Fourier analysis of corneal topography data after photorefractive keratectomy

$Page 1: Fourier analysis of corneal topography data after photorefractive keratectomy$
Fourier analysis of corneal topography data after photorefractive keratectomy

Peter R. Keller, PhD, Charles N.J. McGhee, FRCOphth, Kathryn H. Weed, MSc

More than 100 years since Gullstrand measured the size of photokeratoscopic rings to quantify

corneal curvature, 1 several complex issues involved with the clinical description of the corneal surface remain unresolved. The demand for such information has shifted

From the University oJDundee Department oJOphthalmology, Ninewells Hospital and Medical School, Dundee, Scotland (Keller, McGhee, weed), and Lions Eye Institute, Centre for Ophthalmology and Vision Science, University oj western Australia, Perth, western Australia (Keller).

Reprint requests to Professor Charles N.J McGhee, FRCOphth, Department oJOphthalmology, University oJDundee, Ninewells Hospital and Medical School Dundee, DDI 9SY. Scotland

from the diagnosis of corneal disease,2--4 through contact lens management,5,6 to new forms of keratorefractive surgery7-9 and measures of visual performance.10-12

As the refracting surface responsible for more than two thirds of the optical power of the eye, the anterior corneal surface plays a crucial role in the optical performance of the human visual system. 13 This contribution can be defined in terms of corneal shape, regularity, clarity, and refractive index, and a variety of methods are available to assess these characteristics.

Rec~nt advances in video-imaging and computer technology have facilitated the development of clinical systems for the rapid quantitative assessment of corneal

J CATARACT REFRACT SURG---VOL 24, NOVEMBER 1998 1447

FOURIER ANALYSIS AFTER PRK

topography parameters, advancing our understanding of the corneal surface and its role in retinal image formation. Computer-assisted video keratoscopy has simultaneously expanded our knowledge of the cornea by generating thousands of data points per keratograph, covering much of the corneal surface from near center to limbus, and necessitated the introduction of new data-presentation techniques. The color-coded power map, introduced by Maguire and coauthors14 in 1987, revolutionized the presentation of corneal topography data and remains the clinical standard. Although power maps are quantitative in origin, pattern recognition and classification techniques for evaluating corneal topography data are inherently qualitative. To circumvent interpretation bias and aid diagnosis, many techniques have been used to calculate not only measures of corneal power but other clinically meaningful descriptors from corneal topography data. ll ,15-17

Several methods have been developed to calculate regular corneal astigmatism from corneal topography data, but the most common remains simulated keratometry. Conventional keratometry provides useful information about the image-forming properties of the cornea, although the relation between corneal power and total refractive error is complicated by alignment and noncorneal factors, and measurement is limited to 4 paracentral points under the assumption of orthogonal principal meridians. By comparison, the simulated keratometry (Sim K) methodl7,18 determines corneal

astigmatism from computer-assisted videokeratoscopy using a limited number of data points (only 12 of the several'thousand available) by identifying the full me

ridian with the greatest average power for rings 7, 8, and 9 and subtracting the average of the power values from the same rings for the perpendicular meridian. A more complex best-fit spherocylindrical lens method developed by Maloney and coauthors ll models the central 4.0 mm of the cornea with an ellipsoid and estimates the magnitude and orientation of corneal astigmatism from this surface. The departure of this best-fit surface from the real surface can also be used to measure topographical irregularity.

Traditional spectral analysis techniques are thought to allow a more thorough investigation of regular and irregular astigmatism than the aforementioned methods. Fourier analysis, as first applied by Raasch16 to mathematically model the polar variations in video-

keratographic power values, is a technique that has attracted recent attention. The harmonic components generated via this method have immediate clinical utility; for example, measures of corneal spherical equivalent power, regular astigmatic power and axis, and irregular astigmatism. The decomposition of corneal topography data using fast Fourier transform (FFT) is depicted in Figure 1, which shows how the raw corneal power data can be deconstructed into a series of trigonometric (cosine) functions, each with increasing periodicity. By Fourier series analysis, it is possible to decompose any circumferential fluctuations in corneal power into various components that have direct clinical correlates (Table 1).

