Journal of Refractive Surgery

Original Article Supplemental Data

Incorporating Refraction Measurement Error Into Surgery Planning for Corneal Ablation Procedures, Comparing the Flattening and Correction Indices

David A. Price, BS; Gerald P. Clarke, MD

Abstract

PURPOSE:

To assess the impact of measurement error on cylinder treatment planning in corneal refractive surgery and to compare the reliability of two indices of cylinder change: the Correction Index, based on the surgically induced astigmatism, and the Flattening Index, based on the flattening effect.

METHODS:

Preoperative refractions and surgical outcomes were evaluated using a Monte Carlo simulation. Cylinder change was calculated using vector analysis. The initial distribution of cylinder magnitudes was matched to a population sample of 1,000 eyes prior to LASIK. Refraction measurement error was modeled both preoperatively and postoperatively, with a non-linear optimization solver optimizing treatment effect for each amount of preoperative cylinder. Model assumptions were subsequently tested against LASIK outcomes data.

RESULTS:

The mean Flattening Index was approximately 1 for all levels of preoperative cylinder, whereas the Correction Index was significantly higher than 1 for cylinder treatments of 1.00 diopters (D) or less when preoperative and postoperative measurement errors were taken into consideration. Targeting undercorrection minimized postoperative cylinder compared with targeting full correction, with the optimal target dependent on the amount of random error in refraction measurement. It was optimal to partially treat 0.25 D of cylinder, even if the presumed level of measurement error was relatively high.

CONCLUSIONS:

The Flattening Index outperforms the Correction Index when accounting for errors in preoperative and postoperative cylinder measurement, particularly when pre-operative cylinder is 1.00 D or less. Treatment nomograms should be adjusted to use the Flattening Index to assess cylinder change, and reporting guidelines should be updated to include the Flattening Index.

[J Refract Surg. 2020;36(3):169–174.]

Abstract

PURPOSE:

To assess the impact of measurement error on cylinder treatment planning in corneal refractive surgery and to compare the reliability of two indices of cylinder change: the Correction Index, based on the surgically induced astigmatism, and the Flattening Index, based on the flattening effect.

METHODS:

Preoperative refractions and surgical outcomes were evaluated using a Monte Carlo simulation. Cylinder change was calculated using vector analysis. The initial distribution of cylinder magnitudes was matched to a population sample of 1,000 eyes prior to LASIK. Refraction measurement error was modeled both preoperatively and postoperatively, with a non-linear optimization solver optimizing treatment effect for each amount of preoperative cylinder. Model assumptions were subsequently tested against LASIK outcomes data.

RESULTS:

The mean Flattening Index was approximately 1 for all levels of preoperative cylinder, whereas the Correction Index was significantly higher than 1 for cylinder treatments of 1.00 diopters (D) or less when preoperative and postoperative measurement errors were taken into consideration. Targeting undercorrection minimized postoperative cylinder compared with targeting full correction, with the optimal target dependent on the amount of random error in refraction measurement. It was optimal to partially treat 0.25 D of cylinder, even if the presumed level of measurement error was relatively high.

CONCLUSIONS:

The Flattening Index outperforms the Correction Index when accounting for errors in preoperative and postoperative cylinder measurement, particularly when pre-operative cylinder is 1.00 D or less. Treatment nomograms should be adjusted to use the Flattening Index to assess cylinder change, and reporting guidelines should be updated to include the Flattening Index.

[J Refract Surg. 2020;36(3):169–174.]

Although error in refraction measurement with both machines and phoropters has long been recognized,1–3 study of its effects on observed cylinder change was generally ignored until a study by Bullimore et al.,4 which showed that the Correction Index will be much higher than 1 for low levels of cylinder correction because of refraction measurement error. In this study, we build off of that work to answer two key questions raised in their study: first, what is an appropriate measure for cylinder treatment success for cases where preoperative cylinder is 1.00 diopters (D) or less, which is a majority of patients who have laser-assisted in situ keratomileusis (LASIK), and, second, is it appropriate to treat patients with cylinder of 0.25 D, and, if treating, what target correction should be attempted?

