer laser treatment conforming to Munnerlyn’s paraxial formula, the postoperative theoretical corneal asphericity can be accurately approximated by a best-fit conic section. For initially prolate corneas, there is a discrepancy between the clinically reported topographic trend to oblateness after excimer laser surgery for myopia and the results of these theoretical calculations.


   Introduction
Top
 Abstract
 Introduction
Materials and Methods
 Results
 Discussion
 References
 
In both photorefractive keratectomy (PRK) and laser in situ keratomileusis (LASIK) for myopia, flattening of the central corneal curvature due to tissue photoablation results in decreased refractive power. Although considered proprietary, current excimer laser algorithms rely on the pioneering theoretical work of Munnerlyn et al.1 They predict the change in corneal power by considering the initial unablated and the final ablated corneal surface as two spherical surfaces, with a single but different radius of curvature.

Given that the corneal surface is aspheric, the corneal shape in cross section can be approximated by a conic section.2 3 4 5 6 7 The asphericity of the cornea is then defined by the shape factor of the conic section that approximates it most closely. A high percentage of corneas are prolate.2 3 7 8 9 After PRK for myopia, a change from a prolate conformation to an oblate optical contour has been reported.10 11 12 13 To our knowledge, there have been no reports that address the theoretical change in asphericity induced by excimer laser treatment for myopia.

We attempted to predict the theoretical change of corneal asphericity within the optical zone after myopia treatment, conforming with the work of Munnerlyn et al.1 We developed a mathematical model based on a conic section approximation, enabling prediction of theoretical postoperative asphericity, and investigated the influence of preoperative asphericity, magnitude of correction, and radius of curvature on its outcome.


   Materials and Methods
Top
 Abstract
 Introduction
 Materials and Methods
Results
 Discussion
 References
 
Current excimer laser ablations for treating myopia rely on the pioneering work of Munnerlyn et al.,1 in which the initial and final corneal surfaces are assumed to be spherical. This allows calculation of the ablation profile by the following general formula (Fig. 1)

(1)
where t(y) expresses the depth of tissue removal as a function of the distance y from the center of an optical zone diameter of S when R1 and R2 are the initial and final corneal anterior radii of curvature, respectively. The power of the removed lenticule (D) corresponds to the intended refractive change and is related to R1, R2, and the index of refraction (n) as follows

(2)
where R2 is more than R1 for ablations of myopia.



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  		Figure 1. Model proposed by Munnerlyn et al.1 for the ablation profile in the excimer laser treatment of myopia. The investigators predict the change in corneal power by treating the initial unablated and the final ablated corneal surfaces as two spherical diopters, each having a single but different radius of curvature, respectively R1 and R2. This model enables the general formula (equation 1) to calculate the ablation profile t(y) as a function of distance to the center of the optical zone of diameter S.

 
In our theoretical model, we made the following assumptions: The initial corneal surface is rotationally symmetric, and the plane curve of a corneal meridian is a conic section with its apex located at the origin of rectangular x, y coordinates in a system of Cartesian axes.

A conic section can be described mathematically by Baker’s equation14

(3)
where x and y are the coordinates on a Cartesian system with the axis of revolution, R the apical radius of curvature, and p the shape factor. When p is less than 1, the ellipse is prolate and flattens from the center to the periphery. When p equals 1, the ellipse is a circle. When p is more than 1, the ellipse is oblate and steepens from the center to the periphery.

For our purpose, it is more useful to evaluate x in terms of y. Solving equation 3 for x gives

(4)
When a correction of D diopters is simulated using equation 1 of Munnerlyn et al.1 on a cornea modeled as a conic section of apical radius R1 and shape factor p1, within an optical zone diameter S, the resultant curve X2 is derived from the following equation

(5)

(6)
This equation does not describe a conic section (Fig. 2) . However, the radius of curvature for each point of the curve X2 is given by r2(y), which can be computed as follows

(7)
This formula gives the radius of the osculating circle r2 at any point of the curve (X2). The first derivative X'2(y) is

(8)
and the second derivative X''2(y) is

(9)



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  		Figure 2. Ablation profile for myopia conforming to Munnerlyn’s equation applied on an aspheric surface modeled in cross section as a conic section X1(y) of apical radius equal to R1. The postoperative cross section X2(y) is equal to the addition of the initial aspheric profile X1(y) and the ablation profile t(y).

 
After inserting respective first and second derivatives of function X2(y) into formula r2(y), the radius of curvature can be calculated by substituting X'2(y) and X''2(y) in equation 7 . The apical radius of curvature of X2(y) is r2(0). It is calculated by substituting 0 for y. When

(10)
Thus, the radius of curvature of X2 at the apex (apical radius of curvature) is the same as the final radius of curvature, which is derived from equation 1 (Munnerlyn et al.1 ).