The Fourier spherical equivalent and regular astigmatism components describe the standard clinical parameters. Decentration, in relation to FFT, is a tilt of the cornea with respect to the videokeratoscope axis, and irregular astigmatism has been used to refer to a range of optical imperfections that degrade retinal image quality. Although qualitative keratometric assessment can be made of irregular astigmatism based on mire quality, with Fourier analysis, irregular astigmatism can be quantified as the sinusoidal variation in power that cannot be corrected with conventional lenses (i.e., harmonics with n > 2).12,15,19

Studies of normal, test, and postoperative eyes have demonstrated the clinical utility of this method,2o,21

which has subsequently been shown to be' highly precise, with a 95% confidence interval for interoccasion repeatability of -0.02 diopter (D) ± 0.16 (SD) and -0.5 ± 5.7 degrees for Fourier-derived toricity.22

This paper discusses the application of FFT to corneal topography data and evaluates the relation between the change in corneal power and change in refractive error after photo refractive keratectomy (PRK).

I

Table 1. Common clinical descriptions of the trigonometric components generated by Fourier analysis of corneal topography data,

Spherical equivalent

Decentration

Regular astigmatism

Irregular astigmatism

Zero frequency

One cycle

Two cycle

Higher order

=0

=1

=2

2:3

1448 J CATARACT REFRACT SURG-VOL 24, NOVEMBER 1998


1.0 -45 0.8 c -- c 0.6 ... -Q) ... 0.4 ~44 J 0.2 D. 0

0.0 D. iii iii -0.2 Q) c 43 Q)

-0.4 ... E 0 0 0 -0.6 0

-0.8 42 -1.0

0 90 180 270 360 0 90 180 270 360 Hemimeridian Angle (degree) Hemimeridian Angle (degree)

A B

1.0 1.0 0.8 0.8 - - 0.6 c 0.6 c - -... 0.4 ... 0.4 Q) J ~ 0.2 0.2

0 0 0.0 D. 0.0 D.

iii -0.2 iii -0.2 Q) Q)

-0.4 E -0.4 E 0 -0.6 0 -0.6 0 0

-0.8 -0.8 -1.0 -1.0

0 90 180 270 360 0 90 180 270 360 Hemimeridian Angle (degree) Hemimeridian Angle (degree)

c D

Figure 1. (Keller) Fourier decomposition of corneal topography power data into harmonic components of increasing frequency. A: Axial power (solid line) modified from the raw TMS data and the zero frequency FFT component spherical equivalent (dashed line) plotted against hemimeridian angle. B: The first FFT harmonic (n = 1) decentration (tilt or skewness) demonstrating 1 complete cycle over 360 degrees. C: The second FFT harmonic (n = 2) regular astigmatism completing 2 cycles over 360 degrees. 0: The remaining FFT harmonic components (n > 2) that can be used as a measure of irregular astigmatism.

Patients and Methods

Patients

Data were analyzed retrospectively from the clinical records of consecutive patients attending the Corneal Diseases and Excimer Laser Unit at the Sunderland Eye Infirmary (National Health Service Trust), Sunderland, England, for PRK between May 1993 and May 1995. Surface-based PRK and photoastigmatic refractive keratectomy had been performed by a single surgeon (CN.J.McG.) with a 193 nm VISX Twenty-Twenty excimer laser using a 6.0 mm diameter ablation zone and passive fixation as part of a prospective study of PRK outcome.

All patients had discontinued contact lens wear for at least 2 weeks for soft lenses and 4 weeks for rigid gas-

permeable lenses. Patients with a history of ocular disease, surgery, or trauma or a family history of keratoconus were excluded from treatment. Subjective refraction and videokeratographs were retrieved from data collected at preoperative and 3, 6, and 12 month postoperative assessments. All eyes were routinely treated with a postoperative drug regimen of chloramphenicol drops 4 times a day for 7 days, diclofenac drops 4 times a day for 48 hours, and fluorometholone 0.1 % 4 times a day for 12 weeks.