Cylinder treatment in ablative refractive surgery consists of a two-part decision by the surgeon: what axis to treat and what magnitude of ablation to perform on that axis. These decisions are intertwined in that an error in the axis of treatment versus the actual axis of preoperative cylinder results in a lower optimized cylinder treatment magnitude. This can be illustrated using vector analysis, which is the general standard for cylinder change calculations in the literature,5 with its basis largely developed by Alpins.6–8 This is also the method on which we based our analyses.

In Figure A (available in the online version of this article), a patient with a true preoperative cylinder of 0.25 D @ 0 is refracted as having a preoperative cylinder of 0.25 D @ 30, which is within the standard error of possibility in previous studies on refraction repeatability.1–3 Given that the axis of treatment is different than the true axis of the patient's cylinder, the optimized treatment magnitude, assuming it is successfully on the axis of treatment, is now cut to 0.125 D. This is illustrated graphically (Figure A) and summarized numerically (Table A, available in the online version of this article). Note that in this example, the observed torque is not caused by the treatment and instead is an observed change because of error in preoperative versus postoperative refraction measurement.

For simplicity, postoperative error measurement is not included in this example. Red denotes real values, which are unknown. Black denotes observed values, based on imperfect measurement. The original cylinder 0.25 diopters (D) @ 0 is an unknown that is approximated with error, with the observed cylinder being 0.25 D @ 30 because of the error. Based on the axis of the observed cylinder, a treatment is performed with an effect on the original cylinder resulting in the postoperative cylinder. Based on the observed preoperative and postoperative cylinders, surgically induced astigmatism (SIA) is observed, which can be broken into components of torque and flattening effect. (A) In this case, the treatment effect was −0.25 D @ 30, which resulted in an observed torque and overflattening with an observed SIA and axis of effect of −0.43 D @ 45. (B) In this case, the treatment effect was −0.125 D @ 30, which resulted in an observed torque but a perfect flattening with an observed SIA and axis of effect of −0.33 D @ 50. (C) Although no astigmatic treatment is performed in this case, because of the measurement error there is an observed SIA and axis of effect of −0.25 D @ 60, with an underflattening and observed torque. TIA = target induced astigmatism

Figure A.

For simplicity, postoperative error measurement is not included in this example. Red denotes real values, which are unknown. Black denotes observed values, based on imperfect measurement. The original cylinder 0.25 diopters (D) @ 0 is an unknown that is approximated with error, with the observed cylinder being 0.25 D @ 30 because of the error. Based on the axis of the observed cylinder, a treatment is performed with an effect on the original cylinder resulting in the postoperative cylinder. Based on the observed preoperative and postoperative cylinders, surgically induced astigmatism (SIA) is observed, which can be broken into components of torque and flattening effect. (A) In this case, the treatment effect was −0.25 D @ 30, which resulted in an observed torque and overflattening with an observed SIA and axis of effect of −0.43 D @ 45. (B) In this case, the treatment effect was −0.125 D @ 30, which resulted in an observed torque but a perfect flattening with an observed SIA and axis of effect of −0.33 D @ 50. (C) Although no astigmatic treatment is performed in this case, because of the measurement error there is an observed SIA and axis of effect of −0.25 D @ 60, with an underflattening and observed torque. TIA = target induced astigmatism

Effect of an Optimized Treatment Magnitude (Unknown in Practice) vs Non-optimized Treatment Magnitudes After Accounting for an Error in Axis Measurementa

Table A:

Effect of an Optimized Treatment Magnitude (Unknown in Practice) vs Non-optimized Treatment Magnitudes After Accounting for an Error in Axis Measurement

Historically, the judgment of whether cylinder treatment magnitude is optimized has been based on the Correction Index as defined by Alpins,6,8 which is the standard for refractive outcomes reporting.5 The Correction Index is the ratio of the magnitude of the surgically induced astigmatism change (SIA) to the magnitude of the target induced astigmatism change (TIA).6 The SIA and TIA are illustrated in Figure A, with a more complete description in the original study by Alpins.6 Both the TIA and SIA are magnitudes, with no directional component.

