corneal topography: a review, new ansi standards and ... · corneal topography: a review, new ansi...

Corneal Topography: A review, new ANSI standards and problems to solve

Stanley A. KleinSchool 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 tohow corneal topography instruments work. A quantitative comparison of the relativeaccuracy of slope based and position based instruments is presented. 2. a summary of andcommentary on the newly issued ANSI standards for corneal topography. 3. anexamination of problems still facing corneal topography.OCIS codes: (120.2830) Height measurements; (120.5700) Reflection; (220.1000) Aberrationcompensation; (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 "opticalsectioning corneal topographers" by the ANSI standard), has the advantage of being able to measurecorneal thickness as well as corneal shape. A second method places a coating on the cornea to scatterlight. A projected grid from one or more widely separated angles is used to calculate the depth of thecornea. 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 thecornea than is possible with reflection methods. It is even possible to measure a portion of the scleraltopography. A disadvantage of this method is that if one wants to visualize slope or curvature, first andsecond 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 instrumentsthat directly measure slope (see 1C).B) Instruments based on interferometry. In principle, the most accurate method for measuring cornealshape is achieved by generating an interference pattern using monochromatic light reflected from twosurfaces: the cornea and a matched, known reference shape. The advantage of this method is that inprinciple, one can measure the corneal shape to sub-micron accuracy. The disadvantage is that inpractice the method is probably too sensitive. The slightest eye movements will disrupt the carefulmatching of cornea and reference. The presence of tear film irregularities will distort the interferencepattern. Small deviations in corneal shape away from the reference shape will produce very rapidlychanging interference fringes that are difficult to measure. A corneal topographer based oninterferometry has not yet succeeded in the clinic.C) Instruments that treat the cornea as a mirror and measure corneal slope. The most accurate clinicaltopography instruments are based on measuring the corneal reflection. Anyone who has looked at aminor dent in a car door knows that the dent is much more visible when looking at distorted reflectionsfrom 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 consistingof concentric rings (see figures in Section 2f). We will assume that the cornea is very small compared tothe 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) (1)

OSA TOPS Vol. 35 Vision Science and Its ApplicationsVasudevan Lakshminarayanan, (ed.) 286©2000 Optical Society of America

Vision Science and Its Applications

At the x, y point the camera will image the Placido ring that is at twice the angle of Eq. 1 (the standardlaw of reflection). A contour plot of Eq. 1 gives the Placido image that is seen by the corneal topographycamera (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 thanby 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 whengiven an arbitrary corneal height, z(x, y). The challenge to the topography instruments is the inverseproblem 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 makesan oversimplifying assumption (the 'skew ray error' mentioned following Eq. 3) that the corneal normallies in the meridional plane (the plane containing the corneal point and the CT axis). It then uses a look-up table based on the assumption that the ring image is generated by a sphere whose vertex is at theorigin (definitions of Vertex' and 'CT axis' are given in Section 2). This is the method used by most ofthe early corneal topographers. The output of this method is a map that is close to axial curvature but isin 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 cornealnormal in the meridional plane and does a piecewise integration of corneal height based on the Placidoimage information. An arc step algorithm that allows the corneal normal to deviate from the meridionalplane, avoiding the skew ray error, has been developed2. Finally there are methods based on assuming aparametric 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 fromthe following example of a spherical cornea with a little bump feature that we would like to detect. If thefeature is taken to be a small, centered Gaussian bump then the corneal height, z, as a function ofdistance 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 atthe 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 expressingcurvature in diopters, obtained by multiplying curvature in mm"1 by the conversion factor 337.5. For acornea 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 asimple whole number value for dioptric curvature (45 D). The value of ro=l mm was chosen as being areasonable size for a corneal feature. A 1 mm standard deviation of a Gaussian corresponds to a 2.34mm distance between the half-height points of the Gaussian. The value of A=l micron is chosen becausea 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 byposition-based topographers (category A CTs) and slope-based topographers (category C CTs). Acategory A topographer places planes of light on the cornea and then views those planes from a differentangle (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 basedtopographers, on the other hand, measure corneal slope, which is given by:

slope(r) = z' = r/(R2 - r2)172 + A r/r02exp(-r2/2r0

2) (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 al*f - 5 micron shift in the location of the reflected image. This is about 7 times the position shift for acategory 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 liketo image the full 11 mm cornea using a 1028x1028 pixel CCD camera. That means each pixel coversabout 10 microns. In the preceding paragraph we showed that for a Placido CT to detect the bump in ourshape would require the detection of a 5 micron shift in the Placido image. In order to have confidencein the detection of the bump one needs a resolution of at most 2 rather than 5 microns which implies anaccuracy of better than 1/5 pixel. For a height based topographer an accuracy of better than 1/35 pixel isneeded, illustrating the difficulty of obtaining reliable measurements.

