Analysis of Higher-Order Aberrations in a Large Clinical Population

Abstract

Purpose.: To use a large wave-front database of a clinical population to investigate relationships between refractions and higher-order aberrations and between aberrations of right and left eyes.

Methods.: Third- and fourth-order aberration coefficients and higher-order root-mean-squared aberrations (HO RMS), scaled to a pupil size of 4.5-mm diameter, were analyzed in a population of approximately 24,000 patients from Carl Zeiss Vision's European wave-front database. Correlations were determined between the aberrations and the variables of refraction, near addition, and cylinder.

Results.: Most aberration coefficients were significantly dependent upon these variables, but the proportion of aberrations that could be explained by these factors was less than 2% except for spherical aberration (12%), horizontal coma (9%), and HO RMS (7%). Near addition was the major contributor for horizontal coma (8.5% out of 9.5%) and spherical equivalent was the major contributor for spherical aberration (7.7% out of 11.6%). Interocular correlations were highly significant for all aberration coefficients, varying between 0.16 and 0.81. Anisometropia was a variable of significance for three aberrations (vertical coma, secondary astigmatism, and tetrafoil), but little importance can be placed on this finding because of the small proportion of aberrations that can be explained by refraction (all <1.0%).

Conclusions.: Most third- and fourth-order aberration coefficients were significantly dependent upon spherical equivalent, near addition, and cylinder, but only horizontal coma (9%) and spherical aberration (12%) showed dependencies greater than 2%. Interocular correlations were highly significant for all aberration coefficients, but anisometropia had little influence on aberration coefficients.

Introduction

Measuring and reducing higher-order aberrations has become of considerable interest in the last decade. Higher-order aberrations are measured and taken into account for refractive surgery, ¹ for implantation of intraocular lenses, for corneal reshaping, and in prescribing of contact lenses, ² particularly for keratoconus. ^3,4 Aberrometry data are used for refraction with spectacles, including refinement of subjective refraction. ⁵  

For a better understanding of higher-order aberrations, a detailed knowledge of the distribution of higher-order aberration coefficients in large populations is helpful. These results can be used to detect abnormal ocular conditions and show trends in aberrations with refraction. Salmon and van de Pol ⁶ have summarized higher-order aberration data from 11 studies, six of which had been published, presenting data for approximately 1300 right and 1200 left eyes. The studies use different instruments, some of which are commercial devices and some of which are laboratory devices. Some data sets are restricted in the type of patients (e.g., only navy pilots), and dilation status of pupils is not described for all studies. Recently, Cox et al. ⁷ have presented higher-order aberration data of 1333 eyes. In the studies of Salmon and van de Pol ⁶ and Cox et al., ⁷ issues considered include which aberration coefficients are significantly different from zero, between-eye correlations, and relationships of individual aberrations and root-mean-squared higher-order aberration to refraction and age. 

Anisometropia is interesting for research as the two eyes of an individual have different refraction despite being exposed to the same genetic and environmental conditions. The prevalence of anisometropia, for a particular criterion, increases in parallel with myopia progression. ^8,9 If higher-order aberrations are involved in the development of refractive error, this might be indicated by interocular differences in higher-order aberrations in anisometropia. 

We were able to obtain access to a database of refraction and higher-order aberrations for approximately 24,000 patients in central Europe. The database was part of the refinement of subjective refractions using Carl Zeiss Vision's “i.Scription” scheme (Aalen, Germany). We reported relationships between refractions (sphere, cylinder, and add) and higher-order aberrations and between aberrations of fellow eyes. Because of the considerable interest in the relationships between aberrations with refraction and age, as given by previous but much smaller-scale studies, ^6,7,10–38 we reported relationships between refractions (sphere, cylinder, and add) and higher-order aberrations and also between aberrations of fellow eyes. 