Using the Correction Index for astigmatism analysis is limited in multiple ways. As was shown by Bullimore et al. and illustrated in Table A and Figure A, the Correction Index increases with the error in pre-operative and postoperative cylinder measurements,4 especially at low levels of cylinder. Another key limitation is that the decision of how much cylinder to treat in ablative surgeries such as LASIK is done on the axis of treatment rather than the observed axis of effect, and these differ whenever there is error in the refraction cylinder measurement. Further, if the physician increases or decreases the magnitude of treatment to optimize the Correction Index, it will change the observed axis of effect if there is a refraction measurement error of the axis, with the observed axis of effect changing solely based on the magnitude of the treatment effect changing (Figure A, Table A).

To address these problems, we propose using a different measure called the Flattening Index to analyze all cases of ablative cylinder correction. This was described by Alpins7 but is not included in the most recent standardized Journal of Refractive Surgery (JRS) reporting outcomes5 and to date has been primarily mentioned by Alpins as a means to analyze effects of cylinder correction occurring off of the treatment axis, such as when an incision is made in cataract surgery.7,9 The Flattening Index is the ratio of the achieved cylinder change on the treatment meridian, termed the Flattening Effect versus the intended change (TIA):

FI=FETIA=cos(2*|θ1−θ2|)*|SIA||TIA|

where θ1 = axis of observed cylinder change from preoperatively to postoperatively, and θ2 = treatment axis.

The advantage of using the Flattening Index is that for any preoperative refraction error in cylinder measurement, a Flattening Index of 1 indicates an optimized treatment magnitude decision, even if it does not represent a perfect result. As an example, with a preoperative cylinder of 1.00 D @ 90 and a postoperative cylinder of 0.25 D @ 45, the Flattening Index was 1. This is consistent with an optimized treatment at 90 degrees, with the only remaining cylinder orthogonal to the original treatment axis of ablation. In this case, a smaller cylinder change on the axis of treatment would increase postoperative cylinder, as would a larger one. A key assumption with the Flattening Index is that the ablative effect occurs on the axis of treatment or close to it, so we included several analyses of LASIK cylinder changes to test that assumption.

Patients and Methods

We created simulations of 50,000 cases to determine optimized treatment magnitudes for each amount of measured preoperative cylinder, as well as to assess the performance of the Flattening Index versus the Correction Index for various amounts of cylinder. De-identified preoperative refractions were obtained from 1,000 consecutive virgin eyes with cylinder of 4.00 D or less that underwent LASIK, either topography-guided (n = 896) or wavefront-optimized (n = 104), performed using the EX500 excimer laser (Alcon Laboratories, Inc., Fort Worth, TX) at Price Vision Group (Figure 1). Three simulations were run with different assumed levels of refraction measurement error (normal distributions with a mean of 0 and standard deviations of 0.125, 0.177, and 0.25 D) added on the axes of 0 and 45 degrees both preoperatively and postoperatively. Each simulation included 50,000 trials/procedures, with the distribution of initial cylinder magnitudes matched to the 1,000 patients. Because true cylinder magnitudes are continuous and not limited to the 0.25 D steps of phoropters, a random value from +0.12 to −0.12 D was added to create the final distribution of true preoperative cylinders prior to factoring in measurement error. After measurement error was added, the cylinder was rounded to the nearest quarter diopter to get a measured value. Axis was randomized and not included as an important variable in the simulation beyond being necessary to calculate changes when incorporating measurement error and treatment effects. Optimization was performed using the solver add-in for Excel software (Microsoft Corporation, Redmond, WA), which is a non-linear solver. Optimized treatment magnitudes were calculated for each measured magnitude of cylinder preoperatively, with the objective to minimize average postoperative residual cylinder. The reported residual cylinder, Flattening Index, and Correction Index were averaged after factoring the postoperative measurement error.

Distribution of actual preoperative refractions in 1,000 eyes. Each simulation had 50,000 trials, which was achieved by multiplying the initial distribution by 50. D = diopters

Figure 1.

