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Effect of astigmatic keratotomy on spherical equivalent: results of the astigmatism reduction clinical trial
Effect of Astigmatic Keratotomy on Spherical Equivalent: Results of the Astigmatism Reduction Clinical Trial ELLA G. FAKTOROVICH, MD, ROBERT K. MALONEY, MD, FRANCIS W. PRICE, JR, MD, AND THE ARC-T STUDY GROUP*
● PURPOSE: To determine the effect of astigmatic keratotomy on spherical equivalent, as measured by the coupling ratio and a new quantity, coupling constant. ● METHODS: In a prospective multicenter study, subjects underwent arcuate keratotomy at a 7-mm optical zone by means of the Lindstrom nomogram for correction of astigmatism. One hundred fifty-seven eyes of 95 patients who had a follow-up examination 1 month postoperatively were studied. Mean preoperative refractive cylinder 6 SEM was 2.82 6 1.17 diopters. Coupling ratio was defined as the ratio of the flattening of the incised meridian to the steepening of the opposite meridian. Coupling constant was defined as the ratio of the change in spherical equivalent to the magnitude of the vector change in astigmatism. Coupling ratio, coupling constant, and change in spherical equivalent were calculated on the basis of change in refraction and keratometry. ● RESULTS: On the basis of change in refraction, coupling ratio was 0.95 6 0.10 (mean 6 SEM) and coupling constant was 20.01 6 0.03, consistent with a minor shift in the spherical equivalent of 20.03 6 0.07 diopter. On the basis of change in keratometry, coupling ratio was 0.84 6 0.05 and coupling constant was 20.04 6 0.02, consistent with minor postoperative keratometric steepening of 20.10 6 0.04 diopter. Coupling ratio based on change in refraction was not statistically different from the coupling ratio predicted by the Gauss’ law for inelastic domes (P 5 .370). Incision length and number, amount of achieved cylinder correction, age, and sex had
Accepted for publication Oct 8, 1998. From the Jules Stein Eye Institute and the Department of Ophthalmology, University of California, Los Angeles, California (Drs Faktorovich and Maloney), and the Cornea Research Foundation of America and Corneal Consultants of Indiana, Indianapolis, Indiana (Dr Price). *The members of the Astigmatism Reduction Clinical Trial Study Group are listed at the end of the article. Dr Maloney was supported by a Career Development Award from Research to Prevent Blindness, Inc, New York, New York. Dr Faktorovich was a Heed Fellow and an Elsa and Louis Kelton Fellow. Reprint requests to Robert K. Maloney, MD, 10921 Wilshire Blvd, Suite 900, Los Angeles, CA 90024.
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no statistically significant effect on coupling ratio, coupling constant, and change in spherical equivalent. ● CONCLUSIONS: Cornea behaved as an inelastic surface in response to arcuate keratotomy performed with the Astigmatism Reduction Clinical Trial study nomogram. On average, astigmatic keratotomy had a minimal effect on spherical equivalent refraction. There was variability, however, in coupling ratio, coupling constant, and change in spherical equivalent from eye to eye after astigmatic keratotomy. Caution is therefore advised when simultaneous correction of cylinder and spherical equivalent is planned. (Am J Ophthalmol 1999;127:260–269. © 1999 by Elsevier Science Inc. All rights reserved.)
A
STIGMATIC KERATOTOMY IS A COMMON METHOD
for the surgical correction of astigmatism. Even with the advent of photoastigmatic refractive keratectomy, some protocols for photorefractive keratectomy and laser in situ keratomileusis still combine astigmatic keratotomy and excimer laser keratectomy to simultaneously correct astigmatism and spherical refractive error.1,2 Additionally, astigmatic keratotomy is often combined with cataract surgery and is the most common method of correcting astigmatism after penetrating keratoplasty.3–15 Studies in both human and cadaver eyes have demonstrated that astigmatic keratotomy not only flattens the incised meridian but also induces steepening of the opposite unincised meridian 90 degrees away.7,13,14,16 –20 The ratio of the amount of flattening of the incised meridian to the amount of steepening of the opposite meridian is defined as the coupling ratio.3 The coupling ratio is useful in planning a refractive procedure because it predicts the effect of astigmatic keratotomy on spherical equivalent refraction. If the coupling ratio is 1, the flattening of the incised meridian will equal the steepening of the opposite meridian, and the spherical equivalent will not change after astigmatic keratotomy. A coupling ratio greater than 1 indicates that the flattening of the incised meridian will be greater than the steepening of the opposite meridian,
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FIGURE 1. Change in curvature of the principal corneal meridians induced by arcuate keratotomy. Ksteep is the power of the steep preoperative meridian. Kflat is the power of the flat preoperative meridian. After astigmatic keratotomy, the power of the incised meridian is Ksteep 2 F, where F is flattening of the incised meridian. The power of the opposite, unincised meridian is Kflat 1 S, where S is steepening of the unincised meridian.
there will be overall corneal flattening, and the spherical equivalent will shift toward hyperopia. Conversely, a coupling ratio less than 1 means that the steepening of the incised meridian will be greater than the flattening of the opposite meridian, and the spherical equivalent will shift toward myopia. Therefore, the effect of astigmatic keratotomy on spherical equivalent will depend on the coupling ratio. If the coupling ratio of an astigmatic procedure is known and simultaneous correction of the astigmatism and the spherical equivalent is planned, more accurate correction of the spherical equivalent can be achieved. Coupling ratio was defined by Thornton7 on the basis of the concept of gaussian curvature of a surface.21 Gaussian curvature of a surface (Kgauss) is the product of the curvatures of the two principal meridians, that is, Kgauss 5 K1 3 K2, where K1 is the curvature of one principal meridian and K2 is the curvature of the meridian 90 degrees away. If the curvature of the steep preoperative meridian is Ksteep and the curvature of the flat preoperative meridian is Kflat, then the preoperative gaussian curvature is Ksteep 3 Kflat (Figure 1). Gauss’ law states that when the curvature of a flexible but inelastic surface is changed in one meridian, the curvature of the orthogonal meridian changes in the opposite way so that the gaussian curvature of the surface (Kgauss) remains constant.21 Therefore, if F is flattening of the steep preoperative meridian and S is steepening of the flat preoperative meridian, then the postoperative gaussian curvature is (Ksteep 2 F) 3 (Kflat 1 S) (Figure 1) and Kgauss 5 Ksteep 3 Kflat 5 (Ksteep 2 F) 3 (Kflat 1 S)
[1]
Some authors have interpreted the Gauss’ law to mean that the change in the orthogonal meridian has to be equal to and opposite of the change in the primary meridian.3,4,7,11 Therefore, they argued, if cornea was an inelastic surface, flattening of one meridian would equal steepening of the opposite meridian, coupling ratio would equal 1, and VOL. 127, NO. 3
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spherical equivalent would not change after astigmatic keratotomy. Early studies, however, have shown that coupling ratio is usually not 1. Rather, it ranges from 0.72 to 1.88.18,19 On the basis of this evidence, some have concluded that the cornea must be elastic, it does not obey the Gauss’ law for inelastic domes, and spherical equivalent should change after astigmatic keratotomy.3,7 Biomechanical studies, however, have found that human adult cornea stretches only 20 mm when intraocular pressure increases from 10 to 40 mm Hg and that a Young modulus of corneal elasticity ranges from 3 3 105 to 2 3 107 Nm22, suggesting that human adult cornea is, in fact, quite inelastic.22–25 Most previous studies of the coupling ratio used keratometry to calculate the coupling ratio.18,19 Keratometry, however, does not provide an accurate estimate of corneal power after refractive surgery. Refraction estimates effective corneal power more accurately than keratometry.26,27 In this study, we demonstrate that the coupling ratio of an inelastic surface does not have to equal 1, although it should be close to 1. We then test whether a living cornea behaves like an elastic or an inelastic surface by analyzing coupling ratio calculated on the basis of both change in refraction and keratometry after arcuate keratotomy performed in the Astigmatism Reduction Clinical Trial (ARC-T), a prospective, multicenter, cohort study. Additionally, we define a new quantity, coupling constant, which allows a simple calculation of the change in spherical equivalent expected after astigmatic keratotomy. We then analyze the effect of achieved cylinder correction, arc length, incision number, age, and sex on coupling ratio, coupling constant, and change in spherical equivalent.
