(Investigative Ophthalmology and Visual Science. 2001;42:1736-1742.)
© 2001
by The Association for Research in Vision and Ophthalmology, Inc.
Determination of Corneal Asphericity after Myopia Surgery with the Excimer Laser: A Mathematical Model
Damien Gatinel1,
Thanh Hoang-Xuan1 and
Dimitri T. Azar1,2,3
1 From the Rothschild Foundation, Paris, France; and the
2 Massachusetts Eye and Ear Infirmary and
3 Schepens Eye Research Institute, Harvard Medical School, Boston, Massachusetts.
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Abstract
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PURPOSE. To determine the theoretical change of corneal asphericity within the
zone of laser ablation after a conventional myopia treatment, which
conforms to Munnerlyn’s paraxial formula and in which the initial
corneal asphericity is not taken into consideration.
METHODS. The preoperative corneal shape in cross section was modeled as a conic
section of apical radius R1 and shape factor
p1. A myopia treatment was simulated, and
the equation of the postoperative corneal section within the optical
zone was calculated by subtracting the ablation profile conforming to a
general equation published by Munnerlyn et al. The apical radius of
curvature r2 of the postoperative profile
was calculated analytically. The postoperative corneal shape was fitted
by a conic section, with an apical radius equal to
r2 and a shape factor
p2 equal to the value that induced the
lowest sum of horizontal residuals and the lowest sum of squared
residuals. These calculations were repeated for a range of different
dioptric treatments, initial shape factor values, and radii of
curvature to determine the change of corneal asphericity within the
optical zone of treatment.
RESULTS. Analytical calculation of r2 showed it to be
independent of the initial preoperative shape factor
p1. The determination of
p2 was unambiguous, because the same value
induced both the lowest sum of residuals and the lowest sum of the
squared residuals. For corneas initially prolate
(p1 < 1), prolateness increased
(p2 <
p1 < 1), whereas for oblate corneas
(p1 > 1), oblateness increased
(p2 >
p1 > 1) within the treated zone after
myopia treatment. This trend increased with the increasing magnitude of
treatment and decreased with increasing initial apical radius of
curvature R1.
CONCLUSIONS. After conventional myopic excimer laser treatment conforming to
Munnerlyn’s paraxial formula, the postoperative theoretical corneal
asphericity can be accurately approximated by a best-fit conic section.
For initially prolate corneas, there is a discrepancy between the
clinically reported topographic trend to oblateness after excimer laser
surgery for myopia and the results of these theoretical
calculations.
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Introduction
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In both photorefractive keratectomy (PRK) and laser in situ
keratomileusis (LASIK) for myopia, flattening of the central corneal
curvature due to tissue photoablation results in decreased refractive
power. Although considered proprietary, current excimer laser
algorithms rely on the pioneering theoretical work of Munnerlyn et
al.1
They predict the change in corneal power by
considering the initial unablated and the final ablated corneal surface
as two spherical surfaces, with a single but different radius of
curvature.
Given that the corneal surface is aspheric, the corneal shape in cross
section can be approximated by a conic section.2
3
4
5
6
7
The
asphericity of the cornea is then defined by the shape factor of the
conic section that approximates it most closely. A high percentage of
corneas are prolate.2
3
7
8
9
After PRK for myopia, a
change from a prolate conformation to an oblate optical contour has
been reported.10
11
12
13
To our knowledge, there have been no
reports that address the theoretical change in asphericity induced by
excimer laser treatment for myopia.
We attempted to predict the theoretical change of corneal asphericity
within the optical zone after myopia treatment, conforming with the
work of Munnerlyn et al.1
We developed a mathematical
model based on a conic section approximation, enabling prediction of
theoretical postoperative asphericity, and investigated the influence
of preoperative asphericity, magnitude of correction, and radius of
curvature on its outcome.
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Materials and Methods
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Current excimer laser ablations for treating myopia rely on the
pioneering work of Munnerlyn et al.,1
in which the initial
and final corneal surfaces are assumed to be spherical. This allows
calculation of the ablation profile by the following general formula
(Fig. 1)
 | (1) |
where t(y) expresses the depth of
tissue removal as a function of the distance y from the
center of an optical zone diameter of S when
R1 and
R2 are the initial and final corneal
anterior radii of curvature, respectively. The power of the removed
lenticule (D) corresponds to the intended refractive change
and is related to R1,
R2, and the index of refraction
(n) as follows
 | (2) |
where R2 is more than
R1 for ablations of myopia.
