- Email: [email protected]
Correction of low levels of astigmatism
ARTICLE
Correction of low levels of astigmatism Mark A. Bullimore, MCOptom, PhD, Greg Spooner, PhD, Georg Sluyterman, PhD, Jon G. Dishler, MD
PURPOSE: To reevaluate the analysis of the correction of astigmatism. SETTING: Academia, industry, and private practice. DESIGN: Evaluation of diagnostic test or technology. METHODS: Astigmatic refractive surgery outcomes are based on vector methods, including the correction index (also known as the correction ratio), which is the ratio of the surgically induced astigmatism to the target induced astigmatism (TIA). Mean correction indices substantially greater than 1 have been reported for astigmatic corrections less than 1.00 diopter (D) and as representing systematic overcorrection. We hypothesize that this reflects a limitation of the correction index rather than systematic flaws in treatments. The theoretical mathematic behavior of the correction index was analyzed, accounting for variability in astigmatism measurement. Then, the impact of cylinder measurement variability on the mean correction index was modeled. A Monte Carlo simulation was performed and calculated 10 000 values of correction index for various values of TIA. Finally, correction indices from published and unpublished studies of refractive lasers were compared with the simulations. RESULTS: The mean correction index is always greater than 1 for the case of a perfect refractive correction; however, for astigmatic corrections less than 1.00 D, the mean correction index increases sharply because the measurement variability is similar in magnitude to TIA. Almost all previous studies show the predicted increase in the correction index for low astigmatic corrections. CONCLUSION: The correction index is a useful vector-based metric for the evaluation of refractive procedures, but mean values greater than 1 should be anticipated for lower astigmatic treatments and do not necessarily represent systematic overcorrection. Financial Disclosure: Dr. Bullimore is a consultant to Alcon Surgical, Inc., Carl Zeiss Meditec AG, Digital Vision Systems, Essilor, Innovega, Inc., and Paragon Vision Sciences, Inc. Dr. Spooner is a consultant to Alcon Surgical, Inc., Carl Zeiss Meditec AG, Digital Vision Systems, Thru-Focus Optics LLC, and i2eyediagnostics, Ltd. Dr. Dishler is a consultant to Carl Zeiss Meditec AG and Revision Optics, Inc. Dr. Sluyterman is an employee of Carl Zeiss Meditec AG. J Cataract Refract Surg 2015; 41:1641–1649 Q 2015 ASCRS and ESCRS Supplemental material available at www.jcrsjournal.org.
The presentation of outcomes in refractive surgery has become standardized in the past decade. Leading journals, for example, request that authors use standardized graphs for the reporting of results.1,2 Interpretation of the surgical correction of astigmatism requires vector methods because astigmatism has magnitude and orientation, and some authors have developed numerical and statistical approaches for the management of astigmatic data.3,4 A principal purpose of vector methods in astigmatism treatments is to determine the degree of improvement, generally Q 2015 ASCRS and ESCRS Published by Elsevier Inc.
expressed as a difference between vector quantities characterizing the measured astigmatism.5,6 In 2001, Alpins6 proposed a series of metrics to be used in the evaluation of astigmatic refractive surgery. One such metric was the correction index, defined as the ratio of the surgically induced astigmatism (SIA) vector to the target induced astigmatism (TIA) vector. Subsequently, the American National Standards Institute (ANSI) Z80.117 Working Group on Laser Systems for Corneal Reshaping formed an Astigmatism Project Group made up of experts from academia, http://dx.doi.org/10.1016/j.jcrs.2014.12.060 0886-3350
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government, and industry. The group’s goal was to “identify a common, minimum set of analyses and a presentation format for adequate evaluation of safety and effectiveness of new astigmatism-correcting devices.”5 One of the analysis variables identified by the group was the correction ratio, which is the equivalent of Alpins’ correction index.6 Acknowledging the precedent of Alpins’ work, we use the term correction index throughout this paper. As stated previously and as represented in the ANSI Standard Z80.11-2012,7 the correction index is defined as the ratio of the magnitudes of the SIA vector and the TIA vector. The mean correction index calculated for a cohort of treated eyes could be used to identify systemic overcorrection or undercorrection. For higher astigmatic corrections, mean values of the correction index close to 1 are common, but 2 recent papers reported a correction index substantially greater than 1 for lower astigmatic corrections.8,9 These findings appear to suggest that a refractive laser functions less accurately for small cylindrical corrections than for larger corrections. We hypothesize that it might merely reflect a problem with the interpretation of the metric rather than a flaw with the laser treatment being evaluated. In any thorough analysis, the random errors associated with the measurement must be considered, and these vector quantities are no exception. Measurement error, resulting from the practical limitations to the precision of any physical measurement process, is distinct from systematic or other errors induced by the treatment. When measurement error is included in the analysis, the mean correction index is greater than 1 whenever the size of the TIA is comparable to the inherent measurement error in the subjective refraction measurement.10–13 This behavior will hold even in the case of a perfect refractive treatment. The mean correction index must be interpreted carefully for small cylinder treatments. Findings of mean correction index values greater than 1 for small-cylinder treatments do not necessarily indicate the presence of systemic overcorrection. Submitted: May 9, 2014. Final revision submitted: December 8, 2014. Accepted: December 10, 2014. From the College of Optometry (Bullimore), University of Houston, Houston, Texas, Gain Consulting Services (Spooner), San Francisco, California, and the Dishler Laser Institute (Dishler), Greenwood Village, Colorado, USA; Carl Zeiss Meditec AG (Sluyterman), Jena, Germany. Corresponding author: Mark A. Bullimore, MCOptom, PhD, 356 Ridgeview Lane, Boulder, Colorado 80302, USA. E-mail: [email protected].
MATERIALS AND METHODS The quantity correction index as defined by Alpins6 is a measure of astigmatic refractive surgical correction.5 The correction index is the ratio of the absolute value (or magnitude) of the SIA vector (also called the surgically induced refractive correction vector) to the absolute value of the TIA vector (also called the intended refractive correction vector). CI Z
jSIAj jTIAj
(1)
where vector quantities are represented in bold text. The TIA is the target induced astigmatism vectordthe correction values as entered into the laser device user interface, typically as cylinder (diopters [D]) and axis (degrees). The TIA is usually chosen to be equal to the preoperative measured cylinder minus the target cylinder. To eliminate astigmatism, the TIA must be equal in magnitude to the preoperative cylinder. The SIA is defined as the vector difference between the measured preoperative cylinder and the measured postoperative cylinder. SIA Z Cpre Cpost
(2)
Eydelman et al.5 noted that a particular correction index value for a single eye might indicate an undercorrection, an overcorrection, or a good correction. In a population of treated eyes, the average correction index value might indicate the achieved correction relative to the intended correction; however, the correction index must be carefully interpreted, especially for small astigmatic corrections.8,14,15 The reason for such caution is that the relationship between the intended correction (TIA) and the achieved surgical correction (SIA) is vectorial. The scalar (or purely numerical) definition of the correction index can mask the vector relationship between the TIA and the SIA for small astigmatic corrections. Some random measurement error is always present in the preoperative and postoperative refractions. Measurement error, resulting from the practical limitations to the precision of any physical measurement process, is distinct from systematic or other errors induced by the treatment. This measurement error materially affects the SIA vector for small astigmatic corrections. In the methods of Eydelman et al.5 and Alpins,6 the effects of refraction measurement error are not considered. For small cylinder corrections that are close to the magnitude of the measurement variability, the mean correction index also generally is greater and often is considerably greater than 1.8,14,15 The refraction measurement variability for astigmatism is G0.25 D in orthogonal (90 and 180) and oblique (45 and 135) meridians.10 As a consequence, for TIAs less than approximately 1.0 D, the resulting correction index is substantially greater than 1, assuming there is no systemic undercorrection or overcorrection by the device. That is, in a perfect treatment for the correction of astigmatism, the proper use of the correction index is restricted to astigmatic corrections that are appreciably larger than the refraction measurement accuracy. In practice, the correction index is an unreliable indicator of the effectiveness of cylinder treatments smaller than 1.0 D in magnitude. Below is a mathematical formulation of the mean correction index that properly includes the presence of random measurement errors associated with reported clinical refraction accuracy and repeatability. After that, to visualize the effects of refraction measurement error on the correction
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index, the vector figures of Eydelman et al.5 are recast to include measurement vectors. A numerical simulation of the mean correction index for typical refraction measurement error values follows. Finally, as context for numerical simulation, a range of mean correction index values observed in studies of correction of astigmatism with excimer lasers is provided.