Thus, X2(y) has an apical radius of curvature R2, but the shape factor that describes its asphericity cannot be computed by the foregoing calculations, because X2(y) does not describe a conic section. However, a best-fit conic section, C2(y), with apical radius of curvature R2 and shape factor p2 can be calculated. We plotted multiple conic sections, C(y) with shape factor pc. Substituting C for X in equation 4 (Fig. 3)

(11)
To determine the best-fit conic section with shape factor p2, we used two methods to minimize the sum of the absolute values of the residuals T(pc) and the sum of the squared residuals Ts(pc).



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  		Figure 3. Approximation of the postoperative profile X2i(y) by a conic section C(y) of apical radius of curvature R2 and shape factor pc.

 
We developed a numerical procedure and performed computation on a computer spreadsheet (Excel 97 software; Microsoft, Seattle, WA). We defined 31 points along the hemi y-axis from y = 0 to y = S/2 = 3 mm, equally spaced by 0.01 mm. For a given R1, D, and p1, T(pc) and Ts(pc) were iteratively calculated for values of pc ranging from (p1 - 2) to (p1 + 2) by incremental steps of 0.01. Solutions were represented by the value(s) of pc that induced the smallest T(pc) and the smallest Ts(pc). These sums were recorded and tabulated.

Referring to the geometric model of the laser excimer treatment operation for myopia defined herein, we were able to repeat these calculations for a range of different dioptric treatments, initial shape factor values, and radii of curvature. The numerical values of the selected variables are listed in Table 1 .


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  		Table 1. Representative Variables within the Normal Range

 

   Results
Top
 Abstract
 Introduction
 Materials and Methods
 Results
Discussion
 References
 
In the theoretical conditions that we used to determine the best-fit conic section, the pc that corresponded to the lowest T(pc) was identical with that corresponding to the lowest Ts(pc) value. The minimal value of T(pc) and Ts(pc) corresponding to the same pc for 31 points thus provided a value for p2 within our range of testing of ±0.01. The minimal values of T(pc) and Ts(pc) were always less than 2 and 0.1 µm, respectively.

Figure 4 represents the effect of the myopia treatment on initial corneal asphericity. In corneas that were initially prolate (p1 < 1), we found that prolateness increased (p2 < p1 < 1), whereas in initially oblate corneas (p1 > 1), oblateness increased (p2 > p1 > 1) within the treated zone after myopia treatment. Spherical corneas remained spherical (p2 = p1 = 1) after treatment. The slope of asphericity change was not constant but increased with the magnitude of treatment.



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  		Figure 4. Outcome of asphericity for different magnitudes of correction. The values of p2 are plotted against the magnitude of the correction for different values of p1. For corneas that are initially prolate (p1 < 1), prolateness should increase (p2 < p1 < 1) after myopia treatment, whereas in oblate corneas (p1 > 1), oblateness should increase (p2 > p1 > 1) within the optical zone after myopia treatment. Spherical corneas remain spherical (p1 = p2 = 1) after treatment.

 
Figure 5 shows the effect of the initial radius of curvature on the asphericity outcome after treatment of prolate corneas. For the same treatment parameters, steeper prolate corneas (R = 7.5 mm) tend to become less prolate than flatter prolate corneas (R = 8.1 mm). This effect increases with the magnitude of the treatment. Figure 6 shows the effect of initial radius of curvature in oblate corneas. For the same magnitude of treatment, steeper oblate corneas tend to become less oblate and flatter oblate corneas more oblate.



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  		Figure 5. Influence of the initial apical radius of curvature on p2, which is plotted against the magnitude of treatment for three different apical radii: 7.5, 7.8, and 8.1 mm. The initial corneal surface is prolate. Steeper prolate corneas tend to become less prolate and flatter prolate corneas more prolate.

 


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  		Figure 6. Influence for the initial apical radius of curvature on p2. The initial corneal surface is prolate. Steeper oblate corneas tend to become less oblate and flatter oblate corneas more oblate.

 

   Discussion
Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion
References
 
Defocus correction is the objective of conventional laser refractive procedures based on the formula of Munnerlyn et al.,1 which assumes that preoperative corneal surface has a single radius of curvature. The normal human cornea is not spherical, and, despite its shortcomings, modeling the corneal shape in cross section as a conic section is a better approximation and has been widely used3 4 5 6 15 since its introduction by Mandell and St. Helen in 1971.2 Most human normal corneas conform to a prolate ellipse and flatten from the center to the periphery (negative asphericity; p < 1), but some corneas are oblate and steepen from the center to the periphery (positive asphericity; p > 1).