PATIENTS AND METHODS DETAILS OF STUDY DESIGN, PATIENT SELECTION, SURGICAL
technique, and surgical nomograms of the ARC-T have
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been published previously.9,10 We studied 157 eyes of 95 patients (63 women and 32 men) who had a follow-up examination at 1 month 6 10 days after astigmatic keratotomy. Longer follow-up data were not available because most of the eyes underwent radial keratotomy to correct the spherical equivalent at 1 month 6 10 days after astigmatic keratotomy. Average (6 SD) age was 41 6 11.84 years (range, 18 to 84 years). Mean preoperative keratometric cylinder was 2.74 6 1.12 diopters (range, 0.38 to 5.75 diopters). Mean preoperative refractive cylinder was 2.82 6 1.17 diopters (range, 1.00 to 6.50 diopters). Of the 157 eyes, 144 had had no previous eye surgery except glaucoma or retinal laser surgery. Thirteen eyes underwent cataract extraction with intraocular lens placement 6 months to 2 years before astigmatic keratotomy. All eyes underwent arcuate keratotomy at a 7-mm optical zone. One or two arcuate incisions were performed. Individual arc length ranged from 30 to 90 degrees. Total arc length ranged from 45 to 180 degrees, depending on the amount of desired astigmatism correction and the patient’s age. Lindstrom’s surgical nomograms were used.9,10 The formula for calculating coupling ratio (CR) was derived from the surgically induced refractive change described by Holladay and associates.20,28 Surgically induced refractive change represents the power of a spherocylindrical lens that, when placed before the preoperative eye, would simulate the surgical result. Surgically induced refractive change is S 1 DA @ a degrees, where S is the spherical component of the surgically induced spherocylinder; DA is the magnitude of the vector change in astigmatism in plus cylinder format; and a degrees is the axis of the vector change in astigmatism. When astigmatic keratotomy is performed, the axis a degrees corresponds to the meridian of the corneal incision, so the power of the cylinder DA @ a degrees lies in the opposite meridian. Therefore, the flattening of the steep meridian is 2S and the steepening of the flat meridian is S 1 DA. If refractive change in spherical equivalent, DSEr, is defined as the postoperative spherical equivalent refraction minus the preoperative spherical equivalent refraction, then from the definition of surgically induced refractive change, DSEr 5 2(S 1 DA/2). Flattening of the incised meridian, F, equals DA/2 1 DSEr. Steepening of the opposite meridian, S, equals DA/2 2 )SEr. We express the coupling ratio as the ratio of the flattening of the incised meridian to the steepening of the opposite meridian. Therefore, coupling ratio (CRF/S) is CRF/S 5 (DA/2 1 DSEr) / (DA/2 2 DSEr)
equivalent keratometry because postoperative increase in keratometric spherical equivalent indicates a myopic shift in refraction and postoperative decrease in keratometric spherical equivalent indicates a hyperopic shift in refraction. DA is the magnitude of the vector change in astigmatism calculated on the basis of either change in refraction or keratometry.29 Positive CRF/S values indicate that one meridian flattened whereas the opposite one steepened. Negative CRF/S values indicate that both meridians either flattened or steepened. Equation 2 is useful because it allows calculation of the coupling ratio even when the axis of astigmatism shifts after astigmatic keratotomy. Equation 2 is also useful because it allows calculation of the coupling ratio based on either refraction or keratometry. Coupling ratio of an inelastic gaussian surface (CRgauss) is derived from the equation 1 in the introduction as follows: if Kgauss 5 Ksteep 3 Kflat 5 (Ksteep 2 F) 3 (Kflat 1 S), then Ksteep 3 Kflat 5 Ksteep 3 Kflat 1 Ksteep 3 S 2 Kflat 3 F 2 F 3 S. When coupling ratio is defined as the ratio of the flattening of the steep preoperative meridian to the steepening of the flat preoperative meridian, that is, F/S, and the above equation is solved for F/S, then Coupling Ratio 5 CRgauss 5 F/S 5 (Ksteep 2 F) / Kflat
To facilitate calculation of the change in spherical equivalent after astigmatic keratotomy, we defined the coupling constant (CC). Coupling constant is the ratio of the change in spherical equivalent to the magnitude of the vector change in astigmatism, that is, CC 5 DSE/DA
CC 5 1/2(CRF/S 2 1) / (CR F/S 1 1)
[5]
If flattening of the incised meridian is greater than steepening of the opposite meridian, CRF/S will be greater than 1, CC will be positive, and DSE will be positive, consistent with a hyperopic shift in spherical equivalent and overall corneal flattening. If flattening of the incised meridian is less than steepening of the opposite meridian, CRF/S will be less than 1, CC will be negative, and DSE will be negative as well, consistent with a myopic shift in spherical equivalent and overall corneal steepening. The coupling ratios and the coupling constants in the
[2]
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[4]
Coupling constant is derived from equation 2 as follows: CRF/S 5 (DA/2 1 DSE)/(DA/2 2 DSE). If the numerator and the denominator are divided by DA, then CRF/S 5 (1/2 1 DSE/DA)/(1/2 2 DSE/DA) 5 (1/2 1 CC) / (1/2 2 CC). Solving for CC,
To calculate keratometric coupling ratio by means of equation 2, it is necessary to define the change in keratometric spherical equivalent, DSEk, as preoperative spherical equivalent keratometry minus postoperative spherical 262
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TABLE 1. Results of Astigmatic Keratotomy Calculated Based on Change in Refraction and Change in Keratometry*
Magnitude of vector change in astigmatism, A (diopters) Flattening of incised meridian, F (diopters) Steepening of opposite meridian, S (diopters) Coupling ratio (CRF/S 5 F/S) Coupling ratio (CRG 5 Ksteep 2 F/Kflat) Change in spherical equivalent, DSE (diopters) Coupling constant, CC
Refraction
Keratometry
2.35 6 0.11 1.15 6 0.09 (0.97, 1.33) 1.20 6 0.08 (1.04, 1.36) 0.95 6 0.10 (0.75, 1.15) 1.04 6 0.002 (1.036, 1.044) 20.03 6 0.07 (20.17, 0.11) 20.01 6 0.03 (20.07, 0.05)
2.38 6 0.10 1.08 6 0.07 (0.94, 1.22) 1.29 6 0.06 (1.17, 1.41) 0.84 6 0.05 (0.74, 0.94) 1.04 6 0.002 (1.036, 1.044) 20.10 6 0.04 (20.18, 20.02) 20.04 6 0.02 (20.08, 0.00)
Ksteep 5 the keratometric power of the steep preoperative meridian; Kflat 5 the keratometric power of the flat preoperative meridian; F 5 flattening of the incised meridian calculated based on change in either refraction or keratometry. *All data are mean 6 SEM with 95% confidence intervals.
ARC-T study cohort were not normally distributed. Therefore, to determine mean coupling ratio (mean CRF/S and mean CRgauss) and mean coupling constant (mean CC), maximum likelihood estimators of the means were calculated. The maximum likelihood estimator of the mean CRF/S was calculated by dividing the mean flattening of the incised meridian by the mean steepening of the opposite meridian. The maximum likelihood estimator of the mean CRgauss was calculated by dividing the mean Ksteep 2 F by the mean Kflat. The maximum likelihood estimator of the mean coupling constant was calculated by dividing the mean change in spherical equivalent by the mean magnitude of vector change in astigmatism. Calculations were performed for data based on both keratometry and refraction. Standard error of the mean for the maximum likelihood estimator of the was calculated as follows: SEM 5 ~MEAN 1/MEAN 2)
Î(SEM21/MEAN21) 1 (SEM22/MEAN22) 2 2 r1,2 (SEM1/MEAN1) 3 (SEM2/MEAN2), where MEAN1 is the mean of the values in numerator; MEAN2 is the mean of the values in the denominator; SEM1 is the standard error of MEAN1; SEM2 is the standard error of MEAN2; and r1,2 is a two-tailed Spearman correlation coefficient between MEAN1 and MEAN2. Two-tailed t test for correlated means was used to compare maximum likelihood estimator of the mean CRF/S to the mean CRgauss. Two-tailed t test for two independent VOL. 127, NO. 3
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samples was used to compare the maximum likelihood estimator of the mean coupling ratio, coupling constant, and mean change in spherical equivalent in different age groups, sexes, and groups with different cylinder corrections and different arc lengths. Two-tailed t test for two independent samples was also used to compare the mean age, cylinder correction, and arc length in the group of eyes with 1 diopter or more shift in spherical equivalent to the group of eyes with less than 1 diopter shift in spherical equivalent. All values are mean 6 SEM. All data were analyzed with SPSS software.
RESULTS KERATOMETRIC AND REFRACTIVE FLATTENING OF THE IN-
cised meridian, steepening of the opposite meridian, coupling ratio, coupling constant, change in spherical equivalent, and magnitude of vector change in astigmatism are given in Table 1. On the basis of the change in refraction, mean flattening of the incised meridian was 1.15 6 0.09 diopter. Mean steepening of the opposite meridian was 1.20 6 0.08 diopter. The sum of change in the surgical meridian and the opposite meridian should be equal to the amount of astigmatism corrected.7 Indeed, the sum of mean flattening in the incised meridian and mean steepening in the opposite meridian was 2.35 6 0.12 diopters, consistent with the mean magnitude of vector change in astigmatism of 2.35 6 0.11 diopters. Mean coupling ratio, CRF/S, calculated on the basis of the change in refraction was 0.95 6 0.10. This value was not significantly different from 1 (95% confidence inter-
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vals, 0.75, 1.15) and was consistent with a minor mean myopic shift in refractive spherical equivalent of 20.03 6 0.07 diopter, which was not significantly different from 0 (95% confidence intervals, 20.17, 0.11). Mean keratometric flattening of the incised meridian was 1.08 6 0.07 diopters. Mean keratometric steepening of the opposite meridian was 1.29 6 0.06 diopters. The sum of mean flattening and mean steepening was 2.37 6 0.10 diopters, consistent with the mean magnitude of vector change in keratometric astigmatism of 2.38 6 0.10 diopters. Mean coupling ratio, CRF/S, calculated on the basis of keratometric change was 0.84 6 0.05, consistent with an overall mean corneal steepening of 20.10 6 0.04 diopter. Mean keratometric coupling ratio was statistically different from 1 (95% confidence intervals, 0.74, 0.94). Likewise, mean change in keratometric spherical equivalent was statistically different from 0 (95% confidence intervals, 20.18, 20.02). In the Patients and Methods section, we demonstrated that coupling ratio of an inelastic surface, CRgauss, is (Ksteep 2 F)/Kflat. To test whether cornea behaves like an inelastic surface in response to arcuate keratotomy, we compared the coupling ratio calculated from the ARC-T study data, CRF/S, with the coupling ratio predicted by the Gauss law for inelastic domes, CRgauss. On the basis of refraction, CRF/S was 0.95 6 0.10. CRgauss was 1.04 6 0.002 (Table 1). The difference between CRF/S and CRgauss was not statistically significant (P 5 .370). On the basis of keratometry, CRF/S was 0.84 6 0.05. CRgauss was 1.04 6 0.002 (Table 1). The difference between CRF/S and CRgauss was statistically significant (P , .001). Standard keratometry, however, overestimates effective corneal power after myopic refractive surgery.27,28,30 Overestimation of the effective corneal power will result in underestimation of the true coupling ratio because it will decrease the numerator and increase the denominator of F/S. We hypothesized that the keratometric coupling ratio calculated from the ARC-T study data is probably smaller that the true keratometric coupling ratio and that, like the coupling ratio calculated on the basis of the change in refraction, the true keratometric coupling ratio is probably close to the one predicted by the Gauss law for inelastic domes. To test this hypothesis, we assumed that postoperative keratometry overestimated true corneal power by a small amount of 0.125 diopter. If this were the case, true mean flattening of the incised meridian would have been 1.21 6 0.07 diopters and true mean steepening of the opposite meridian would have been 1.17 6 0.06 diopters. Therefore, true mean keratometric coupling ratio, CRF/S, would have been 1.03 6 0.06 (95% confidence intervals, 0.91, 1.15). True mean keratometric coupling ratio predicted by the Gauss law for inelastic domes, CRgauss, would have not changed appreciably if a 0.125-diopter keratometric error was made. The coupling ratio would have still been 1.04 6 0.002 (95% confidence intervals, 1.036, 264
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FIGURE 2. Achieved refractive cylinder correction vs change in refractive spherical equivalent. Slope 6 SES of the linear regression line plotted through 0 is 0.01 6 0.03 (95% confidence intervals, 20.04, 0.06). There is no significant effect of achieved refractive cylinder correction on refractive spherical equivalent.