Mathematical Behavior of the Mean Correction Index with Refraction Measurement Error The SIA is most properly represented as SIA Z SIA0 þ DC
(3)
where DC represents the randomly varying measurement error vector from the “true” or error-free value SIA0. In the simplest case, DC can be considered to vary randomly in magnitude and direction, within some limits. The statistical average of DC in this simple model of the error behavior therefore is zero. This SIA that includes random error in the definition of the correction index can be used to find an expression for the mean correction index (represented as !Correction IndexO in equations) for a fixed value of the TIA. hCorrection Indexi Z
hjSIAji jTIAj
(4)
It is convenient to represent the SIA in Cartesian coordinates denoted with subscripts x and y (shown later as J0 and J45). The mean correction index can then be represented as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðSIA0x þDCx Þ2 þ SIA0y þDCy hCorrection IndexiZ jTIAj (5)
hCorrection Indexi Z
above averages the final term on the right side of the equation to zero because the quantities DCx and DCy can carry a positive or negative sign, and a random variation of linear products of these symmetrically varying quantities produces an average of zero. Quantities DC2x and DC2y are always positive, and the quantity SIA0 has a positive magnitude. Therefore, both the first and second terms in equation 7 do not average to zero. + * DC2x þ DC2y SIA0 (9) þ hCorrection Indexiz jTIAj jTIAj 2SIA0 It is clear by inspection that all quantities in this simple analytic expression are positive. Further, because the expression was developed for a perfect treatment, the magnitude of the TIA equals the mean of the SIA magnitude. The term on the left of equation 9 is therefore equal to 1. Because the term on the right is the average of a sum of (positive) squared numbers divided by a positive magnitude, the average is a positive number: * + DC2x þ DC2y hCorrection Indexiz1 þ (10) jTIAj 2SIA0 The mean correction index is therefore greater than 1 for the special case of a perfect SIA correction (perfect being defined as a SIA correction that is exactly equal in magnitude to the target astigmatism magnitude, with no deviceintroduced error). In a perfect treatment, the mean correction index should always be greater than 1 unless the physical measurements could somehow be performed with zero measurement error. Further, when the measurement error DC is comparable in size to the SIA, the mean correction index will be significantly greater than 1. It is important to note that if systematic error were present due to a non-perfect treatment, then the average correction index could be different from 1 due to the combination of
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE 1 D SIA20x þ DC2x þ2SIA0x DCx þSIA20y þ DC2y þ2SIA0y DCy jTIAj
Computing the statistical average of this overall quantity requires specific knowledge of the individual quantities; however, for DC components that are smaller than the SIA components (ie, relatively small measurement error), the square root can be approximated by a binomial expansion keeping only the first-order terms to give an approximate or limiting value of the mean correction index. * 1 DC2x þ DC2y 2 1 hCorrection Indexiz SIA0 þ SIA0 jTIAj (7) + SIA0x DCx þSIA0y DCy þ SIA0 where the term SIA0 is defined as the quantity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SIA0 Z SIA20x þSIA20y (8) Assuming that the measurement errors are normally distributed, performing a statistical average of equation 7
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(6)
the measurement error and the systematic error from the treatment. Therefore, a non-unity value for the average correction index does not necessarily indicate an overcorrection or undercorrection for small astigmatic corrections. Only when considering TIA values that are much larger than DC (ie, cylinder values that are larger than the measurement error) can the correction index be considered an indicator of a real overcorrection or undercorrection.
Vector Nature of the Correction Index A major factor contributing to the higher correction indices reported in excimer laser clinical trials is the challenge of measuring low amounts of cylinder reliably.10 Obtaining reliable measurements might be even more difficult when a low level of cylinder is in the presence of a high levels of myopia, reduced visual acuity, or both. We have demonstrated analytically that the mean correction index will be greater than 1 for any real measurements of refraction. The impact of the variability in cylinder
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Figure 1. Sample vector plot illustrating the relationship among measured preoperative cylinder, measured postoperative cylinder, and SIA (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism).
measurement on the correction index can also be demonstrated graphically, based on the definitions found in Eydelman et al.5 and Alpins.6 Figure 1 shows the relationship among the measured preoperative cylinder (Cpre), the measured postoperative cylinder (Cpost), and the SIA. As in the previous section, the SIA is defined as the vector difference between the preoperative cylinder and the postoperative cylinder (equation 2). Refractive cylinder amounts at oblique orientations may be combined as vectors but must be plotted on a
Figure 3. True preoperative cylinder and true postoperative cylinder (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
Figure 2. Vectors used in the calculation of SIA and TIA (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
2-dimensional double-angle plot.3 The angles must be doubled because whereas vectors of equal magnitude 180 degrees apart cancel each other out, equal cylinders 90 degrees apart cancel each other out. The axes of the plot correspond to the orientation of the commonly used variables, J0 and J45.4 In Figure 1, the measured cylinder preoperatively might be 2.50 108, and the measured cylinder postoperatively might be 1.13 40. Therefore, the SIA in this example would equal 3.41 115. As in any real measurement, the preoperative cylinder and the postoperative cylinder can vary. Figure 2 shows how the measured postoperative cylinder is calculated with a series of vectors. The dotted gray lines for each measurement error represent the horizontal and vertical components, corresponding to J0 and J45, respectively. First, there is error associated with measurement of the preoperative cylinder. Second, the excimer laser alters the cylinder by the TIA. Note that the magnitude and orientation of the TIA are equal to the measured preoperative cylinder, thus assuming a perfect treatment. Finally, there is error associated with measurement of the postoperative cylinder. The correction index is defined as the ratio of the absolute of SIA to the absolute TIA (equation 1). In the example shown in Figure 2, the correction index is 1.6. Figure 3 shows 2 additional variables: the true preoperative cylinder and the true postoperative cylinder. For both preoperative and postoperative variables, the true cylinder and the measured cylinder radiate from the origin and are connected by the error. In Figures 1 to 3, the measurement errors are similar in magnitude to the cylinder being corrected. This is frequently the case in eyes with small amounts of cylinder. For small cylinder amounts, the intended correction will be similar in magnitude to the measurement error. For large cylinder amounts, the intended correction will be substantially larger than the measurement error. Figure 4 illustrates the vectors when a higher amount of cylinder is treated. The measured preoperative cylinder
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Figure 4. Vectors used in the calculation of SIA and TIA at a higher level of astigmatism (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
and hence the TIA are nearly 2.5 times larger than in Figures 1 to 3 but with the same orientation. The magnitudes and orientations of the errors are also the same as in Figures 1 to 3. As a result, the measured postoperative cylinder is the same but the SIA is closer in magnitude to the TIA, resulting in a lower correction index, 1.2 in this example. It should be noted that in an individual correction, the SIA might be smaller in magnitude than the TIA, leading to a correction index of less than 1. Figure 5 shows 1 such example. The direction of both the preoperative and postoperative error vectors (ie, the orientation of the cylinder axis) relative to the preoperative cylinder ultimately determines whether the SIA is greater than or less than the TIA. In Figure 5, the error in the preoperative cylinder markedly reduces the SIA. In contrast, the error in the postoperative cylinder increases the SIA slightly. This results in a correction index of 0.7 in this example. Vector analysis is extended in the supplementary material (Supplement A, available at http:// jcrsjournal.org) to show why the average correction index is necessarily larger than 1 for a perfect treatment and substantially larger than 1 when small cylinder corrections are performed.