Based on topographic observations that the central cornea becomes flatter than the untreated peripheral cornea after excimer laser surgery for myopia, it could be erroneously predicted that the postoperative outcome in all myopic corneas after excimer laser surgery would be increased oblateness. Such a prediction does not take into consideration the changes of corneal curvature within the treatment zone. The focus of our study was to examine the postoperative changes of corneal asphericity within the treatment zone without consideration of the transition zone or the peripheral zone outside the treatment.

We have demonstrated that in prolate and oblate corneas, laser treatments for myopia based on the equation of Munnerlyn et al.1 result in a final apical radius of curvature that is independent of the initial asphericity (equation 10) . Furthermore, we have demonstrated that in prolate corneas these laser treatments result in more prolate configuration within the area of treatment (Fig. 5) and conversely in oblate corneas, the outcome is increased oblateness (Fig. 6) .

In initially prolate corneas, there is a discrepancy between the topographic observations and the theoretical predictions of our model with regard to the change in asphericity. This discrepancy may be due to several factors.

First, there are limitations in the elliptical model used in our study. The true corneal section does not conform exactly to an ellipse. The ideal model may have to incorporate additional factors (i.e., higher polynomial) to match the human cornea. The elliptical model is, however, a good approximation of the corneal profile over the central 8 mm of its approximately 12-mm diameter.16 This area represents the central optical zone, which is flattened after excimer treatment for myopia and through which light passes to form the foveal image. In conventional optics, conic sections are frequently used to model the corneal surface. Furthermore, topographic evaluation of cornea asphericity has been estimated from the conicoid that best fits the keratoscopic or keratometric data.5 6 7 17 18 19 20 21 22 23

Our analysis assumes a rotationally symmetric mathematical model, but the human cornea may exhibit toricity. Several groups have investigated the meridional variations of corneal asphericity.3 7 21 The difference between the maximal and minimal values of asphericity is low, ranging from 0.13 to 0.50 in 80% of the corneas.8 This difference in the overall asphericity change after laser treatments for myopia does not seem to be significant.

Second, the ablation profile of current lasers for myopia does not follow Munnerlyn’s formula. The effects of lasers may differ according to the homogeneity and the location of the laser beam, and laser manufacturers may have altered initial nomograms to improve clinical outcomes and eliminate central islands. Differences in ablation rate of Bowman’s membrane and stroma or within the stroma may also contribute to this discrepancy. In addition, the applied fluence at the cornea, even when the laser beam is homogeneous, decreases with the distance from the center because of reflection of the UV light and because of the curvature of the cornea. The achieved ablation pattern could thus differ from the one attempted. However, no published data confirm this possibility. Because the ablation patterns are proprietary, we cannot confidently accept this possibility as the major explanation for the discrepancy.

Third, wound healing (epithelial hyperplasia, stromal remodeling) could be the source of the shape discrepancy. Topography patterns have been shown to change with time.24 Variations in epithelial thickness and curvature of the epithelial–stromal interface have been implicated in the refractive regression occurring after LASIK and PRK.25 26 27 28 They may modify the specific effect induced by the ablation of myopia and could account for the observed trend to oblateness observed by Hersh et al.,10 who used corneal topography.

Fourth, photokeratoscopic or videokeratoscopic instruments do not properly assess the shape of the cornea when spherical algorithm assumptions are used.29 These assumptions have been thought to be responsible for the differences observed between the measured corneal power and the magnitude of change in manifest refraction.13 30 31 Douthwaite32 used calibrated convex ellipsoidal surfaces of known apical radius (R) and asphericity (p) to assess the accuracy of the EyeSys videokeratscope (Premier Laser Systems, Irvine, CA). This device appeared to overestimate both p and R, especially for asphericities outside the 0.8 to 1.0 region (i.e., the near-spherical zone). In their study of corneal asphericity after PRK, Hersch et al.10 acknowledge that idiosyncrasies in their Placido-based topography system could have affected their results. These considerations may be important for our purposes. The information provided by keratoscopes after laser refractive surgery is subject to cautious interpretation, and current devices may not be sensitive enough to quantify or assess precisely the postoperative corneal asphericity within the ablated zone.

After the correction of myopia, one of the typical topographical aspects on chromatic maps displays a central circular area of uniform colder color with regard to the surrounding surface. If the overall shape of the cornea after treatment is considered, the central flattening contrasting with the unchanged peripheral contour may lead to the subjective assessment of postoperative oblateness. Our study focused on the determination of the theoretical change of asphericity within the optical zone—that is, the corneal surface of the cornea receiving the laser treatment. Other typical topographical features after PRK have been described, relating nonhomogeneous power within the treated zone.33 The "keyhole," the semi-circular ablative patterns, and the central islands represent three entities with different clinical issues, but all are characterized by the presence of a higher dioptric power area inside the ablation zone. These features may represent increased prolateness of the cornea. The cause of the central island has not yet been clarified with certainty, although many hypotheses have been offered. Our model suggests that preoperative asphericity could be another factor, especially in patients with preoperative marked prolateness.

