Determination of corneal asphericity after myopia surgery with the excimer laser: a mathematical model

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.

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.

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).

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.