1.044). The difference between CRF/S and CRgauss would then not have been statistically significant (P 5 .881). We defined coupling constant as the ratio of the change in spherical equivalent to the magnitude of vector change in astigmatism. Coupling constant was calculated from the coupling ratio as follows: CC 5 1/2(CR21)/(CR11). Mean coupling constant calculated on the basis of change in refraction was 20.01 6 0.03, consistent with a minimal myopic shift in refractive spherical equivalent. Mean coupling constant calculated on the basis of keratometric change was 20.04 6 0.02, consistent with mild keratometric corneal steepening postoperatively (Table 1). By its definition, coupling constant should be equal to the slope of the linear regression line relating the change in spherical equivalent to the magnitude of the achieved cylinder correction. Indeed, the slope in Figure 2 was 0.01 6 0.03, which was not significantly different from the mean coupling constant calculated on the basis of change in refraction (P 5 .680). The results of the effect of attempted and achieved cylinder correction, arc length, incision number, age, and sex on coupling ratio, coupling constant, and change in spherical equivalent are presented in Table 2. In eyes with achieved refractive cylinder correction of less than 2.00 diopters, mean refractive coupling ratio was 0.65 6 0.15. Larger achieved cylinder corrections tended to result in larger coupling ratios. In eyes with achieved refractive cylinder correction of 2.00 diopters or greater, mean refractive coupling ratio was 1.06 6 0.12. The difference between the mean coupling ratios, however, was not statistically significant when the data were analyzed by means of the Bonferroni correction for multiple compariOF
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TABLE 2. Effect of Surgical and Demographic Factors on Outcome Variables* Refraction (r)
No. of Eyes
Achieved cylinder correction ,2.00 diopters $2.00 diopters P‡ Total arc length ,90 degrees $90 degrees P Incision number 1 3 90 degrees 2 3 45 degrees P Age ,40 yrs $40 yrs P Sex Male Female P
Keratometry (k)
CRF/S(r)
CCr
DSEr
CRF/S(k)
CCk
DSEk
68 (73)† 89 (84)
0.65 6 0.15 1.06 6 0.12 .035
20.11 6 0.06 0.02 6 0.03 .054
20.13 6 0.07 0.05 6 0.10 .165
0.74 6 0.09 0.87 6 0.06 .232
20.08 6 0.03 20.03 6 0.18 .780
20.10 6 0.04 20.11 6 0.60 .825
62 95
0.87 6 0.16 0.99 6 0.12 .549
20.04 6 0.04 20.003 6 0.03 .454
20.06 6 0.07 20.01 6 0.10 .707
0.76 6 0.09 0.87 6 0.06 .319
20.07 6 0.03 20.03 6 0.02 .273
20.11 6 0.05 20.10 6 0.05 .987
33 20
1.07 6 0.22 0.93 6 0.25 .604
0.02 6 0.08 20.02 6 0.06 .690
0.04 6 0.19 20.05 6 0.17 .737
0.88 6 0.09 0.97 6 0.21 .690
20.03 6 0.03 20.01 6 0.06 .735
20.08 6 0.07 20.02 6 0.13 .645
84 73
0.80 6 0.09 1.14 6 0.21 .130
20.05 6 0.03 0.03 6 0.05 .660
20.12 6 0.06 0.08 6 0.13 .146
0.82 6 0.07 0.86 6 0.08 .706
20.05 6 0.02 20.04 6 0.02 .725
20.11 6 0.05 20.10 6 0.05 .809
51 106
1.01 6 0.19 0.92 6 0.12 .689
0.003 6 0.05 20.02 6 0.03 .689
0.01 6 0.14 20.05 6 0.07 .720
0.88 6 0.08 0.82 6 0.06 .549
20.03 6 0.03 20.05 6 0.02 .583
20.09 6 0.07 20.11 6 0.04 .799
Ksteep 5 the keratometric power of the steep preoperative meridian; Kflat 5 the keratometric power of the flat preoperative meridian; F 5 flattening of the incised meridian calculated based on change in either refraction or keratometry. *All data are mean 6 SEM. † Number in parentheses indicates the number of eyes in each group of the achieved keratometric cylinder correction. Number outside parentheses represents the number of eyes in each group of the achieved refractive cylinder correction. ‡ Two-tailed t test for two independent samples. With Bonferroni correction for multiple comparisons, P , .01 was considered statistically significant.
sons (P 5 .035). In eyes with achieved refractive cylinder correction of less than 2.00 diopters, mean refractive coupling constant was 20.11 6 0.06. In eyes with achieved refractive cylinder correction of 2.00 diopters or greater, mean refractive coupling constant was 0.02 6 0.03. The difference between the means was not statistically significant (P 5 .054). Smaller achieved refractive cylinder correction resulted in a minor mean myopic shift in refractive spherical equivalent of 20.13 6 0.07 diopter. Larger achieved refractive cylinder correction resulted in a small mean hyperopic shift in refractive spherical equivalent of 0.05 6 0.10 diopter. The difference between the means was not statistically significant (P 5 .165). Figure 2 confirms the minimal effect of the achieved refractive cylinder correction on the change in spherical equivalent refraction. The slope of the linear regression line was 0.01 6 0.03, which was not significantly different from 0 (95% confidence intervals, 20.04, 0.06). Smaller achieved keratometric cylinder correction resulted in smaller mean keratometric coupling ratio, coupling constant, and change in spherical equivalent than VOL. 127, NO. 3
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did larger achieved keratometric cylinder correction. The differences between the means, however, were not statistically significant. Mean keratometric and refractive coupling ratio, coupling constant, and mean change in keratometric and refractive spherical equivalent in eyes with total arc length of less than 90 degrees were compared with those in the eyes with total arc length of 90 degrees or longer. Longer arcs tended to result in larger coupling ratios, coupling constants, and flatter corneas than shorter arcs, but the difference between the means was not statistically significant. Linear regression analysis confirmed the minimal correlation between total arc length and change in refractive spherical equivalent (Figure 3). The slope of the linear regression line was 20.0002 6 0.001, which was not significantly different from 0 (95% confidence intervals, 20.002, 0.001). Number of arcuate incisions (one or two) also had no effect on coupling ratio, coupling constant, or the change in spherical equivalent. Mean coupling ratio, coupling constant, and change in spherical equivalent in eyes with
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refraction or the change in keratometry (equation 2). Use of the vector change in astigmatism allows one to determine the coupling ratio even when the axis of astigmatism shifts after the procedure. To simplify the calculation of the expected change in spherical equivalent after astigmatic keratotomy, we defined a new entity, the coupling constant, which is calculated from the coupling ratio. Coupling constant is the ratio of the change in spherical equivalent to the magnitude of the vector change in astigmatism. The change in spherical equivalent expected after astigmatism correction is, then, the coupling constant of the procedure multiplied by the intended astigmatism correction. For example, if a 2-diopter astigmatism correction is planned with the use of astigmatic keratotomy incisions with a coupling constant of 0.2, a 0.4-diopter hyperopic shift in spherical equivalent refraction should be expected. The residual spherical equivalent can then be determined and corrected with either photorefractive keratectomy or laser in situ keratomileusis. Previous reports of coupling ratios have been limited to a small sample size, cadaver eyes, or studies of combined transverse and radial incisions.12–20,31 The ARC-T study is the largest study of arcuate keratotomy performed at a single optical zone. Coupling ratio calculated on the basis of the change in refraction was 0.95 6 0.10. This value was not statistically different from 1, consistent with a small coupling constant of 20.01 6 0.03, which was not significantly different from 0, and a minor shift in spherical equivalent of 20.03 6 0.07, which was also not statistically different from 0. Chavez and associates13 also reported coupling ratios calculated on the basis of change in refraction after arcuate keratotomy. In their study, mean coupling ratio was 1.16 6 0.36, consistent with a hyperopic shift in spherical equivalent of 10.18 6 0.34 (Table 3). The study, however, was limited to 15 eyes, and there was considerable variability in the outcome. Moreover, Chavez and associates13 studied arcuate incisions at a 5-mm optical zone. Transverse incisions performed at smaller optical zones result in larger coupling ratios and more corneal flattening than incisions performed at larger optical zones.14,16,19 Coupling ratio calculated on the basis of change in keratometry in this study was 0.85 6 0.05, which was significantly different from 1. Keratometric coupling constant was 20.04 6 0.02, consistent with the mean change in keratometric spherical equivalent of 20.10 6 0.04, which was significantly different from 0. This result was in general agreement with other studies.13,14,18 Standard keratometry, however, overestimates corneal power after refractive surgery.27,28,30 For example, after radial keratotomy, keratometric values measured by standard keratometry are 0.83 to 5.01 diopters steeper than the ideal and the refraction-derived values predicted for an accurate intraocular lens power calculation.27,28 Because astigmatic keratotomy is performed at a larger optical zone than radial keratotomy, one might
FIGURE 3. Total arc length vs change in refractive spherical equivalent. Slope 6 SES of the linear regression line plotted through 0 is 20.0002 6 0.001 (95% confidence intervals, 20.002, 0.001). There is no significant effect of total arc length on refractive spherical equivalent.
one arc 90 degrees long were compared to those in the eyes with two 45-degree arcs. The difference between the means was not statistically significant. Mean coupling ratio and coupling constant tended to be larger in the eyes of older patients and in men. The difference between the means, however, was again not statistically significant. Most eyes in the ARC-T study had a minor shift in spherical equivalent of less than 1 diopter. Twenty-four of 157 eyes, however, had a 1-diopter or more change in spherical equivalent in the hyperopic or myopic direction (Figure 2). Table 3 compares surgical and demographic variables in the eyes with 1 diopter or more shift in spherical equivalent with those in the eyes with less than 1 diopter of shift in spherical equivalent after astigmatic keratotomy. Eyes whose spherical equivalent changed 1 diopter or more tended to be in older patients, had larger cylinder corrections, and had longer arcuate incisions than the eyes whose spherical equivalent changed less than 1 diopter. The difference between the groups, however, was not statistically significant.
