Corneal Topography

Corneal Topography: A review, new ANSI standards and problems to solve
Stanley A. Klein
School of Optometry, University of California at Berkeley, Berkeley, CA 94720-2020
klein@ spectacle.berkeley.edu
Abstract: This review of corneal topography has three sections: 1. a brief introduction to
how corneal topography instruments work. A quantitative comparison of the relative
accuracy of slope based and position based instruments is presented. 2. a summary of and
commentary on the newly issued ANSI standards for corneal topography. 3. an
examination of problems still facing corneal topography.
OCIS codes: (120.2830) Height measurements; (120.5700) Reflection; (220.1000) Aberration
compensation; (330.4300) Noninvasive assessment of the visual system; (330.4460) Ophthalmic optics
1. Operating principles of corneal topography instruments.
Three types of instruments are being used for measuring corneal shape:
A) Instruments that directly measure corneal position. One method, using a scanning slit (called "optical
sectioning corneal topographers" by the ANSI standard), has the advantage of being able to measure
corneal thickness as well as corneal shape. A second method places a coating on the cornea to scatter
light. A projected grid from one or more widely separated angles is used to calculate the depth of the
cornea. These instruments are called "luminous surface corneal topographers" by the ANSI standard.
One advantage of these position measuring methods is that one can measure a larger region of the
cornea than is possible with reflection methods. It is even possible to measure a portion of the scleral
topography. A disadvantage of this method is that if one wants to visualize slope or curvature, first and
second derivatives respectively are needed. The process of taking derivatives exaggerates the noise.
Much of this noise can be eliminated by starting from the raw slope data that is produced by instruments
that directly measure slope (see 1C).
B) Instruments based on interferometry. In principle, the most accurate method for measuring corneal
shape is achieved by generating an interference pattern using monochromatic light reflected from two
surfaces: the cornea and a matched, known reference shape. The advantage of this method is that in
principle, one can measure the corneal shape to sub-micron accuracy. The disadvantage is that in
practice the method is probably too sensitive. The slightest eye movements will disrupt the careful
matching of cornea and reference. The presence of tear film irregularities will distort the interference
pattern. Small deviations in corneal shape away from the reference shape will produce very rapidly
changing interference fringes that are difficult to measure. A corneal topographer based on
interferometry has not yet succeeded in the clinic.
C) Instruments that treat the cornea as a mirror and measure corneal slope. The most accurate clinical
topography instruments are based on measuring the corneal reflection. Anyone who has looked at a
minor dent in a car door knows that the dent is much more visible when looking at distorted reflections
from the dent than by looking at the distorted shape itself (visible when the door is dusty).
Placido image. Suppose the cornea is positioned in front of a bullseye (Placido ring) target consisting
of concentric rings (see figures in Section 2f). We will assume that the cornea is very small compared to
the distance from the cornea to the rings and to the camera looking at the reflected image. Klein1' 2
showed that in this small cornea limit the reflected pattern is simple to specify mathematically (see Eqs.
1-3). Suppose the corneal shape is specified by the function z(x, y). Then the gradient is (dz/dx, dz/dy).
The slope angle that the surface normal makes with the z-axis is given by:
angle(x,y) = atan([(dz/dx)2+ (dz/dy)2]1/2)
OSA TOPS Vol. 35 Vision Science and Its Applications
Vasudevan Lakshminarayanan, (ed.)
©2000 Optical Society of America
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At the x, y point the camera will image the Placido ring that is at twice the angle of Eq. 1 (the standard
law of reflection). A contour plot of Eq. 1 gives the Placido image that is seen by the corneal topography
camera (Klein1, and see examples in Section 2f):
Placido(x, y) = contour(2 angle(x, y))
(2)
For an axially symmetric cornea, the Placido image is1:
Placido(r) = contour(2 atan(dz/dr))
(3)
Present Placido topographers make the assumption that the ring locations are given by Eq. 3 rather than
by Eq. 2. For corneas that are not axially symmetric, this assumption results in an error1.
Inverse problem. Eq. 2, sometimes called the forward problem, calculates the reflected image when
given an arbitrary corneal height, z(x, y). The challenge to the topography instruments is the inverse
problem where one is given the Placido image and the challenge is to calculate the corneal height.