Manifest subjective refractions had been performed by a single investigator using a trial frame and fogging technique in a darkened (average room illuminance 2.5 lux) 6 meter refracting lane and converted to the corneal plane. Uncorrected visual acuity and best spectacle-corrected visual acuity (BSCVA) had been

J CATARACf REFRACf SURG--VOL 24, NOVEMBER 1998 1449

FOURIER ANALYSIS AFTER PRK -----------------------------------------------

measured under the same conditions. Corneal powers had been measured using a 2-stage keratometer, and video keratoscopy had been performed in all cases using the Topographic Modeling System (TMS-1, Computed Anatomy Inc.) that had been calibrated at regular intervals throughout the study. No topical lubricants or eyelid speculums were used to obtain videokeratographs, and videokeratographs were rejected on the basis of tear layer abnormalities, obscured ring images, or poor image quality.23 Only the left eyes of patients with a complete set of information at each assessment were included; 26 left eyes with complete data sets and videokeratographs met the criteria and were analyzed in this study.

Fourier Analysis Fourier series are trigonometric sine and cosme

functions with increasing periodicity; thus a function 1(<1» with a period of 2n can be transformed into a Fourier harmonic series

N

1(<1» = L [an cos(n<l» + bn sin(n<l»] (1) n=O

which can be rewritten as a cosine function including a phase shift angle (ex) as follows:

N

1(<1» = L [en cos n(<I> + aJ] (2) n = 0

where <1> is hemimeridian angle; an, bn, and Cn are Fourier coefficients; ex is a phase shift angle, N is the total number of elements sampled per function; n is the frequency of the harmonic component in the series.

Corneal topography data generated by the TMS-1 were converted from binary to ASCII format, containing the 256 axial power values for each mire, which were then downloaded into Microsoft Excel (version 7.0) for Fourier analysis. The FFT requires a 2X element data set (where x is an integer), limiting this technique to videokeratographs with 2x data points per ring or methods that sample at this frequency (e.g., 256 data points per function). Although interpolation techniques have been suggested to overcome missing data points and correct for spurious readings, these were not used in this investigation. An average of videokeratograph rings 5 to 12 was used, which correlates to an annular region extending from approximately 1.0 mm from the corneal vertex at the inner margin to a 2.2 mm radius at the outer margin, providing a good representation of the corneal surface overlying the entrance pupil for

correlation to visual performance while ignoring the less reliable data provided by the central 4 rings.

Statistical Analysis The multivariate nature of measures of refractive

status presents certain difficulties when statistics are applied to aggregated refractive data because of the interdependence between the conventional representations of spherocylindrical power by the 3 parameters: sphere, cylinder power, and cylinder axis. Transforming these 3 parameters using a vector approach into 3 independent terms-spherical equivalent, cosine astigmatism, and sine astigmatism-provides a more complete analysis of the relation between corneal and total refractive power. 24-26

It has been suggested that because of enantiomorphism, the vector sine terms (corresponding to oblique astigmatism) are equal but opposite for fellow eyes,27 and that changing the sign of the Fourier harmonic coefficients mirrors the corneal topography data about a vertical axis, permitting statistical analysis of left and right eye data.20 However, this assumption is true only as a generality and cannot be used to convert right to left data. This study was, therefore, constrained to the analysis of left-eye data only.