DISCUSSION COUPLING RATIO DESCRIBES THE EFFECT OF ASTIGMATIC
keratotomy on spherical equivalent. If the coupling ratio of an astigmatic keratotomy procedure is known, spherical equivalent remaining after the astigmatism correction can be calculated. We developed a simple formula for calculating coupling ratio based on either the change in 266
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TABLE 3. Surgical and Demographic Factors in Eyes With 1 Diopter or Greater Change in Spherical Equivalent (DSEr) vs Eyes With Less Than 1 Diopter Change After Astigmatic Keratotomy*
Achieved cylinder correction (diopters) Total arc length (degrees) Age (yrs)
$1 Diopter DSEr (n 5 24 Eyes)
,1 Diopter DSEr (n 5 133 eyes)
P value†
2.83 6 0.31 106.04 6 8.92 43.83 6 2.85
2.27 6 0.11 87.11 6 2.93 40.26 6 0.99
.101 .053 .245
*All data are mean 6 SEM. † Two-tailed t test for two independent samples. With Bonferroni correction for multiple comparisons, P , .02 was considered statistically significant.
expect that keratometry would be more accurate after astigmatic keratotomy. In the ARC-T study, astigmatic keratotomy was performed with a 7-mm optical zone. However, even a small overestimation of the effective corneal power will result in underestimation of the coupling ratio. We found that if keratometry after arcuate incisions in the ARC-T study overestimated true corneal power by 0.125 diopter, true coupling ratio would have been 1.03 6 0.06, which would not have been significantly different from 1. Coupling ratios reported in the early studies were not 1. These results led some authors7 to conclude that the cornea does not follow the Gauss’ law for inelastic domes, and spherical equivalent should be expected to change after astigmatic keratotomy. We have demonstrated that coupling ratio of an inelastic surface should not be expected to equal 1. Rather, it should be (Ksteep 2 F)/Kflat. However, because Ksteep is typically close to 44 diopters, Kflat is typically close to 42 diopters, and F is typically 1 to 2 diopters, coupling ratio of an inelastic surface should be between 1.00 and 1.03 for virtually all corneas. Indeed, the mean coupling ratio calculated on the basis of change in refraction in the ARC-T study was not significantly different from 1 and from the mean coupling ratio predicted by the Gauss’ law for inelastic domes. The mean coupling ratio calculated on the basis of change in keratometry, however, was significantly different from the mean coupling ratio for inelastic domes predicted by Gauss. This discrepancy could have resulted from an inaccurate estimation of the true central corneal power by keratometry. We have demonstrated that if keratometry in the ARC-T study overestimated true corneal power by 0.125 diopter, true keratometric coupling ratio would not have been significantly different than the keratometric coupling ratio predicted by the Gauss law for inelastic domes. Coupling ratio results of the early studies could have, likewise, been flawed because coupling ratios were calculated solely on the basis of keratometry. Previously, it has been reported that surgical factors, such as incision length and number, and demographic factors, such as age and sex, strongly influence the amount of astigmatism corrected.9 In this study, we have shown VOL. 127, NO. 3
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that none of these factors significantly affects the coupling ratio, coupling constant, or change in spherical equivalent after astigmatic keratotomy. Most of the published series do report a correlation between arc length and coupling ratio (Table 4).3,4,11,14,18 –20,31 Some authors have even stated that incision length is the major factor that affects the coupling ratio of transverse incisions.3,4 Several studies note a hyperopic shift with increasing arc length, whereas most others report a myopic shift associated with increasing arc length. Most studies, however, are limited to a small sample size. We noted a tendency toward mild hyperopia with the increasing arc length and the amount of achieved cylinder correction. However, the effect of arc length and achieved cylinder correction on coupling ratio, coupling constant, and change in spherical equivalent was not statistically significant. One of the limitations of this study is a relatively short mean follow-up time of 1 month. Perhaps if the follow-up time were extended, a statistically significant effect of the arc length and the amount of the achieved cylinder correction on coupling ratio, coupling constant, and change in spherical equivalent would have been observed. Lindquist and Lindstrom,4 however, found that the 1-month results after astigmatic keratotomy correlate well with the 1-year results. On the basis of change in refraction after astigmatic keratotomy, the human cornea behaves as an inelastic surface and responds to arcuate incisions at a 7-mm optical zone with an insignificant shift in spherical equivalent refraction in most eyes. Spherical equivalent remaining after arcuate keratotomy at a 7-mm optical zone was not significantly affected by the patient’s age, sex, the incision length and number, and the amount of achieved cylinder correction. However, coupling ratio, coupling constant, and change in spherical equivalent after astigmatic keratotomy varied from eye to eye. Caution is, therefore, advised when the results of this study are applied to individual eyes. Surgeons who combine astigmatic keratotomy with a procedure to correct the spherical equivalent should consider planning to undercorrect the spherical equivalent, especially when large cylinder correction is planned.
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TABLE 4. Comparison of Coupling Ratio (CR) and Change in Spherical Equivalent (DSE) After Arcuate Keratotomy Reported in Literature* No. of Eyes
Study, yr
3 Cadaver
Lundergan and Rowsey,18 1985
Merlin,11 1987
Procedure
CRF/S(r)
CRF/S(k)
DSEr
7.5-mm optical zone; single and paired arcs; 30 to 180 degrees total arc length
N/A
0.72
N/A
5.0- to 7.0-mm optical zone; paired arcs; 200 to 300 degrees total arc length
N/A
N/A
10.50
Source
205 Human
Duffey and associates,19 1988
25 Cadaver
5.0- to 9.0-mm optical zone
N/A
1.47 6 0.41
N/A
Lipshitz and associates,15 1994
11 Human
N/A
N/A
10.58
Chavez and associates,13 1996 Buzard and associates,14 1996
15 Human
5.0- to 6.0-mm optical zone; 9 of 11 eyes had arcuate incisions 5.0-mm optical zone; arcuate incisions 7.0-mm optical zone; paired arcs; 90 to 180 degrees total arc length
1.16 6 0.36
0.97
N/A
0.97 6 0.90
46 Human
Effect of Arc Length on Coupling Ratio
Arc length ,90 degrees: CR 5 20.22; arc length .90 degrees: CR 5 1.60 For all optical zones, hyperoptic shift increased with increasing arc length For 7.0-mm optical zone, arc length ,90 degrees: CR 5 2.2; arc length .90 degrees: CR 5 1.80 N/A
10.18 6 0.34 N/A 20.18
CR decreased with increasing arc length
Ksteep 5 the keratometric power of the steep preoperative meridian; Kflat 5 the keratometric power of the flat preoperative meridian; F 5 flattening of the incised meridian calculated based on change in either refraction or keratometry; N/A 5 not applicable. *Mean 6 SD CRF/S and DSE data are shown. Positive CRF/S values indicate that change in one meridian was in the opposite direction from change in the other meridian. Negative CRF/S values indicate change in both meridians was in the same direction.
THE ARC-T STUDY GROUP FRANCIS W. PRICE, JR, MD (STUDY CHAIRMAN), INDIANAP-
3.
olis, Indiana; Perry S. Binder, MD, La Jolla, California; Bruce I. Bodner, MD, Norfolk, Virginia; Daniel S. Durrie, MD, Kansas City, Missouri; Henry Gelender, MD, Dallas, Texas; Bruce Grene, MD, Wichita, Kansas; Kenneth R. Kenyon, MD, Boston, Massachusetts; Richard L. Lindstrom, MD, Minneapolis, Minnesota; William E. Whitson, MD, Indianapolis, Indiana.
4.
5. 6.
ACKNOWLEDGMENTS
We thank Peter Petersen, PhD, Department of Mathematics, UCLA, Los Angeles, California, for mathematical assistance and Jeffrey A. Gornbein, PhD, Department of Biomathematics, UCLA, for statistical consultation.
7.
8.
REFERENCES
9.
1. Hadden OB, Morris AT, Ring CP. Excimer laser surgery for myopia and myopic astigmatism. Aust N Z J Ophthalmol 1995;23:183–188. 2. Lipshitz I, Loewenstein A, Lazar M. Astigmatic keratotomy
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followed by photorefractive keratectomy in the treatment of compound myopic astigmatism. J Refract Corneal Surg 1994; 10:S282–S284. Binder PS, Waring GO III. Keratotomy for astigmatism. In: Waring GO III, editor. Refractive keratotomy for myopia and astigmatism. St Louis: Mosby-Year Book, 1992:1101–1119. Lindquist TD, Lindstrom RL. Surgical correction of astigmatism. In: Tasman W, editor. Duane’s clinical ophthalmology. Volume 6, Chapter 42. Philadelphia: Lippincott Raven, 1994:1–17. Krachmer JL, Fenzl RE. Surgical correction of high postkeratoplasty astigmatism. Arch Ophthalmol 1980;98:1400 –1402. Lavery GW, Lindstrom RL, Hofer LA, Doughman DJ. The surgical management of corneal astigmatism after penetrating keratoplasty. Ophthalmic Surg 1985;16:165–169. Thornton SP. Astigmatic keratotomy: a review of basic concepts with case reports. J Cataract Refract Surg 1990;16: 430 – 435. Lindstrom RL, Agapitos PJ, Koch DD. Cataract surgery and astigmatic keratotomy. Int Ophthalmol Clin 1994;34:145– 164. Price FW, Grene BR, Marks RG, Gonzales JS, the ARC-T Study Group. Astigmatism reduction clinical trial: a multicenter prospective evaluation of the predictability of arcuate keratotomy. Arch Ophthalmol 1995;113:277–282. Price FW, Grene BR, Marks RG, Gonzales JS, the ARC-T Study Group. Arcuate transverse keratotomy for astigmatism
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11. 12. 13. 14. 15.
16. 17.