Various algorithms have been developed for solving this inverse problem. The simplest method makes
an oversimplifying assumption (the 'skew ray error' mentioned following Eq. 3) that the corneal normal
lies in the meridional plane (the plane containing the corneal point and the CT axis). It then uses a lookup table based on the assumption that the ring image is generated by a sphere whose vertex is at the
origin (definitions of Vertex' and 'CT axis' are given in Section 2). This is the method used by most of
the early corneal topographers. The output of this method is a map that is close to axial curvature but is
in error because even for axially symmetric surfaces the center of the sphere is improperly located3.
More sophisticated algorithms use an arc step method that assumes a smooth cornea with the corneal
normal in the meridional plane and does a piecewise integration of corneal height based on the Placido
image information. An arc step algorithm that allows the corneal normal to deviate from the meridional
plane, avoiding the skew ray error, has been developed2. Finally there are methods based on assuming a
parametric shape for the cornea based on a Taylor's expansion4, Zernike expansion5 or spline expansion6'
' 8 These latter methods do a least squares search for the optimal parameters by repeatedly generating
forward solutions and matching them to the Placido image.
An example. Insight into the quantitative aspects of reflection-based topographers can be obtained from
the following example of a spherical cornea with a little bump feature that we would like to detect. If the
feature is taken to be a small, centered Gaussian bump then the corneal height, z, as a function of
distance from the axis, r, is given by:
z(r) = R - (R2 - r2)172 - A exp(-r2/2r02)
(4)
where R is the radius of the base sphere and ro is the standard deviation of the bump. The curvature at
the origin (useful for gaining insight into the relative magnitude of the bump) is:
z"(0) = 1/R + A/r02
(5)
As will be discussed in connection with the ANSI standard, clinicians are familiar with expressing
curvature in diopters, obtained by multiplying curvature in mm"1 by the conversion factor 337.5. For a
cornea specified by R=7.5 mm, ro=l mm and A=l micron, the curvature at the center becomes:
central curvature (D) = 337.5 (1/7.5 + .001/12) = 45 + .3375 D
(6)
The value of 7.5 mm was chosen since it is a reasonable value for corneal curvature and since it gives a
simple whole number value for dioptric curvature (45 D). The value of ro=l mm was chosen as being a
reasonable size for a corneal feature. A 1 mm standard deviation of a Gaussian corresponds to a 2.34
mm distance between the half-height points of the Gaussian. The value of A=l micron is chosen because
a 0.34 D change in curvature is just barely clinically relevant.
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Now that our model cornea is fully specified it is useful to compare what would be measured by
position-based topographers (category A CTs) and slope-based topographers (category C CTs). A
category A topographer places planes of light on the cornea and then views those planes from a different
angle (like a slit lamp). The maximum deviation that is caused by our bump would have a height of .7A
(obtained by viewing the bump from a 45 deg angle relative to the projected grid). Reflection based
topographers, on the other hand, measure corneal slope, which is given by:
slope(r) = z' = r/(R2 - r2)172 + A r/r02exp(-r2/2r02)
(7)
The maximum effect of the small bump occurs near r- r0 where the slope is approximately:
slope(r0)~ r0/R + A/r0 exp(-l/2)
The bump changes the slope by approximately the fractional amount f = A R exp(-l/2)/ r02 = .0045 for
our chosen parameters. At the radius of r0 - 1 mm this fractional change of slope corresponds to about a
l*f - 5 micron shift in the location of the reflected image. This is about 7 times the position shift for a
category A topographer, thereby illustrating the advantage of reflection (slope) based topographers.
It is important to point out the difficulty of making these measurements. Suppose we would like
to image the full 11 mm cornea using a 1028x1028 pixel CCD camera. That means each pixel covers
about 10 microns. In the preceding paragraph we showed that for a Placido CT to detect the bump in our
shape would require the detection of a 5 micron shift in the Placido image. In order to have confidence
in the detection of the bump one needs a resolution of at most 2 rather than 5 microns which implies an
accuracy of better than 1/5 pixel. For a height based topographer an accuracy of better than 1/35 pixel is
needed, illustrating the difficulty of obtaining reliable measurements.