Numerical Modeling of Correction Index The impact of the variability of cylinder measurement on the mean correction index can be numerically modeled for a given cylinder correction. For this, a Monte Carlo simulation was performed in which 10 000 correction index values were calculated for various values of TIA. To perform this simulation, the results of Bullimore et al.10 were used to develop a realistic, reliable value to characterize the variability of cylinder measurement. They assessed the repeatability of subjective refraction in 86 visually normal subjects. A mixed model of variance, accounting for the correlation between pairs of eyes, enabled data from both eyes of each subject to be included in the analyses. Using this model, the 95% limits of agreement were G0.38 D for J0
1645
Figure 5. Illustration of how direction of the preoperative and postoperative error vectors can lead to a correction index of less than 1 (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
and G0.31 D for J45. These values were generated for paired comparisons; the variability of a single measurement should be smaller by a factor of O2 (ie, the variance is halved). This adjustment produces values of 0.27 D for J0 and 0.22 D for J45. For simplicity, the simulation described below used 0.25 D for both values. (A simulation using values of 0.27 D for J0 and 0.22 D for J45 produced similar outcomes.) In the Monte Carlo simulation, J0 and J45 were independently and equally distributed between G0.25 D. For each simulation, the preoperative cylinder was added to the randomly generated values of J0 and J45 to give the true preoperative cylinder. Although the random values could have been added to the true cylinder to obtain the measured cylinder, the goal was to present the results in terms of the measured cylinder. Next, the TIA (equal to the measured preoperative cylinder) was added to the true preoperative cylinder to give the true postoperative cylinder. Then, 2 new randomly generated values of J0 and J45 between G0.25 D were added to the true postoperative cylinder to obtain the measured postoperative cylinder. The SIA was calculated as the vector difference between the measured preoperative cylinder and the measured postoperative cylinder, and the correction index was calculated as the ratio of the absolute value of SIA and the absolute value of TIA (equation 1). This sequence was repeated 10 000 times for each specified value of the preoperative cylinder power (ie, the TIA magnitude). For each TIA value, the mean and standard deviation (SD) were calculated for the 10 000 simulated correction index values.
RESULTS Figure 6 shows the results of the Monte Carlo simulation, plotting the mean correction index as a function of the magnitude of the cylinder correction. The error bars represent G 1 SD. For TIAs greater than 1.00 D,
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Figure 6. The mean G 1 SD correction index of 10 000 simulations at specified values of TIA (TIA Z target induced astigmatism).
the mean correction index approaches 1, as would be expected based on the mathematical and vector results presented in Figures 1 to 4 and equation 10. This is because the measurement variability is much smaller than the cylinder correction. The mean correction index is 1.08 at 1.00 D and 1.01 at 2.50 D. At low levels of cylinder correction, the mean correction index increases sharply because the measurement variability becomes similar in magnitude to the cylinder correction. The SDs are large for all values of TIA, particularly for low astigmatic corrections. Although the mean correction index is more than 1 for low astigmatic corrections, a number of individual subjects have correction indices less than 1. Only when the TIA is 0.25 D is the mean correction index more than 1 SD from 1. These values include variability in preoperative and postoperative cylinder measurement and assume a perfect refractive treatment. They do not account for other sources of variability, including intraoperative variables such as fluctuations in fluence, fixation stability, or post-treatment healing and remodeling. Therefore, correction indices observed in practice might be higher than those modeled here. DISCUSSION Comparison of Correction Indices from Published Studies with Simulation Results Figure 7 superimposes the median7 or mean14,16 measured correction indices from 3 published studies of astigmatic laser correction on the results from the
Figure 7. The mean13,15 or median7 correction indices from 3 astigmatic laser studies overlaid with Monte Carlo simulation results for the mean correction index with variance (TIA Z target induced astigmatism).
present study’s Monte Carlo simulation. The data observed in all 3 studies follow the same general trend as the data generated by the simulation. The increase in the mean measured correction index at low cylinder corrections is particularly obvious in Frings et al.8 wherein each datapoint represents over 100 treated eyes. The data from Labiris et al.16 are not connected because they represent 4 subsets: laser in situ keratomileusis (LASIK), photorefractive keratectomy, and high and low proportions of residual astigmatism. Figure 8 superimposes the mean measured correction indices from 5 unpublished studies of United States Food and Drug Administration (FDA)-approved lasersA on the results from our Monte Carlo simulation. Among the approximately 25 Premarket Approval Summaries that could be accessed in FDA’s database, only these had correction indices that provided for at least 2 ranges of cylinder of 1.00 D or less. Four of the 5 datasets follow the same general trend as the data generated by the simulation with mean correction indices close to 1 for TIA values above 1.00 D and greater than 1 for low cylinders, sometimes markedly so. Data for 1 approved laser showed correction indices of less than 1 for all cylinder values. Note that almost all of the published mean correction index values in Figures 7 and 8 fall within 1 SD of the Monte Carlo simulations. No trend of systematic overcorrection is therefore required to explain the high mean correction indices for low levels of cylinder correction, and the expected behavior of the mean
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Figure 8. The mean correction indices reported for 5 FDA-approved lasersA overlaid with Monte Carlo simulation results for the mean correction index with variance (EC-5000, Nidek Inc.; Ladarvision 4000, Wavelight Allegretto Wave, and Wavelight Allegretto Wave Eye-Q, Alcon Research, Ltd.; Mel 80, Carl Zeiss Meditec) (TIA Z target induced astigmatism).
correction index is sufficient to describe the true behavior of the mean correction index as found in these publications. Implications for Future Studies and Analyses The ANSI standard Laser Systems for Corneal Reshaping7 referenced the paper by Eydelman et al.,5 although the FDA guidance for investigational device exemptions for refractive surgery lasers does not.B The refractive community has adopted the use of vector analysis to interpret the outcomes of refractive procedures such as LASIK treatments and intraocular lenses (IOLs) and, in particular, to assess the outcomes of astigmatic treatments; however, because the original vector methods of Eydelman et al.5 and of Alpins6 did not consider the effects of refraction measurement error on the vector and scalar quantities associated with cylinder measurements, the vector analysis quantities are not always properly interpreted. For example, some published reports of astigmatism treatments concluded that this indicates systematic overcorrection of low levels of astigmatism. At times, overcorrection was asserted, even when all other measures of refractive outcomes were otherwise positive.8,9 In such reports, it is likely that positive safety and effectiveness outcomes indicated that the high mean correction index values observed at low levels of cylinder reflect the impact of measurement variability on correction index rather than the presence
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of any actual cylinder overcorrection. Kunert et al.14 suggested that interpretation of the mean correction index and related vector analysis quantities is in general problematic for small astigmatic corrections dnot because of any tendency to overcorrect (or undercorrect) but because of the inherent variability or scatter in the refraction measurements. As we have shown in a numerical model, a series of vector diagrams, and a mathematical analysis of the correction index, the intrinsic variability in the measurement of cylinder power axis dominates the mean correction index for low levels of astigmatism. We have shown that the interpretation of a mean correction index greater than 1 as being necessarily indicative of systematic cylinder overcorrection in the treatment of low cylinder is incorrect. For this reason, a mean correction index greater than 1 for small astigmatic treatments cannot by itself be interpreted to denote the presence of a systematic cylinder overcorrection because the mean correction index is expected to be greater than 1 for small amounts of cylinder. Given these considerations, correction indices should be used with caution for low levels of astigmatism. Alternative methods of characterizing astigmatic corrections should be considered, such as the proposed cylinder predictability plots of Kunert et al.14 Use of the actual vector quantities developed by Eydelman et al.5 or Alpins,6 such as the error vector or normalized error vector, is also advised. One might ask whether the trends observed for low levels of cylinder would occur at low levels of sphere. Assuming no systematic bias in the measurement of preoperative or postoperative sphere (eg, over-plussing), the distribution of errors should be centered on zero. Furthermore, spherical power is linear; in other words, the TIA and SIA will always be parallel vectors. These factors will drive a hypothetical correction index for spherical corrections toward 1, assuming that a laser ablates the prescribed profile and there are no biomechanical or remodeling considerations. How Much Astigmatism Should Be Treated? Our model and the data presented prompt the question, What is the minimum amount of cylinder corrections that should be attempted in a refractive procedure?17,18 With regard to other modes of refractive correction, astigmatic soft contact lenses are manufactured on a large scale but only at cylinder powers of 0.75 D and above. This circumstance might arise through management of product inventory as well as the marginal improvement in visual quality when correcting low levels of astigmatism.