18. 19. 20. 21. 22.
followed by subsequent radial or transverse keratotomy. J Refract Surg 1996;12:68 –76. Merlin U. Curved keratotomy procedure for congenital astigmatism. J Refract Surg 1987;3:92–97. Agapitos PJ, Lindstrom RL, Williams PA, Sanders DR. Analysis of astigmatic keratotomy. J Cataract Refract Surg 1989;15:13–18. Chavez S, Chayet A, Celikkol L, Parker J, Celikkol G, Feldman ST. Analysis of astigmatic keratotomy with a 5.0-mm optical clear zone. Am J Ophthalmol 1996;121:65–76. Buzard KA, Laranjeira E, Fundingsland BR. Clinical results of arcuate incisions to correct astigmatism. J Cataract Refract Surg 1996;22:1062–1069. Lipshitz I, Loewenstein A, Lazar M. Astigmatic keratotomy followed by photorefractive keratectomy in the treatment of compound myopic astigmatism. J Refract Corneal Surg 1994; 10:S282–S284. Lavery GW, Lindstrom RL. Trapezoidal astigmatic keratotomy in human cadaver eyes. J Refract Surg 1985;1:18 –24. Sabates MA, Buzard KA, Friedlander MH, Cortinas MB. Induction of astigmatism by straight transverse corneal incisions, 450 long, at different clear zones in human cadaver eyes. J Refract Corneal Surg 1994;10:327–332. Lundergan MK, Rowsey JJ. Relaxing incisions: corneal topography. Ophthalmology 1985;92:1226 –1236. Duffey RJ, Jain VN, Tchah H, Hofmann RF, Lindstrom RL. Paired arcuate keratotomy: a surgical approach to mixed and myopic astigmatism. Arch Ophthalmol 1988;106:1130– 1135. Nordan LT. Quantifiable astigmatism correction: concepts and suggestions, 1986. J Cataract Refract Surg 1986;12:507– 518. Krumeich JH, Knuelle A. Circular keratotomy for the correction of astigmatism. Refract Corneal Surg 1992;8:204 – 210. Woo SL-Y, Kobayashi AS, Schlegel WA, Lawrence C.
23.
24. 25. 26. 27.
28. 29.
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Nonlinear material properties of intact cornea and sclera. Exp Eye Res 1972;14:29 –39. Hjortdal JO, Koch-Jensen P. In situ mechanical behavior of the human cornea as evaluated by simultaneous measurements of corneal strain, corneal surface contour, and corneal thickness. ARVO abstracts. Invest Ophthalmol Vis Sci 1992;33(,suppl):1022. Wang H, Prendville PL, McDonnell PJ, Chang WV. An ultrasonic technique for the measurement of the elastic moduli of human cornea. J Biomech 1996;29:1633–1636. Hjortdal JO. Regional elastic performance of the human cornea. J Biomech 1996;29:931–942. Koch DD, Liu JF, Hyde LL, Rock RL, Emery JM. Refractive complications of cataract surgery after radial keratotomy. Am J Ophthalmol 1989;108:676 – 682. Celikkol L, Pavlopoulos G, Weistein B, Celikkol G, Feldman ST. Calculation of intraocular lens power after radial keratotomy with computerized videokeratography. Am J Ophthalmol 1995;120:739 –750. Holladay JT, Cravy TV, Koch DD. Calculating the surgically induced refractive change following ocular surgery. J Cataract Refract Surg 1992;18:429 – 443. Waring GO III, Holladay JT. Optics and topography of corneal astigmatism. In: Waring GO III, editor. Refractive keratotomy for myopia and astigmatism. St Louis: MosbyYear Book, 1992:1078. Waring GO III, Lynn MJ, Gelender H, PERK Study Group. Results of the Prospective Evaluation of Radial Keratotomy (PERK) study one year after surgery. Ophthalmology 1985; 92:177–198. Ibrahim O, Hussein HA, El-Sahn MF, El-Nawawy S, Kassem A, Waring GO. Trapezoidal keratotomy for the correction of naturally occurring astigmatism. Arch Ophthalmol 1991; 109:1374 –1381.
Authors Interactivet We encourage questions and comments regarding this article via the Internet on Authors Interactivet at http://www.ajo.com/ Questions, comments, and author responses are posted.
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● PURPOSE: To determine the effect of astigmatic keratotomy on spherical equivalent, as measured by the coupling ratio and a new quantity, coupling constant. ● METHODS: In a prospective multicenter study, subjects underwent arcuate keratotomy at a 7-mm optical zone by means of the Lindstrom nomogram for correction of astigmatism. One hundred fifty-seven eyes of 95 patients who had a follow-up examination 1 month postoperatively were studied. Mean preoperative refractive cylinder 6 SEM was 2.82 6 1.17 diopters. Coupling ratio was defined as the ratio of the flattening of the incised meridian to the steepening of the opposite meridian. Coupling constant was defined as the ratio of the change in spherical equivalent to the magnitude of the vector change in astigmatism. Coupling ratio, coupling constant, and change in spherical equivalent were calculated on the basis of change in refraction and keratometry. ● RESULTS: On the basis of change in refraction, coupling ratio was 0.95 6 0.10 (mean 6 SEM) and coupling constant was 20.01 6 0.03, consistent with a minor shift in the spherical equivalent of 20.03 6 0.07 diopter. On the basis of change in keratometry, coupling ratio was 0.84 6 0.05 and coupling constant was 20.04 6 0.02, consistent with minor postoperative keratometric steepening of 20.10 6 0.04 diopter. Coupling ratio based on change in refraction was not statistically different from the coupling ratio predicted by the Gauss’ law for inelastic domes (P 5 .370). Incision length and number, amount of achieved cylinder correction, age, and sex had
Accepted for publication Oct 8, 1998. From the Jules Stein Eye Institute and the Department of Ophthalmology, University of California, Los Angeles, California (Drs Faktorovich and Maloney), and the Cornea Research Foundation of America and Corneal Consultants of Indiana, Indianapolis, Indiana (Dr Price). *The members of the Astigmatism Reduction Clinical Trial Study Group are listed at the end of the article. Dr Maloney was supported by a Career Development Award from Research to Prevent Blindness, Inc, New York, New York. Dr Faktorovich was a Heed Fellow and an Elsa and Louis Kelton Fellow. Reprint requests to Robert K. Maloney, MD, 10921 Wilshire Blvd, Suite 900, Los Angeles, CA 90024.
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no statistically significant effect on coupling ratio, coupling constant, and change in spherical equivalent. ● CONCLUSIONS: Cornea behaved as an inelastic surface in response to arcuate keratotomy performed with the Astigmatism Reduction Clinical Trial study nomogram. On average, astigmatic keratotomy had a minimal effect on spherical equivalent refraction. There was variability, however, in coupling ratio, coupling constant, and change in spherical equivalent from eye to eye after astigmatic keratotomy. Caution is therefore advised when simultaneous correction of cylinder and spherical equivalent is planned. (Am J Ophthalmol 1999;127:260–269. © 1999 by Elsevier Science Inc. All rights reserved.)
A
STIGMATIC KERATOTOMY IS A COMMON METHOD
for the surgical correction of astigmatism. Even with the advent of photoastigmatic refractive keratectomy, some protocols for photorefractive keratectomy and laser in situ keratomileusis still combine astigmatic keratotomy and excimer laser keratectomy to simultaneously correct astigmatism and spherical refractive error.1,2 Additionally, astigmatic keratotomy is often combined with cataract surgery and is the most common method of correcting astigmatism after penetrating keratoplasty.3–15 Studies in both human and cadaver eyes have demonstrated that astigmatic keratotomy not only flattens the incised meridian but also induces steepening of the opposite unincised meridian 90 degrees away.7,13,14,16 –20 The ratio of the amount of flattening of the incised meridian to the amount of steepening of the opposite meridian is defined as the coupling ratio.3 The coupling ratio is useful in planning a refractive procedure because it predicts the effect of astigmatic keratotomy on spherical equivalent refraction. If the coupling ratio is 1, the flattening of the incised meridian will equal the steepening of the opposite meridian, and the spherical equivalent will not change after astigmatic keratotomy. A coupling ratio greater than 1 indicates that the flattening of the incised meridian will be greater than the steepening of the opposite meridian,
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FIGURE 1. Change in curvature of the principal corneal meridians induced by arcuate keratotomy. Ksteep is the power of the steep preoperative meridian. Kflat is the power of the flat preoperative meridian. After astigmatic keratotomy, the power of the incised meridian is Ksteep 2 F, where F is flattening of the incised meridian. The power of the opposite, unincised meridian is Kflat 1 S, where S is steepening of the unincised meridian.
there will be overall corneal flattening, and the spherical equivalent will shift toward hyperopia. Conversely, a coupling ratio less than 1 means that the steepening of the incised meridian will be greater than the flattening of the opposite meridian, and the spherical equivalent will shift toward myopia. Therefore, the effect of astigmatic keratotomy on spherical equivalent will depend on the coupling ratio. If the coupling ratio of an astigmatic procedure is known and simultaneous correction of the astigmatism and the spherical equivalent is planned, more accurate correction of the spherical equivalent can be achieved. Coupling ratio was defined by Thornton7 on the basis of the concept of gaussian curvature of a surface.21 Gaussian curvature of a surface (Kgauss) is the product of the curvatures of the two principal meridians, that is, Kgauss 5 K1 3 K2, where K1 is the curvature of one principal meridian and K2 is the curvature of the meridian 90 degrees away. If the curvature of the steep preoperative meridian is Ksteep and the curvature of the flat preoperative meridian is Kflat, then the preoperative gaussian curvature is Ksteep 3 Kflat (Figure 1). Gauss’ law states that when the curvature of a flexible but inelastic surface is changed in one meridian, the curvature of the orthogonal meridian changes in the opposite way so that the gaussian curvature of the surface (Kgauss) remains constant.21 Therefore, if F is flattening of the steep preoperative meridian and S is steepening of the flat preoperative meridian, then the postoperative gaussian curvature is (Ksteep 2 F) 3 (Kflat 1 S) (Figure 1) and Kgauss 5 Ksteep 3 Kflat 5 (Ksteep 2 F) 3 (Kflat 1 S)
[1]
Some authors have interpreted the Gauss’ law to mean that the change in the orthogonal meridian has to be equal to and opposite of the change in the primary meridian.3,4,7,11 Therefore, they argued, if cornea was an inelastic surface, flattening of one meridian would equal steepening of the opposite meridian, coupling ratio would equal 1, and VOL. 127, NO. 3
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spherical equivalent would not change after astigmatic keratotomy. Early studies, however, have shown that coupling ratio is usually not 1. Rather, it ranges from 0.72 to 1.88.18,19 On the basis of this evidence, some have concluded that the cornea must be elastic, it does not obey the Gauss’ law for inelastic domes, and spherical equivalent should change after astigmatic keratotomy.3,7 Biomechanical studies, however, have found that human adult cornea stretches only 20 mm when intraocular pressure increases from 10 to 40 mm Hg and that a Young modulus of corneal elasticity ranges from 3 3 105 to 2 3 107 Nm22, suggesting that human adult cornea is, in fact, quite inelastic.22–25 Most previous studies of the coupling ratio used keratometry to calculate the coupling ratio.18,19 Keratometry, however, does not provide an accurate estimate of corneal power after refractive surgery. Refraction estimates effective corneal power more accurately than keratometry.26,27 In this study, we demonstrate that the coupling ratio of an inelastic surface does not have to equal 1, although it should be close to 1. We then test whether a living cornea behaves like an elastic or an inelastic surface by analyzing coupling ratio calculated on the basis of both change in refraction and keratometry after arcuate keratotomy performed in the Astigmatism Reduction Clinical Trial (ARC-T), a prospective, multicenter, cohort study. Additionally, we define a new quantity, coupling constant, which allows a simple calculation of the change in spherical equivalent expected after astigmatic keratotomy. We then analyze the effect of achieved cylinder correction, arc length, incision number, age, and sex on coupling ratio, coupling constant, and change in spherical equivalent.