An alternative derivation of the needed accuracy of localizing the Placido images can be
obtained by considering the desired accuracy of curvature. The radius of curvature, R, is given by9:
R = Ar/Asin(0)
(8)
where r is the radius distance from the corneal point to the CT axis and 0 is the corneal slant angle. We
assume that the slant angle is known exactly since it is specified by the ring number of the Placido
target. The uncertainty in the measurement is the uncertainty of Ar associated with the change in ring
number. The fractional accuracy that we desire for R is about 1 part in 200 (based on a curvature
accuracy of 0.2D/40D, since 0.2D is the desired curvature accuracy and 40D is a typical corneal
curvature). If we desire to sample the cornea every Ar = 0.3 mm, then the needed accuracy in measuring
Ar is Ar/200 = 300/200 =1.5 microns. This value is similar to the 2 microns estimated above.
Empirical validation of CT accuracy has been examined by a number of authors10' n. Further
details on the operations of different CTs and on the types of maps can be found in the review article by
Fowler and Dave12.
2. ANSI Standard for Corneal Topography Systems.
On October 18, 1999, the American National Standards Institute (ANSI) approved a standard for
corneal topography instruments. Although these standards are voluntary, they are meant to encourage
some common usage of terminology and methods for testing the accuracy of the instruments. In this
review I will discuss a few of the items that I think are especially noteworthy. Charles Campbell, the
Working Group chair, did almost all the work of organizing the committee, writing the standard and
bringing order to a fairly contentious group of individuals. I cherish the many discussions and arguments
we had on nomenclature, test shapes and instrument specifications, and I expect that the ANSI corneal
topography standard will clarify a host of previous confusions.
2a. Simple terms. A few examples of terms that have now become standardized are:
i. A corneal topographer will be abbreviated as CT. This solves the problem of deciding between the
previous, more cumbersome and narrower contending names: keratograph vs keratoscope.
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ii. The CT axis as defined by the instrument axis will be the z axis for defining corneal shape. The CT
alignment procedure makes the CT axis normal to the cornea. It is displaced from the pupil center13
which can be important to know for refractive surgery.
iii. The corneal vertex is the corneal point closest to the camera along the CT axis,
iv. The corneal apex is the corneal point that has the largest mean curvature.
v. A corneal meridian is the curve created by the intersection of the corneal surface and a plane
containing the CT axis. Slopes and curvatures are typically calculated for a corneal meridian.
2b. Power vs. curvature. One of the confusing aspects of corneal topography is that most of the maps
are presently presented as dioptric power. This gets refractive properties of the cornea (i.e. power)
confused with shape properties (i.e. curvature). The ANSI standard now advises that shape properties be
called curvature. The normal units of curvature are mm"1. Since that is not a commonly used unit in the
ophthalmic community, a new unit called the keratometric diopter (D) is introduced. The committee
allowed the letter D to be used for keratometric diopters. The conversion between the two units is:
Curvature (D) = 337.5 Curvature (mm"1).
(9)
where the factor of 337.5 comes from a desire to match shape curvature and refractive power near the
corneal vertex. Thus a curvature of 1/7.5 mm"1 is the same as a curvature of 45 D. The main point here is
that people involved with corneal topography need to replace the word 'power'with the word 'curvature'.
2c. Specifying eccentricity of ellipse. The Working Group had lengthy struggles on how to deal with
specifying the eccentricity of an ellipse. This is an important issue because the cornea is often portrayed
as a prolate ellipsoid (each meridian is an ellipse whose curvature decreases away from the apex) with
an eccentricity of ecc - 0.5. For an ellipse given by:
b x2 + c z 2 = 1
(10)
where z is the sag or depth of the cornea as a function of corneal location, x. The eccentricity (ecc) and
apical radius of curvature (ra) are:
ecc = (l-b/c)1/2
ra = c1/2/b
(11)
where ra is the apical radius of curvature. The problem with using this definition of eccentricity is that
for an oblate ellipse (a cornea that steepens away from the apex) b>c, so the eccentricity becomes
imaginary. That predicament is often avoided by switching b and c in the oblate case, corresponding to a
90 deg rotation that ensures the ellipse is always prolate. One reason that the use of eccentricity has
survived for so long is that most corneas are prolate so the problem is avoided. Only recently, with the
advent of refractive surgery, have oblate corneas become common. An improved method for describing
oblate corneas is needed.