Distribution of actual preoperative refractions in 1,000 eyes. Each simulation had 50,000 trials, which was achieved by multiplying the initial distribution by 50. D = diopters

Following the simulation, several tests were run to verify that treatment effects from LASIK were on the axis programmed into the laser. First, eyes that met the criteria of having some preoperative cylinder and completion of the 1-month postoperative examination at Price Vision Group (n = 662) were included in a regression of observed torque at 1 month postoperatively compared to TIA to assess whether there was a relationship between observed torque and TIA. This analysis was done to confirm that the true surgical treatment was actually on the intended axis, the axis programmed into the laser by the physician. If the true surgical treatment effect was off the intended axis, it would manifest as a linear relationship between treatment magnitude and observed torque. As a control for this analysis, average meridional and torque changes based on the axis of original preoperative cylinder were compared between the 1- and 3-month postoperative appointments for all of the patients with preoperative cylinder of 4.00 D or less and 1- and 3-month appointments (n = 348). These control tests were done to confirm that the amount of observed torque following LASIK surgery was close to what would be observed by simply repeating refractions months apart in the same patients. The Student's t test, run in SAS software (SAS Institute, Cary, NC), was used for all significance testing with a two-sided cut-off of 5%. This study was determined to be exempt from institutional review board approval under Category 4 by IRB Company, Inc. (Buena Park, CA), and no identifiable patient data were used.

Results

The results of the analysis are summarized in Table 1. It was optimal to treat less than the full refraction for all measured preoperative cylinder amounts, and the optimal reduction in treatment increased with increased average measurement error. Residual cylinder was almost entirely determined by the refraction measurement error levels and not by the preoperative refraction using optimized treatments. The Correction Index was greater than 1 for patients initially measured to have cylinder of 0.25 D and approached 1 as preoperative cylinder increased. The Correction Index was higher with increasing measurement error. With optimized treatments, the Flattening Index was approximately 1 for all levels of preoperative cylinder and refraction measurement error.

Summary of Results per Simulation, With the Initial Measured Refraction and for Each Group the Optimized Treatment Effect Magnitude, Average Measured Postoperative Astigmatism, FI, and CI

Table 1:

Summary of Results per Simulation, With the Initial Measured Refraction and for Each Group the Optimized Treatment Effect Magnitude, Average Measured Postoperative Astigmatism, FI, and CI

It was optimal to treat 0.25 D of preoperative cylinder if the correction could be programmed to the hundredths. However, when only allowed 0.25 D steps, the benefit from treating versus not treating 0.25 D of cylinder was little to none (Figure 2). With the lowest measurement error assumption, there was a slight benefit to treating a full 0.25 D versus no treatment, whereas with the higher measurement error assumptions, there was a slight benefit to not treating versus a full 0.25 D treatment. Partially treating only improved residual cylinder by 0.04 D on average versus no treatment at all with the lowest measurement error assumption, and the benefit was reduced as the assumed measurement error increased.

Residual astigmatism (not factoring in postoperative measurement error) in eyes initially measured to have 0.25 diopters (D) of cylinder preoperatively. When able to program treatments to the hundredths, a partial treatment is optimal, with the magnitude of the optimal treatment decreasing as measurement error increases. SD = standard deviation

Figure 2.

Residual astigmatism (not factoring in postoperative measurement error) in eyes initially measured to have 0.25 diopters (D) of cylinder preoperatively. When able to program treatments to the hundredths, a partial treatment is optimal, with the magnitude of the optimal treatment decreasing as measurement error increases. SD = standard deviation

In the actual clinical data at 1 month postoperatively, absolute observed torque was almost non-dependent on TIA (R2 = 3%, Figure 3). The slope of 0.05 D of torque per diopter of targeted change was consistent with ablation misalignment from intended axis of 1 to 2 degrees. The average and standard deviation of absolute observed torque were 0.15 ± 0.19 D. When separating into positive (counter-clockwise) and negative (clockwise) torque,8 there was not a significant trend (0.01 ± 0.23, P = .33).

Induced torque regressed on target induced astigmatism (TIA) in 664 eyes. The weak relationship is consistent with the laser effect being on or near the intended axis, with most of the observed torque likely being from sources other than the ablation, such as measurement error. D = diopters

Figure 3.