Results

The change in subjective refraction spherical equivalent power over the 12 month interval for excimer PRK is shown in Figure 2. The mean preoperative spherical equivalent of -6.03 ± 3.04 D (range -1.75 to -13.25 D) at the spectacle plane was corrected toward emmetropia but demonstrated a small myopic regres-

m 2,-------------r----, ,~o t __ ~--~T~~~T==±===~T~_! 0>_ 0

~~ -2 /1 I 1 e~ -4

w"E -6 (])~ ,s ~ -8 o's ~ cr -10 QjW a: -12 -'--------------------'

-3 o 3 6 9 12 15

Time (months postoperative)

Figure 2. (Keller) Change in mean refractive error spherical equivalent over 12 months postoperatively (error bars::': 1 SO; n =

26 eyes),



9:45 tijt 43 .Q ~

~~ 41 0._ ene: 39

t"* LJ.. .~ 37

CT W 35

·3 0 3 6 9 12 15

lime (months postoperative)

Figure 3. (Keller) Change in mean FFT·derived corneal spherical equivalent over 12 months postoperatively (error bars:!:: 1 SO; n = 26 eyes).

sion from 3 months (-0.13 ::!: 1.64 D) to 6 months (-0.64 ::!: 1.70 D), with a stabilizing of refractive error between 6 and 12 months (-0.52 ::!: 1.78 D). The changes in mean refractive spherical equivalent from 3 to 6, 6 to 12, and 3 to 12 months were not statistically significant.

The FFT corneal spherical equivalent shows the same trend, with a large reduction in corneal power followed by a slight regression (Figure 3). Preoperative corneal spherical equivalent (mean::!: 1 SD) decreased from 43.52 ::!: 1.73 to 39.03 ::!: 2.51 D at 3 months, 39.49 ::!: 2.32 D at 6 months, and 39.68 ::!: 2.14 D at 12 months postoperatively. The changes from preoperatively to 3 months postoperatively (P = 2.2 X 10-9)

and from 3 to 6 months postoperatively were significant (P = .007); the change from 6 to 12 months was not significant.

The change in refractive spherical equivalent over the 12 month follow-up versus the change in FFT corneal spherical equivalent is shown in Figure 4. Although highly correlated (Pearson correlation coefficient = 0.9402, r2 = 0.8839), the FFT-derived corneal change underestimated the total manifest refractive change, as indicated by the linear regression slope of -0.745 when constrained to intercept at the origin. The probability that the mean change in refractive spherical equivalent equaled the mean change in FFT corneal spherical equivalent over the 12 months is less than 0.0001 (Student t test).

Keratometric and FFT-derived corneal spherical equivalents were highly correlated at each time interval and followed a 1-to-1 relationship (Figure 5).

The mean cosine (Figure 6) and sine (Figure 7) values determined by both subjective refraction and FFT showed no significant change over the 12 month

period. The FFT-determined astigmatic cosine values overestimated the astigmatic cosine values calculated from subjective refraction by approximately 0.3 D, and the 2 measures were significantly different at each time interval (P < .01, a = 5%).

There were no significant differences between the sine values calculated by the 2 techniques (i.e., from refractive data or from corneal topography data) at each interval.

Irregular astigmatism did not change over the 12 month follow-up. Decentration, as measured by the first Fourier harmonic from corneal topography data, demonstrated a significant increase in mean (P < .0002) and variability from preoperative to all postoperative intervals (Figure 8). (This should not be confused with ablation zone decentration relative to the pupil.) The direction of this decentration was more likely to be superonasal (12 of 26 cases) at 12 months postoperatively. Decentration can be quantified in units of displacement as well as prism diopters by a simple conversion using the formula

decentration (il) = displacement (em)

X lens power (0) (3)

Substituting the mean preoperative measures for decentration and corneal spherical equivalent yields the following: 0.512t. = displacement (cm) X 43.52 D; displacement = 0.12 ::!: 0.07 mm (mean::!: SD).

Using this conversion, the mean displacement at 12 months postoperatively was 0.51 ::!: 0.35 mm.

~ .~

.c 0. en tijo Ol ~ e:_ ... e: OOl utij Ii: .~ LJ..CT

.sW Ol en e: <11 .c U

0 0

-2

-4

-6

-8

2 4

y = -0.7449x (2 = 0.8839

6 8 10 12

-10 ..L.-_____________ --J

Change in Refractive Error Spherical Equivalent (D)

Figure 4. (Keller) Comparison of change from preoperative to 12 montAs postoperative refractive error spherical equivalent versus change in FFT-derived corneal spherical equivalent power (n = 26).