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Atchison and Mathur19 placed crossed cylinder lenses of G0.12, G0.25, and G0.37 over the correcting lenses of the right eyes of 8 subjects at 15 to 180 degrees in 15-degree steps. For measures of both high- and low-contrast visual acuity, the 0.50 and 0.75 D cylinder lenses significantly reduced visual acuity, but the 0.25 D cylinder lenses did not. Furthermore, the reductions were dependent on the cylinder axis. Villegas et al.17 recently evaluated how small amounts of astigmatism affect visual acuity and found that in most subjects, astigmatism less than 0.50 D did not degrade visual acuity. They proposed that the visual benefit of a precise correction of astigmatism values smaller than 0.50 D would be limited. But what of the prescribing habits of practitioners and patients’ tolerance to small errors? O’Leary and Evans20 reported that their sample of surveyed optometrists (n Z 38) would prescribe 1.50 D of cylinder 70% of the time to adults in the absence of asthenopic symptoms and 0.75 D of cylinder 60% of the time for symptomatic patients. Note that this was for patients with “no significant spherical refractive error.” We hypothesize that clinicians would rarely convert a refraction containing a small cylindrical component to a purely spherical spectacle prescription. In a simple but elegant study, Miller et al.21 quantified the effects of small refractive errors on the perceived quality of vision. Patients wore 8 different pairs of spectacles with small errors deliberately included, each for a 2-day trial wear period. For C0.50 D cylinder at an axis of 180, 70% of wearers regarded the prescription as unacceptable and nearly 100% thought the same when the axis was 90 or 45. Note that no adjustment was made to the spherical equivalent. Based on these considerations, there appears to be consensus that attempting to correct 0.50 D of cylinder is worthwhile, but attempting to correct 0.25 D is not. Of course, one could argue that there is no potential harm in treating 0.25 D. Although the motivation for this paper was laser refractive correction and the data presented are predominantly from studies of the excimer laser, the concepts can be extended to toric IOLs. The use of a correction index or correction ratio is rare in published reports of toric IOLs,22,23 and other factors must be considered, including surgically induced corneal astigmatism. Traditional teaching might have warned against flipping the axis of astigmatism, but a recent study suggested that this is well tolerated by patients.22 In summary, the correction index (correction ratio) is a useful vector-based metric for the evaluation of refractive procedures. Most published studies and 5 unpublished studies of FDA-approved lasers reported mean correction indices greater than 1 for small astigmatic corrections, which has been
misinterpreted as systematic overcorrection. We have shown that these higher indices should be expected because of the variability in preoperative and postoperative measurement of astigmatism and do not represent a systematic overcorrection. Therefore, we urge caution when judging the clinical significance of correction index analyses applied to treatments of low amounts of cylindrical refractive error. The decision to surgically correct small amounts of astigmatism should consider the visual demands of patients, the knowledge of the effects of small cylindrical errors on vision, and our ability to precisely measure astigmatism preoperatively and postoperatively. WHAT WAS KNOWN The correction index (correction ratio) is a useful vectorbased metric for the evaluation of refractive procedures and should be close to 1. Published and unpublished studies of FDA-approved lasers report mean correction indices much greater than 1 for low astigmatic corrections. WHAT THIS PAPER ADDS Mathematical modeling of the correction index, accounting for variability in astigmatism measurement, predicted sharp increases when the TIA is less than 1.00 D. Data from published and unpublished studies showed the predicted increase in the correction index for cylinder correction. Correction indices greater than 1 should be anticipated for lower astigmatic treatments and do not necessarily represent systematic overcorrection.
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analyses of correction of astigmatism by laser systems that reshape the cornea. J Refract Surg 2006; 22:81–95 Alpins N. Astigmatism analysis by the Alpins method. J Cataract Refract Surg 2001; 27:31–49 American National Standards Institute, Inc. American National Standard for Ophthalmics – Laser Systems for Corneal Reshaping. New York, NY, ANSI Z80.11, 2012 Frings A, Katz T, Richard G, Druchkiv V, Linke SJ. Efficacy and predictability of laser in situ keratomileusis for low astigmatism of 0.75 diopter or less. J Cataract Refract Surg 2013; 39:366–377 Katz T, Frings A, Linke SJ, Richard G, Druchkiv V, Steinberg J. Laser in situ keratomileusis for astigmatism %0.75 diopter combined with low myopia: a retrospective data analysis. BMC Ophthalmol 2014; 14:1. Available at: http://www.ncbi.nlm.nih.gov/ pmc/articles/PMC3890523/pdf/1471-2415-14-1.pdf. Accessed May 19, 2015 Bullimore MA, Fusaro RE, Adams CW. The repeatability of automated and clinician refraction. Optom Vis Sci 1998; 75:617–622. Available at: http://journals.lww.com/optvissci/Abstract/1998/ 08000/The_Repeatability_of_Automated_and_Clinician.28.aspx. Accessed May 19, 2015 MacKenzie GE. Reproducibility of sphero-cylindrical prescriptions. Ophthalmic Physiol Opt 2008; 28:143–150 Shah R, Edgar DF, Rabbetts R, Harle DE, Evans BJW. Standardized patient methodology to assess refractive error reproducibility. Optom Vis Sci 2009; 86:517–528. Available at: http://journals.lww. com/optvissci/Fulltext/2009/05000/Standardized_Patient_Method ology_to_Assess.14.aspx. Accessed May 19, 2015 Sheedy J, Schanz P, Bullimore M. Evaluation of an automated subjective refractor. Optom Vis Sci 2004; 81:334–340 Kunert KS, Russmann C, Blum M, Sluyterman VLG. Vector analysis of myopic astigmatism corrected by femtosecond refractive lenticule extraction. J Cataract Refract Surg 2013; 39:759–769 Reinstein DZ, Couch DG, Archer TJ. LASIK for hyperopic astigmatism and presbyopia using micro-monovision with the Carl Zeiss Meditec MEL80 platform. J Refract Surg 2009; 25:37–58 Labiris G, Gatzioufas Z, Giarmoukakis A, Sideroudi H, Kozobolis V. Evaluation of the efficacy of the Allegretto Wave and the Wavefront-optimized ablation profile in non-anterior astigmatisms. Acta Ophthalmol 2012; 90:e442–e446. Available at: http://onlinelibrary.wiley.com/doi/10.1111/j.1755-3768.2012. 02463.x/pdf. Accessed May 19, 2015 n E, Artal P. Minimum amount of astigmatism Villegas EA, Alco that should be corrected. J Cataract Refract Surg 2014; 40:13–19
18. Wolffsohn JS, Bhogal G, Shah S. Astigmatism and vision: should all astigmatism always be corrected? [editorial] Br J Ophthalmol 2014; 98:2–3 19. Atchison DA, Mathur A. Visual acuity with astigmatic blur. Optom Vis Sci 2011; 88:E798–E805. Available at: http://jour nals.lww.com/optvissci/Fulltext/2011/07000/Visual_Acuity_ with_Astigmatic_Blur.6.aspx. Accessed May 19, 2015 20. O’Leary CI, Evans BJW. Criteria for prescribing optometric interventions: literature review and practitioner survey. Ophthalmic Physiol Opt 2003; 23:429–439 21. Miller AD, Kris MJ, Griffiths AC. Effect of small focal errors on vision. Optom Vis Sci 1997; 74:521–526. Available at: http:// pdfs.journals.lww.com/optvissci/1997/07000/Effect_of_Small_ Focal_Errors_on_Vision.20.pdf. Accessed May 19, 2015 €tz WW. Results of higher power toric 22. Hoffmann PC, Auel S, Hu intraocular lens implantation. J Cataract Refract Surg 2011; 37:1411–1418 23. Visser N, Berendschot TTJM, Bauer NJC, Nuijts RMMA. Vector analysis of corneal and refractive astigmatism changes following toric pseudophakic and toric phakic IOL implantation. Invest Ophthalmol Vis Sci 2012; 53:1865–1873. Available at: http:// iovs.arvojournals.org/article.aspx?articleidZ2188106. Accessed May 19, 2015