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Figure 1. Model proposed by Munnerlyn et al.1
for the ablation
profile in the excimer laser treatment of myopia. The investigators
predict the change in corneal power by treating the initial unablated
and the final ablated corneal surfaces as two spherical diopters, each
having a single but different radius of curvature, respectively
R1 and R2. This
model enables the general formula (equation 1)
to calculate the
ablation profile t(y) as a function of
distance to the center of the optical zone of diameter S.
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In our theoretical model, we made the following assumptions: The
initial corneal surface is rotationally symmetric, and the plane curve
of a corneal meridian is a conic section with its apex located at the
origin of rectangular x, y coordinates in a
system of Cartesian axes.
A conic section can be described mathematically by Baker’s
equation14
 | (3) |
where x and y are the coordinates on a Cartesian system with the
axis of revolution, R the apical radius of curvature, and
p the shape factor. When p is less than 1, the
ellipse is prolate and flattens from the center to the periphery. When
p equals 1, the ellipse is a circle. When p is
more than 1, the ellipse is oblate and steepens from the center to the
periphery.
For our purpose, it is more useful to evaluate x in terms of
y. Solving equation 3
for x gives
 | (4) |
When a correction of D diopters is simulated using
equation 1
of Munnerlyn et al.1
on a cornea modeled as a
conic section of apical radius R1 and
shape factor p1, within an optical
zone diameter S, the resultant curve
X2 is derived from the following
equation
 | (5) |
 | (6) |
This equation does not describe a conic section (Fig. 2)
. However, the radius of curvature for each point of the curve
X2 is given by
r2(y), which can be
computed as follows
 | (7) |
This formula gives the radius of the osculating circle
r2 at any point of the curve
(X2). The first derivative
X'2(y) is
 | (8) |
and the second derivative
X''2(y) is
 | (9) |
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Figure 2. Ablation profile for myopia conforming to Munnerlyn’s equation applied
on an aspheric surface modeled in cross section as a conic section
X1(y) of apical radius equal
to R1. The postoperative cross section
X2(y) is equal to the
addition of the initial aspheric profile
X1(y) and the
ablation profile t(y).
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After inserting respective first and second derivatives of
function X2(y) into
formula r2(y), the
radius of curvature can be calculated by substituting
X'2(y) and
X''2(y) in
equation 7
. The apical radius of curvature of
X2(y) is
r2(0). It is calculated by
substituting 0 for y. When
 | (10) |
Thus, the radius of curvature of
X2 at the apex (apical radius of
curvature) is the same as the final radius of curvature, which is
derived from equation 1
(Munnerlyn et al.1
).
Thus, X2(y) has an
apical radius of curvature R2, but the
shape factor that describes its asphericity cannot be computed by the
foregoing calculations, because
X2(y) does not
describe a conic section. However, a best-fit conic section,
C2(y), with apical
radius of curvature R2 and shape
factor p2 can be calculated. We
plotted multiple conic sections, C(y) with
shape factor pc. Substituting
C for X in equation 4
(Fig. 3)
 | (11) |
To determine the best-fit conic section with shape factor
p2, we used two methods to minimize
the sum of the absolute values of the residuals
T(pc) and the sum of the
squared residuals
Ts(pc).
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Figure 3. Approximation of the postoperative profile
X2i(y) by a conic
section C(y) of apical radius of
curvature R2 and shape factor
pc.
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We developed a numerical procedure and performed computation on a
computer spreadsheet (Excel 97 software; Microsoft, Seattle, WA). We
defined 31 points along the hemi y-axis from
y = 0 to y = S/2 = 3
mm, equally spaced by 0.01 mm. For a given
R1, D, and
p1,
T(pc) and
Ts(pc)
were iteratively calculated for values of
pc ranging from
(p1 - 2) to
(p1 + 2) by incremental steps of 0.01.
Solutions were represented by the value(s) of
pc that induced the smallest
T(pc) and the smallest
Ts(pc).
These sums were recorded and tabulated.
