The importance in understanding and recognizing angle kappa in corneal refractive surgery is well established. Although angle kappa has been defined in different ways,^{1} the most agreed upon definition states that angle kappa is the angular separation between the visual axis and the pupillary axis. The pupillary axis is defined as the line that passes through the center of the entrance pupil, perpendicular to the cornea, and therefore has a clear anatomical marker.^{2} The visual axis is a theoretical line joining the fixation point and the fovea via the anterior and posterior nodal points.^{1} Therefore, applying the visual axis in a clinical setting is difficult because it has no physical indication of position on the cornea itself.
In 1987, Uozato and Guyton^{3} recommended that corneal ablations should be centered on the entrance pupil center, whereas Pande and Hillman^{4} proposed in 1993 that the best optical results would be achieved by centering corneal ablations on the coaxially sighted corneal light reflex (CSCLR) or corneal vertex, which best approximates the corneal intercept of the visual axis (because it is not currently possible to measure the visual axis itself). Centration on the CSCLR or corneal vertex was studied by different groups during the 2000s, demonstrating improved results, particularly for hyperopia.^{5–9} More recently, review articles by various groups have shown that the consensus is converging on CSCLR or corneal vertex centration rather than entrance pupil centration.^{1,10–12} The use of CSCLR was recently supported by the results of a study by Manzanera et al^{13} that showed the CSCLR was within 0.2 mm of the achromatic axis in 55% of eyes, and within 0.4 mm in 95% of eyes. Centration of corneal refractive surgery is important because of the potential for inducing visual side effects if the treatment zone is decentered.^{14,15}
When angle kappa is small, the entrance pupil and CSCLR are relatively aligned, so the outcome is unaffected by the fiducial point used for centration. However, when angle kappa is large, the entrance pupil and CSCLR are separated, meaning centration has a significant influence on the outcome. Therefore, it is useful to be aware of the distribution of angle kappa in the population to appreciate the incidence of patients who may be affected by centration choice. Angle kappa can be clinically measured by topography, such as with the Orbscan II (Bausch & Lomb, Inc).^{1} Orbscan II angle kappa measurements have been shown to have good correlation (r = 0.932) with synoptophore measurements (documented to measure the exact angle kappa), albeit with the Orbscan measurement being larger (P < .0001).^{16} The Orbscan output (which we define as the pupil offset) is calculated as the difference between the corneal vertex normal and the entrance pupil center. This parameter acts as a proxy measurement for the angle kappa, the pupil offset being a distance measurement rather than an angle.
There are only a few studies^{16–22} that show the distribution of angle kappa. Angle kappa is considered to be slightly positive due to the position of the fovea, which is situated marginally temporal with respect to the pupillary axis trajectory onto the retina. Angle kappa has been reported to be greater in hyperopia than myopia.^{16,17,19–22}
This aim of the current study was to report the Orbscan II pupil offset to describe the distribution within a normal population and investigate correlations with refraction and between eyes.
Patients and Methods
Patients
This study was a retrospective analysis of consecutive eyes screened for corneal laser refractive surgery at the London Vision Clinic, London, United Kingdom, between January 2006 and February 2013. To be included in the study, the patient had to have undergone a preoperative Orbscan II measurement and match the following additional criteria: both eyes falling within the same refractive subgroup, a complete ocular examination, no pathology such as keratoconus (ie, unsuitable for corneal laser refractive surgery), corneal scarring, corneal dystrophy, and no previous ocular surgery or trauma. Corrected distance visual acuity (CDVA) had to be 20/32 to ensure the patient was able to maintain proper fixation during testing. The eyes were divided into three groups based on the manifest refraction spherical equivalent (SEQ). The emmetropic group included eyes with SEQ between −0.25 and +0.50 diopters (D) and cylinder of 1.00 D or less. The myopic group included eyes with SEQ greater than −0.50 D and no mixed cylinder. The hyperopic group included eyes with SEQ greater than +0.50 D and no mixed cylinder. Both eyes of consecutive patients were included and analyzed until there was a total of 250 eyes of 125 patients for each group, to perform mirrorsymmetry (superior/inferior/nasal/temporal) and withinsubject contralateral symmetry analysis.