OTHER CITED MATERIAL A. U.S. Food and Drug Administration. Premarket approval (PMA). Available at: http://www.accessdata.fda.gov/scripts/cdrh/cfdocs/ cfPMA/pma.cfm. Accessed May 19, 2015 B. U.S. Food and Drug Administration. Checklist of Information Usually Submitted in an Investigational Device Exemptions (IDE) Application for Refractive Surgery Lasers, 1996. Available at: http://www.fda.gov/downloads/MedicalDevices/DeviceRegu lationandGuidance/GuidanceDocuments/ucm080250.pdf. Accessed May 19, 2015
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First author: Mark A. Bullimore, MCOptom, PhD College of Optometry, University of Houston, Houston, Texas, USA
Correction of low levels of astigmatism Mark A. Bullimore, MCOptom, PhD, Greg Spooner, PhD, Georg Sluyterman, PhD, Jon G. Dishler, MD
PURPOSE: To reevaluate the analysis of the correction of astigmatism. SETTING: Academia, industry, and private practice. DESIGN: Evaluation of diagnostic test or technology. METHODS: Astigmatic refractive surgery outcomes are based on vector methods, including the correction index (also known as the correction ratio), which is the ratio of the surgically induced astigmatism to the target induced astigmatism (TIA). Mean correction indices substantially greater than 1 have been reported for astigmatic corrections less than 1.00 diopter (D) and as representing systematic overcorrection. We hypothesize that this reflects a limitation of the correction index rather than systematic flaws in treatments. The theoretical mathematic behavior of the correction index was analyzed, accounting for variability in astigmatism measurement. Then, the impact of cylinder measurement variability on the mean correction index was modeled. A Monte Carlo simulation was performed and calculated 10 000 values of correction index for various values of TIA. Finally, correction indices from published and unpublished studies of refractive lasers were compared with the simulations. RESULTS: The mean correction index is always greater than 1 for the case of a perfect refractive correction; however, for astigmatic corrections less than 1.00 D, the mean correction index increases sharply because the measurement variability is similar in magnitude to TIA. Almost all previous studies show the predicted increase in the correction index for low astigmatic corrections. CONCLUSION: The correction index is a useful vector-based metric for the evaluation of refractive procedures, but mean values greater than 1 should be anticipated for lower astigmatic treatments and do not necessarily represent systematic overcorrection. Financial Disclosure: Dr. Bullimore is a consultant to Alcon Surgical, Inc., Carl Zeiss Meditec AG, Digital Vision Systems, Essilor, Innovega, Inc., and Paragon Vision Sciences, Inc. Dr. Spooner is a consultant to Alcon Surgical, Inc., Carl Zeiss Meditec AG, Digital Vision Systems, Thru-Focus Optics LLC, and i2eyediagnostics, Ltd. Dr. Dishler is a consultant to Carl Zeiss Meditec AG and Revision Optics, Inc. Dr. Sluyterman is an employee of Carl Zeiss Meditec AG. J Cataract Refract Surg 2015; 41:1641–1649 Q 2015 ASCRS and ESCRS Supplemental material available at www.jcrsjournal.org.
The presentation of outcomes in refractive surgery has become standardized in the past decade. Leading journals, for example, request that authors use standardized graphs for the reporting of results.1,2 Interpretation of the surgical correction of astigmatism requires vector methods because astigmatism has magnitude and orientation, and some authors have developed numerical and statistical approaches for the management of astigmatic data.3,4 A principal purpose of vector methods in astigmatism treatments is to determine the degree of improvement, generally Q 2015 ASCRS and ESCRS Published by Elsevier Inc.
expressed as a difference between vector quantities characterizing the measured astigmatism.5,6 In 2001, Alpins6 proposed a series of metrics to be used in the evaluation of astigmatic refractive surgery. One such metric was the correction index, defined as the ratio of the surgically induced astigmatism (SIA) vector to the target induced astigmatism (TIA) vector. Subsequently, the American National Standards Institute (ANSI) Z80.117 Working Group on Laser Systems for Corneal Reshaping formed an Astigmatism Project Group made up of experts from academia, http://dx.doi.org/10.1016/j.jcrs.2014.12.060 0886-3350
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government, and industry. The group’s goal was to “identify a common, minimum set of analyses and a presentation format for adequate evaluation of safety and effectiveness of new astigmatism-correcting devices.”5 One of the analysis variables identified by the group was the correction ratio, which is the equivalent of Alpins’ correction index.6 Acknowledging the precedent of Alpins’ work, we use the term correction index throughout this paper. As stated previously and as represented in the ANSI Standard Z80.11-2012,7 the correction index is defined as the ratio of the magnitudes of the SIA vector and the TIA vector. The mean correction index calculated for a cohort of treated eyes could be used to identify systemic overcorrection or undercorrection. For higher astigmatic corrections, mean values of the correction index close to 1 are common, but 2 recent papers reported a correction index substantially greater than 1 for lower astigmatic corrections.8,9 These findings appear to suggest that a refractive laser functions less accurately for small cylindrical corrections than for larger corrections. We hypothesize that it might merely reflect a problem with the interpretation of the metric rather than a flaw with the laser treatment being evaluated. In any thorough analysis, the random errors associated with the measurement must be considered, and these vector quantities are no exception. Measurement error, resulting from the practical limitations to the precision of any physical measurement process, is distinct from systematic or other errors induced by the treatment. When measurement error is included in the analysis, the mean correction index is greater than 1 whenever the size of the TIA is comparable to the inherent measurement error in the subjective refraction measurement.10–13 This behavior will hold even in the case of a perfect refractive treatment. The mean correction index must be interpreted carefully for small cylinder treatments. Findings of mean correction index values greater than 1 for small-cylinder treatments do not necessarily indicate the presence of systemic overcorrection. Submitted: May 9, 2014. Final revision submitted: December 8, 2014. Accepted: December 10, 2014. From the College of Optometry (Bullimore), University of Houston, Houston, Texas, Gain Consulting Services (Spooner), San Francisco, California, and the Dishler Laser Institute (Dishler), Greenwood Village, Colorado, USA; Carl Zeiss Meditec AG (Sluyterman), Jena, Germany. Corresponding author: Mark A. Bullimore, MCOptom, PhD, 356 Ridgeview Lane, Boulder, Colorado 80302, USA. E-mail: [email protected].