One method of avoiding imaginary numbers is to square the eccentricity to get the shape factor,
E, or the asphericity, Q=-E or the conic constant K=Q. Rather than any of these quantities the committee
recommends using the p-value given by: p = b/c = 1 - E = 1 - ecc2. In terms of p, the conic sections are:
oblate ellipse (p>l), sphere (p=l), prolate ellipse (l>p>0), parabola (p=0), hyperbola (p<0).
A convenient formula for an ellipse is:
z = x 2 /(r a + (r a 2 -px 2 ) 172 )
(12)
For a parabola (p=0) the equation becomes the familiar 'sag'formula:
z = x 2 /2r a
(13)
The committee chose p over E and Q because p is the quantity that appears in the simplest equations for
ellipses and because it is easy to forget the sign convention and the common names for E and Q (p is
called 'p-value' in the ANSI document). The reader would be amazed at the voluminous e-mailings and
discussions that took place to come up with this recommendation on eccentricity. We hope it is used.
2c. Meridional vs. normal curvature. In the process of discussing the various maps that are used by
different topographers, the ANSI working committee became aware that different companies had been
using different definitions of curvature. Slopes and curvatures that are defined based on a corneal
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meridian (see 2a.v.) will be called axial curvature and meridional curvature. For an axially symmetric
shape, 'axial' curvature corresponds to the Seidel sagittal curvature and 'meridional' curvature
corresponds to the Seidel tangential curvature (it was previously called instantaneous curvature).
Alternative definitions of slope and curvature, not based on the corneal meridian1 are possible.
For systems that are not axially symmetric, it is more common among mathematicians to define
curvature in a set of planes that contain the surface normal (e.g., ANSI defines the transverse plane as
the plane perpendicular to the meridional plane while including the normal to the surface point). These
curvatures were given the name normal curvatures by the ANSI standards. The principal curvatures are
the maximum and minimum normal curvatures. The mean curvature and surface toricity are the
arithmetic mean and the difference of the two principal curvatures respectively. On the other hand,
meridional definitions of curvature have the advantage of simplicity and invertability over normal
curvatures. That is, from knowledge of meridional curvature one can calculate the surface shape.
However, knowledge of the principal curvatures at all points of the cornea is not sufficient to generate
the surface; one also needs to know the angle of the local toricity.
To go from meridional curvature to surface height requires three steps14. Step 1 is an integration
to go from meridional curvature to axial curvature, Ca, since axial curvature is the average of the
meridional curvature along a given meridian from the axial point to the corneal point (even for nonaxially symmetric shapes9). Step 2 involves the conversion of axial curvature to meridional slope:
sin(ang) = rC a
(14)
dz/dr = tan(ang)
(15)
r is the distance from the CT axis to the corneal point and ang is the angle of the normal in the meridional plane. Step 3 involves integration of dz/dr from the axial point to the corneal point to obtain z.
2d. Reporting performance. An important feature of the ANSI standards are specifications for how to
measure the accuracy and the repeatability of CT instruments. Accuracy is determined by measuring the
topography of a number of known test surfaces. Repeatability (precision) is determined by repeated
measurements on a number of human eyes. One of the nice features of the ANSI recommendations is
that values should be reported separately for three zones: central (diameter < 3 mm), middle (3 <
diameter < 6 mm), and outer (6 mm < diameter). The goal of this section is commendable. However, it
is my belief that the committee may have spent insufficient time on this aspect of the standard and I
would like to offer suggestions for future modifications.