Referring to the geometric model of the laser excimer treatment
operation for myopia defined herein, we were able to repeat these
calculations for a range of different dioptric treatments, initial
shape factor values, and radii of curvature. The numerical values of
the selected variables are listed in Table 1
.
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Results
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In the theoretical conditions that we used to determine the
best-fit conic section, the pc that
corresponded to the lowest
T(pc) was identical with
that corresponding to the lowest
Ts(pc)
value. The minimal value of
T(pc) and
Ts(pc)
corresponding to the same pc for 31
points thus provided a value for p2
within our range of testing of ±0.01. The minimal values of
T(pc) and
Ts(pc)
were always less than 2 and 0.1 µm, respectively.
Figure 4
represents the effect of the myopia treatment on initial corneal
asphericity. In corneas that were initially prolate
(p1 < 1), we found that prolateness
increased (p2 <
p1 < 1), whereas in initially oblate
corneas (p1 > 1), oblateness
increased (p2 >
p1 > 1) within the treated zone after
myopia treatment. Spherical corneas remained spherical
(p2 =
p1 = 1) after treatment. The slope of
asphericity change was not constant but increased with the magnitude of
treatment.
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Figure 4. Outcome of asphericity for different magnitudes of correction. The
values of p2 are plotted against the
magnitude of the correction for different values of
p1. For corneas that are initially prolate
(p1 < 1), prolateness should increase
(p2 <
p1 < 1) after myopia treatment,
whereas in oblate corneas (p1 > 1),
oblateness should increase (p2 >
p1 > 1) within the optical zone after
myopia treatment. Spherical corneas remain spherical
(p1 =
p2 = 1) after treatment.
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Figure 5
shows the effect of the initial radius of curvature on the asphericity
outcome after treatment of prolate corneas. For the same treatment
parameters, steeper prolate corneas (R = 7.5 mm) tend
to become less prolate than flatter prolate corneas (R = 8.1 mm). This effect increases with the magnitude of the treatment.
Figure 6
shows the effect of initial radius of curvature in oblate corneas. For
the same magnitude of treatment, steeper oblate corneas tend to become
less oblate and flatter oblate corneas more oblate.
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Figure 5. Influence of the initial apical radius of curvature on
p2, which is plotted against the magnitude
of treatment for three different apical radii: 7.5, 7.8, and 8.1 mm.
The initial corneal surface is prolate. Steeper prolate corneas tend to
become less prolate and flatter prolate corneas more prolate.
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Figure 6. Influence for the initial apical radius of curvature on
p2. The initial corneal surface is prolate.
Steeper oblate corneas tend to become less oblate and flatter oblate
corneas more oblate.
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Discussion
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Defocus correction is the objective of conventional laser
refractive procedures based on the formula of Munnerlyn et
al.,1
which assumes that preoperative corneal surface has
a single radius of curvature. The normal human cornea is not spherical,
and, despite its shortcomings, modeling the corneal shape in cross
section as a conic section is a better approximation and has been
widely used3
4
5
6
15
since its introduction by Mandell and
St. Helen in 1971.2
Most human normal corneas conform to a
prolate ellipse and flatten from the center to the periphery (negative
asphericity; p < 1), but some corneas are oblate and
steepen from the center to the periphery (positive asphericity;
p > 1).
Based on topographic observations that the central cornea becomes
flatter than the untreated peripheral cornea after excimer laser
surgery for myopia, it could be erroneously predicted that the
postoperative outcome in all myopic corneas after excimer laser surgery
would be increased oblateness. Such a prediction does not take into
consideration the changes of corneal curvature within the treatment
zone. The focus of our study was to examine the postoperative changes
of corneal asphericity within the treatment zone without consideration
of the transition zone or the peripheral zone outside the treatment.
We have demonstrated that in prolate and oblate corneas, laser
treatments for myopia based on the equation of Munnerlyn et
al.1
result in a final apical radius of curvature that is
independent of the initial asphericity (equation 10)
. Furthermore, we
have demonstrated that in prolate corneas these laser treatments result
in more prolate configuration within the area of treatment (Fig. 5)
and
conversely in oblate corneas, the outcome is increased oblateness (Fig. 6)
.
In initially prolate corneas, there is a discrepancy between the
topographic observations and the theoretical predictions of our model
with regard to the change in asphericity. This discrepancy may be due
to several factors.