An alternative derivation of the needed accuracy of localizing the Placido images can beobtained 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. Weassume that the slant angle is known exactly since it is specified by the ring number of the Placidotarget. The uncertainty in the measurement is the uncertainty of Ar associated with the change in ringnumber. The fractional accuracy that we desire for R is about 1 part in 200 (based on a curvatureaccuracy of 0.2D/40D, since 0.2D is the desired curvature accuracy and 40D is a typical cornealcurvature). If we desire to sample the cornea every Ar = 0.3 mm, then the needed accuracy in measuringAr 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. Furtherdetails on the operations of different CTs and on the types of maps can be found in the review article byFowler 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 encouragesome common usage of terminology and methods for testing the accuracy of the instruments. In thisreview I will discuss a few of the items that I think are especially noteworthy. Charles Campbell, theWorking Group chair, did almost all the work of organizing the committee, writing the standard andbringing order to a fairly contentious group of individuals. I cherish the many discussions and argumentswe had on nomenclature, test shapes and instrument specifications, and I expect that the ANSI cornealtopography 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 CTalignment 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 mapsare 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 becalled curvature. The normal units of curvature are mm"1. Since that is not a commonly used unit in theophthalmic community, a new unit called the keratometric diopter (D) is introduced. The committeeallowed 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 thecorneal vertex. Thus a curvature of 1/7.5 mm"1 is the same as a curvature of 45 D. The main point here isthat 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 withspecifying the eccentricity of an ellipse. This is an important issue because the cornea is often portrayedas a prolate ellipsoid (each meridian is an ellipse whose curvature decreases away from the apex) withan eccentricity of ecc - 0.5. For an ellipse given by:

b x2 + c z2 = 1 (10)where z is the sag or depth of the cornea as a function of corneal location, x. The eccentricity (ecc) andapical 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 thatfor an oblate ellipse (a cornea that steepens away from the apex) b>c, so the eccentricity becomesimaginary. That predicament is often avoided by switching b and c in the oblate case, corresponding to a90 deg rotation that ensures the ellipse is always prolate. One reason that the use of eccentricity hassurvived for so long is that most corneas are prolate so the problem is avoided. Only recently, with theadvent of refractive surgery, have oblate corneas become common. An improved method for describingoblate 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 committeerecommends 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 = x2/(ra + (ra2-px2)172) (12)

For a parabola (p=0) the equation becomes the familiar 'sag'formula:z = x2/2ra (13)

The committee chose p over E and Q because p is the quantity that appears in the simplest equations forellipses and because it is easy to forget the sign convention and the common names for E and Q (p iscalled 'p-value' in the ANSI document). The reader would be amazed at the voluminous e-mailings anddiscussions 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 bydifferent topographers, the ANSI working committee became aware that different companies had beenusing 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 symmetricshape, 'axial' curvature corresponds to the Seidel sagittal curvature and 'meridional' curvaturecorresponds 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 definecurvature in a set of planes that contain the surface normal (e.g., ANSI defines the transverse plane asthe plane perpendicular to the meridional plane while including the normal to the surface point). Thesecurvatures were given the name normal curvatures by the ANSI standards. The principal curvatures arethe maximum and minimum normal curvatures. The mean curvature and surface toricity are thearithmetic 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 normalcurvatures. 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 generatethe 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 integrationto go from meridional curvature to axial curvature, Ca, since axial curvature is the average of themeridional curvature along a given meridian from the axial point to the corneal point (even for non-axially symmetric shapes9). Step 2 involves the conversion of axial curvature to meridional slope:

sin(ang) = rCa (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 merid-ional 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 tomeasure the accuracy and the repeatability of CT instruments. Accuracy is determined by measuring thetopography of a number of known test surfaces. Repeatability (precision) is determined by repeatedmeasurements on a number of human eyes. One of the nice features of the ANSI recommendations isthat 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, itis my belief that the committee may have spent insufficient time on this aspect of the standard and Iwould like to offer suggestions for future modifications.2e. Comments on repeatabilityi) Number of repetitions. The ANSI standard suggests taking the difference of two measurements as anestimate of the repeatability. This difference happens to be sqrt(2) times the standard deviation of themeasurement. My suggestion is to change the standard by replacing the difference of just twomeasurements with the standard deviation of multiple measurements. The motivation for this suggestionis that two measurements aren't sufficient for obtaining an accurate estimate of repeatability. One mightsay that two measurements are enough because multiple points are being measured. This argument isflawed 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 obtainan accurate measurement of the distance from the CT to the cornea: the focusing problem. Suppose thecornea is 1 % further away than intended. In that case the Placido image would be about 1 % smaller andthe 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 capturethe image when a narrow sideways beam is interrupted by the cornea. Other instruments have one of thePlacido rings in a different depth plane so that the Placido image has information about the cornealdistance. These different methods may have different degrees of repeatability and they may havedifferent accuracies when a real eye is used in place of the model eye (due for example to differentreflection intensities between real and model eyes). One instrument might be excellent at analyzing thePlacido 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 inaddition to the three values for the three zones mentioned in Section 2d. An example may be helpfulhere. Suppose the curvature (in keratometric diopters) at four corneal points were measured three timeseach 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 ofeach of the measurements. The means are 40.3, 41.3, 39.3 which have a standard deviation of ID. Oncethe defocus (the mean) is subtracted out the remaining standard deviation at each of the corneal pointsis: SD = 0.115, 0.100, 0.100, 0.153, giving an rms SD=0.119 D. For this example the defocusrepeatability would be reported as having SD=1.0 D, and the curvature repeatability would be reportedas having an SD=0.12 D. If the defocus term hadn't been removed then the four corneal points would bereported as having SD=0.92, 1.05, 0.95, 1.11 with an rms SD=1.01 D. This is much larger than the rmsSD = 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 thering location by the radial distance of the corneal point, a very small number for the inner rings. Thusany error in measuring the ring location gets magnified. Points in the outer zones are unreliable forreasons 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 9.00mm; an ellipsoid with apical radius of 7.80 mm and eccentricity=0.5 (p=.75); and three axiallysymmetric 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 (adouble toric for hyperopic PRK). The difference between a sphere and a torus is that in the former thecenter of curvature lies on the symmetry axis. The specifications of latter two shapes are given by:

Name

keratoconushyperopic PRK1hyperopic PRK2

innerdiameter

2 mm66

outerdiameter

6 mm96.61

inner radiusof curvature

5 mm7.347.34

transition radiusof curvature

11.5 mm-5.51, +5.51infinite

outer radius ofcurvature

8.04 mm8.04 (fix center)8.04 (float center)

Placido image of keratoconus shape Placido image of hyperopic PRK shape

- 5 0 5cornea location (mm)

Placido image of hyperopic PRK shape5

-5 0cornea location (mm)

- 5 0 5cornea location (mm)