Induced torque regressed on target induced astigmatism (TIA) in 664 eyes. The weak relationship is consistent with the laser effect being on or near the intended axis, with most of the observed torque likely being from sources other than the ablation, such as measurement error. D = diopters

In the control analyses from 1 to 3 months postoperatively, absolute observed torque change was 0.14 ± 0.18 D and not significantly different from the average between preoperative and 1-month refractions (P = .17). Directional cylinder changes on the preoperative meridional (−0.01 ± 0.21, P = .42) and torque (−0.01 ± 0.22, P = .35) axes were not significantly different from zero.

Discussion

The Flattening Index Outperforms the Correction Index

Based on the clinical data, it is a safe assumption in LASIK calculations that the surgical effect is occurring on or close to the intended axis, and therefore the Flattening Index is an appropriate measure for judging the magnitude of cylinder change. This was shown first in the regression analysis, where the slope of observed torque was only consistent with ablation misalignment from intended axis of 1 to 2 degrees. If the LASIK treatment effect was occurring off the axis of treatment and better measured by the Correction Index, the linear relationship would have been stronger and the slope steeper. Second, in the t test analysis, the absolute change in observed torque from the 1- to 3-month refractions (0.14 ± 0.18 D), when no intervening surgery was performed, was neither statistically nor practically different from the change observed preoperatively to 1 month (0.15 ± 0.19 D). If the treatment was misaligned, these would have been different. Both of these tests used hundreds of LASIK cases, minimizing the possibility of missing a significant relationship.

Within the simulation, the Flattening Index was close to 1 for all levels of cylinder and all tested levels of error in measuring refractions, and it was clearly superior to the Correction Index for adapting to the effects of measurement errors when using vector analysis. This raises the question of whether standard refractive reporting measures5 should be updated to include the Flattening Index. Surgeons are making the decision of axis to treat and magnitude to treat in combination. Ultimately, the surgeon's decision prior to surgery is how much to increase or decrease a magnitude of cylinder treatment programmed into a laser on a given axis, which is a question better answered by his or her historical average Flattening Index rather than Correction Index. Furthermore, as shown by this model, the Flattening Index is a better approximation of optimized treatment than the Correction Index.

However, future comparisons using different lasers may find that treatments are misaligned from the intended axis, in which case the Flattening Index may only be a better measure for smaller treatment magnitudes, such as 1.00 D or less, which is when pre-operative refraction measurement error is a relatively greater consideration, and the Correction Index may be better for larger treatment magnitudes, such as 3.00 D or greater, which is when misalignment would be a more important consideration.

Targeting Undercorrection Minimizes Postoperative Cylinder

Although in practice there are additional factors that come into play when evaluating a tendency to overrefract or underrefract cylinder, there are three principal mathematical factors for why random error leads to targeting undercorrection being the optimal approach. The first is “the tendency to over-measure astigmatism,” as illustrated in Figure B (available in the online version of this article). Bullimore et al.4 showed this algebraically, and it can be summarized by observing that any measurement error in either direction on the axis of torque will result in an increase in the measured cylinder.

The patient's actual preoperative astigmatism is 0.25 diopters (D) @ 22.5 degrees (black dot). Error on the principal axis (black) can either decrease or increase astigmatism and will likely balance out, unless the initial astigmatism is low, in which case it will tend to increase measured astigmatism versus the actual. In contrast, torque error (red) will only increase the amount of measured astigmatism, and is unbalanced toward increasing measured astigmatism.

Figure B.

The patient's actual preoperative astigmatism is 0.25 diopters (D) @ 22.5 degrees (black dot). Error on the principal axis (black) can either decrease or increase astigmatism and will likely balance out, unless the initial astigmatism is low, in which case it will tend to increase measured astigmatism versus the actual. In contrast, torque error (red) will only increase the amount of measured astigmatism, and is unbalanced toward increasing measured astigmatism.

Second, “low levels of astigmatism are more common,” increasing the number of opportunities for a measured cylinder to be an overestimation versus an underestimation. For example, 5,650 patients in the simulation had 1.00 D of cylinder, whereas 2,650 had 1.50 D, leading to twice as many opportunities for 1.00 D to be measured as 1.25 versus 1.50 D to be measured as 1.25 D, making it more likely that cylinder was overestimated versus underestimated (Table B, available in the online version of this article). Given that the initial distribution of refractions came from measured data of patients prior to LASIK surgery, this is representative of what is observed in practice.