PATIENTS AND METHODS DETAILS OF STUDY DESIGN, PATIENT SELECTION, SURGICAL
technique, and surgical nomograms of the ARC-T have
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been published previously.9,10 We studied 157 eyes of 95 patients (63 women and 32 men) who had a follow-up examination at 1 month 6 10 days after astigmatic keratotomy. Longer follow-up data were not available because most of the eyes underwent radial keratotomy to correct the spherical equivalent at 1 month 6 10 days after astigmatic keratotomy. Average (6 SD) age was 41 6 11.84 years (range, 18 to 84 years). Mean preoperative keratometric cylinder was 2.74 6 1.12 diopters (range, 0.38 to 5.75 diopters). Mean preoperative refractive cylinder was 2.82 6 1.17 diopters (range, 1.00 to 6.50 diopters). Of the 157 eyes, 144 had had no previous eye surgery except glaucoma or retinal laser surgery. Thirteen eyes underwent cataract extraction with intraocular lens placement 6 months to 2 years before astigmatic keratotomy. All eyes underwent arcuate keratotomy at a 7-mm optical zone. One or two arcuate incisions were performed. Individual arc length ranged from 30 to 90 degrees. Total arc length ranged from 45 to 180 degrees, depending on the amount of desired astigmatism correction and the patient’s age. Lindstrom’s surgical nomograms were used.9,10 The formula for calculating coupling ratio (CR) was derived from the surgically induced refractive change described by Holladay and associates.20,28 Surgically induced refractive change represents the power of a spherocylindrical lens that, when placed before the preoperative eye, would simulate the surgical result. Surgically induced refractive change is S 1 DA @ a degrees, where S is the spherical component of the surgically induced spherocylinder; DA is the magnitude of the vector change in astigmatism in plus cylinder format; and a degrees is the axis of the vector change in astigmatism. When astigmatic keratotomy is performed, the axis a degrees corresponds to the meridian of the corneal incision, so the power of the cylinder DA @ a degrees lies in the opposite meridian. Therefore, the flattening of the steep meridian is 2S and the steepening of the flat meridian is S 1 DA. If refractive change in spherical equivalent, DSEr, is defined as the postoperative spherical equivalent refraction minus the preoperative spherical equivalent refraction, then from the definition of surgically induced refractive change, DSEr 5 2(S 1 DA/2). Flattening of the incised meridian, F, equals DA/2 1 DSEr. Steepening of the opposite meridian, S, equals DA/2 2 )SEr. We express the coupling ratio as the ratio of the flattening of the incised meridian to the steepening of the opposite meridian. Therefore, coupling ratio (CRF/S) is CRF/S 5 (DA/2 1 DSEr) / (DA/2 2 DSEr)
equivalent keratometry because postoperative increase in keratometric spherical equivalent indicates a myopic shift in refraction and postoperative decrease in keratometric spherical equivalent indicates a hyperopic shift in refraction. DA is the magnitude of the vector change in astigmatism calculated on the basis of either change in refraction or keratometry.29 Positive CRF/S values indicate that one meridian flattened whereas the opposite one steepened. Negative CRF/S values indicate that both meridians either flattened or steepened. Equation 2 is useful because it allows calculation of the coupling ratio even when the axis of astigmatism shifts after astigmatic keratotomy. Equation 2 is also useful because it allows calculation of the coupling ratio based on either refraction or keratometry. Coupling ratio of an inelastic gaussian surface (CRgauss) is derived from the equation 1 in the introduction as follows: if Kgauss 5 Ksteep 3 Kflat 5 (Ksteep 2 F) 3 (Kflat 1 S), then Ksteep 3 Kflat 5 Ksteep 3 Kflat 1 Ksteep 3 S 2 Kflat 3 F 2 F 3 S. When coupling ratio is defined as the ratio of the flattening of the steep preoperative meridian to the steepening of the flat preoperative meridian, that is, F/S, and the above equation is solved for F/S, then Coupling Ratio 5 CRgauss 5 F/S 5 (Ksteep 2 F) / Kflat
To facilitate calculation of the change in spherical equivalent after astigmatic keratotomy, we defined the coupling constant (CC). Coupling constant is the ratio of the change in spherical equivalent to the magnitude of the vector change in astigmatism, that is, CC 5 DSE/DA
CC 5 1/2(CRF/S 2 1) / (CR F/S 1 1)
[5]
If flattening of the incised meridian is greater than steepening of the opposite meridian, CRF/S will be greater than 1, CC will be positive, and DSE will be positive, consistent with a hyperopic shift in spherical equivalent and overall corneal flattening. If flattening of the incised meridian is less than steepening of the opposite meridian, CRF/S will be less than 1, CC will be negative, and DSE will be negative as well, consistent with a myopic shift in spherical equivalent and overall corneal steepening. The coupling ratios and the coupling constants in the
[2]
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[4]
Coupling constant is derived from equation 2 as follows: CRF/S 5 (DA/2 1 DSE)/(DA/2 2 DSE). If the numerator and the denominator are divided by DA, then CRF/S 5 (1/2 1 DSE/DA)/(1/2 2 DSE/DA) 5 (1/2 1 CC) / (1/2 2 CC). Solving for CC,
To calculate keratometric coupling ratio by means of equation 2, it is necessary to define the change in keratometric spherical equivalent, DSEk, as preoperative spherical equivalent keratometry minus postoperative spherical 262
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TABLE 1. Results of Astigmatic Keratotomy Calculated Based on Change in Refraction and Change in Keratometry*
Magnitude of vector change in astigmatism, A (diopters) Flattening of incised meridian, F (diopters) Steepening of opposite meridian, S (diopters) Coupling ratio (CRF/S 5 F/S) Coupling ratio (CRG 5 Ksteep 2 F/Kflat) Change in spherical equivalent, DSE (diopters) Coupling constant, CC
Refraction
Keratometry
2.35 6 0.11 1.15 6 0.09 (0.97, 1.33) 1.20 6 0.08 (1.04, 1.36) 0.95 6 0.10 (0.75, 1.15) 1.04 6 0.002 (1.036, 1.044) 20.03 6 0.07 (20.17, 0.11) 20.01 6 0.03 (20.07, 0.05)
2.38 6 0.10 1.08 6 0.07 (0.94, 1.22) 1.29 6 0.06 (1.17, 1.41) 0.84 6 0.05 (0.74, 0.94) 1.04 6 0.002 (1.036, 1.044) 20.10 6 0.04 (20.18, 20.02) 20.04 6 0.02 (20.08, 0.00)
Ksteep 5 the keratometric power of the steep preoperative meridian; Kflat 5 the keratometric power of the flat preoperative meridian; F 5 flattening of the incised meridian calculated based on change in either refraction or keratometry. *All data are mean 6 SEM with 95% confidence intervals.
ARC-T study cohort were not normally distributed. Therefore, to determine mean coupling ratio (mean CRF/S and mean CRgauss) and mean coupling constant (mean CC), maximum likelihood estimators of the means were calculated. The maximum likelihood estimator of the mean CRF/S was calculated by dividing the mean flattening of the incised meridian by the mean steepening of the opposite meridian. The maximum likelihood estimator of the mean CRgauss was calculated by dividing the mean Ksteep 2 F by the mean Kflat. The maximum likelihood estimator of the mean coupling constant was calculated by dividing the mean change in spherical equivalent by the mean magnitude of vector change in astigmatism. Calculations were performed for data based on both keratometry and refraction. Standard error of the mean for the maximum likelihood estimator of the was calculated as follows: SEM 5 ~MEAN 1/MEAN 2)
Î(SEM21/MEAN21) 1 (SEM22/MEAN22) 2 2 r1,2 (SEM1/MEAN1) 3 (SEM2/MEAN2), where MEAN1 is the mean of the values in numerator; MEAN2 is the mean of the values in the denominator; SEM1 is the standard error of MEAN1; SEM2 is the standard error of MEAN2; and r1,2 is a two-tailed Spearman correlation coefficient between MEAN1 and MEAN2. Two-tailed t test for correlated means was used to compare maximum likelihood estimator of the mean CRF/S to the mean CRgauss. Two-tailed t test for two independent VOL. 127, NO. 3
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samples was used to compare the maximum likelihood estimator of the mean coupling ratio, coupling constant, and mean change in spherical equivalent in different age groups, sexes, and groups with different cylinder corrections and different arc lengths. Two-tailed t test for two independent samples was also used to compare the mean age, cylinder correction, and arc length in the group of eyes with 1 diopter or more shift in spherical equivalent to the group of eyes with less than 1 diopter shift in spherical equivalent. All values are mean 6 SEM. All data were analyzed with SPSS software.