A routine preoperative assessment was performed as previously described.^{23} Informed consent and permission to use their data for analysis and publication was obtained from all patients. Because this was a retrospective study, an exemption from full institutional review board approval was obtained from the United Kingdom Health Research Authority.
Pupil Offset Measurement
An Orbscan measurement was obtained in a dark room with the only source of light being the Orbscan measurement head. The pupil offset was measured as the distance in millimeters at the corneal plane between the entrance pupil center and the corneal vertex. The pupil offset was measured manually using the Orbscan II software as follows:

In the whole eye view, the white pupil boundary overlay was displayed and checked to ensure it was a correct fit to the pupil (Figure A, available in the online version of this article). Patients were excluded from the study if pupil boundary detection by Orbscan II was not accurate.
Figure A.
Orbscan II whole eye view with pupil recognition software (Bausch & Lomb). The white line is displayed automatically to match the circumference of the pupil.

In the EyeMetrics software module, the entrance pupil center was displayed as a green dot and the corneal vertex normal as a white dot. The distance between the two points was manually measured using the caliper tool (Figure B, available in the online version of this article).

The (X, Y) coordinates of the entrance pupil center and the corneal vertex normal were recorded and pupil offset distance was calculated as the magnitude of the vectorial difference.

The data for left eyes were reflected in the vertical axis to account for enantiomorphism, so that nasal/temporal characteristics could be combined.

The angle kappa and axis values that are automatically calculated by the Orbscan II software were also recorded.
The manual assessment was done by the same observer for all eyes (ELR).
Statistical Analysis
First, each group was analyzed individually. The pupil offset was divided into 0.05mm steps and plotted as a histogram with the percentage of eyes for each pupil offset bin.
A radial plot was created to analyze the location of the pupil center relative to the corneal vertex. The radial plot was divided into eight sectors with the origin representing the corneal vertex. The percentage of eyes within each sector was calculated to evaluate the pupil offset orientation in terms of inferior/superior and nasal/temporal.
A comparison of pupil offset magnitude and spherical equivalent refraction for all three groups was plotted over a continuous refractive range (from −14.00 to +8.00 D). A standard linear regression equation was derived for the myopic and hyperopic groups. This analysis was repeated for the X (horizontal) and Y (vertical) components of the pupil offset. A similar analysis was performed to investigate the influence of patient age, Atlas (Carl Zeiss Meditec AG) average keratometry, and scotopic pupil diameter. Univariate multiple linear regression analysis using stepwise backward elimination was performed to investigate which parameters were correlated to pupil offset magnitude and for the Xand Ycomponents.
Finally, contralateral correlation was evaluated using a scatter plot of the pupil offset for right eyes against the pupil offset for left eyes. A standard linear regression equation and a zero intercept line were derived.
Microsoft Excel 2010 software (Microsoft Corporation) was used for data entry and statistical analysis and a P value of .05 was considered statistically significant. Nonparametric tests were used where the data was not normally distributed.
Results
The study included 250 eyes (125 patients) for each refractive group. The patient demographics, preoperative SEQ, and cylinder for each group are shown in Table 1.
Table 1: Demographics, Angle Kappa, and Pupil Offset Data for the Myopic, Emmetropic, and Hyperopic Groups 
Distribution of Pupil Offset
The mean pupil offset is included in Table 1, showing that the pupil offset was smallest in the myopic group (0.27 mm), followed by the emmetropic group (0.34 mm), and largest in the hyperopic group (0.39 mm). The pupil offset was statistically significantly different between all group pairs (P < .0001). Figure 1A shows the distribution of pupil offset for each group. The tendency of small pupil offset in the myopic group can be appreciated because the pupil offset was less than 0.15 mm in 20.8% of myopic eyes, 9.6% of emmetropic eyes, and 3.2% of hyperopic eyes. Similarly, the pupil offset was greater than 0.45 mm in 8.8% of myopic eyes, 18.8% of emmetropic eyes, and 28.0% of hyperopic eyes.