2e. Comments on repeatability
i) Number of repetitions. The ANSI standard suggests taking the difference of two measurements as an
estimate of the repeatability. This difference happens to be sqrt(2) times the standard deviation of the
measurement. My suggestion is to change the standard by replacing the difference of just two
measurements with the standard deviation of multiple measurements. The motivation for this suggestion
is that two measurements aren't sufficient for obtaining an accurate estimate of repeatability. One might
say that two measurements are enough because multiple points are being measured. This argument is
flawed because the values at the multiple points can be correlated as will be discussed next.
ii) Isolation of defocus term. One of the prime technical problems that a CT must solve is how to obtain
an accurate measurement of the distance from the CT to the cornea: the focusing problem. Suppose the
cornea is 1 % further away than intended. In that case the Placido image would be about 1 % smaller and
the curvatures would be 1% greater. For a 45 D cornea this error would be 0.45 D, a significant amount.
Some instruments have extra cameras on the side to look at corneal position. Other instruments capture
the image when a narrow sideways beam is interrupted by the cornea. Other instruments have one of the
Placido rings in a different depth plane so that the Placido image has information about the corneal
distance. These different methods may have different degrees of repeatability and they may have
different accuracies when a real eye is used in place of the model eye (due for example to different
reflection intensities between real and model eyes). One instrument might be excellent at analyzing the
Placido image and in reconstructing corneal shape but might be poor at measuring the corneal distance.
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I believe the CT standards should add a fourth value specifying the focusing repeatability in
addition to the three values for the three zones mentioned in Section 2d. An example may be helpful
here. Suppose the curvature (in keratometric diopters) at four corneal points were measured three times
each with the results given by: Ml = 40.1, 40.3, 40.5, 40.3; M2 = 41.1, 41.2, 41.3, 41.6; M3=39.3, 39.1,
39.4, 39.4. The defocus repeatability can be estimated by looking at the repeatability of the mean of
each of the measurements. The means are 40.3, 41.3, 39.3 which have a standard deviation of ID. Once
the defocus (the mean) is subtracted out the remaining standard deviation at each of the corneal points
is: SD = 0.115, 0.100, 0.100, 0.153, giving an rms SD=0.119 D. For this example the defocus
repeatability would be reported as having SD=1.0 D, and the curvature repeatability would be reported
as having an SD=0.12 D. If the defocus term hadn't been removed then the four corneal points would be
reported as having SD=0.92, 1.05, 0.95, 1.11 with an rms SD=1.01 D. This is much larger than the rms
SD = 0.12 D that is found if the dominant defocus error is removed.
When calculating the defocus term, it is advisable to only use the data from the middle zone.
That is because the inner zone is unreliable since the estimate of axial curvature involves dividing the
ring location by the radial distance of the corneal point, a very small number for the inner rings. Thus
any error in measuring the ring location gets magnified. Points in the outer zones are unreliable for
reasons I don't fully understand, but possibly due to sensitivity to decentration, depth of field problems,
and inaccuracies in the arc step reconstruction algorithm.
2f. Comments on accuracy.
i. The ANSI shapes. Seven shapes were recommended: three spheres with radii of 6.50, 8.00, and 9.00
mm; an ellipsoid with apical radius of 7.80 mm and eccentricity=0.5 (p=.75); and three axially
symmetric shapes simulating myopic refractive surgery keratoconus and hyperopic refractive surgery.
Each of the latter three shapes consisted of an inner and outer sphere with a toric transition zone (a
double toric for hyperopic PRK). The difference between a sphere and a torus is that in the former the
center of curvature lies on the symmetry axis. The specifications of latter two shapes are given by:
Name
keratoconus
hyperopic PRK1
hyperopic PRK2
inner
diameter
outer
diameter
inner radius
of curvature
transition radius
of curvature
outer radius
curvature
2 mm
6
6
6 mm
9
6.61
5 mm
7.34
7.34
11.5 mm
-5.51, +5.51
infinite
8.04 mm
8.04 (fix center)
8.04 (float center)
Placido image of keratoconus shape
Placido image of hyperopic PRK shape
of
Placido image of hyperopic PRK shape
5
-
5
0
5
-5
0
-
5
0
cornea location (mm)
cornea location (mm)
Fig. 1 Placido image for three axially symmetric test shapes with a rotated CT.
cornea location (mm)
5
The above figures show several Placido image rings for the keratoconic shape and two variants of the
hyperbolic PRK shape (to be discussed). For ease of reading the plot we have selected contour lines
every 10 deg of corneal slant (the Placido target rings are at twice the indicated angle). To make the
images more interesting the CT has been rotated 0.2 radians to the right while leaving the shapes
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unrotated. The transition zones are indicated by the dot-dashed white circles. The symmetry axis of the
shapes is the center of the white circles. The CT axis is at the center of the inner and outer contours.