RESULTS KERATOMETRIC AND REFRACTIVE FLATTENING OF THE IN-
cised meridian, steepening of the opposite meridian, coupling ratio, coupling constant, change in spherical equivalent, and magnitude of vector change in astigmatism are given in Table 1. On the basis of the change in refraction, mean flattening of the incised meridian was 1.15 6 0.09 diopter. Mean steepening of the opposite meridian was 1.20 6 0.08 diopter. The sum of change in the surgical meridian and the opposite meridian should be equal to the amount of astigmatism corrected.7 Indeed, the sum of mean flattening in the incised meridian and mean steepening in the opposite meridian was 2.35 6 0.12 diopters, consistent with the mean magnitude of vector change in astigmatism of 2.35 6 0.11 diopters. Mean coupling ratio, CRF/S, calculated on the basis of the change in refraction was 0.95 6 0.10. This value was not significantly different from 1 (95% confidence inter-
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vals, 0.75, 1.15) and was consistent with a minor mean myopic shift in refractive spherical equivalent of 20.03 6 0.07 diopter, which was not significantly different from 0 (95% confidence intervals, 20.17, 0.11). Mean keratometric flattening of the incised meridian was 1.08 6 0.07 diopters. Mean keratometric steepening of the opposite meridian was 1.29 6 0.06 diopters. The sum of mean flattening and mean steepening was 2.37 6 0.10 diopters, consistent with the mean magnitude of vector change in keratometric astigmatism of 2.38 6 0.10 diopters. Mean coupling ratio, CRF/S, calculated on the basis of keratometric change was 0.84 6 0.05, consistent with an overall mean corneal steepening of 20.10 6 0.04 diopter. Mean keratometric coupling ratio was statistically different from 1 (95% confidence intervals, 0.74, 0.94). Likewise, mean change in keratometric spherical equivalent was statistically different from 0 (95% confidence intervals, 20.18, 20.02). In the Patients and Methods section, we demonstrated that coupling ratio of an inelastic surface, CRgauss, is (Ksteep 2 F)/Kflat. To test whether cornea behaves like an inelastic surface in response to arcuate keratotomy, we compared the coupling ratio calculated from the ARC-T study data, CRF/S, with the coupling ratio predicted by the Gauss law for inelastic domes, CRgauss. On the basis of refraction, CRF/S was 0.95 6 0.10. CRgauss was 1.04 6 0.002 (Table 1). The difference between CRF/S and CRgauss was not statistically significant (P 5 .370). On the basis of keratometry, CRF/S was 0.84 6 0.05. CRgauss was 1.04 6 0.002 (Table 1). The difference between CRF/S and CRgauss was statistically significant (P , .001). Standard keratometry, however, overestimates effective corneal power after myopic refractive surgery.27,28,30 Overestimation of the effective corneal power will result in underestimation of the true coupling ratio because it will decrease the numerator and increase the denominator of F/S. We hypothesized that the keratometric coupling ratio calculated from the ARC-T study data is probably smaller that the true keratometric coupling ratio and that, like the coupling ratio calculated on the basis of the change in refraction, the true keratometric coupling ratio is probably close to the one predicted by the Gauss law for inelastic domes. To test this hypothesis, we assumed that postoperative keratometry overestimated true corneal power by a small amount of 0.125 diopter. If this were the case, true mean flattening of the incised meridian would have been 1.21 6 0.07 diopters and true mean steepening of the opposite meridian would have been 1.17 6 0.06 diopters. Therefore, true mean keratometric coupling ratio, CRF/S, would have been 1.03 6 0.06 (95% confidence intervals, 0.91, 1.15). True mean keratometric coupling ratio predicted by the Gauss law for inelastic domes, CRgauss, would have not changed appreciably if a 0.125-diopter keratometric error was made. The coupling ratio would have still been 1.04 6 0.002 (95% confidence intervals, 1.036, 264
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FIGURE 2. Achieved refractive cylinder correction vs change in refractive spherical equivalent. Slope 6 SES of the linear regression line plotted through 0 is 0.01 6 0.03 (95% confidence intervals, 20.04, 0.06). There is no significant effect of achieved refractive cylinder correction on refractive spherical equivalent.
1.044). The difference between CRF/S and CRgauss would then not have been statistically significant (P 5 .881). We defined coupling constant as the ratio of the change in spherical equivalent to the magnitude of vector change in astigmatism. Coupling constant was calculated from the coupling ratio as follows: CC 5 1/2(CR21)/(CR11). Mean coupling constant calculated on the basis of change in refraction was 20.01 6 0.03, consistent with a minimal myopic shift in refractive spherical equivalent. Mean coupling constant calculated on the basis of keratometric change was 20.04 6 0.02, consistent with mild keratometric corneal steepening postoperatively (Table 1). By its definition, coupling constant should be equal to the slope of the linear regression line relating the change in spherical equivalent to the magnitude of the achieved cylinder correction. Indeed, the slope in Figure 2 was 0.01 6 0.03, which was not significantly different from the mean coupling constant calculated on the basis of change in refraction (P 5 .680). The results of the effect of attempted and achieved cylinder correction, arc length, incision number, age, and sex on coupling ratio, coupling constant, and change in spherical equivalent are presented in Table 2. In eyes with achieved refractive cylinder correction of less than 2.00 diopters, mean refractive coupling ratio was 0.65 6 0.15. Larger achieved cylinder corrections tended to result in larger coupling ratios. In eyes with achieved refractive cylinder correction of 2.00 diopters or greater, mean refractive coupling ratio was 1.06 6 0.12. The difference between the mean coupling ratios, however, was not statistically significant when the data were analyzed by means of the Bonferroni correction for multiple compariOF
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TABLE 2. Effect of Surgical and Demographic Factors on Outcome Variables* Refraction (r)
No. of Eyes
Achieved cylinder correction ,2.00 diopters $2.00 diopters P‡ Total arc length ,90 degrees $90 degrees P Incision number 1 3 90 degrees 2 3 45 degrees P Age ,40 yrs $40 yrs P Sex Male Female P
Keratometry (k)
CRF/S(r)
CCr
DSEr
CRF/S(k)
CCk
DSEk
68 (73)† 89 (84)
0.65 6 0.15 1.06 6 0.12 .035
20.11 6 0.06 0.02 6 0.03 .054
20.13 6 0.07 0.05 6 0.10 .165
0.74 6 0.09 0.87 6 0.06 .232
20.08 6 0.03 20.03 6 0.18 .780
20.10 6 0.04 20.11 6 0.60 .825
62 95
0.87 6 0.16 0.99 6 0.12 .549
20.04 6 0.04 20.003 6 0.03 .454
20.06 6 0.07 20.01 6 0.10 .707
0.76 6 0.09 0.87 6 0.06 .319
20.07 6 0.03 20.03 6 0.02 .273
20.11 6 0.05 20.10 6 0.05 .987
33 20
1.07 6 0.22 0.93 6 0.25 .604
0.02 6 0.08 20.02 6 0.06 .690
0.04 6 0.19 20.05 6 0.17 .737
0.88 6 0.09 0.97 6 0.21 .690
20.03 6 0.03 20.01 6 0.06 .735
20.08 6 0.07 20.02 6 0.13 .645
84 73
0.80 6 0.09 1.14 6 0.21 .130
20.05 6 0.03 0.03 6 0.05 .660
20.12 6 0.06 0.08 6 0.13 .146
0.82 6 0.07 0.86 6 0.08 .706
20.05 6 0.02 20.04 6 0.02 .725
20.11 6 0.05 20.10 6 0.05 .809
51 106
1.01 6 0.19 0.92 6 0.12 .689
0.003 6 0.05 20.02 6 0.03 .689
0.01 6 0.14 20.05 6 0.07 .720
0.88 6 0.08 0.82 6 0.06 .549
20.03 6 0.03 20.05 6 0.02 .583
20.09 6 0.07 20.11 6 0.04 .799
Ksteep 5 the keratometric power of the steep preoperative meridian; Kflat 5 the keratometric power of the flat preoperative meridian; F 5 flattening of the incised meridian calculated based on change in either refraction or keratometry. *All data are mean 6 SEM. † Number in parentheses indicates the number of eyes in each group of the achieved keratometric cylinder correction. Number outside parentheses represents the number of eyes in each group of the achieved refractive cylinder correction. ‡ Two-tailed t test for two independent samples. With Bonferroni correction for multiple comparisons, P , .01 was considered statistically significant.
sons (P 5 .035). In eyes with achieved refractive cylinder correction of less than 2.00 diopters, mean refractive coupling constant was 20.11 6 0.06. In eyes with achieved refractive cylinder correction of 2.00 diopters or greater, mean refractive coupling constant was 0.02 6 0.03. The difference between the means was not statistically significant (P 5 .054). Smaller achieved refractive cylinder correction resulted in a minor mean myopic shift in refractive spherical equivalent of 20.13 6 0.07 diopter. Larger achieved refractive cylinder correction resulted in a small mean hyperopic shift in refractive spherical equivalent of 0.05 6 0.10 diopter. The difference between the means was not statistically significant (P 5 .165). Figure 2 confirms the minimal effect of the achieved refractive cylinder correction on the change in spherical equivalent refraction. The slope of the linear regression line was 0.01 6 0.03, which was not significantly different from 0 (95% confidence intervals, 20.04, 0.06). Smaller achieved keratometric cylinder correction resulted in smaller mean keratometric coupling ratio, coupling constant, and change in spherical equivalent than VOL. 127, NO. 3
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did larger achieved keratometric cylinder correction. The differences between the means, however, were not statistically significant. Mean keratometric and refractive coupling ratio, coupling constant, and mean change in keratometric and refractive spherical equivalent in eyes with total arc length of less than 90 degrees were compared with those in the eyes with total arc length of 90 degrees or longer. Longer arcs tended to result in larger coupling ratios, coupling constants, and flatter corneas than shorter arcs, but the difference between the means was not statistically significant. Linear regression analysis confirmed the minimal correlation between total arc length and change in refractive spherical equivalent (Figure 3). The slope of the linear regression line was 20.0002 6 0.001, which was not significantly different from 0 (95% confidence intervals, 20.002, 0.001). Number of arcuate incisions (one or two) also had no effect on coupling ratio, coupling constant, or the change in spherical equivalent. Mean coupling ratio, coupling constant, and change in spherical equivalent in eyes with
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refraction or the change in keratometry (equation 2). Use of the vector change in astigmatism allows one to determine the coupling ratio even when the axis of astigmatism shifts after the procedure. To simplify the calculation of the expected change in spherical equivalent after astigmatic keratotomy, we defined a new entity, the coupling constant, which is calculated from the coupling ratio. Coupling constant is the ratio of the change in spherical equivalent to the magnitude of the vector change in astigmatism. The change in spherical equivalent expected after astigmatism correction is, then, the coupling constant of the procedure multiplied by the intended astigmatism correction. For example, if a 2-diopter astigmatism correction is planned with the use of astigmatic keratotomy incisions with a coupling constant of 0.2, a 0.4-diopter hyperopic shift in spherical equivalent refraction should be expected. The residual spherical equivalent can then be determined and corrected with either photorefractive keratectomy or laser in situ keratomileusis. Previous reports of coupling ratios have been limited to a small sample size, cadaver eyes, or studies of combined transverse and radial incisions.12–20,31 The ARC-T study is the largest study of arcuate keratotomy performed at a single optical zone. Coupling ratio calculated on the basis of the change in refraction was 0.95 6 0.10. This value was not statistically different from 1, consistent with a small coupling constant of 20.01 6 0.03, which was not significantly different from 0, and a minor shift in spherical equivalent of 20.03 6 0.07, which was also not statistically different from 0. Chavez and associates13 also reported coupling ratios calculated on the basis of change in refraction after arcuate keratotomy. In their study, mean coupling ratio was 1.16 6 0.36, consistent with a hyperopic shift in spherical equivalent of 10.18 6 0.34 (Table 3). The study, however, was limited to 15 eyes, and there was considerable variability in the outcome. Moreover, Chavez and associates13 studied arcuate incisions at a 5-mm optical zone. Transverse incisions performed at smaller optical zones result in larger coupling ratios and more corneal flattening than incisions performed at larger optical zones.14,16,19 Coupling ratio calculated on the basis of change in keratometry in this study was 0.85 6 0.05, which was significantly different from 1. Keratometric coupling constant was 20.04 6 0.02, consistent with the mean change in keratometric spherical equivalent of 20.10 6 0.04, which was significantly different from 0. This result was in general agreement with other studies.13,14,18 Standard keratometry, however, overestimates corneal power after refractive surgery.27,28,30 For example, after radial keratotomy, keratometric values measured by standard keratometry are 0.83 to 5.01 diopters steeper than the ideal and the refraction-derived values predicted for an accurate intraocular lens power calculation.27,28 Because astigmatic keratotomy is performed at a larger optical zone than radial keratotomy, one might
FIGURE 3. Total arc length vs change in refractive spherical equivalent. Slope 6 SES of the linear regression line plotted through 0 is 20.0002 6 0.001 (95% confidence intervals, 20.002, 0.001). There is no significant effect of total arc length on refractive spherical equivalent.