J CATARACT REFRACT SURG-VOL 24, NOVEMBER 1998 1451


~ E C])

Iii > ·s 0"

W

Iii () .~

.r:: Q. (f)

l-LL LL

50

y = 0.9983x 48 (2 = 0.9714

46

44

42

40+-----+-----+-----+-----+---~

40 42 44 46 48 Keratometric Spherical Equivalent (0)

A

50

~ E C])

Iii .~ ::::J 0" w Iii ()

"ai .r:: Q. (f)

t LL

45

43 y = 0.9995x (2 = 0.9833

41

39

37

35

33+----r--~----r---~--_r--~

33 35 37 39 41 43 Keratometric Spherical Equivalent (0)

B

45

Figure 5. (Keller) The relation between keratometric spherical equivalent and FFT-derived corneal spherical equivalent preoperatively (A) and 12 months postoperatively (8) (n = 26 eyes).

1.0 --+- Refractive Cosine

~ 0.8

C]) 0.6

-0- FFT Cosine

;.-::::J

0.4 Iii > C]) 0.2 c

~ - r---'iii 0.0 0 <.)

~.2

~.4

·3 o 3 6 9 12 15


Figure 6. (Keller) Change in mean astigmatic cosine value over time; comparison of values obtained by FFT with those by subjective refraction (error bars::':: 1 SO; n = 26 eyes).

1.0

0.8

~ 0.6

~ 0.4

~ 0.2 C]) a 0.0

·0.2

.0.4

·3

___ FFTSine

-0-Refractive Sine

..----:

o 3 6 9 12 15 Time (months postoperative)

Figure 7. (Keller) Change in mean astigmatic sine value over

time; comparison values obtained by FFT with those by subjective refraction (error bars::':: 1 SO; n = 26 eyes).

Unless otherwise stated, all statistical analyses were Student t tests with a = 0.05.

Discussion

The change in spherical equivalent refractive power after excimer laser PRK follows the typical pattern of a small regression of effect for 3 to 6 months, followed by a stabilization of refractive effect between 6 and 12 months (Figure 2).28,29 This trend is duplicated in

Figure 3, showing stabilization of the change in FFTderived corneal spherical equivalent power by 12 months postoperatively.

I!? 5 ~ 0

4 is E en 3 ~ /~ c: 2

0

~ 1 E C]) () 0 C])

Cl -3 0 3 6 9 12 15


Figure 8. (Keller) Change in mean FFT decentration relative to the corneal vertex over time (error bars::':: 1 SO).



Of particular interest is that the change in corneal spherical equivalent underestimated the change in refractive spherical equivalent by 25.5% as indicated by a regression line slope of -0.745. Not all of this discrepancy can be explained by the use of the keratometric index of 1.3375 instead of 1.376 (the actual refractive index for corneal epithelium and stroma), which produces a 10.2% error (1/1.114) as described by Mandell.30

Other potential sources of error have been suggested, including instrument alignment and sampling limitations,30 but these are unlikely to cause such a large systematic difference. Video keratoscopy sampling limitations include both an area sampling bias from the reduced spacing between data points for smaller rings and the exclusion of the unreliable central rings.

The arguments supporting the use of weighting factors12 for analyzing videokeratoscope data are countered by the simple observation that the center of the entrance pupil and the corneal vertex are rarely superimposed and that the location of the entrance pupil center may shift with changing pupil diameterY Not only does the application of the Stiles-Crawford effect (SCE) become more complex in these circumstances, but because the area sampling bias (which is directly proportional to radius) is opposite in effect to the SCE and the combined weighting function varies from 0.87 to 1.13 over rings 5 to 12, we believe it is reasonable to analyze the data without recourse to weighting factors.