First, there are limitations in the elliptical model used in our study.
The true corneal section does not conform exactly to an ellipse. The
ideal model may have to incorporate additional factors (i.e., higher
polynomial) to match the human cornea. The elliptical model is,
however, a good approximation of the corneal profile over the central 8
mm of its approximately 12-mm diameter.16
This area
represents the central optical zone, which is flattened after excimer
treatment for myopia and through which light passes to form the foveal
image. In conventional optics, conic sections are frequently used to
model the corneal surface. Furthermore, topographic evaluation of
cornea asphericity has been estimated from the conicoid that best fits
the keratoscopic or keratometric data.5
6
7
17
18
19
20
21
22
23
Our analysis assumes a rotationally symmetric mathematical model, but
the human cornea may exhibit toricity. Several groups have investigated
the meridional variations of corneal asphericity.3
7
21
The difference between the maximal and minimal values of asphericity is
low, ranging from 0.13 to 0.50 in 80% of the corneas.8
This difference in the overall asphericity change after laser
treatments for myopia does not seem to be significant.
Second, the ablation profile of current lasers for myopia does not
follow Munnerlyn’s formula. The effects of lasers may differ according
to the homogeneity and the location of the laser beam, and laser
manufacturers may have altered initial nomograms to improve clinical
outcomes and eliminate central islands. Differences in ablation rate of
Bowman’s membrane and stroma or within the stroma may also contribute
to this discrepancy. In addition, the applied fluence at the cornea,
even when the laser beam is homogeneous, decreases with the distance
from the center because of reflection of the UV light and because of
the curvature of the cornea. The achieved ablation pattern could thus
differ from the one attempted. However, no published data confirm this
possibility. Because the ablation patterns are proprietary, we cannot
confidently accept this possibility as the major explanation for the
discrepancy.
Third, wound healing (epithelial hyperplasia, stromal remodeling) could
be the source of the shape discrepancy. Topography patterns have been
shown to change with time.24
Variations in epithelial
thickness and curvature of the epithelial–stromal interface have been
implicated in the refractive regression occurring after LASIK and
PRK.25
26
27
28
They may modify the specific effect induced by
the ablation of myopia and could account for the observed trend to
oblateness observed by Hersh et al.,10
who used corneal
topography.
Fourth, photokeratoscopic or videokeratoscopic instruments do not
properly assess the shape of the cornea when spherical algorithm
assumptions are used.29
These assumptions have been
thought to be responsible for the differences observed between the
measured corneal power and the magnitude of change in manifest
refraction.13
30
31
Douthwaite32
used
calibrated convex ellipsoidal surfaces of known apical radius
(R) and asphericity (p) to assess the accuracy of
the EyeSys videokeratscope (Premier Laser Systems, Irvine, CA). This
device appeared to overestimate both p and R,
especially for asphericities outside the 0.8 to 1.0 region (i.e., the
near-spherical zone). In their study of corneal asphericity after PRK,
Hersch et al.10
acknowledge that idiosyncrasies in their
Placido-based topography system could have affected their
results. These considerations may be important for our purposes. The
information provided by keratoscopes after laser refractive surgery is
subject to cautious interpretation, and current devices may not be
sensitive enough to quantify or assess precisely the postoperative
corneal asphericity within the ablated zone.
After the correction of myopia, one of the typical topographical
aspects on chromatic maps displays a central circular area of uniform
colder color with regard to the surrounding surface. If the overall
shape of the cornea after treatment is considered, the central
flattening contrasting with the unchanged peripheral contour may lead
to the subjective assessment of postoperative oblateness. Our study
focused on the determination of the theoretical change of asphericity
within the optical zone—that is, the corneal surface of the cornea
receiving the laser treatment. Other typical topographical features
after PRK have been described, relating nonhomogeneous power within the
treated zone.33
The "keyhole," the semi-circular
ablative patterns, and the central islands represent three entities
with different clinical issues, but all are characterized by the
presence of a higher dioptric power area inside the ablation zone.
These features may represent increased prolateness of the cornea. The
cause of the central island has not yet been clarified with certainty,
although many hypotheses have been offered. Our model suggests that
preoperative asphericity could be another factor, especially in
patients with preoperative marked prolateness.
