Using a statistical description of keratoconus (KC) topography, schematic eye models of various KC conditions are constructed to study their optical influence on visual performance. The cone shape, protruding height and extent, and distance from the visual zone are independently investigated with the three-dimensional optical eye-modeling and ray-tracing techniques. The subsequent spherical equivalent (SE), cylinder, together with residual high-order ocular aberrations, are examined and related to each separated variable. The results show that myopic nature of SE is greatly dominated by the location of the cone. The cylinder is determined by the cone shape when the cone is inside the visual zone. It is dominated by the cone location when the cone is away from visual axis. The least myopic meridian always falls on the cone direction, and the high-order aberrations strongly relate to the cone dimension. This study investigates KC cone effect on optical quality and provides comprehension of clinical observations.

*Z*

_{ n}

^{ m}} and then eliminating the low-order polynomials that represent the defocus (near- and farsightedness) and cylindrical power (astigmatism).

*C*

_{2}

^{0}

*Z*

_{2}

^{0}) and the cylindrical (

*C*

_{2}

^{+2}

*Z*

_{2}

^{+2}and

*C*

_{2}

^{−2}

*Z*

_{2}

^{−2}) components were eliminated to yield a residual height map (Schwiegerling, 1997; Schwiegerling et al., 1995). Corneas with normal refractive errors appear to have relatively flat residual maps. In contrast, a KC cornea's residual map reveals more significant high-order Zernike terms, which represent the irregular surface of the KC cone. After the cones' surfaces were obtained, they were fitted to two-dimensional Gaussian surfaces to define the sizes and positions of the assumed right elliptical cones. This allows an accurate optical KC cornea model to be constructed based on the 5 cone parameters, (

*x*

_{o},

*y*

_{o},

*σ*

_{x},

*σ*

_{y},

*h*

_{o}), from the Gaussian expression (Equation 1):

*h*

_{o}is the peak height of the cone, (

*x*

_{o},

*y*

_{o}) is the cone's center location with respect to the visual axis, and (

*σ*

_{x},

*σ*

_{y}) are the corresponding dimensions where the height drops to

*e*

^{−1/2}of the cone's peak height. The full width at half maximum of a Gaussian function is equal to 2.35

*σ*. The 56 clinically diagnosed KC corneas' residual height maps were processed and each parameter's statistical distribution was reported (Schwiegerling, 1997).

*x*

_{o},

*y*

_{o}), (b) cone shape that needs no less than 2 variables, (

*σ*

_{x},

*σ*

_{y}), and (c) cone dimension that requires at least one additional variable, (

*h*

_{o}).

KC cones | # | h _{o} (mm) | σ _{ x} (mm) | σ _{ y} (mm) | Volume, V (mm ^{3}) | V ^{1/3} (mm) | Eccentricity e |
---|---|---|---|---|---|---|---|

Mild V < 0.02 mm ^{3} | 1 | 0.0051 | 0.4183 | 0.4729 | 0.0063 | 0.1848 | 0.467 |

2 | 0.0087 | 0.4348 | 0.5718 | 0.0136 | 0.2389 | 0.649 | |

3 | 0.0090 | 0.5170 | 0.4960 | 0.0146 | 0.2442 | 0.282 | |

Moderate Volume .02–0.1 mm ^{3} | 4 | 0.0101 | 0.7323 | 0.6944 | 0.0323 | 0.3185 | 0.317 |

5 | 0.0118 | 0.6581 | 0.7755 | 0.0380 | 0.3361 | 0.529 | |

6 | 0.0156 | 0.6417 | 0.6008 | 0.0377 | 0.3354 | 0.351 | |

7 | 0.0200 | 0.8000 | 0.8000 | 0.0804 | 0.4316 | 0.000 | |

Advanced Volume 0.1–0.4 mm ^{3} | 8 | 0.0246 | 1.1821 | 0.8553 | 0.1561 | 0.5385 | 0.690 |

9 | 0.0269 | 0.9700 | 0.8823 | 0.1447 | 0.5249 | 0.415 | |

10 | 0.0296 | 1.1606 | 0.8822 | 0.1907 | 0.5756 | 0.650 | |

11 | 0.0400 | 1.2000 | 1.2000 | 0.3619 | 0.7126 | 0.000 | |

Severe V > 0.4 mm ^{3} | 12 | 0.0410 | 1.7380 | 1.0590 | 0.4746 | 0.7800 | 0.793 |

13 | 0.0507 | 1.7013 | 1.0280 | 0.5568 | 0.8227 | 0.797 | |

14 | 0.0541 | 1.7629 | 1.0309 | 0.6180 | 0.8518 | 0.811 |

*e*< 1. An eccentricy

*e*= 0 corresponds to a circular cone, and as

*e*increases the cone becomes more elliptical. The synthetic anterior KC corneal surface is generated by superimposing the Gaussian surface onto a normal corneal surface of the emmetropic eye model (Escudero-Sanz & Navarro, 1999). Although the posterior corneal surface is also affected in KC patients, the posterior irregularity was omitted in the modeling. The optical influence of irregular posterior surface was estimated 10–20% of the anterior influence due to the smaller refractive index difference.