Methods

Participants

This study was conducted in accordance with the Declaration of Helsinki. Retrospective analysis of higher-order aberration data was based on the Carl Zeiss Vision database. For optimization of spectacle lens prescriptions (i.Scription), Carl Zeiss Vision received aberration data from patients whose vision was measured in private optometry practices in Austria, Belgium, Germany, The Netherlands, and Switzerland. All data were de-identified. The database contained not only right and left eye higher-order aberrations of 40,850 subjects but also subjective refraction data in terms of sphere, cylinder, axis, and near addition (where applicable). Of these subjects, 24,604 (60%) had pupil sizes that were equal to or exceeded 4.5 mm, and we used this reasonably large size to conduct our analyses. 

There were 16,647 myopic right eyes and 16,663 myopic left eyes (subjective spherical equivalent <0 diopter [D]), with 7957 hyperopic right eyes and 7941 hyperopic left eyes (subjective spherical equivalent ≥0 D). Subjective spherical equivalent corrections for right eyes were −17.4 to +9.6 D (mean ± SD: −1.2 ± 2.6 D) and for left eyes were −20.1 to +10.3 D (mean ± SD: −1.2 ± 2.6 D). The mean absolute difference between right and left eyes in spherical equivalent was 0.4 ± 0.4 D (median: 0.3 D). There were 13,038 cases with near additions, which ranged from 0.75 to 4.0 D. 

The database did not contain information about age, sex, ocular pathology, and binocular vision anomalies. Cases of refractive surgery (IOL implantation and corneal surgery), which is known to affect higher-order aberrations, could not be distinguished. 

For clarification, the term cylinder was used for astigmatism, based on the subjective refraction, and the term astigmatism was used for higher-order aberration coefficients such as secondary astigmatism. 

Protocol

Measurements were conducted with the i.Profiler (Carl Zeiss Vision), which uses the Hartman-Shack principle. After an approximate manual alignment, the i.Profiler performs the measurements automatically. The process involves improvement of the alignment. The i.Profiler changes automatically to the contralateral eye once the measurement in the first eye is finished. Aberration coefficients of each subject were scaled to those for a 4.5-mm pupil in a Matlab program (The Mathworks Inc., Natick, MA). Higher-order aberrations are provided up to the seventh order by the i.Profiler, but analysis was performed up to the fourth order only because the coefficients in higher orders than this were expected to be small for a 4.5-mm pupil. For the presentation of higher-order aberrations the American National Standard Institute/International Organization for Standardization standard was used. ^39,40 To take the nasal–temporal asymmetry of right and left eyes into account, the signs of the left eye coefficients were inverted for Zernike polynomials with either negative, even m indices (oblique tetrafoil and oblique secondary astigmatism) or with positive, odd m indices (horizontal coma and trefoil). ⁴⁰ This assumption was checked for its validity in “Results: Comparison of Right and Left Eyes.” 

Spherical equivalent (SE) was calculated by adding half the cylinder to the spherical component based on subjective refraction data and used for classification of hyperopia and myopia. 

SPSS 16.0 (SPSS Inc., Chicago, IL) was used for statistical analysis. Pearson correlations compared various parameters, and independent samples t-tests compared interocular differences in higher-order aberrations. The statistical significance level was set to P ≤ 0.05. 

We considered that the important factors likely to affect higher-order aberrations were spherical equivalent, cylinder, and age. As we did not have information about age, we used the near addition as a de facto measure for this. We performed simple linear regressions of a higher-order aberration with each of these factors. The factors were arranged in order of decreasing r ² values, and they were introduced into multiple regressions in this order. 

Except for correlations between the eyes or for anisometropia analyses, we presented right eye data in order to avoid interocular correlations confounding the results. 

Results

Refraction Distributions

Figure 1 shows subjective refraction distributions of right and left eyes in terms of spherical equivalent and cylinder. The distributions of spherical equivalent and cylinder were similar for the two eyes. Spherical equivalent data were not normally distributed (tested by Kolmogorov-Smirnov, P < 0.001), as might be expected in a clinical population. Figures 2 and 3 show the distributions of near addition and anisometropia, respectively. Slightly more than half of the population (53%) had near additions. 