Actual Astigmatism Magnitude vs Measured Astigmatism Magnitude (D)a

Table B:

Actual Astigmatism Magnitude vs Measured Astigmatism Magnitude (D)

The third factor is “axis measurement error.” Whenever there is some axis measurement error, treating a smaller amount of cylinder will result in an optimized treatment. This was illustrated in Figure A and Table A, where an error of 30 degrees in the axis of measurement results in the optimized magnitude of treatment being cut in half and the residual cylinder magnitude being approximately 87% of the original cylinder magnitude even with an optimized treatment. Figure C (available in the online version of this article) shows the change in the optimized magnitude as a proportion of the whole cylinder magnitude based on measurement error in the axis and residual cylinder with an optimized treatment. For small differences between measured and true axis, the decrease in the optimized treatment is minimal, but for larger shifts, as may be seen for patients with only 0.25 D of cylinder to begin with, significant percentage reductions are needed to optimize treatment. Axis measurement error likely explains why the Flattening Index has been found to be significantly elevated for 0.25 D cylinder treatments when full correction is attempted.10

The circles and solid line show the optimized treatment magnitude based on the difference between the true cylinder axis (unknown) and measured cylinder axis (known). Based on making an optimized treatment, the final residual cylinder (dashed and triangles) is also shown as a percentage of the magnitude of cylinder. Of note, while small errors in axis (< 10 degrees) do not lead to much change in the optimized treatment amount, they do lead to residual astigmatism. In optimized treatments, residual astigmatism will end up at 45 degrees from the axis of treatment.

Figure C.

The circles and solid line show the optimized treatment magnitude based on the difference between the true cylinder axis (unknown) and measured cylinder axis (known). Based on making an optimized treatment, the final residual cylinder (dashed and triangles) is also shown as a percentage of the magnitude of cylinder. Of note, while small errors in axis (< 10 degrees) do not lead to much change in the optimized treatment amount, they do lead to residual astigmatism. In optimized treatments, residual astigmatism will end up at 45 degrees from the axis of treatment.

It is optimal to target undercorrections to minimize postoperative cylinder because there is a tendency to overmeasure astigmatism, low levels of astigmatism are more common, and any axis measurement error decreases the optimal treatment.

Applying These Results to Current Practice

The study findings have two practical applications: updating corneal refractive surgery reporting outcomes and improving nomograms. First, reporting outcomes for corneal refractive surgery should be updated to include the Flattening Index because the Flattening Index is an effective means of judging whether treatment magnitudes are optimized for cylinder correction less than 1.00 D and the Correction Index is not. Second, treatment nomograms should be updated to use the Flattening Index to calculate how much cylinder change was achieved in surgeons' historical cases and use those results to optimize future treatments. For example, if a surgeon's average Flattening Index for 0.25 D treatments is 1.25, reducing the treatment magnitude 20% would likely optimize cylinder outcomes for those patients. A limitation of the Correction Index is that it does not approximate to 1 when there is a low amount of preoperative cylinder, limiting analysis in the past. It may be helpful to include data on who is performing the refractions in nomograms, because measurement error in the preoperative refraction will affect the optimized treatment magnitude. Although it is tempting to look for a specific amount of undercorrection to target based on the knowledge of measurement error existing, the reality is that the optimal amount of undercorrection is dependent on refractive measurement error, which will likely vary based on who is performing refractions. Further, other considerations will affect the optimized treatment, such as specific providers having general tendencies to overmeasure or undermeasure cylinder non-randomly. Therefore, nomogram analysis is required to determine the optimized treatment adjustments. However, as measurement techniques improve and better devices are developed to measure astigmatism, the significance of refraction measurement error in cylinder treatment planning may eventually decline.