The three plots were produced by the Matlab function 'contourf(angle)' where 'angle' is the angle
the corneal normal makes with the CT axis:
angle = acos(cos(.2) cos(sl) - sin(.2) sin(sl) cos(phi)
(16)
where 0.2 and 'si' are the CT axis tilt and the corneal slope with respect to the corneal symmetry axis
(the plot axis), and phi = atan2(x,y) is the azimuthal angle of the corneal point at location x, y. For the
keratoconic shape (Fig. la) the two centered dot-dashed circles correspond to the transition zones at 2
and 6 mm diameters. Note that the Placido image curves are closer together in the central zone and
farther apart in the transition zone corresponding to the 5 and 11.5 mm radii of curvature. In the first
hyperopic PRK shape (Fig. Ib) the transition region consists of two zones (demarcated by the three dotdashed circles) with the inner and outer transition zones having curvatures of -5.519 and +5.519 mm
respectively. The transition between the ±5.519 mm radii occurs at a diameter of 6.667 mm. The
negative curvature is indicated by the Placido ring lines having opposite slope.
One of the features of the three shapes in the above tables is that there are discontinuities in
curvature at each transition. For example, at the first transition point, the keratoconic shape has a jump
from a 5.00 to a 11.50 mm radius. If the shape axis coincides with the CT axis then the Placido image
will be centered circles, and the separation between circles will have a large change at the transition
point. The precise location of that discontinuity will not be available to the CT, due to the sparse Placido
ring sampling, and the reconstruction algorithms may not do well. If the shape axis is rotated relative to
the CT axis then the Placido image will be centered circles in the central zone and displaced circles in
the other two zones as shown in the figures. At the transition points the Placido image circles will have a
sharp discontinuity in their slopes, seen most clearly in the hyperopic shape. The discontinuity is barely
visible in the outer transition of Fig. Ib where the curvature changes from 5.51 to 8.04 mm.
Present Placido CT algorithms would fail for the ANSI hyperopic shape. For the tilted shape
shown in Fig. Ib present algorithms fail at the early step of tracing the jagged rings. They would also
have trouble with specifying ring radius as a function of angle since there are regions where the function
is triple valued (rather than single valued). Even if that shape weren't tilted the algorithms will have
problems since the rings in the negative curvature zone will be triplicates of the rings in the central zone.
It might be interesting to test the CT algorithms with this extreme shape to see how well they do, but it is
also important to have a shape that the CTs can handle, to be able to check the algorithm accuracy. With
that in mind I suggest a variant on the hyperopic PRK shape that is specified in the bottom row of the
above table. It produces the Placido image shown in Fig. Ic. The new shape has a transition zone with
zero curvature. Since there are no negative curvatures, the ring locations as a function of angle are single
valued (except at a few discrete points). The outer transition diameter is calculated by r0uter*dinner/rinner=
8.04*6/7.34 = 6.61 mm. The center of the outer sphere must be shifted along the symmetry axis for
continuity. In the ANSI shape (row 2 of the Table) the center of the outer zone was fixed at a distance of
8.04 mm from the vertex. A side benefit of this new shape (row 3) is that the flattening to zero curvature
is quite similar to the rapid flattening of normal corneas near the limbus.
ii. Tilting the surface. The ANSI document recommends tilting several of the test shapes to test how
well the CT handles shapes without axial symmetry. This brings up the question of how to guarantee
that the test shape holder is oriented properly. There is no problem with centering the holder since the
procedure of using a CT allows for the alignment of the vertex normal with the CT axis. A simple
suggestion for validating the holder's proper orientation would be to generate a bicurve test shape with
very large radii of curvature. The central zone could be a sphere with a radius of curvature of say 100
mm. The outer zone would be a torus with a similar radius. There would be a discontinuity of slope at
the transition point, such that the slope in the middle of the toric region would be the desired slope of the
holder. This calibration shape would not be used with the CT software but would be used by merely
visual inspection of the Placido image. The low curvature of the surfaces would result in very large
displacements of the Placido image if the tilt of the holder is incorrect.