Figure 1. 

 

Histograms for subjective refraction data in diopters. Left column for right eye (OD) data and right column for left eye (OS) data; first row represents spherical equivalent (1-diopter interval) and second row represents cylinder (0.5-diopter interval).

Figure 2. 

 

Histogram for near addition in diopters (0.25-diopter interval).

Figure 3. 

 

Histogram for anisometropia (right minus left eye) in diopter (0.5-diopter interval). Note the logarithmic scale for the frequency.

Higher-Order Aberrations and Refraction in Terms of Spherical Equivalent, Cylinder, and Near Addition

Means of all coefficients were significantly different from zero for the myopic eyes, for the hyperopic eyes, and for all participants analyzed together. Third- and fourth-order aberration coefficients and higher-order root-mean-squared aberrations (HO RMS) had significantly different means for myopes and hyperopes (independent samples t-tests, P < 0.001), except for trefoil and secondary astigmatism. The largest effects occurred for vertical coma, horizontal coma, and spherical aberration (Fig. 4). In detail, the means and standard deviations for these coefficients were as follows: vertical coma, +0.025 ± 0.103 μm in myopes and +0.001 ± 0.104 μm in hyperopes; horizontal coma, −0.012 ± 0.072 μm in myopes and −0.047 ± 0.079 μm in hyperopes; and spherical aberration, +0.001 ± 0.056 μm in myopes and +0.035 ± 0.056 μm in hyperopes. 

Figure 4. 

 

Higher-order aberration coefficients and HO RMS data for right eyes separated for myopes and hyperopes. Error bars represent ±1 standard deviation. Except for trefoil and secondary astigmatism, the coefficients were significantly different between myopes and hyperopes.

The multiple linear regressions showed that HO RMS and most aberration coefficients were significantly dependent upon spherical equivalent, near addition, and cylinder (only four combinations of factors and coefficients were not significant). However, the proportion of aberrations that could be explained by these factors was usually small, varying between 0.1% (oblique secondary astigmatism) and 11.6% (spherical aberration). The models explained more than 2% of the variation for only spherical aberration, horizontal coma (9.5%), and HO RMS (6.5%). While all three parameters contributed significantly to the models for the two aberration coefficients, the cylinder contribution was less than 0.1%. Spherical equivalent was the major contributor for spherical aberration (7.7% out of 11.6%), and near addition was the major contributor for horizontal coma (8.5% out of 9.5%) and HO RMS (3.9 out of 6.5%). 

Omitting the cylinder gave the following regression equations: 

Horizontal coma: −0.0043 (±0.0008) – 0.0031 (±0.0002) SE – 0.0183 (±0.0004) Add (1) 

Spherical aberration: +0.0040 (±0.0007) + 0.0051 (±0.0001) SE + 0.0105 (±0.0003) Add (2) 

HO RMS: +0.1694 (±0.0008) + 0.0011 (±0.0002) SE + 0.0137 (±0.0005) Add (3), 

where the numbers in brackets are the standard errors of the constant, spherical equivalent (SE), and near addition (Add). Figures 5 and 6 show plots of horizontal coma and spherical aberration coefficients, respectively, against spherical equivalent. 

Figure 5. 

 

Horizontal coma coefficient for right eyes as a function of subjective spherical equivalent. The scale on right shows the number of cases in an area 1.25 * 10⁻⁴ μm.D (0.125-D width and 0.001-μm height). The white lines are linear fits for no add (dashed dot), 1.25-D add (solid line), and 2.5-D add (dot).

Figure 6. 

 

Spherical aberration coefficient of right eyes as a function of subjective spherical equivalent. The scale on the right shows the number of cases in an area 1.25 * 10⁻⁴ μm.D (0.125-D width and 0.001-μm height). The white lines show the linear fits for no add (dashed dot), 1.25 D (solid), and 2.5 D (dot).