The values for both FFT and refractive astigmatism vectors (cosine and sine terms) are relatively small compared with the spherical equivalent values, and they demonstrate substantial variability. It is not surprising that the with-the-rule/against-the-rule astigmatic component, the cosine term, was larger than the oblique astigmatism component (sine term). A significant difference was found between the preoperative refractive and corneal cosine terms, which was repeated at each postoperative interval. This 0.3 D (approximate) difference between cosine terms is consistent with the residual astigmatism measures reported elsewhere (i.e., of the order of 0.5 D).32

The large, highly significant increase in average FFT decentration after excimer laser PRK in this study can be attributed to the ablation zone alignment strategy, wherein the laser was centered on the corneal intercept of the line of sight; that is, the corneal point overlying the center of the virtual entrance pupil rather

than the corneal vertex. In contrast, Fourier-derived terms, including decentration, are calculated relative to the videokeratoscope axis, which when correctly positioned is centered on, and normal to, the corneal vertex. (It follows that videokeratoscope alignment errors during image capture may influence this parameter.) The vertex and the corneal intercept of the line of sight do not coincide. Thus, centering the ablation on the latter creates corneal asymmetry when subsequently measured by the videokeratoscope.

The correct interpretation of this decentration term is further complicated by the accuracy to which the ablation is centered and to a lesser extent the shift in vertex location due to this ablation strategy. In a study by Webber and coauthors,33 average ablation zone decentration relative to the center of the entrance pupil was 0.46 ± 0.27 mm, which compares favorably with other studies. It has been demonstrated by computer simulation that a 5.00 D 6.0 mm diameter ablation for myopia located 1.0 mm from the corneal vertex results in a shift of the vertex by 0.13 mm from its original position (in the direction opposite to the ablation decentration).34

The preoperative FFT decentration value can, therefore, be interpreted as a measure of the distance between the corneal apex and vertex. The preoperative mean of 0.12 mm calculated in this study suggests that if the decentration value measures the distance between the apex and the vertex, the 2 points are closer together than the 0.62 mm reported by Mandell and coauthors,35 who used a very different technique.

Several methods have been proposed for quantifYing irregular astigmatism from corneal topography datay,12,20 Fourier analysis permits decomposition of

corneal topography data into various harmonic components that can be used to measure the variation in corneal power that cannot be corrected by standard lenses; namely, irregular astigmatism. Subtraction of the 0-, 1-, and 2-cycle components from the raw data provides a measurement of irregular astigmatism that may have significant clinical utility. Unfortunately, the relation between irregular astigmatism and BSCVA could not be further investigated in this study because of the possibility that corneal haze after PRK would decrease BSCVA because it cannot be isolated from decreased BSCVA caused by irregular astigmatism.

An understanding of the corneal contribution to



retinal image formation is of great importance given the

recent advances in keratorefractive surgery, and new

methods are now available to quantifY the relation.

Fourier analysis is a powerful mathematical tool for

determining meaningful clinical descriptors from cor

neal topography data after excimer laser photoablative

surgery. The correct interpretation of the Fourier

decentration component requires an understanding of

the complex issues of instrument alignment and ocular

reference axes, and further research is needed to eluci

date the contribution of the central corneal area to

FFT-derived measures of various corneal parameters.

This study overcomes the methodological limita

tions of earlier work36 in the determination of corneal

spherical equivalent power and demonstrates a highly

correlated relation between the change in corneal power

and the change in manifest refraction following excimer

laser PRK. As the results from this study suggest, this

relation is more complex than a simple I-to-l ratio, and

caution must be exercised when interpreting refractive

power changes from corneal topography data after

excimer laser PRK or laser in situ keratomileusis,

particularly when such data are used to direct surgical

retreatment.

References

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Presented in part at the United Kingdom dr Ireland Society of Cataract dr Refractive Surgeons Satellite Meeting, Glasgow, Scotland, April 199B.

Supported by a research travel grant from Lions Eye Institute and an ad hoc grant from the University of Dundee Department of Ophthalmology.


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Fourier analysis of corneal topography data after photorefractive keratectomy