MATERIALS AND METHODS The quantity correction index as defined by Alpins6 is a measure of astigmatic refractive surgical correction.5 The correction index is the ratio of the absolute value (or magnitude) of the SIA vector (also called the surgically induced refractive correction vector) to the absolute value of the TIA vector (also called the intended refractive correction vector). CI Z
jSIAj jTIAj
(1)
where vector quantities are represented in bold text. The TIA is the target induced astigmatism vectordthe correction values as entered into the laser device user interface, typically as cylinder (diopters [D]) and axis (degrees). The TIA is usually chosen to be equal to the preoperative measured cylinder minus the target cylinder. To eliminate astigmatism, the TIA must be equal in magnitude to the preoperative cylinder. The SIA is defined as the vector difference between the measured preoperative cylinder and the measured postoperative cylinder. SIA Z Cpre Cpost
(2)
Eydelman et al.5 noted that a particular correction index value for a single eye might indicate an undercorrection, an overcorrection, or a good correction. In a population of treated eyes, the average correction index value might indicate the achieved correction relative to the intended correction; however, the correction index must be carefully interpreted, especially for small astigmatic corrections.8,14,15 The reason for such caution is that the relationship between the intended correction (TIA) and the achieved surgical correction (SIA) is vectorial. The scalar (or purely numerical) definition of the correction index can mask the vector relationship between the TIA and the SIA for small astigmatic corrections. Some random measurement error is always present in the preoperative and postoperative refractions. Measurement error, resulting from the practical limitations to the precision of any physical measurement process, is distinct from systematic or other errors induced by the treatment. This measurement error materially affects the SIA vector for small astigmatic corrections. In the methods of Eydelman et al.5 and Alpins,6 the effects of refraction measurement error are not considered. For small cylinder corrections that are close to the magnitude of the measurement variability, the mean correction index also generally is greater and often is considerably greater than 1.8,14,15 The refraction measurement variability for astigmatism is G0.25 D in orthogonal (90 and 180) and oblique (45 and 135) meridians.10 As a consequence, for TIAs less than approximately 1.0 D, the resulting correction index is substantially greater than 1, assuming there is no systemic undercorrection or overcorrection by the device. That is, in a perfect treatment for the correction of astigmatism, the proper use of the correction index is restricted to astigmatic corrections that are appreciably larger than the refraction measurement accuracy. In practice, the correction index is an unreliable indicator of the effectiveness of cylinder treatments smaller than 1.0 D in magnitude. Below is a mathematical formulation of the mean correction index that properly includes the presence of random measurement errors associated with reported clinical refraction accuracy and repeatability. After that, to visualize the effects of refraction measurement error on the correction
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index, the vector figures of Eydelman et al.5 are recast to include measurement vectors. A numerical simulation of the mean correction index for typical refraction measurement error values follows. Finally, as context for numerical simulation, a range of mean correction index values observed in studies of correction of astigmatism with excimer lasers is provided.
Mathematical Behavior of the Mean Correction Index with Refraction Measurement Error The SIA is most properly represented as SIA Z SIA0 þ DC
(3)
where DC represents the randomly varying measurement error vector from the “true” or error-free value SIA0. In the simplest case, DC can be considered to vary randomly in magnitude and direction, within some limits. The statistical average of DC in this simple model of the error behavior therefore is zero. This SIA that includes random error in the definition of the correction index can be used to find an expression for the mean correction index (represented as !Correction IndexO in equations) for a fixed value of the TIA. hCorrection Indexi Z
hjSIAji jTIAj
(4)
It is convenient to represent the SIA in Cartesian coordinates denoted with subscripts x and y (shown later as J0 and J45). The mean correction index can then be represented as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ðSIA0x þDCx Þ2 þ SIA0y þDCy hCorrection IndexiZ jTIAj (5)
hCorrection Indexi Z
above averages the final term on the right side of the equation to zero because the quantities DCx and DCy can carry a positive or negative sign, and a random variation of linear products of these symmetrically varying quantities produces an average of zero. Quantities DC2x and DC2y are always positive, and the quantity SIA0 has a positive magnitude. Therefore, both the first and second terms in equation 7 do not average to zero. + * DC2x þ DC2y SIA0 (9) þ hCorrection Indexiz jTIAj jTIAj 2SIA0 It is clear by inspection that all quantities in this simple analytic expression are positive. Further, because the expression was developed for a perfect treatment, the magnitude of the TIA equals the mean of the SIA magnitude. The term on the left of equation 9 is therefore equal to 1. Because the term on the right is the average of a sum of (positive) squared numbers divided by a positive magnitude, the average is a positive number: * + DC2x þ DC2y hCorrection Indexiz1 þ (10) jTIAj 2SIA0 The mean correction index is therefore greater than 1 for the special case of a perfect SIA correction (perfect being defined as a SIA correction that is exactly equal in magnitude to the target astigmatism magnitude, with no deviceintroduced error). In a perfect treatment, the mean correction index should always be greater than 1 unless the physical measurements could somehow be performed with zero measurement error. Further, when the measurement error DC is comparable in size to the SIA, the mean correction index will be significantly greater than 1. It is important to note that if systematic error were present due to a non-perfect treatment, then the average correction index could be different from 1 due to the combination of
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE 1 D SIA20x þ DC2x þ2SIA0x DCx þSIA20y þ DC2y þ2SIA0y DCy jTIAj
Computing the statistical average of this overall quantity requires specific knowledge of the individual quantities; however, for DC components that are smaller than the SIA components (ie, relatively small measurement error), the square root can be approximated by a binomial expansion keeping only the first-order terms to give an approximate or limiting value of the mean correction index. * 1 DC2x þ DC2y 2 1 hCorrection Indexiz SIA0 þ SIA0 jTIAj (7) + SIA0x DCx þSIA0y DCy þ SIA0 where the term SIA0 is defined as the quantity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SIA0 Z SIA20x þSIA20y (8) Assuming that the measurement errors are normally distributed, performing a statistical average of equation 7
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(6)
the measurement error and the systematic error from the treatment. Therefore, a non-unity value for the average correction index does not necessarily indicate an overcorrection or undercorrection for small astigmatic corrections. Only when considering TIA values that are much larger than DC (ie, cylinder values that are larger than the measurement error) can the correction index be considered an indicator of a real overcorrection or undercorrection.
Vector Nature of the Correction Index A major factor contributing to the higher correction indices reported in excimer laser clinical trials is the challenge of measuring low amounts of cylinder reliably.10 Obtaining reliable measurements might be even more difficult when a low level of cylinder is in the presence of a high levels of myopia, reduced visual acuity, or both. We have demonstrated analytically that the mean correction index will be greater than 1 for any real measurements of refraction. The impact of the variability in cylinder
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Figure 1. Sample vector plot illustrating the relationship among measured preoperative cylinder, measured postoperative cylinder, and SIA (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism).