one arc 90 degrees long were compared to those in the eyes with two 45-degree arcs. The difference between the means was not statistically significant. Mean coupling ratio and coupling constant tended to be larger in the eyes of older patients and in men. The difference between the means, however, was again not statistically significant. Most eyes in the ARC-T study had a minor shift in spherical equivalent of less than 1 diopter. Twenty-four of 157 eyes, however, had a 1-diopter or more change in spherical equivalent in the hyperopic or myopic direction (Figure 2). Table 3 compares surgical and demographic variables in the eyes with 1 diopter or more shift in spherical equivalent with those in the eyes with less than 1 diopter of shift in spherical equivalent after astigmatic keratotomy. Eyes whose spherical equivalent changed 1 diopter or more tended to be in older patients, had larger cylinder corrections, and had longer arcuate incisions than the eyes whose spherical equivalent changed less than 1 diopter. The difference between the groups, however, was not statistically significant.
DISCUSSION COUPLING RATIO DESCRIBES THE EFFECT OF ASTIGMATIC
keratotomy on spherical equivalent. If the coupling ratio of an astigmatic keratotomy procedure is known, spherical equivalent remaining after the astigmatism correction can be calculated. We developed a simple formula for calculating coupling ratio based on either the change in 266
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TABLE 3. Surgical and Demographic Factors in Eyes With 1 Diopter or Greater Change in Spherical Equivalent (DSEr) vs Eyes With Less Than 1 Diopter Change After Astigmatic Keratotomy*
Achieved cylinder correction (diopters) Total arc length (degrees) Age (yrs)
$1 Diopter DSEr (n 5 24 Eyes)
,1 Diopter DSEr (n 5 133 eyes)
P value†
2.83 6 0.31 106.04 6 8.92 43.83 6 2.85
2.27 6 0.11 87.11 6 2.93 40.26 6 0.99
.101 .053 .245
*All data are mean 6 SEM. † Two-tailed t test for two independent samples. With Bonferroni correction for multiple comparisons, P , .02 was considered statistically significant.
expect that keratometry would be more accurate after astigmatic keratotomy. In the ARC-T study, astigmatic keratotomy was performed with a 7-mm optical zone. However, even a small overestimation of the effective corneal power will result in underestimation of the coupling ratio. We found that if keratometry after arcuate incisions in the ARC-T study overestimated true corneal power by 0.125 diopter, true coupling ratio would have been 1.03 6 0.06, which would not have been significantly different from 1. Coupling ratios reported in the early studies were not 1. These results led some authors7 to conclude that the cornea does not follow the Gauss’ law for inelastic domes, and spherical equivalent should be expected to change after astigmatic keratotomy. We have demonstrated that coupling ratio of an inelastic surface should not be expected to equal 1. Rather, it should be (Ksteep 2 F)/Kflat. However, because Ksteep is typically close to 44 diopters, Kflat is typically close to 42 diopters, and F is typically 1 to 2 diopters, coupling ratio of an inelastic surface should be between 1.00 and 1.03 for virtually all corneas. Indeed, the mean coupling ratio calculated on the basis of change in refraction in the ARC-T study was not significantly different from 1 and from the mean coupling ratio predicted by the Gauss’ law for inelastic domes. The mean coupling ratio calculated on the basis of change in keratometry, however, was significantly different from the mean coupling ratio for inelastic domes predicted by Gauss. This discrepancy could have resulted from an inaccurate estimation of the true central corneal power by keratometry. We have demonstrated that if keratometry in the ARC-T study overestimated true corneal power by 0.125 diopter, true keratometric coupling ratio would not have been significantly different than the keratometric coupling ratio predicted by the Gauss law for inelastic domes. Coupling ratio results of the early studies could have, likewise, been flawed because coupling ratios were calculated solely on the basis of keratometry. Previously, it has been reported that surgical factors, such as incision length and number, and demographic factors, such as age and sex, strongly influence the amount of astigmatism corrected.9 In this study, we have shown VOL. 127, NO. 3
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that none of these factors significantly affects the coupling ratio, coupling constant, or change in spherical equivalent after astigmatic keratotomy. Most of the published series do report a correlation between arc length and coupling ratio (Table 4).3,4,11,14,18 –20,31 Some authors have even stated that incision length is the major factor that affects the coupling ratio of transverse incisions.3,4 Several studies note a hyperopic shift with increasing arc length, whereas most others report a myopic shift associated with increasing arc length. Most studies, however, are limited to a small sample size. We noted a tendency toward mild hyperopia with the increasing arc length and the amount of achieved cylinder correction. However, the effect of arc length and achieved cylinder correction on coupling ratio, coupling constant, and change in spherical equivalent was not statistically significant. One of the limitations of this study is a relatively short mean follow-up time of 1 month. Perhaps if the follow-up time were extended, a statistically significant effect of the arc length and the amount of the achieved cylinder correction on coupling ratio, coupling constant, and change in spherical equivalent would have been observed. Lindquist and Lindstrom,4 however, found that the 1-month results after astigmatic keratotomy correlate well with the 1-year results. On the basis of change in refraction after astigmatic keratotomy, the human cornea behaves as an inelastic surface and responds to arcuate incisions at a 7-mm optical zone with an insignificant shift in spherical equivalent refraction in most eyes. Spherical equivalent remaining after arcuate keratotomy at a 7-mm optical zone was not significantly affected by the patient’s age, sex, the incision length and number, and the amount of achieved cylinder correction. However, coupling ratio, coupling constant, and change in spherical equivalent after astigmatic keratotomy varied from eye to eye. Caution is, therefore, advised when the results of this study are applied to individual eyes. Surgeons who combine astigmatic keratotomy with a procedure to correct the spherical equivalent should consider planning to undercorrect the spherical equivalent, especially when large cylinder correction is planned.
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TABLE 4. Comparison of Coupling Ratio (CR) and Change in Spherical Equivalent (DSE) After Arcuate Keratotomy Reported in Literature* No. of Eyes
Study, yr
3 Cadaver
Lundergan and Rowsey,18 1985
Merlin,11 1987
Procedure
CRF/S(r)
CRF/S(k)
DSEr
7.5-mm optical zone; single and paired arcs; 30 to 180 degrees total arc length
N/A
0.72
N/A
5.0- to 7.0-mm optical zone; paired arcs; 200 to 300 degrees total arc length
N/A
N/A
10.50
Source
205 Human
Duffey and associates,19 1988
25 Cadaver
5.0- to 9.0-mm optical zone
N/A
1.47 6 0.41
N/A
Lipshitz and associates,15 1994
11 Human
N/A
N/A
10.58
Chavez and associates,13 1996 Buzard and associates,14 1996
15 Human
5.0- to 6.0-mm optical zone; 9 of 11 eyes had arcuate incisions 5.0-mm optical zone; arcuate incisions 7.0-mm optical zone; paired arcs; 90 to 180 degrees total arc length
1.16 6 0.36
0.97
N/A
0.97 6 0.90
46 Human
Effect of Arc Length on Coupling Ratio
Arc length ,90 degrees: CR 5 20.22; arc length .90 degrees: CR 5 1.60 For all optical zones, hyperoptic shift increased with increasing arc length For 7.0-mm optical zone, arc length ,90 degrees: CR 5 2.2; arc length .90 degrees: CR 5 1.80 N/A
10.18 6 0.34 N/A 20.18
CR decreased with increasing arc length
Ksteep 5 the keratometric power of the steep preoperative meridian; Kflat 5 the keratometric power of the flat preoperative meridian; F 5 flattening of the incised meridian calculated based on change in either refraction or keratometry; N/A 5 not applicable. *Mean 6 SD CRF/S and DSE data are shown. Positive CRF/S values indicate that change in one meridian was in the opposite direction from change in the other meridian. Negative CRF/S values indicate change in both meridians was in the same direction.
THE ARC-T STUDY GROUP FRANCIS W. PRICE, JR, MD (STUDY CHAIRMAN), INDIANAP-
3.
olis, Indiana; Perry S. Binder, MD, La Jolla, California; Bruce I. Bodner, MD, Norfolk, Virginia; Daniel S. Durrie, MD, Kansas City, Missouri; Henry Gelender, MD, Dallas, Texas; Bruce Grene, MD, Wichita, Kansas; Kenneth R. Kenyon, MD, Boston, Massachusetts; Richard L. Lindstrom, MD, Minneapolis, Minnesota; William E. Whitson, MD, Indianapolis, Indiana.
4.
5. 6.
ACKNOWLEDGMENTS
We thank Peter Petersen, PhD, Department of Mathematics, UCLA, Los Angeles, California, for mathematical assistance and Jeffrey A. Gornbein, PhD, Department of Biomathematics, UCLA, for statistical consultation.
7.
8.
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