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Corneal Asphericity Change after Excimer Laser Hyperopic Surgery: Theoretical Effects on Corneal Profiles and Corresponding Zernike Expansions

¹From the Rothschild Foundation, Paris University, Paris, France; the ²Institute of Statistics, Pierre et Marie Curie University, Paris, France; and the ³Massachusetts Eye and Ear Infirmary and Schepens Eye Research Institute, Harvard Medical School, Boston, Massachusetts.

PURPOSE. To determine the theoretical relationships between the changes in corneal paraxial power, asphericity, and the corresponding Zernike polynomial expansion after conventional and customized excimer laser correction of hyperopia.

METHODS. The initial corneal profile was modeled as a conic section of apical radius of curvature R₁ and asphericity Q₁. The theoretical value of the postoperative apical radius of curvature R₂ was computed by using a paraxial formula from the value of R₁ and hyperopic defocus D. The postoperative asphericity Q₂ of the corneal surface was computed within the optical zone of diameter S after the delivery of a Munnerlyn-based profile of ablation for hyperopia using conic section-fitting and minimization of the squared residuals. These calculations were repeated for different values of defocus, initial apical radius of curvature, and asphericity. Taylor series expansions were also used to provide an approximation aimed at predicting change in asphericity. The coefficients of a Zernike polynomial expansion of the rotationally symmetrical corneal profile (defocus C₂⁰, spherical aberration C₄⁰, secondary spherical aberration C₆⁰) were also computed, by using scalar products applied to the considered corneal profile modeled as a conic section and were expressed as a function of both its apical radius and asphericity. This allowed approximation of the variations of the Zernike polynomial expansion of the corneal profiles by subtracting the postoperative coefficient weighting a particular aberration from that of the preoperative one in different theoretical situations, after both conventional and customized hyperopia treatments aimed at controlling the postoperative corneal asphericity and delivered over a normalized pupil diameter.

RESULTS. Conical least-squares fitting was unambiguous, allowing approximation of the postoperative corneal profile as a conic section of apical radius R₂. After a Munnerlyn-based hyperopia treatment, the sign of the asphericity of this profile remains theoretically unchanged, but its value decreased for initially oblate and increased for initially prolate corneas, respectively. A similar trend was noted with the approximation obtained by the Taylor series expansion. The alteration of the apical radius and/or of the asphericity of the corneal surface resulted in variations of both the corneal profile Zernike coefficients C₂⁰ and C₄⁰. The former was essentially dependent on the variation of the apical radius and the latter essentially on the variation of both apical radius and asphericity.

CONCLUSIONS. Conventional and customized profiles of ablation for hyperopia alter the postoperative corneal asphericity and the Zernike coefficients of the corneal profile. The results of this study may be useful in the interpretation of the postoperative variations of the corneal profile and their impact on corneal wavefront expansion variations after both conventional and customized profiles of ablation.