measurement on the correction index can also be demonstrated graphically, based on the definitions found in Eydelman et al.5 and Alpins.6 Figure 1 shows the relationship among the measured preoperative cylinder (Cpre), the measured postoperative cylinder (Cpost), and the SIA. As in the previous section, the SIA is defined as the vector difference between the preoperative cylinder and the postoperative cylinder (equation 2). Refractive cylinder amounts at oblique orientations may be combined as vectors but must be plotted on a
Figure 3. True preoperative cylinder and true postoperative cylinder (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
Figure 2. Vectors used in the calculation of SIA and TIA (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
2-dimensional double-angle plot.3 The angles must be doubled because whereas vectors of equal magnitude 180 degrees apart cancel each other out, equal cylinders 90 degrees apart cancel each other out. The axes of the plot correspond to the orientation of the commonly used variables, J0 and J45.4 In Figure 1, the measured cylinder preoperatively might be 2.50 108, and the measured cylinder postoperatively might be 1.13 40. Therefore, the SIA in this example would equal 3.41 115. As in any real measurement, the preoperative cylinder and the postoperative cylinder can vary. Figure 2 shows how the measured postoperative cylinder is calculated with a series of vectors. The dotted gray lines for each measurement error represent the horizontal and vertical components, corresponding to J0 and J45, respectively. First, there is error associated with measurement of the preoperative cylinder. Second, the excimer laser alters the cylinder by the TIA. Note that the magnitude and orientation of the TIA are equal to the measured preoperative cylinder, thus assuming a perfect treatment. Finally, there is error associated with measurement of the postoperative cylinder. The correction index is defined as the ratio of the absolute of SIA to the absolute TIA (equation 1). In the example shown in Figure 2, the correction index is 1.6. Figure 3 shows 2 additional variables: the true preoperative cylinder and the true postoperative cylinder. For both preoperative and postoperative variables, the true cylinder and the measured cylinder radiate from the origin and are connected by the error. In Figures 1 to 3, the measurement errors are similar in magnitude to the cylinder being corrected. This is frequently the case in eyes with small amounts of cylinder. For small cylinder amounts, the intended correction will be similar in magnitude to the measurement error. For large cylinder amounts, the intended correction will be substantially larger than the measurement error. Figure 4 illustrates the vectors when a higher amount of cylinder is treated. The measured preoperative cylinder
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Figure 4. Vectors used in the calculation of SIA and TIA at a higher level of astigmatism (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
and hence the TIA are nearly 2.5 times larger than in Figures 1 to 3 but with the same orientation. The magnitudes and orientations of the errors are also the same as in Figures 1 to 3. As a result, the measured postoperative cylinder is the same but the SIA is closer in magnitude to the TIA, resulting in a lower correction index, 1.2 in this example. It should be noted that in an individual correction, the SIA might be smaller in magnitude than the TIA, leading to a correction index of less than 1. Figure 5 shows 1 such example. The direction of both the preoperative and postoperative error vectors (ie, the orientation of the cylinder axis) relative to the preoperative cylinder ultimately determines whether the SIA is greater than or less than the TIA. In Figure 5, the error in the preoperative cylinder markedly reduces the SIA. In contrast, the error in the postoperative cylinder increases the SIA slightly. This results in a correction index of 0.7 in this example. Vector analysis is extended in the supplementary material (Supplement A, available at http:// jcrsjournal.org) to show why the average correction index is necessarily larger than 1 for a perfect treatment and substantially larger than 1 when small cylinder corrections are performed.
Numerical Modeling of Correction Index The impact of the variability of cylinder measurement on the mean correction index can be numerically modeled for a given cylinder correction. For this, a Monte Carlo simulation was performed in which 10 000 correction index values were calculated for various values of TIA. To perform this simulation, the results of Bullimore et al.10 were used to develop a realistic, reliable value to characterize the variability of cylinder measurement. They assessed the repeatability of subjective refraction in 86 visually normal subjects. A mixed model of variance, accounting for the correlation between pairs of eyes, enabled data from both eyes of each subject to be included in the analyses. Using this model, the 95% limits of agreement were G0.38 D for J0
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Figure 5. Illustration of how direction of the preoperative and postoperative error vectors can lead to a correction index of less than 1 (Rx Z cylindrical prescription; SIA Z surgically induced astigmatism; TIA Z target induced astigmatism).
and G0.31 D for J45. These values were generated for paired comparisons; the variability of a single measurement should be smaller by a factor of O2 (ie, the variance is halved). This adjustment produces values of 0.27 D for J0 and 0.22 D for J45. For simplicity, the simulation described below used 0.25 D for both values. (A simulation using values of 0.27 D for J0 and 0.22 D for J45 produced similar outcomes.) In the Monte Carlo simulation, J0 and J45 were independently and equally distributed between G0.25 D. For each simulation, the preoperative cylinder was added to the randomly generated values of J0 and J45 to give the true preoperative cylinder. Although the random values could have been added to the true cylinder to obtain the measured cylinder, the goal was to present the results in terms of the measured cylinder. Next, the TIA (equal to the measured preoperative cylinder) was added to the true preoperative cylinder to give the true postoperative cylinder. Then, 2 new randomly generated values of J0 and J45 between G0.25 D were added to the true postoperative cylinder to obtain the measured postoperative cylinder. The SIA was calculated as the vector difference between the measured preoperative cylinder and the measured postoperative cylinder, and the correction index was calculated as the ratio of the absolute value of SIA and the absolute value of TIA (equation 1). This sequence was repeated 10 000 times for each specified value of the preoperative cylinder power (ie, the TIA magnitude). For each TIA value, the mean and standard deviation (SD) were calculated for the 10 000 simulated correction index values.
RESULTS Figure 6 shows the results of the Monte Carlo simulation, plotting the mean correction index as a function of the magnitude of the cylinder correction. The error bars represent G 1 SD. For TIAs greater than 1.00 D,
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Figure 6. The mean G 1 SD correction index of 10 000 simulations at specified values of TIA (TIA Z target induced astigmatism).
the mean correction index approaches 1, as would be expected based on the mathematical and vector results presented in Figures 1 to 4 and equation 10. This is because the measurement variability is much smaller than the cylinder correction. The mean correction index is 1.08 at 1.00 D and 1.01 at 2.50 D. At low levels of cylinder correction, the mean correction index increases sharply because the measurement variability becomes similar in magnitude to the cylinder correction. The SDs are large for all values of TIA, particularly for low astigmatic corrections. Although the mean correction index is more than 1 for low astigmatic corrections, a number of individual subjects have correction indices less than 1. Only when the TIA is 0.25 D is the mean correction index more than 1 SD from 1. These values include variability in preoperative and postoperative cylinder measurement and assume a perfect refractive treatment. They do not account for other sources of variability, including intraoperative variables such as fluctuations in fluence, fixation stability, or post-treatment healing and remodeling. Therefore, correction indices observed in practice might be higher than those modeled here. DISCUSSION Comparison of Correction Indices from Published Studies with Simulation Results Figure 7 superimposes the median7 or mean14,16 measured correction indices from 3 published studies of astigmatic laser correction on the results from the
Figure 7. The mean13,15 or median7 correction indices from 3 astigmatic laser studies overlaid with Monte Carlo simulation results for the mean correction index with variance (TIA Z target induced astigmatism).
present study’s Monte Carlo simulation. The data observed in all 3 studies follow the same general trend as the data generated by the simulation. The increase in the mean measured correction index at low cylinder corrections is particularly obvious in Frings et al.8 wherein each datapoint represents over 100 treated eyes. The data from Labiris et al.16 are not connected because they represent 4 subsets: laser in situ keratomileusis (LASIK), photorefractive keratectomy, and high and low proportions of residual astigmatism. Figure 8 superimposes the mean measured correction indices from 5 unpublished studies of United States Food and Drug Administration (FDA)-approved lasersA on the results from our Monte Carlo simulation. Among the approximately 25 Premarket Approval Summaries that could be accessed in FDA’s database, only these had correction indices that provided for at least 2 ranges of cylinder of 1.00 D or less. Four of the 5 datasets follow the same general trend as the data generated by the simulation with mean correction indices close to 1 for TIA values above 1.00 D and greater than 1 for low cylinders, sometimes markedly so. Data for 1 approved laser showed correction indices of less than 1 for all cylinder values. Note that almost all of the published mean correction index values in Figures 7 and 8 fall within 1 SD of the Monte Carlo simulations. No trend of systematic overcorrection is therefore required to explain the high mean correction indices for low levels of cylinder correction, and the expected behavior of the mean
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Figure 8. The mean correction indices reported for 5 FDA-approved lasersA overlaid with Monte Carlo simulation results for the mean correction index with variance (EC-5000, Nidek Inc.; Ladarvision 4000, Wavelight Allegretto Wave, and Wavelight Allegretto Wave Eye-Q, Alcon Research, Ltd.; Mel 80, Carl Zeiss Meditec) (TIA Z target induced astigmatism).
