4.1 POPULATION-BASED EYE MODELING
4.1.4 General Keratoconus Models
KC is a degenerative non-inflammatory disorder of the eye for which structural changes occur within the cornea that result in thinning of the cornea and change to a more conical shape than its normal gradual curve. KC can cause substantial distortion of vision, with multiple images, streaking and sensitivity to light all often reported by the patients. KC is the most common dystrophy of the cornea and affects around one person in a thousand. It seems to occur in populations throughout the world, although some ethnic groups experience a greater prevalence than others. It is typically diagnosed in the patient's adolescent years and presents as a more severe state in the twenties and thirties. KC is a little-understood disease with an uncertain cause, and its progression following diagnosis is unpredictable. If both eyes are affected, the deterioration in vision can affect the patient's ability to drive a car or read normal print. In most cases, special corrective lenses are effective enough to allow the patient to continue to drive legally and likewise function more normally. Further progression of the disease may require surgery including transplantation of the cornea. However, despite its uncertainties, KC can be successfully managed with a variety of clinical and surgical techniques to lessen significantly the impairment to the patient's quality of life.
One purpose of KC modeling is to study and understand the influence of the properties of the KC cone(s) on the optical performance of human eyes. With the general KC eye models, the effects and visual impacts of different parameters of KC cone, such as the cone location, volume, and shape, were investigated. The research results of this subject have been published in the online journal, Journal of Vision in 2008 [Tan 2008]. In the general KC eye modeling in this paper, the anterior cornea surface is assumed to be the only affected ocular element. The diseased condition of cornea is engineered on the Navarro healthy adult model. Although the thickness of cornea and the posterior surface are also influenced by this disease, they are ignored in the pioneer modeling because of the relatively weak optical impact owing to the smaller refractive index difference on the interface.The optical influence of irregular posterior surface was estimated 10–20% of the anterior influence due to the smaller refractive index difference.
The corneal topography of patients can be measured clinically as described in Chapter 2. The elevation maps of anterior corneal surface can be exported from the ophthalmic devices and used for mathematical analysis. This is described in Appendix B for one of the most common topography device, the Humphrey. The similar method was used by Schwiegerling et al. to examine the resulting KC cone from the topographical map [Schwiegerling 1995]. The height maps from 56 KC eyes were decomposed into Zernike polynomials. Then the parabolic (C2
0 Z2
0
) and the cylindrical (C2 +2 Z2 +2 and C2 -2 Z2 -2 ) components were eliminated to yield a residual height map [Schwiegerling 1997, and Schwiegerling 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, (xo, yo, σx,
σy, ho), from the Gaussian expression,
}
2
)
(
2
)
(
{
exp
)
,
(
2 2 0 2 2 0 0 y xy
y
x
x
h
y
x
f
, where ho is the peakheight of the cone, (xo, yo) is the cone‘s center location with respect to the visual axis, and (σx, σy) are the corresponding dimensions where the height drops to e1/2 of the cone‘s peak height. The full width at half
136
maximum of a Gaussian function is equal to 2.35σ. The 56 clinically diagnosed KC corneas‘ residual height maps were processed and each parameter‘s statistical distribution was reported [Schwiegerling 1997].
The 5-parameter elliptical Gaussian elevation is a simple assumption on KC cone structure. Although many KC cones have more complex shapes, the Gaussian surface fits well to a very good portion of KC cases. The more particular and complex cones that are not well modeled in this general KC cone modeling include significant asymmetric cones and cones with multiple peaks. These complex shapes can be mathematically modeled by adopting more shape parameters. Here we use the least number of parameters to enable the study on the comprehension of the optical influences of (a) cone location that requires at least 2 variables, (xo, yo), (b) cone shape that needs no less than 2 variables, (σx, σy), and (c) cone dimension that requires at least one additional variable, (ho).
To corroborate the KC statistics of Schwiegerling‘s 56 eyes, 15 additional KC topography maps from the Wang Vision Institute at Nashville, TN, were examined. These 15 KC cases include two cases with steepest corneal curvature less than 45 diopter, nine cases between 45 and 52 diopter, and four greater than 52 diopter. The statistical distributions of the five cone parameters from measurement and reported data were then adopted to model various KC cone dimensions and locations.
Four degrees of KC cones (mild, moderate, advanced, and severe) are created based on the statistical distribution of measured cone volumes. The volume enclosed by the two-dimensional Gaussian surface is given by
V
2
h
0 x y. The shape-correlated eccentricity e of the cross-sectional ellipse of semi-majorand minor axis, a and b, respectively, is 2 2
1
a
b
e
. The eccentricity always lies between 0≤ 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 1999]. The importing of user-defined surface to ZEMAX is described in Appendix B.Subsequent to the construction of the general KC models, three-dimensional ray tracing on KC eye models was performed to determine the resulting optical imaging quality. The spherical equivalent (SE), cylinder, together with residual high-order ocular aberrations, are examined and related to each separated variable.
Determination of the subsequent refractive error is achieved using optimization in ZEMAX. Similarly to the step in general ametropic eye model construction, a Gaussian thin lens with three variables, spherical equivalent, cylindrical power, and astigmatic axis, is placed in front of the optical model eye. These three values are set to be the iteration variables in the ray-tracing program to achieve optimized optical performance. It is noted that the wavefront aberration maps of KC patients are very irregular and that the high-order Zernike coefficients, including the m≠0 terms, are pronounced. Because of this, the simplified Zernike derivation methods (―Paraxial curvature matching‖ discussed in Chapter 3) [Atchison 2004, and Dorsch 1998] that use only the ρ2
- 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. In addition to the sphero-cylindrical prescription, the residual RMS wavefront aberration provides the measure of the high-order ocular aberration that causes the higher level of difficulty for the KC patient.
In the study and the published paper, the cone shape, protruding height and extent, and distance from the visual zone are independently investigated for how they influence the patient vision. This study demonstrates a novel and contemporary research application using the general population-based eye modeling technique.