These considerations point out the ambiguity in the definition of the
asphericity of the corneal surface after refractive surgery for myopia.
To clarify the term, provide an accurate baseline description of the
corneal profile, and model optical errors such as spherical
aberrations, the terms oblate and prolate should refer only to
continuous conical shapes.
Our mathematical procedure for finding the best-fitting conic section
to our given set of points consisted of minimizing the sum of the
offset absolute values and the sum of the square of the offsets. The
latter did not allow the residuals to be treated as a continuous
differentiable quantity, but may have given to the outlying points a
disproportionate effect on the fit, which was not the case. Using two
fitting criteria raises the question of which method should have been
adopted if there had been discrepancies between the best-fit results of
optimizing p2. The sum of the squared
residuals is the most commonly used method. However, we used the
sum-of-residual-fitting method to confirm our findings. We found that
the determination of p2 was
unambiguous, given the small value of both the sum of the residuals and
the sum of the squared residuals. Therefore, the conic section
approximation can also be successfully used to describe the corneal
profile within the optical zone after a myopia laser treatment
conforming to Munnerlyn’s equation.
Equations 7 through 10
allow calculation of the apical radius of
curvature of the best-fitting conic section to our set of points. This
radius is independent of the initial value of the asphericity. This is
contradictory to the findings of Patel and
Marshall,34
who found that corneal asphericity
could marginally affect the initial refractive outcome of PRK. As in
this study, their mathematical model assumed the corneal surface to be
a conic section and, for a given correction, the amount of corneal
tissue removed by PRK was computed based on spherical corneal optics.
However, they arbitrarily assumed that initial and final corneal shapes
were typically prolate and oblate, respectively. This assumption led to
computation of a different radius of curvature from the one that was
expected, according to the spherical model.
There is general agreement that negative corneal asphericity has direct
optical significance. It is thought to influence visual performance
directly by lowering spherical aberration,12
16
although
this finding remains to be demonstrated. Patel et al.,17
using optical raytracing of finite schematic eyes, found that the value
of p required to eliminate spherical aberration at the
anterior surface is -0.528, given a refractive index of 1.376.
Despite the elimination of spherocylindrical errors, refractive surgery
may decrease visual performance by altering the corneal shape and
inducing unwanted changes in corneal asphericity. Many investigators
have noted that after radial keratotomy the cornea becomes oblate,
because the paracentral cornea is relatively steeper than the central
cornea.35
36
37
38
Seiler et al.39
proposed an
aspheric nomogram for PRK, designed to preserve a negative asphericity.
Their clinical results were encouraging, but the aspheric nomogram used
did not take into account the patient’s preoperative asphericity.
In summary, given that the anterior surface of the human cornea is the
main refractive element of the eye, its shape may contribute to optical
aberrations. For the normal, untreated, central corneal zone, a conic
section can accurately approximate the profile of the corneal zone
remodeled by ablation of myopia. To limit or treat optical aberrations
after refractive surgery, new profiles including customized aspherical
treatments must be developed. The major point of this study—that the
modeling of asphericity compares poorly with reported asphericity
outcomes—warrants better representation and collection of data before
and after treatment. One future rationale may be to use exact
raytracing to determine retinal image quality, given a certain ablation
profile and a certain shape factor, and then look for ways to alter
given asphericities to a desirable level. To reach this goal, further
theoretical and clinical studies of corneal shape after laser
refractive surgery are needed.
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Footnotes
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Supported by the Research to Prevent Blindness Lew R. Wasserman Merit
Award (DTA); the Corneal Transplantation Research Fund, Boston,
Massachusetts (DTA); and the Massachusetts Lions Eye Research Award,
Northborough, Massachusetts (DTA).
Submitted for publication November 28, 2000; revised January 26, 2001;
accepted February 7, 2001.
Commercial relationships policy: N.
The publication costs of this article were defrayed in part by page
charge payment. This article must therefore be marked
"advertisement" in accordance with 18 U.S.C. 
1734
solely to indicate this fact.
Corresponding author: Dimitri T. Azar, Corneal, External Disease, and
Refractive Surgery Service, Massachusetts Eye and Ear Infirmary, 243
Charles Street, Boston, MA 02114.
dazar{at}meei.harvard.edu
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Abstract
Introduction