Limitations

This study has several limitations. First, the distribution of true refractions was based on measured data, with some random measurement error already built in, which would affect the shape of the distribution. Second, refraction measurement error was simplified with no dependency on cylinder magnitude, which may exist and should be studied. This analysis included data from one laser (the EX500) and may not hold true for other laser systems. Measured cylinder is a correction being fit to approximate the true eye and is not distinguishing between anterior corneal astigmatism, posterior corneal astigmatism, and lenticular astigmatism, True corrections likely do not correlate exactly to what is predicted by vector analysis, which is an approximation.

Conclusion

This study is a start to incorporating refraction measurement error into treatment planning and an extension of the work of Bullimore et al.4 to examine the effect of refraction measurement error on the Correction Index. Targeting some undercorrection of cylinder results in minimizing postoperative cylinder as compared with targeting a full correction, whenever some random error exists in the preoperative refraction, as it always does. Refractive reporting outcomes and nomograms for surgery planning should be updated to include the Flattening Index. The Flattening Index is superior to the Correction Index when preoperative measured cylinder is 1.00 D or less, because measurement error in preoperative refraction affects the efficacy of the Correction Index in that range. In this study, the Correction Index and Flattening Index perform similarly when cylinder is greater than 1.00 D, although this may not be the case with all laser systems.

References

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Summary of Results per Simulation, With the Initial Measured Refraction and for Each Group the Optimized Treatment Effect Magnitude, Average Measured Postoperative Astigmatism, FI, and CI

Measured Cyl (D) 0.125 SD 0.177 SD 0.25 SD



Optimized Treatment Average Postop FI CI Optimized Treatment Average Postop FI CI Optimized Treatment Average Postop FI CI
0.00 −0.19 −0.26 −0.39
−0.25 0.13 −0.23 0.98 1.31 0.09 −0.30 1.01 1.54 0.07 −0.40 1.00 1.85
−0.50 0.48 −0.22 1.00 1.07 0.44 −0.32 1.01 1.15 0.36 −0.43 1.00 1.26
−0.75 0.69 −0.23 0.99 1.02 0.65 −0.32 1.00 1.06 0.59 −0.44 0.99 1.11
−1.00 0.96 −0.23 1.00 1.02 0.94 −0.32 1.00 1.03 0.85 −0.44 1.01 1.07
−1.25 1.21 −0.23 1.00 1.01 1.18 −0.31 1.00 1.02 1.12 −0.44 0.99 1.03
−1.50 1.45 −0.23 1.00 1.01 1.41 −0.32 1.00 1.01 1.35 −0.44 1.00 1.02
−1.75 1.72 −0.23 1.00 1.00 1.68 −0.32 1.00 1.01 1.62 −0.45 1.00 1.02
−2.00 1.97 −0.24 1.00 1.00 1.95 −0.32 1.00 1.01 1.89 −0.46 1.00 1.02
−2.25 2.24 −0.24 0.99 1.00 2.21 −0.34 1.00 1.00 2.18 −0.46 1.00 1.01
−2.50 2.49 −0.23 1.00 1.00 2.48 −0.32 1.00 1.01 2.44 −0.43 1.00 1.01
−2.75 2.70 −0.23 1.00 1.00 2.67 −0.31 1.00 1.01 2.63 −0.44 1.01 1.01
−3.00 2.95 −0.24 1.00 1.00 2.91 −0.31 1.00 1.00 2.82 −0.44 1.00 1.02
−3.25 3.23 −0.23 0.99 1.00 3.20 −0.34 1.00 1.00 3.10 −0.46 0.99 1.00
−3.50 3.49 −0.23 1.00 1.00 3.48 −0.31 1.00 1.00 3.46 −0.41 1.01 1.01
−3.75 3.69 −0.22 1.00 1.00 3.70 −0.33 1.00 1.01 3.64 −0.42 1.00 1.01
−4.00 3.96 −0.21 1.00 1.00 3.87 −0.31 0.99 1.00 3.83 −0.43 1.00 1.00

Effect of an Optimized Treatment Magnitude (Unknown in Practice) vs Non-optimized Treatment Magnitudes After Accounting for an Error in Axis Measurementa

