Corneal elevation topography: best fit sphere, elevation distance, asphericity, toricity, and clinical implications

Abstract

Purpose: To describe the effect of the corneal asphericity and toricity on the map patterns and best fit sphere (BFS) characteristics in elevation topography.

Methods: The corneal surface was modeled as a biconic surface of principal radii and asphericity values of r1 and r2 and Q1 and Q2, respectively. The apex of the biconic surface corresponded to the origin of a polar coordinates system. Minimization of the squared residuals was used to calculate the values of the radii of the BFSs and apex distance (A-values: z distance between the corneal apex and the BFS) of the modeled corneal surface for various configurations relating to commonly clinically measured values of apical radius, asphericity, and toricity.

Results: Increased apical radius of curvature and increased prolateness (negative asphericity) led to an increase in BFS radius but had opposite effects on the A-value. Increased prolateness resulted in increased BFS radius and A-value. Increasing toricity did not alter these findings. Color-plot elevation maps of the modeled corneal surface showed complete ridge patterns when toricity was increased and showed incomplete ridge and island patterns when prolateness was increased.

Conclusions: High A-values in patients with corneal astigmatism may result from steep apical curvature and/or high prolateness (negative asphericity). The BFS radius may help in distinguishing between these 2 causes of increased A-values. Increased prolateness and decreased apical radius of curvature (often seen in keratoconus) have opposite effects on the BFS radius but similar effects on the apex distance.

Figure 1

Determination of the Best Fit radius (R) and distance to the apex (A) for a corneal profile modeled as a conic section of apical radius r and asphericity A, respectively.

Figure 2

Representation of the elevation map of a corneal surface modeled as a biconic. The flatter and steeper meridians have apical radii of r1 and r2 (r1>r2) and asphericity of Q1 and Q2, respectively. The color scale has a 5 micron steps. This scale is used for the Figures R5 and R6.

Figure 3

A: Influence of the apical radius of the corneal surface modeled as a revolution conic on the distance between the BFS to the apex. There is a non linear increase of the distance to the apex (anterior elevation) of the BFS radius with the decrease of the apical radius (apical steepening) of the surface. B: Influence of the asphericty of the corneal surface modeled as a revolution conic on the distance between the BFS to the apex. There is a non linear increase of the distance to the apex (anterior elevation) of the BFS radius with the increase of negative asphericity (increased prolateness) of the surface.

Figure 4

A: Influence of the apical radius of the corneal surface modeled as a revolution conic on the BFS radius. There is a linear increase of the BFS radius with the increase of the apical radius of the surface. B: Influence of the asphericty of the corneal surface modeled as a revolution conic on radius of the BFS. There is a linear increase of BFS radius of the BFS radius with the increase of negative asphericity (increased prolateness) of the surface.

Figure 5

Elevation plots of corneal surface of increasing negative asphericty (X axis) and apical curvature (Y axis). The top row represents surface plots corresponding to steep corneas (apical radius of curvature = 5.75 mm) Q= -0.2 (left), Q= -0.4, Q= -0.6, and Q= -0.8 (right). The plots in rows 2-4 represent surface plots of flatter corneas. All surfaces are rotationally symmetrical. The value of the distance of the apex to the sphere (microns, black font) and the radius of the BFS (mm, green font) are shown to the right of each surface plot.

Figure 6

Elevation plots of corneal surface of increasing negative asphericty (X axis) and apical toricity (Y axis). r2 is the apical radius of the vertical meridian and is expressed in mm (the corresponding dioptric power change from the horizontal meridian is shown in parenthesis) along the Y axis. The top row represents corneas with high astigmatism of 3.35 Diopters (r2=7.25 mm). The plots in rows 2-4 represent surface plots of corneas with less astigmatism (2D, 1.3D, and 0 D). The value of the distance of the apex to the sphere (microns, black font) and the radius of the BFS (mm, green font) are shown to the right of each surface plot.