Comparison of Right and Left Eyes

We analyzed interocular relationships of higher-order aberration coefficients and HO RMS. Correlation coefficients varied considerably between +0.16 and +0.81 and all were highly significant (P < 0.001). The coefficients were as follows: oblique trefoil +0.58, vertical coma +0.71, horizontal coma +0.47, trefoil +0.35, oblique tetrafoil +0.16, oblique secondary astigmatism +0.22, spherical aberration +0.81, secondary astigmatism +0.42, tetrafoil +0.30, and HO RMS +0.55. 

Anisometropia Analysis

To determine how anisometropia might affect aberrations, we determined differences in aberration coefficients between right and left eyes as a function of the differences in spherical equivalents, and compared this with the aberration coefficients of the right eye as a function of its spherical equivalent. If anisometropia has an influence on aberrations, the two regressions should have significantly different slopes. There were significant differences for the three aberrations of vertical coma, secondary astigmatism, and tetrafoil. However, little importance can be placed on these findings as the respective r ² values for the right eye aberration versus right spherical equivalent were only 0.9%, 0.02%, and 0.2%. 

To explore the anisometropia issue further, we compared aberration coefficients of the two eyes for which the anisometropia was >2.0 D. This was done separately for myopes, comparing the more myopic eyes with the less myopic eyes (n = 614), and for hyperopes, comparing the more hyperopic eyes with the less hyperopic eyes (n = 115). Mixed anisometropes were excluded from the analysis. t-tests compared slopes and intercepts between higher and lower refraction groups. While significant effects were found for trefoil for the myopic groups for both the slopes and intercepts, no more than 0.03% of the coefficient could be explained by the refraction. A marginally significant larger intercept for spherical aberration for the higher hyperopic eye group than for the lower hyperopic eyes (difference 0.041 μm, P = 0.05) was also found. 

Figure 7 shows the regressions for spherical aberration coefficients for the four groups. Interestingly, the slopes for the hyperopes were negative despite the fact that the whole group of hyperopes had a positive slope (−0.008 ± 0.004 and −0.009 ± 0.004 μm/D, compared with +0.005 ± 0.001 μm/D estimate and its standard error). The slopes for the myopic groups were similar to those for the whole group of myopes (+0.004 ± 0.001 and +0.004 ± 0.001 μm/D compared with +0.004 ± 0.000 μm/D). 

Figure 7. 

 

Spherical aberration coefficients for eyes where anisometropia was >2 D. Fits are as follows: more myopic eyes +0.0213 (±0.0074) + 0.0038 (±0.0010) SE; less myopic eyes +0.0174 (±0.0037) + 0.0039 (±0.0010) SE; more hyperopic eyes +0.0774 (±0.0081) − 0.0091 (±0.0042) SE; less hyperopic eyes +0.0368 (±0.0189) − 0.0081 (±0.0043) SE. All constants apart from those for the less hyperopic eyes are significantly different from zero. Dotted lines are plotted to indicate the intercept (spherical aberration for emmetropia).

Discussion

Third- and fourth-order aberration coefficients and HO RMS, at a pupil size of 4.5-mm diameter, were analyzed in a large population of approximately 24,000 subjects to determine correlations between these and the variables of refraction, near addition, and cylinder. Most aberration coefficients were significantly dependent upon the variables, but the proportion of aberrations that could be explained by these factors was less than 2% except for spherical aberration (12%), horizontal coma (9%), and HO RMS (7%). Near addition was the major contributor for horizontal coma (8.5% out of 9.5%) and spherical equivalent was the major contributor for spherical aberration (7.7% out of 11.6%). Interocular correlations were highly significant for all aberration coefficients, varying between 0.16 and 0.81. While anisometropia was a variable of significance for three aberrations (vertical coma, secondary astigmatism, and tetrafoil), little importance can be placed on this finding because of the small proportion of aberrations that can be explained by refraction (all <1.0 %). 

In the following subsections, we make comparisons with previous studies. 