*m*≠ 0 terms, are pronounced. Because of this, the simplified Zernike derivation methods (Atchison, 2004; Dorsch, Haimerl, & Esser, 1998) that use only the

*ρ*

^{2}- Zernike terms do not provide adequate results for KC cases. We find that the optimization method provides stable, converged results that are significantly different from the Zernike-derived prediction. The iteration is carefully examined over the 180-deg meridians to prevent convergence of local minimum.

*RMS*_

*W*

^{2–8}is related to best-corrected visual acuity and is used, in addition to the refractive error, to evaluate the optical quality of KC vision.

*μ*m. The cross in each image indicates

*σ*

_{ x}and

*σ*

_{ y}. Cones #7 and #11 represent the circular cones.

*y*

_{o}= −0.9 ± 0.5 mm) in the temporal quadrant (

*x*

_{o}= 0.4 ± 0.7 mm) as indicated in green. A far location at (

*x*

_{o}= 1.1 mm,

*y*

_{o}= −1.4 mm), is also indicated (in blue). The KC cones on the three marked locations are investigated as will be discussed later.

*σ*distance for the 3.5- and 5-mm pupil diameters. At the very far locations, KC cones result in mild hyperopic conditions for this same pupil-size range. This occurs because the KC cone curvature (the second derivative of elevation) determines the supplementary power on the cornea surface. The contribution from outside the

*2σ*zone of a cone is hyperopic instead of myopic. This effect is clearly seen for the 3.5- and 5-mm pupil diameters, and the trend is noted for the 7-mm pupil case of Figure 3. Also, as Figure 3 shows, the cylinder power increases and then decreases with the cone distance for all pupil sizes studied. For all but the largest pupil diamneters, the locations of maximum cylinder error correspond to approximately the zero-SE locations.

*σ,*and twice that size is a good measure of the extent of the cone base on the cornea. The HOA results indicate that the maximum aberration exists when the cone is, of course, off-center and the curvature of the cone overlaps most of the visual zone. It is seen that the HOA is on the order of 0.1–0.6 to 0.3–1.5 and 0.3–2.0

*μ*m for 3.5–, 5-, and 7-mm pupil, respectively. This level of KC aberration is significant compared to that for normal healthy adults (n = 2,560) of 0.10 ± 0.044

*μ*m at 4-mm pupil and 0.19 ± 0.08

*μ*m at 5-mm pupil, and 0.50

*μ*m at 7-mm pupil (Salmon & van de Pol, 2006).

_{//}. The computation results prove the situation. The power difference (S

_{//}−S

_{⊥}) between the major and minor meridians represents the cylindrical error. Intuitively, one might expect that both meridians would contribute positive power (myopic) and that the power along the cone direction is greater than along the perpendicular direction, that is, ∣S

_{//}∣ > ∣S

_{⊥}∣. Figure 4 illustrates S

_{//}, S

_{⊥}, and SE = (S

_{//}+ S

_{⊥})/2 of the three circular cones. The three KC cones exhibit similar behavior. When located at the visual center, S

_{//}= S

_{⊥}. For the cones located peripherally, the myopic S

_{//}reduces to zero and becomes hyperopic. As the cones are moved farther from the center, S

_{//}exhibits a maximum of hyperopic condition and then reduces again and approaches zero according to the corneal curvature contribution. On the other hand, S

_{⊥}is a monotonic myopic function of cone distance. Note that the myopic power along the cone direction is always less than the power in the perpendicular meridian for all circumstances. This theoretical prediction is in sound agreement with our clinical observation in all 15 KC cases.

*h*

_{o}:

*σ*

_{ x}:

*σ*

_{ y}= 0.1: 4.29: 3.26 (Schwiegerling, 1997) over the range of volume from 0 to 8.78 mm

^{3}are studied. Three vertical lines in the center plots define the mild, moderate, advanced, and severe KC dimensions as volume values given in Table 1. The vertical lines correspond to the dimension (

*Volume*

^{1/3}) = 0.27, 0.46, and 0.74 mm.

^{3}and a constant cone height of 24

*μ*m (medium size) were used for all calculated cases while the ratio

*σ*

_{ x}/

*σ*

_{ y}of cones varied from 2.5/1 to 1/2.5. As we observed before, the on-axis cones (red line) have the worst myopic conditions and the greatest shape influence on their cylinder. However, their high-order aberrations are considerably smaller. For the distantly located cones (blue line), the dioptric contribution to the visual central area is negative and farsightedness results from the cone in this specified size range, regardless of its shape. Due to the far distance from visual center, cone shape does not significantly affect cylindrical powers and the high-order aberrations. For the cones at the average location (green line), the spherical equivalent power has a strong coupling effect between the shape and location. The shape slightly affects the cylindrical power. Within the worst cone volume range and at a bad location, the high-order aberration is dramatic. Moderate-to-advanced-sized cones located at about average locations with irregular shape have the worst high-order aberrations.