[A review of mathematical descriptors of corneal asphericity]

Corneal asphericity may be modeled on a conic section which can be described by the apical radius of curvature in the meridian studied and by a measure of the degree of asphericity.

Through an extensive review of the literature, we expose the principles, the population variations and report the application of such corneal modeling.

The aspheric anterior corneal surface can be described by a conic section, defined by its radius of curvature and by a parameter measuring asphericity. We analyse the various parameters used in the literature to determine their usefulness. Conic sections, obtained by cutting a cone by a plane, include ellipses, hyperbolas and parabolas. Two useful parameters are the apical radius of the ellipse and its eccentricity defined in Cartesian terms by a second order equation where the apical radius is R and the eccentricity is e: The apical radius is that of the circle tangent to the apex of the conic section and e describes the variation of this curve with distance from the corneal apex. Baker introduced the form factor p making the equation: with It is easier to understand the effect of alteration of p than of e on corneal curvature: There is a relation between the horizontal, a, and the vertical, b, hemi-axes and R The advantage of this notation is that e(2) can be greater than 1 When p=0 the conic section is a parabola, when p<0 it is a hyperbola. Kiely et al. studied corneal asphericity by photokeratoscopy and introduced the parameter Q, where Q=p-1. Q, the asphericity factor, is used by the Eyesis and Orbscan systems; when Q=0 the cornea is spherical. Thus different parameters describe variations in corneal curvature along any meridian. Average anterior corneal asphericity using various keratometric systems is p=0.8, making the corneal section a prolate ellipse. However there is great individual variation, 20% of normals exhibiting oblate (p>1), paraboloid (p=0) or hyperbolic (p<0) corneas. all becoming more spherical with age. Little connection between asphericity and ametropia is reported, except for a tendency to flattening in myopia and towards oblateness in progressive myopia. Direct measurement of denuded cadaver corneas gave a prolate elliptical profile although calculation after deduction of epithelial thickness measured by ultrasonic biomicroscopy suggested p=-0.22, a hyperbolic profile. The few reports on the posterior surface suggest it to be hyperbolic or prolate. Increasing distance from the corneal apex worsens the comparison to a conic section as flattening increases. Precision can be improved by adding polynomial coefficients above the second degree to the equation of the section: The non-toric 3D corneal surface can be described by the following equation for the revolution of a conic section about the optic axis: x(2)+y(2)+pz(2)-2rz=0 where z is the axis of revolution. Since the mean value of p is 0.8 this corresponds to a sphere stretched along one axis, as is a rugby ball. Each meridian has the same radius of curvature and the value of p is constant. For a toric cornea the radius and value of p must be defined for two meridia at right angles. This corresponds to an elongation on an axis different from that of revolution. Similarly a toric ellipsoid is generated by rotation of an arc around an axis at right angles to its elongation. Because of its asphericity, representation of the corneal surface depends on the direction in which its curvature is measured: In the ellipsoidal model this depends on the principal meridians, the tangential, in the plane of the axis of symmetry and the saggittal, perpendicular to this. These may define two radii of curvature, the saggital (axial) and the tangential. Most algorithms assume these properties of ellipsoids. Asphericity is translated into variations in radius of curvature from apex to periphery, increasing for a flat periphery, decreasing for a steep one. Associated to toricity, it gives rise to the common butterfly pattern. Spherical aberration is minimal through a small pupil but becomes significant the larger the aperture, with deterioration of image quality. Raytracing allows analysis of the effects of non-axial rays. The mean value of Q, at -0.26 thanks to the naturally prolate asphericity of the cornea reduces spherical aberration by half. The relaxed form of the crystalline lens further reduces it by inducing the opposite spherical aberation to that of the cornea. This is important in accommodation and presbyopia. The use of an aspheric corneal surface in the schematic eye allows calculation of the ideal asphericity, which corresponds quite well with clinical findings. Radial keratotomy reverses the natural asphericity of the cornea. Photorefractive keratotomy (PRK) also changes asphericity, Q increasing to an oblate value. These changes might increase spherical aberration, explaining some postoperative deficiencies. Current excimer laser protocols ignore asphericity, relying on paraxial algorithms alone. New strategies to control asphericity in order to diminish spherical aberration may offer solutions. The original conic section models were made to improve the geometry of contact lenses. Understanding of asphericity is important in adaptation after refractive surgery. Modification of spherical aberration by contact lenses and corneal warpage induced by rigid lenses have also been studied.

The approximation of the corneal surface by a conic section allows understanding of corneal asphericity and offers a quantitative description. This allows a more precise description of the corneal surface and of the genesis of certain optical aberrations of the eye.