Mean Levels of Aberrations and Their Significance

Previous studies have found one or more higher-order ocular aberrations to be significantly different from zero, ^{12–17,41–43} with spherical aberration being common to all studies with positive values. In their consideration of previous studies, Salmon and van de Pol ⁶ have noted that only oblique tetrafoil, vertical coma, and spherical aberration do not cluster near zero. With the benefit of a large number of subjects, we found that the means of all coefficients were significantly different from zero, with the main ones being oblique trefoil, vertical coma, horizontal coma, and spherical aberration (at least for hyperopes) (Fig. 4). 

Aberrations and Refraction

There have been several previous studies that have considered the influence of refraction on aberrations. Some studies have reported higher HO RMS in myopes than in emmetropes or a trend of increasing HO RMS with increase in myopia, ^11,18,37 while others have found no effect. ^19–22,38 Kwan et al. ²³ have found increasing HO RMS with decrease in myopia, and Philip et al. ¹⁶ have found higher levels in hyperopes than in emmetropes and myopes. Two studies of the first group had an artifact in which the use of correcting lenses during the measurement increased the effective pupil size that is being considered in myopia. ³⁸ One study ²⁴ has found that spherical aberration becomes increasingly negative with decrease in myopia; this study is different from most other studies in that most subjects had negative spherical aberration. Some studies ^7,19,38 have found no significant effect of refraction upon spherical aberration, but Bao et al. ²⁵ have found less spherical aberration in a myopic group than in an emmetropic group and Kwan et al. ²³ have found that spherical aberration increases as myopia is reduced. Philip et al. ¹⁶ have found more positive spherical aberration with hyperopes than with other groups, with moderate myopes having less spherical aberration than low myopes, emmetropes, and low hyperopes. Llorente et al. ²⁶ also have found that hyperopes have higher spherical aberration than myopes, but this has not been found by Artal et al. ²⁷ Martinez et al. ²⁸ have found greater HO RMS and spherical aberration in hyperopes than in emmetropes. 

The present study supports most previous studies in that HO RMS aberrations were higher in the hyperopic group than in the myopic group (Fig. 4) and spherical aberration became more positive as refraction became less negative or more positive (Fig. 6). The regressions predict that most of the higher myopes will have negative spherical aberration, but this tends toward positive spherical aberration as the add increases. All aberration coefficients were affected significantly by refraction (although some correlations were small), particularly horizontal coma and spherical aberration for which the refraction explained 2.8% and 7.7% of the variation, respectively. 

There is no clear understanding as to why the aberrations change with refraction as reported here. Atchison ⁴⁴ has presented optical models of myopic eyes, from his own data, in which both vitreous chamber depth and anterior corneal curvature increase with increase in myopia. As these changes predict increase in positive spherical aberration, rather than decrease as reported here, we do not have sufficient understanding about the optics of myopic eyes. Cylinder refractions will be associated with tilts and decentration of surfaces, which will affect aberrations. 

Unlike other studies, ^11,19,37,38 we found that many myopes have negative rather than positive spherical aberration. Some further analysis indicates that there is slight effect due to a smaller pupil size: an analysis of the 3469 persons with pupil sizes between 6 and 7 mm showed a reduction of positive spherical aberration from 55% to 46% with a reduction in analyzed pupil size from 6.0 to 4.5 mm. 

Aberrations and Age

Several studies ^{10,13,29–32} have found that HO RMS aberrations increase significantly with age, although one study ³³ has found that aberrations are at a minimum in the third decade of life. Spherical aberration becomes more positive with increase in age. ^13,29,32,34 However, studies with populations containing limited ranges of refractive errors have found only moderate effects of age on HO RMS. ^14,15 These studies have not found a significant effect of age on spherical aberration, but Atchison and Markwell ¹⁵ have found a significant effect for horizontal coma. 

The present study used the near addition as a proxy for age. All third-order aberrations, spherical aberration, and secondary astigmatism were affected significantly by the addition (although some correlations were small), particularly horizontal coma and spherical aberration, for which the near addition explained 8.5% and 6.6% of the variation, respectively. 