Actual Astigmatism Measured Astigmatism Treatment Effect Final Cylinder Observed Flattening Effect Observed Torque Change Observed SIA + Axis of Effect CI FI
0.25 D @ 0 0.25 D @ 30 None 0.25 @ 0 −0.125 D −0.22 D −0.25 D @ 60 1.00 0.50
0.25 D @ 0 0.25 D @ 30 −0.125 D @ 30 0.22 D @ 165 −0.25 D −0.22 D −0.33 D @ 50 1.3 1
0.25 D @ 0 0.25 D @ 30 −0.25 D @ 30 0.25 D @ 150 −0.375 D −0.22 D −0.43 D @ 45 1.7 1.5

Actual Astigmatism Magnitude vs Measured Astigmatism Magnitude (D)a

Measured Astigmatism, Simulation with 0.177 SD Error at 0 and 45 Degrees (No. of Eyes/Trials)

0.00 −0.25 −0.50 −0.75 −1.00 −1.25 −1.50 −1.75 −2.00 −2.25 −2.50 −2.75 −3.00 −3.25 −3.50 −3.75 −4.00 −4.25 −4.50 Total
Actual astigmatism (no. of eyes/trials) Actual Avg −0.12 −0.22 −0.46 −0.68 −0.95 −1.19 −1.43 −1.69 −1.95 −2.22 −2.48 −2.68 −2.92 −3.21 −3.48 −3.70 −3.87
0.00 1,357 4,185 787 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6,350
−0.25 494 2,848 1,644 210 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5,200
−0.50 104 2,159 5,697 3,166 366 8 0 0 0 0 0 0 0 0 0 0 0 0 0 11,500
−0.75 2 116 1,336 3,328 1,699 215 4 0 0 0 0 0 0 0 0 0 0 0 0 6,700
−1.00 0 4 115 1,202 2,668 1,490 167 4 0 0 0 0 0 0 0 0 0 0 0 5,650
−1.25 0 0 1 81 1,000 2,218 1,110 140 0 0 0 0 0 0 0 0 0 0 0 4,550
−1.50 0 0 0 1 65 585 1,311 620 68 0 0 0 0 0 0 0 0 0 0 2,650
−1.75 0 0 0 0 3 25 394 813 410 54 1 0 0 0 0 0 0 0 0 1,700
−2.00 0 0 0 0 0 0 27 322 707 362 32 0 0 0 0 0 0 0 0 1,450
−2.25 0 0 0 0 0 0 0 15 216 465 230 24 0 0 0 0 0 0 0 950
−2.50 0 0 0 0 0 0 0 1 22 271 637 284 35 0 0 0 0 0 0 1,250
−2.75 0 0 0 0 0 0 0 0 0 20 153 351 157 19 0 0 0 0 0 700
−3.00 0 0 0 0 0 0 0 0 0 0 15 99 228 102 6 0 0 0 0 450
−3.25 0 0 0 0 0 0 0 0 0 0 0 4 40 99 53 4 0 0 0 200
−3.50 0 0 0 0 0 0 0 0 0 0 0 0 6 78 175 85 6 0 0 350
−3.75 0 0 0 0 0 0 0 0 0 0 0 0 1 7 35 105 50 2 0 200
−4.00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 42 54 44 5 150
Total 1,957 9,312 9,580 8,009 5,805 4,541 3,013 1,915 1,423 1,172 1,068 762 467 305 274 236 110 46 5
Authors

From Cornea Research Foundation of America, Indianapolis, Indiana (DAP); Price Vision Group, Indianapolis, Indiana (DAP); Indiana School of Medicine, Indianapolis, Indiana (DAP); and OptiVision EyeCare, Neenah/Oshkosh, Wisconsin (GPC).

The authors have no financial or proprietary interest in the materials presented herein.

AUTHOR CONTRIBUTIONS

Study concept and design (DAP); data collection (DAP); analysis and interpretation of data (DAP, GPC); writing the manuscript (DAP); critical revision of the manuscript (GPC); statistical expertise (DAP, GPC)

Correspondence: David A. Price, BS, Cornea Research Foundation of America, 9002 N. Meridian St., Suite 212, Indianapolis, IN 46260. E-mail: daprice200@gmail.com

Received: September 12, 2019
Accepted: January 20, 2020

10.3928/1081597X-20200121-01

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