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2g. Standardized displays. The ANSI committee discussed the need for standardized color maps for
displaying the data. We agreed upon the need to have standardized colors so that clinicians could
quickly understand the meaning of the map without needing to scrutinize the color scale. The ANSI
recommendation suggested that one of three dioptric intervals be used: 0.5, 1.0 and 1.5 D centered at 44
D, and the number of intervals must be between 21 and 25.
I have just two minor suggestions to make regarding the color scale typically found on the side
of the color map. First, some CT maps have discrete colors on the scale, with the numerical scale values
placed in the middle of the color region. My suggestion is to have the scale values placed next to the
transition points of the color scale. Thus if the green-yellow transition is at 46 D, then the value 46
would be placed right next to the green-yellow discrete transition on the scale. Second, the color scale
should display only the range of colors shown on the map (while including the central 44 D contour) to
make the transitions easier to read. The gray 'color' maps and 'color' scale for Figs la-c provide an
example of what I mean (but not using one of the ANSI recommended intervals).
3. Problems still to be solved. This final section considers a number of problems that must be solved
before corneal topography becomes commonly used. CT instruments have evolved to become an
accurate tool for measuring corneal shape. There is general agreement that a CT assessment is needed
before refractive surgery in order to screen out patients with corneal anomalies such as keratoconus.
Refractive surgeons already have CTs. This section asks whether there are additional uses for CTs that
will make them more accepted by clinicians who are not refractive surgeons. For that to happen, CTs
need to be shown to be clearly helpful in some realm of clinical practice. There have been claims that
they are useful for contact lens fitting, for evaluating corneal anomalies such as keratoconus or corneal
warpage, for improving refractive surgery outcomes, and for predicting optical limitations to visual
acuity. We will consider each of these claims.
3a. Contact lens fitting. CT instruments can potentially change how contact lenses are fit 15 ' 16 ' 17 ' 18 . A
sizeable fraction of rigid gas permeable (RGP) lens patients are difficult to fit because the lens doesn't
center well or move well. Even the more easily fit soft lenses may benefit from CT assistance because
the new silicone-hydrogel materials are making extended wear more successful. With extended wear
there is concern that there may be insufficient tear mixing, and improved CT-based custom fitting may
be needed to ensure proper motion for tear mixing. However, the use of CTs for contact lens fitting is
still the exception. I believe that before CTs can become indispensable for contact lens fitting four things
must happen:
i. Expanded coverage of the cornea. Topography of the full 11 mm of cornea (and possibly the near
sclera for soft lenses) is needed for contact lens fitting because the points of bearing for RGP lenses are
often in the peripheral cornea. Similarly, the comfort and motion of soft lenses are affected by the
peripheral regions where corneal curvature changes most rapidly. Unfortunately, present Placido based
topographers are limited to the central 8 mm or so. CTs that are based on measuring the height directly
(see Section 1) may have problems achieving the required accuracy, but improvements should be
possible. Placido CTs may also be used to get expanded coverage, either by splicing multiple shots taken
with different gaze directions, or by placing a high power lens in front of the eye to that the light rays
come in with close to normal incidence even for peripheral cornea19. This could expand the coverage to
the full cornea.
ii. Tracing the Placido image rings. There is a further problem with expanding the Placido image to the
limbus. Because of the rapid flattening of peripheral cornea, present CTs do a poor job of tracing the
image rings in peripheral cornea for off-axis shots (needed to expand corneal coverage). It is not
uncommon to have Placido images that have the zig-zag shape similar to the hyperopic PRK test shape
Placido image shown in Fig. Ib.