Fig. 1 Placido image for three axially symmetric test shapes with a rotated CT.The above figures show several Placido image rings for the keratoconic shape and two variants of thehyperbolic PRK shape (to be discussed). For ease of reading the plot we have selected contour linesevery 10 deg of corneal slant (the Placido target rings are at twice the indicated angle). To make theimages 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 theshapes 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 anglethe 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 thekeratoconic shape (Fig. la) the two centered dot-dashed circles correspond to the transition zones at 2and 6 mm diameters. Note that the Placido image curves are closer together in the central zone andfarther apart in the transition zone corresponding to the 5 and 11.5 mm radii of curvature. In the firsthyperopic PRK shape (Fig. Ib) the transition region consists of two zones (demarcated by the three dot-dashed circles) with the inner and outer transition zones having curvatures of -5.519 and +5.519 mmrespectively. The transition between the ±5.519 mm radii occurs at a diameter of 6.667 mm. Thenegative 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 incurvature at each transition. For example, at the first transition point, the keratoconic shape has a jumpfrom a 5.00 to a 11.50 mm radius. If the shape axis coincides with the CT axis then the Placido imagewill be centered circles, and the separation between circles will have a large change at the transitionpoint. The precise location of that discontinuity will not be available to the CT, due to the sparse Placidoring sampling, and the reconstruction algorithms may not do well. If the shape axis is rotated relative tothe CT axis then the Placido image will be centered circles in the central zone and displaced circles inthe other two zones as shown in the figures. At the transition points the Placido image circles will have asharp discontinuity in their slopes, seen most clearly in the hyperopic shape. The discontinuity is barelyvisible 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 shapeshown in Fig. Ib present algorithms fail at the early step of tracing the jagged rings. They would alsohave trouble with specifying ring radius as a function of angle since there are regions where the functionis triple valued (rather than single valued). Even if that shape weren't tilted the algorithms will haveproblems 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 isalso important to have a shape that the CTs can handle, to be able to check the algorithm accuracy. Withthat in mind I suggest a variant on the hyperopic PRK shape that is specified in the bottom row of theabove table. It produces the Placido image shown in Fig. Ic. The new shape has a transition zone withzero curvature. Since there are no negative curvatures, the ring locations as a function of angle are singlevalued (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 forcontinuity. In the ANSI shape (row 2 of the Table) the center of the outer zone was fixed at a distance of8.04 mm from the vertex. A side benefit of this new shape (row 3) is that the flattening to zero curvatureis 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 howwell the CT handles shapes without axial symmetry. This brings up the question of how to guaranteethat the test shape holder is oriented properly. There is no problem with centering the holder since theprocedure of using a CT allows for the alignment of the vertex normal with the CT axis. A simplesuggestion for validating the holder's proper orientation would be to generate a bicurve test shape withvery large radii of curvature. The central zone could be a sphere with a radius of curvature of say 100mm. The outer zone would be a torus with a similar radius. There would be a discontinuity of slope atthe transition point, such that the slope in the middle of the toric region would be the desired slope of theholder. This calibration shape would not be used with the CT software but would be used by merelyvisual inspection of the Placido image. The low curvature of the surfaces would result in very largedisplacements 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 fordisplaying the data. We agreed upon the need to have standardized colors so that clinicians couldquickly understand the meaning of the map without needing to scrutinize the color scale. The ANSIrecommendation suggested that one of three dioptric intervals be used: 0.5, 1.0 and 1.5 D centered at 44D, 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 sideof the color map. First, some CT maps have discrete colors on the scale, with the numerical scale valuesplaced in the middle of the color region. My suggestion is to have the scale values placed next to thetransition points of the color scale. Thus if the green-yellow transition is at 46 D, then the value 46would be placed right next to the green-yellow discrete transition on the scale. Second, the color scaleshould display only the range of colors shown on the map (while including the central 44 D contour) tomake the transitions easier to read. The gray 'color' maps and 'color' scale for Figs la-c provide anexample 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 solvedbefore corneal topography becomes commonly used. CT instruments have evolved to become anaccurate tool for measuring corneal shape. There is general agreement that a CT assessment is neededbefore 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 thatwill make them more accepted by clinicians who are not refractive surgeons. For that to happen, CTsneed to be shown to be clearly helpful in some realm of clinical practice. There have been claims thatthey are useful for contact lens fitting, for evaluating corneal anomalies such as keratoconus or cornealwarpage, for improving refractive surgery outcomes, and for predicting optical limitations to visualacuity. We will consider each of these claims.3a. Contact lens fitting. CT instruments can potentially change how contact lenses are fit15'16'17'18. Asizeable fraction of rigid gas permeable (RGP) lens patients are difficult to fit because the lens doesn'tcenter well or move well. Even the more easily fit soft lenses may benefit from CT assistance becausethe new silicone-hydrogel materials are making extended wear more successful. With extended wearthere is concern that there may be