Bill,

I am using the Google format.  If you would look at this thread as a
tree, you would see that I was replying to your post.  I simply hit the
"reply" to your post, a window appears, and I dutifully type away.  So,
don't blame it on me, blame it on Google.

DrG

J Refract Surg. 2004 Sep-Oct;20(5):S581-5. Related Articles, Links  

Corneal asphericity and retinal image quality: a case study and
simulations.

Somani S, Tuan KA, Chernyak D.

VISX Incorporated, Santa Clara, CA 95051, USA. seemas@visx.com

PURPOSE: The optical quality of retinal images is dependent on the
refracting elements of the eye including the nominally aspheric cornea
and crystalline lens. This paper presents a retrospective theoretical
analysis of the impact of corneal asphericity on the quality of
retinal images. Clinical data are from the VISX Incorporated CustomVue
IDE. METHOD: Topography, contrast sensitivity, and visual acuity data
were collected from 278 myopic eyes before and after wavefront-guided
laser surgery. The measured corneal surface of each eye was fitted to
a conic, and a Q-value was computed for a 5.5-mm pupil. A model eye
was used to simulate various amounts of optical asphericity. RESULTS:
Preoperatively, most corneas exhibited negative conic shape constants.
Postoperatively, corneas were about equally divided between positive
and negative conics. There was no statistically significant
correlation between the shape of the cornea and the subjects'
perceptions of image quality including contrast sensitivity and visual
acuity. Simulations showed that the corneal Q-value can vary from more
to less prolate depending upon the shape of the internal surface.
CONCLUSION: Following wavefront-guided laser in situ keratomileusis
(LASIK), contrast sensitivity is usually good and is not dependent
upon the corneal conic shape. Better visual outcomes are more likely
with a customized shape than a standard best conic shape.

PMID: 15523979 [PubMed - indexed for MEDLINE]

~~~~~~~~~~~~~~~~~~~~

Arch Soc Esp Oftalmol. 2004 Aug;79(8):385-92. Related Articles, Links


[Corneal asphericity in a young adult population. Clinical
implications]

[Article in Spanish]

Yebra-Pimentel E, Gonzalez-Jeijome JM, Cervino A, Giraldez MJ,
Gonzalez-Perez J, Parafita MA.

Universidad de Santiago de Compostela, A Coruna, Espana.

PURPOSE: To determine the relevance of the different ocular optical
components in the refractive state of young adults, paying special
attention to the corneal topography represented by the asphericity
value. SUBJECTS AND METHODS: Corneal topographies and ultrasonic
biometries were obtained from 109 university students with different
refractive errors (spherical equivalent range: +3.25 D to -11.00 D). A
regression study was performed in order to establish the relationships
between corneal asphericity and refractive error, as well as other
ocular optical components related to the emmetropization mechanism of
the eye. RESULTS: The mean asphericity values were -0.23 (SD 0.08,
range: -0.42 to -0.03). All the values correspond to the mathematical
description of the prolate ellipse, most commonly accepted for the
normal human cornea. The statistical correlation between asphericity
and equivalent refractive error was not significant, but a significant
correlation was found for the asphericity with respect to the radius
of curvature, vitreous chamber depth and axial length. CONCLUSIONS: 1)
The asphericity values support the generalised morphology of the
prolate cornea as the standard. The influence of this configuration on
the contact lens fit, refractive surgery or the visual performance of
the eye are discussed. 2) Results suggest that, although a
relationship between axial length and corneal topography actually
exists, it is not likely that the latter has implications for the
emmetropization mechanisms which determine the refractive state of the
adult eye.

PMID: 15306965 [PubMed - indexed for MEDLINE]

~~~~~~~~~~~~~~~~~~~~

J Fr Ophtalmol. 2002 Jan;25(1):81-90. Related Articles, Links  


[A review of mathematical descriptors of corneal asphericity]

[Article in French]

Gatinel D, Haouat M, Hoang-Xuan T.

Service d'Ophthalmologie, Fondation Ophtalmologique A. de Rothschild,
Hopital Bichat Claude-Bernard, Universite Paris VII.

PURPOSE: 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. MATERIAL AND
METHODS: Through an extensive review of the literature, we expose the
principles, the population variations and report the application of
such corneal modeling. RESULTS: 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. CONCLUSION:
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.

Publication Types:
Review
Review, Tutorial

PMID: 11965125 [PubMed - indexed for MEDLINE]

Glenn Hagele
Executive Director
USAEyes.org

"Consider and Choose With Confidence"

Email to glenn dot hagele at usaeyes dot org

http://www.USAEyes.org
http://www.ComplicatedEyes.org

I am not a doctor.

No.  Prolate just means the (solid) geometry of the surface is more
closely approximated by rotating an ellipse around it's major axis (the
longer one), while oblate means it is approximated by rotating the
ellipse around its minor (shorter) axis.  It really doesn't say much
about the spherical aberration involved, which really depends on
curvature, pupil size, and other factors.
And just saying "prolate" is pretty meaningless, IMO, without
specifying  the mathematical details.

I wish they'd never started using the terms.  But it makes people sound
like they're smart.

Suffice it to say that the human eye is an amazing design, and yes, the
cornea is normally somewhat prolate, providing just about the right
amount of spherical aberration to be neutralized by the crystalline
lens.  i.e., an almost perfect design.  Mess with the curvatures, and
you almost always end up with more aberration than you started with  
(the promises of super vision with wave front analysis haven't really
materialized).

clear as mud?

w.stacy, o.d.