Many changes in the biometry of the eye are associated with change in age, mainly related to the lens, and include its surface curvature and asphericities, central thickness, and refractive index distribution. ^45–51 Any pupil center shift with age will affect most higher-order aberration coefficients. It must be remembered that pupil size decreases with increase in age, which will reduce natural aberration magnitudes. ²⁹  

Correlations of Aberrations between Eyes

Previous studies ^13,14,16,25 have found significant correlations between the magnitudes of aberrations on a pair of eyes for several aberrations (see also Porter et al. ⁴² and Castejón-Mochón et al. ⁵² ). The present study found that all third- and fourth-order aberrations show significant interocular correlations with the expected mirror symmetry between eyes. 

Anisometropia

Tian et al. ²⁴ have not found any significant interocular differences for higher-order aberrations for either anisomyopes or isometropes, Vincent et al. ³⁶ have not found any significant interocular differences for any higher-order aberration coefficients between the more myopic and the less myopic eyes of myopic anisometropes. However, Kwan et al. ²³ have found spherical aberration to be less positive in the more myopic eyes than in the less myopic eyes of their anisometrope population, and Vincent et al. ³⁵ have found significant correlations between the interocular spherical aberration differences and the interocular refraction differences of amblyopes. 

To separate effects of anisometropia from those of refraction, we determined the interocular difference in aberration coefficients as a function of interocular differences in spherical equivalents and compared this with the aberrations of right eyes as a function of its spherical equivalent. Significantly different rates of change were found for vertical coma, secondary astigmatism, and tetrafoil, but we believe that little importance can be placed on these findings as the respective r ² values for the influence of refractions were less than 1.0%. Further analysis for anisometropic cases (>2 D interocular differences in refraction) showed only minor effects. 

In view of the generally low correlations that we found in this large-scale study, it is likely that some previous studies have failed to find significant interocular aberration differences in anisometropes because of insufficient numbers of subjects. ^24,36 When significant interocular differences have been found for spherical aberration, ^23,35 they may be explained by refraction of the two eyes without the need to invoke the additional influence of anisometropia. 

Shortcomings of Study

This is by far the largest population study of higher-order aberrations, but we acknowledge several weaknesses. Most importantly, the database did not contain information about age, and we used the near addition as a proxy for this. We also had no information on sex, race (although the population can be assumed to be mainly Caucasian), ocular pathology, or binocular vision anomalies, and the population may include people who had undergone refractive surgery. Usually, positive spherical aberration increases in corneal surgery for myopia and is reduced in corneal surgery for hyperopia, ^53–55 but the magnitude will be influenced by “wave-front–guided” and “wave-front–optimized” refractive surgery. With intraocular lens implantation spherical aberration usually increases, but this is influenced by the asphericity of intraocular lens surfaces. ^56–58  

There was no standardization of lighting between various clinics. The population study is likely to suffer from the usual clinical population issue of underrepresentation of young subjects with small refractive errors, and we were selective in excluding people with measured pupil diameters of less than 4.5 mm, which is likely to mainly exclude elderly people. This pupil size was smaller than that reported for most other studies. 

Given the relative small pupil size, we did not pay any attention to aberration terms higher than the fourth order. Given the nature of the study, we were not able to investigate the component contributions to aberrations and how these are affected by factors such as refraction and age. 

Conclusions

In a large clinical population, most third- and fourth-order aberration coefficients were significantly dependent upon spherical equivalent, near addition (proxy for age), and cylinder, but only horizontal coma (9%) and spherical aberration (12%) showed dependencies of greater than 2%. Interocular correlations were highly significant for all aberration coefficients, varying between 0.16 and 0.81. Anisometropia had little influence on aberration coefficients. 

Acknowledgments

The authors thank Carl Zeiss, Aalen, Germany, for access to its wave front database and for useful advice on the manuscript. 

Footnotes

 Supported by ARC Discovery Grant DP110102018 and ARC Linkage Grant LP100100575 (DAA).

Footnotes

 Disclosure: A. Hartwig, None; D.A. Atchison, None