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 *R* _{1} and shape factor *p* _{1}. 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 *r* _{2} of the postoperative profile
was calculated analytically. The postoperative corneal shape was fitted
by a conic section, with an apical radius equal to *r* _{2} and a shape factor *p* _{2} 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 *r* _{2} showed it to be
independent of the initial preoperative shape factor *p* _{1}. The determination of *p* _{2} 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
(*p* _{1} < 1), prolateness increased
(*p* _{2} < *p* _{1} < 1), whereas for oblate corneas
(*p* _{1} > 1), oblateness increased
(*p* _{2} > *p* _{1} > 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 *R* _{1}.

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.

^{ 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.

^{ 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.

^{ 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.

^{ 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)

*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

*R*

_{1}and

*R*

_{2}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

*R*

_{1},

*R*

_{2}, and the index of refraction (

*n*) as follows

*R*

_{2}is more than

*R*

_{1}for ablations of myopia.

*x*,

*y*coordinates in a system of Cartesian axes.

^{ 14 }

*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.

*x*in terms of

*y*. Solving equation 3 for

*x*gives

*D*diopters is simulated using equation 1 of Munnerlyn et al.

^{ 1 }on a cornea modeled as a conic section of apical radius

*R*

_{1}and shape factor

*p*

_{1}, within an optical zone diameter

*S*, the resultant curve

*X*

_{2}is derived from the following equation

*X*

_{2}is given by

*r*

_{2}(

*y*), which can be computed as follows

*r*

_{2}at any point of the curve (

*X*

_{2}). The first derivative

*X*

^{′}

_{2}(

*y*) is

*X*

^{″}

_{2}(

*y*) is

*X*

_{2}(

*y*) into formula

*r*

_{2}(

*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

*X*

_{2}(

*y*) is

*r*

_{2}(0). It is calculated by substituting 0 for

*y*. When

*X*

_{2}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 }).

*X*

_{2}(

*y*) has an apical radius of curvature

*R*

_{2}, but the shape factor that describes its asphericity cannot be computed by the foregoing calculations, because

*X*

_{2}(

*y*) does not describe a conic section. However, a best-fit conic section,

*C*

_{2}(

*y*), with apical radius of curvature

*R*

_{2}and shape factor

*p*

_{2}can be calculated. We plotted multiple conic sections,

*C*(

*y*) with shape factor

*p*

_{c.}Substituting

*C*for

*X*in equation 4 (Fig. 3)

*p*

_{2}, we used two methods to minimize the sum of the absolute values of the residuals

*T*(

*p*

_{c}) and the sum of the squared residuals

*T*

_{s}(

*p*

_{c}).

*y*-axis from

*y*= 0 to

*y*=

*S*/2 = 3 mm, equally spaced by 0.01 mm. For a given

*R*

_{1,}

*D*, and

*p*

_{1},

*T*(

*p*

_{c}) and

*T*

_{s}(

*p*

_{c}) were iteratively calculated for values of

*p*

_{c}ranging from (

*p*

_{1}− 2) to (

*p*

_{1}+ 2) by incremental steps of 0.01. Solutions were represented by the value(s) of

*p*

_{c}that induced the smallest

*T*(

*p*

_{c}) and the smallest

*T*

_{s}(

*p*

_{c}). These sums were recorded and tabulated.

*p*

_{c}that corresponded to the lowest

*T*(

*p*

_{c}) was identical with that corresponding to the lowest

*T*

_{s}(

*p*

_{c}) value. The minimal value of

*T*(

*p*

_{c}) and

*T*

_{s}(

*p*

_{c}) corresponding to the same

*p*

_{c}for 31 points thus provided a value for

*p*

_{2}within our range of testing of ±0.01. The minimal values of

*T*(

*p*

_{c}) and

*T*

_{s}(

*p*

_{c}) were always less than 2 and 0.1 μm, respectively.

*p*

_{1}< 1), we found that prolateness increased (

*p*

_{2}<

*p*

_{1}< 1), whereas in initially oblate corneas (

*p*

_{1}> 1), oblateness increased (

*p*

_{2}>

*p*

_{1}> 1) within the treated zone after myopia treatment. Spherical corneas remained spherical (

*p*

_{2}=

*p*

_{1}= 1) after treatment. The slope of asphericity change was not constant but increased with the magnitude of treatment.

*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.

^{ 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 used

^{ 3 }

^{ 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).

^{ 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) .

^{ 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 }

^{ 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.

^{ 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.

^{ 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 }Douthwaite

^{ 32 }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.

^{ 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.

*p*

_{2}. 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

*p*

_{2}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.

^{ 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.

^{ 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.

^{ 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.

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

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