Figure 1. Distribution of (A) pupil offset magnitude, (B) Xcomponent, and (C) Ycomponent measured in millimeters for the myopic, emmetropic, and hyperopic groups. 
Location of Pupil Center Relative to Corneal Vertex
The orientation of the pupil offset can be appreciated by the histograms for the pupil offset Xcomponent (Figure 1B) and Ycomponent (Figure 1C). The distribution for the Xcomponent was skewed in the temporal direction (ie, the entrance pupil center was temporal to the corneal vertex, or in other words the corneal vertex was nasal to the entrance pupil center) for all three groups, with myopia, hyperopia, and emmetropia showing increasing temporal skew (mean Xoffset difference with zero was statistically significant, P < .001). The Xoffset was more than 0.05 mm nasal in only 8.4% of eyes in the myopic group, 2.0% of eyes in the emmetropic group, and 1.2% of eyes in the hyperopic group. The distribution of the Ycomponent was more symmetrical about the corneal vertex in all three groups, demonstrating that the majority of the pupil offset was in the horizontal direction. The mean Yoffset difference with zero was statistically significant for the myopic and emmetropic groups (P < .001), but not for the hyperopic group (P = .293).
Figure 2 shows a radial plot of the location of the pupil center relative to the corneal vertex for each group. The pupil offset, in most cases, was found to be in the temporal direction (ie, the center of the entrance pupil was temporal to the corneal vertex). The pupil offset was in the temporal hemisphere in 85.2% of myopic eyes, 96.0% of emmetropic eyes, and 98.8% of hyperopic eyes. The myopic group displayed the largest number of negative pupil offset measurements, with 14.8% of eyes having the pupil center nasal to the corneal vertex.
Correlation of Pupil Offset with Manifest Refraction
Figure 3 shows scatter plots of the pupil offset magnitude, pupil offset Xcomponent (horizontal), and pupil offset Ycomponent (vertical) against SEQ for each group. The pupil offset trend line runs from myopia to hyperopia in a positive linear fashion, demonstrating the more myopic the patient, the smaller the pupil offset, and the more hyperopic the patient, the larger the pupil offset (P < .001). However, there was a large amount of scatter within all three populations, as demonstrated by the R^{2} values being 0.0533 for the myopic group and 0.0489 for the hyperopic group. Looking at the X and Ycomponents separately showed that this trend was due to a deviation in the horizontal meridian (P < .001), with no correlation apparent in the vertical meridian (P = .06 for the myopic group and P = .35 for the hyperopic group).
Correlation of Pupil Offset with Age
Figure C (available in the online version of this article) shows a scatter plot of the pupil offset magnitude against age for each group. The pupil offset showed a trend for larger pupil offset for younger age for all three groups, but only by 0.015 mm per decade (slope was −0.0015 for the myopic group, −0.0019 for the emmetropic group, and −0.0016 for the hyperopic group). There was also a large amount of scatter within all three populations, as demonstrated by the R^{2} values being less than 0.0166 for all groups, and the correlation was statistically significant only for the hyperopic group (P = .09 for the myopic group, P = .20 for the emmetropic group, and P = .045 for the hyperopic group).
Correlation of Pupil Offset with Average Keratometry
Figure D (available in the online version of this article) shows a scatter plot of the pupil offset magnitude against average keratometry for each group. The pupil offset showed a trend for larger pupil offset for flatter keratometry for all three groups, but only by 0.01 mm/D (slope was −0.0169 for the myopic group, −0.0075 for the emmetropic group, and −0.0119 for the hyperopic group). There was also a large amount of scatter within all three populations, as demonstrated by the R^{2} values being less than 0.03 for all groups. The correlation was statistically significant for the myopic (P = .005) and hyperopic (P = .040) groups, but not for the emmetropic group (P = .23).