insufficient tear mixing, and improved CT-based custom fitting maybe needed to ensure proper motion for tear mixing. However, the use of CTs for contact lens fitting isstill the exception. I believe that before CTs can become indispensable for contact lens fitting four thingsmust happen:i. Expanded coverage of the cornea. Topography of the full 11 mm of cornea (and possibly the nearsclera for soft lenses) is needed for contact lens fitting because the points of bearing for RGP lenses areoften in the peripheral cornea. Similarly, the comfort and motion of soft lenses are affected by theperipheral regions where corneal curvature changes most rapidly. Unfortunately, present Placido basedtopographers 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 bepossible. Placido CTs may also be used to get expanded coverage, either by splicing multiple shots takenwith different gaze directions, or by placing a high power lens in front of the eye to that the light rayscome in with close to normal incidence even for peripheral cornea19. This could expand the coverage tothe full cornea.ii. Tracing the Placido image rings. There is a further problem with expanding the Placido image to thelimbus. Because of the rapid flattening of peripheral cornea, present CTs do a poor job of tracing theimage rings in peripheral cornea for off-axis shots (needed to expand corneal coverage). It is notuncommon to have Placido images that have the zig-zag shape similar to the hyperopic PRK test shapePlacido image shown in Fig. Ib.iii. Improved understanding of RGP fitting is needed. For RGP lenses, corneal topography has beenused to generate a simulated fluorescein picture. This is a useful picture, especially if it can be extendedto the full area under the contact lens. However, more than that is needed. We need to be able to predictthe 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 lensproperties and corneal shapes in a manner relevant to lens motion. New tools of this sort need to bedeveloped to enable improved simulations of how the lenses will act on real corneas. With thesesimulations numerous contact lens designs could then be tested to see which gives the best fit to theparticular patient's cornea. These simulations will not only facilitate the choice of the best available lenstype, 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 withthe new silicone-hydrogel lenses we will need to learn more about the complexity of the forces thataffect soft lens motion. With extended wear lenses, tear mixing becomes an important factor for washingout debris that could cause problems. Similar to the RGP case, having CT assistance may becomeimportant 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 indicesmeasure the items like the symmetry, spherical aberration and raggedness of the cornea. There havebeen a number of successes in showing that various indices are able to discriminate keratoconus fromother anomalies24. What would really be nice is to have the computer provide a detailed specification ofthe anomaly. Thus, once a cornea has been diagnosed as being keratoconic it should provide the fulldetails about the location, size, severity and shape of the cone25. With that information the clinicianwould be able to follow changes in the cone as it is affected by items such as contact lens wear. Forrefractive surgery and orthokeratology the shape descriptors would include a characterization of theoptic 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 reshapingfollowing the surgery. In addition to the need for improved numerical shape descriptors, there is a needfor improved visualizations26. Present maps that are based on meridional corneal sections often maskunderlying corneal properties because of the singularity at the origin. For example, keratoconus is mosteasily seen and diagnosed in a map of minimum principal curvature at each point. Refractive surgery, onthe other hand is best illustrated by a map of maximum principal curvature.3c. Improving refractive surgery outcomes. It is sometimes thought that knowledge of cornealtopography can help design improved ablation profiles. That is not the case27. The main information fordetermining the ablation profile is knowledge of the wavefront aberration. Since this aberration iscalculated in a plane (often the entrance pupil) it needs to be slightly shifted to determine the aberrationat the cornea. This correction term, based on corneal topography, is quite minimal, at least for areasonable choice of corneal shapes. However, when one considers the biomechanical properties of thecorneal response to refractive surgery, corneal shape near the limbus may play an important role. Thereare 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 tohow 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 bestcorrected vision is worse than 20/20. The clinician usually wants to determine whether the cause of thedegraded performance is due to optical or to neural factors. It had been hoped that corneal topographycould be used to see if optical factors are causing the problem. There are many cases such askeratoconus, corneal warpage28' 29 and refractive surgery with large pupils30'31, where a visual loss canbe predicted by corneal topography. If this article had been written one or two years ago this topic wouldhave been more important than it is at present; new instruments for measuring the full aberrations of theeye will soon become commercially available and will replace CTs for this purpose. However, even withknowledge of the full aberrations of the eye there remains a major challenge of getting a high correlationbetween the optics and predicted acuity for near-normal corneas. Of course, if one includes badlydegraded optics one will get good correlations between optical quality and acuity. We need to do betteron those individuals with moderately degraded wavefronts who still have 20/20 acuity. For individualswith severely degraded wavefronts, improved predictors of the quality of vision is needed. The problemof reliably predicting acuity based on the wavefront is a tantalizing challenge to researchers concernedwith optics and vision.

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Acknowledgments. I thank John Corzine, Robert Mandell and Charles Campbell for helpful commentsand Neurometrics Institute for partial financial support of this research.References

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