correction index is sufficient to describe the true behavior of the mean correction index as found in these publications. Implications for Future Studies and Analyses The ANSI standard Laser Systems for Corneal Reshaping7 referenced the paper by Eydelman et al.,5 although the FDA guidance for investigational device exemptions for refractive surgery lasers does not.B The refractive community has adopted the use of vector analysis to interpret the outcomes of refractive procedures such as LASIK treatments and intraocular lenses (IOLs) and, in particular, to assess the outcomes of astigmatic treatments; however, because the original vector methods of Eydelman et al.5 and of Alpins6 did not consider the effects of refraction measurement error on the vector and scalar quantities associated with cylinder measurements, the vector analysis quantities are not always properly interpreted. For example, some published reports of astigmatism treatments concluded that this indicates systematic overcorrection of low levels of astigmatism. At times, overcorrection was asserted, even when all other measures of refractive outcomes were otherwise positive.8,9 In such reports, it is likely that positive safety and effectiveness outcomes indicated that the high mean correction index values observed at low levels of cylinder reflect the impact of measurement variability on correction index rather than the presence
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of any actual cylinder overcorrection. Kunert et al.14 suggested that interpretation of the mean correction index and related vector analysis quantities is in general problematic for small astigmatic corrections dnot because of any tendency to overcorrect (or undercorrect) but because of the inherent variability or scatter in the refraction measurements. As we have shown in a numerical model, a series of vector diagrams, and a mathematical analysis of the correction index, the intrinsic variability in the measurement of cylinder power axis dominates the mean correction index for low levels of astigmatism. We have shown that the interpretation of a mean correction index greater than 1 as being necessarily indicative of systematic cylinder overcorrection in the treatment of low cylinder is incorrect. For this reason, a mean correction index greater than 1 for small astigmatic treatments cannot by itself be interpreted to denote the presence of a systematic cylinder overcorrection because the mean correction index is expected to be greater than 1 for small amounts of cylinder. Given these considerations, correction indices should be used with caution for low levels of astigmatism. Alternative methods of characterizing astigmatic corrections should be considered, such as the proposed cylinder predictability plots of Kunert et al.14 Use of the actual vector quantities developed by Eydelman et al.5 or Alpins,6 such as the error vector or normalized error vector, is also advised. One might ask whether the trends observed for low levels of cylinder would occur at low levels of sphere. Assuming no systematic bias in the measurement of preoperative or postoperative sphere (eg, over-plussing), the distribution of errors should be centered on zero. Furthermore, spherical power is linear; in other words, the TIA and SIA will always be parallel vectors. These factors will drive a hypothetical correction index for spherical corrections toward 1, assuming that a laser ablates the prescribed profile and there are no biomechanical or remodeling considerations. How Much Astigmatism Should Be Treated? Our model and the data presented prompt the question, What is the minimum amount of cylinder corrections that should be attempted in a refractive procedure?17,18 With regard to other modes of refractive correction, astigmatic soft contact lenses are manufactured on a large scale but only at cylinder powers of 0.75 D and above. This circumstance might arise through management of product inventory as well as the marginal improvement in visual quality when correcting low levels of astigmatism.
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Atchison and Mathur19 placed crossed cylinder lenses of G0.12, G0.25, and G0.37 over the correcting lenses of the right eyes of 8 subjects at 15 to 180 degrees in 15-degree steps. For measures of both high- and low-contrast visual acuity, the 0.50 and 0.75 D cylinder lenses significantly reduced visual acuity, but the 0.25 D cylinder lenses did not. Furthermore, the reductions were dependent on the cylinder axis. Villegas et al.17 recently evaluated how small amounts of astigmatism affect visual acuity and found that in most subjects, astigmatism less than 0.50 D did not degrade visual acuity. They proposed that the visual benefit of a precise correction of astigmatism values smaller than 0.50 D would be limited. But what of the prescribing habits of practitioners and patients’ tolerance to small errors? O’Leary and Evans20 reported that their sample of surveyed optometrists (n Z 38) would prescribe 1.50 D of cylinder 70% of the time to adults in the absence of asthenopic symptoms and 0.75 D of cylinder 60% of the time for symptomatic patients. Note that this was for patients with “no significant spherical refractive error.” We hypothesize that clinicians would rarely convert a refraction containing a small cylindrical component to a purely spherical spectacle prescription. In a simple but elegant study, Miller et al.21 quantified the effects of small refractive errors on the perceived quality of vision. Patients wore 8 different pairs of spectacles with small errors deliberately included, each for a 2-day trial wear period. For C0.50 D cylinder at an axis of 180, 70% of wearers regarded the prescription as unacceptable and nearly 100% thought the same when the axis was 90 or 45. Note that no adjustment was made to the spherical equivalent. Based on these considerations, there appears to be consensus that attempting to correct 0.50 D of cylinder is worthwhile, but attempting to correct 0.25 D is not. Of course, one could argue that there is no potential harm in treating 0.25 D. Although the motivation for this paper was laser refractive correction and the data presented are predominantly from studies of the excimer laser, the concepts can be extended to toric IOLs. The use of a correction index or correction ratio is rare in published reports of toric IOLs,22,23 and other factors must be considered, including surgically induced corneal astigmatism. Traditional teaching might have warned against flipping the axis of astigmatism, but a recent study suggested that this is well tolerated by patients.22 In summary, the correction index (correction ratio) is a useful vector-based metric for the evaluation of refractive procedures. Most published studies and 5 unpublished studies of FDA-approved lasers reported mean correction indices greater than 1 for small astigmatic corrections, which has been
misinterpreted as systematic overcorrection. We have shown that these higher indices should be expected because of the variability in preoperative and postoperative measurement of astigmatism and do not represent a systematic overcorrection. Therefore, we urge caution when judging the clinical significance of correction index analyses applied to treatments of low amounts of cylindrical refractive error. The decision to surgically correct small amounts of astigmatism should consider the visual demands of patients, the knowledge of the effects of small cylindrical errors on vision, and our ability to precisely measure astigmatism preoperatively and postoperatively. WHAT WAS KNOWN The correction index (correction ratio) is a useful vectorbased metric for the evaluation of refractive procedures and should be close to 1. Published and unpublished studies of FDA-approved lasers report mean correction indices much greater than 1 for low astigmatic corrections. WHAT THIS PAPER ADDS Mathematical modeling of the correction index, accounting for variability in astigmatism measurement, predicted sharp increases when the TIA is less than 1.00 D. Data from published and unpublished studies showed the predicted increase in the correction index for cylinder correction. Correction indices greater than 1 should be anticipated for lower astigmatic treatments and do not necessarily represent systematic overcorrection.
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18. Wolffsohn JS, Bhogal G, Shah S. Astigmatism and vision: should all astigmatism always be corrected? [editorial] Br J Ophthalmol 2014; 98:2–3 19. Atchison DA, Mathur A. Visual acuity with astigmatic blur. Optom Vis Sci 2011; 88:E798–E805. Available at: http://jour nals.lww.com/optvissci/Fulltext/2011/07000/Visual_Acuity_ with_Astigmatic_Blur.6.aspx. Accessed May 19, 2015 20. O’Leary CI, Evans BJW. Criteria for prescribing optometric interventions: literature review and practitioner survey. Ophthalmic Physiol Opt 2003; 23:429–439 21. Miller AD, Kris MJ, Griffiths AC. Effect of small focal errors on vision. Optom Vis Sci 1997; 74:521–526. Available at: http:// pdfs.journals.lww.com/optvissci/1997/07000/Effect_of_Small_ Focal_Errors_on_Vision.20.pdf. Accessed May 19, 2015 €tz WW. Results of higher power toric 22. Hoffmann PC, Auel S, Hu intraocular lens implantation. J Cataract Refract Surg 2011; 37:1411–1418 23. Visser N, Berendschot TTJM, Bauer NJC, Nuijts RMMA. Vector analysis of corneal and refractive astigmatism changes following toric pseudophakic and toric phakic IOL implantation. Invest Ophthalmol Vis Sci 2012; 53:1865–1873. Available at: http:// iovs.arvojournals.org/article.aspx?articleidZ2188106. Accessed May 19, 2015
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First author: Mark A. Bullimore, MCOptom, PhD College of Optometry, University of Houston, Houston, Texas, USA