iii. Improved understanding of RGP fitting is needed. For RGP lenses, corneal topography has been
used to generate a simulated fluorescein picture. This is a useful picture, especially if it can be extended
to the full area under the contact lens. However, more than that is needed. We need to be able to predict
the motion and centration of the lenses. One suggestion20' 21 is to develop maps of total tear volume
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under the contact lens for different lens positions. These maps summarize a multitude of contact lens
properties and corneal shapes in a manner relevant to lens motion. New tools of this sort need to be
developed to enable improved simulations of how the lenses will act on real corneas. With these
simulations numerous contact lens designs could then be tested to see which gives the best fit to the
particular patient's cornea. These simulations will not only facilitate the choice of the best available lens
type, but it can also aid the manufacturers in developing new lens designs.
iv. Understand soft lens flexing and eyelid interactions. In order to obtain improved tear mixing with
the new silicone-hydrogel lenses we will need to learn more about the complexity of the forces that
affect soft lens motion. With extended wear lenses, tear mixing becomes an important factor for washing
out debris that could cause problems. Similar to the RGP case, having CT assistance may become
important for speeding up the fitting procedure and for aiding the design of improved lenses.
3b. Diagnosing corneal anomalies. Starting with the first computer based CTs by Computed Anatomy,
numerous summary indices have been developed to detect corneal anomalies22' 23. These indices
measure the items like the symmetry, spherical aberration and raggedness of the cornea. There have
been a number of successes in showing that various indices are able to discriminate keratoconus from
other anomalies24. What would really be nice is to have the computer provide a detailed specification of
the anomaly. Thus, once a cornea has been diagnosed as being keratoconic it should provide the full
details about the location, size, severity and shape of the cone25. With that information the clinician
would be able to follow changes in the cone as it is affected by items such as contact lens wear. For
refractive surgery and orthokeratology the shape descriptors would include a characterization of the
optic zone (sphere, cylinder, eccentricity), and transition zone (inner and outer radii, peak curvature,
uniformity, centration). This information would allow the clinician to follow the corneal reshaping
following the surgery. In addition to the need for improved numerical shape descriptors, there is a need
for improved visualizations26. Present maps that are based on meridional corneal sections often mask
underlying corneal properties because of the singularity at the origin. For example, keratoconus is most
easily seen and diagnosed in a map of minimum principal curvature at each point. Refractive surgery, on
the other hand is best illustrated by a map of maximum principal curvature.
3c. Improving refractive surgery outcomes. It is sometimes thought that knowledge of corneal
topography can help design improved ablation profiles. That is not the case27. The main information for
determining the ablation profile is knowledge of the wavefront aberration. Since this aberration is
calculated in a plane (often the entrance pupil) it needs to be slightly shifted to determine the aberration
at the cornea. This correction term, based on corneal topography, is quite minimal, at least for a
reasonable choice of corneal shapes. However, when one considers the biomechanical properties of the
corneal response to refractive surgery, corneal shape near the limbus may play an important role. There
are large individual differences in the shape of peripheral cornea and these differences, measured by CT,
might be related to the organization of the underlying collagen molecules that in turn might be related to
how the cornea responds to the refractive surgery. Further research in this area is needed.
3d. Predicting optical limitations to visual acuity. There are a substantial fraction of patients whose best
corrected vision is worse than 20/20. The clinician usually wants to determine whether the cause of the
degraded performance is due to optical or to neural factors. It had been hoped that corneal topography
could be used to see if optical factors are causing the problem. There are many cases such as
keratoconus, corneal warpage28' 29 and refractive surgery with large pupils30'31, where a visual loss can
be predicted by corneal topography. If this article had been written one or two years ago this topic would
have been more important than it is at present; new instruments for measuring the full aberrations of the
eye will soon become commercially available and will replace CTs for this purpose. However, even with
knowledge of the full aberrations of the eye there remains a major challenge of getting a high correlation
between the optics and predicted acuity for near-normal corneas. Of course, if one includes badly
degraded optics one will get good correlations between optical quality and acuity. We need to do better
on those individuals with moderately degraded wavefronts who still have 20/20 acuity. For individuals
with severely degraded wavefronts, improved predictors of the quality of vision is needed. The problem
of reliably predicting acuity based on the wavefront is a tantalizing challenge to researchers concerned
with optics and vision.
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Acknowledgments. I thank John Corzine, Robert Mandell and Charles Campbell for helpful comments
and Neurometrics Institute for partial financial support of this research.
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