These considerations point out the ambiguity in the definition of the asphericity of the corneal surface after refractive surgery for myopia. To clarify the term, provide an accurate baseline description of the corneal profile, and model optical errors such as spherical aberrations, the terms oblate and prolate should refer only to continuous conical shapes.

Our mathematical procedure for finding the best-fitting conic section to our given set of points consisted of minimizing the sum of the offset absolute values and the sum of the square of the offsets. The latter did not allow the residuals to be treated as a continuous differentiable quantity, but may have given to the outlying points a disproportionate effect on the fit, which was not the case. Using two fitting criteria raises the question of which method should have been adopted if there had been discrepancies between the best-fit results of optimizing p2. The sum of the squared residuals is the most commonly used method. However, we used the sum-of-residual-fitting method to confirm our findings. We found that the determination of p2 was unambiguous, given the small value of both the sum of the residuals and the sum of the squared residuals. Therefore, the conic section approximation can also be successfully used to describe the corneal profile within the optical zone after a myopia laser treatment conforming to Munnerlyn’s equation.

Equations 7 through 10 allow calculation of the apical radius of curvature of the best-fitting conic section to our set of points. This radius is independent of the initial value of the asphericity. This is contradictory to the findings of Patel and Marshall,34 who found that corneal asphericity could marginally affect the initial refractive outcome of PRK. As in this study, their mathematical model assumed the corneal surface to be a conic section and, for a given correction, the amount of corneal tissue removed by PRK was computed based on spherical corneal optics. However, they arbitrarily assumed that initial and final corneal shapes were typically prolate and oblate, respectively. This assumption led to computation of a different radius of curvature from the one that was expected, according to the spherical model.

There is general agreement that negative corneal asphericity has direct optical significance. It is thought to influence visual performance directly by lowering spherical aberration,12 16 although this finding remains to be demonstrated. Patel et al.,17 using optical raytracing of finite schematic eyes, found that the value of p required to eliminate spherical aberration at the anterior surface is -0.528, given a refractive index of 1.376.

Despite the elimination of spherocylindrical errors, refractive surgery may decrease visual performance by altering the corneal shape and inducing unwanted changes in corneal asphericity. Many investigators have noted that after radial keratotomy the cornea becomes oblate, because the paracentral cornea is relatively steeper than the central cornea.35 36 37 38 Seiler et al.39 proposed an aspheric nomogram for PRK, designed to preserve a negative asphericity. Their clinical results were encouraging, but the aspheric nomogram used did not take into account the patient’s preoperative asphericity.

In summary, given that the anterior surface of the human cornea is the main refractive element of the eye, its shape may contribute to optical aberrations. For the normal, untreated, central corneal zone, a conic section can accurately approximate the profile of the corneal zone remodeled by ablation of myopia. To limit or treat optical aberrations after refractive surgery, new profiles including customized aspherical treatments must be developed. The major point of this study—that the modeling of asphericity compares poorly with reported asphericity outcomes—warrants better representation and collection of data before and after treatment. One future rationale may be to use exact raytracing to determine retinal image quality, given a certain ablation profile and a certain shape factor, and then look for ways to alter given asphericities to a desirable level. To reach this goal, further theoretical and clinical studies of corneal shape after laser refractive surgery are needed.


   Footnotes
 
Supported by the Research to Prevent Blindness Lew R. Wasserman Merit Award (DTA); the Corneal Transplantation Research Fund, Boston, Massachusetts (DTA); and the Massachusetts Lions Eye Research Award, Northborough, Massachusetts (DTA).

Submitted for publication November 28, 2000; revised January 26, 2001; accepted February 7, 2001.

Commercial relationships policy: N.

The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

Corresponding author: Dimitri T. Azar, Corneal, External Disease, and Refractive Surgery Service, Massachusetts Eye and Ear Infirmary, 243 Charles Street, Boston, MA 02114. dazar{at}meei.harvard.edu


   References
Top
 Abstract
 Introduction
 Materials and Methods
 Results
 Discussion
 References
 

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D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar
Corneal Asphericity Change after Excimer Laser Hyperopic Surgery: Theoretical Effects on Corneal Profiles and Corresponding Zernike Expansions
Invest. Ophthalmol. Vis. Sci., May 1, 2004; 45(5): 1349 - 1359.
[Abstract] [Full Text] [PDF]

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D. Gatinel, J. Malet, T. Hoang-Xuan, and D. T. Azar
Analysis of Customized Corneal Ablations: Theoretical Limitations of Increasing Negative Asphericity
Invest. Ophthalmol. Vis. Sci., April 1, 2002; 43(4): 941 - 948.
[Abstract] [Full Text] [PDF]

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