Correlation of Pupil Offset With Scotopic Pupil Diameter
Figure EA (available in the online version of this article) shows a scatter plot of the pupil offset magnitude against scotopic pupil diameter for each group. The pupil offset showed a weak trend for larger pupil offset for larger pupil diameter for all three groups. There was a large amount of scatter within all three populations, as demonstrated by the R^{2} values being less than 0.110 for all groups. The correlation was statistically significant for the emmetropic (P < .001) and hyperopic (P = .027) groups, but not for the myopic group (P = .25). When the correlation was made with the pupil offset Xcomponent (Figure EB), this was found to be statistically significant for all three groups (P < .01), although again with R^{2} values less than 0.116. There were no statistically significant correlations for the pupil offset Ycomponent (Figure EC).
Univariate Multiple Linear Regression Analysis
The results of the univariate multiple linear regression analysis using stepwise backward elimination are presented in Table 2, for pupil offset magnitude and for X and Y pupil offset components separately. For pupil offset magnitude, all variables were found to be statistically significant except for age and cylinder Ycomponent. The significant variables for the pupil offset Xcomponent were spherical equivalent, cylinder Xcomponent, and scotopic pupil diameter. The significant variables for the pupil offset Ycomponent were spherical equivalent and scotopic pupil diameter.
Table 2: Multivariate Regression Analysis Results 
Contralateral Correlation
The mean pupil offset for right and left eyes within each group is included in Table 1, showing that there was no difference in pupil offset between right and left eyes for the myopic and emmetropic groups, but interestingly the pupil offset was 0.03 mm larger (P = .001) in left eyes for the hyperopic group. Figure F (available in the online version of this article) shows a scatter plot of the pupil offset X and Ycomponents for right eyes against the pupil offset X and Ycomponents for left eyes. The regression lines deviated from the line of equality, indicating a potential systematic difference between right and left eyes. The correlations were statistically significant for both the Xcomponent (R^{2} = 0.331 in the myopic group, 0.416 in the emmetropic group, and 0.375 in the hyperopic group, P < .001 for all groups) and the Ycomponent (R^{2} = 0.507 in the myopic group, 0.483 in the emmetropic group, and 0.561 in the hyperopic group, P < .001 for all groups). However, there were a few outliers with a magnitude difference of up to 0.38 mm between eyes.
Orbscan II Angle Kappa
The mean angle kappa as measured by the Orbscan II is included in Table 3. As for pupil offset, the angle kappa was largest in the hyperopia group, followed by the emmetropic group, and was smallest in the myopia group. There was a statistically significant difference between all three groups (P < .01). The angle kappa is not a linear estimate of the pupil offset, and the differences in angle kappa were found to be relatively smaller than the differences in the pupil offset. Figure G (available in the online version of this article) shows the correlation between Orbscan angle kappa and the pupil offset. As expected, there was a positive correlation between these values (P < .001), demonstrating that pupil offset is associated with angle kappa.
Table 3: Literature Review for Pupil Offset Magnitude and X and Ycomponents for the Myopic, Emmetropic, and Hyperopic Groups 
Discussion
Results of this study show that both the pupil offset magnitude and angle kappa were higher in the hyperopic and emmetropic groups compared with the myopic group. However, even in the myopic group 10% of eyes had a pupil offset magnitude greater than 0.46 mm. The pupil offset Xcomponent was skewed temporally in all three groups, whereas the Ycomponent was more symmetrical. The entrance pupil center was in the temporal quadrant relative to the corneal vertex in the majority of eyes. A larger pupil offset was found to be correlated with increasing hyperopia, higher cylinder, larger scotopic pupil diameter, and flatter keratometry.
The current study found a close agreement with the results of previous studies that have reported the pupil offset, all of which had found a larger pupil offset magnitude in the hyperopic group, as set out in Table 3.^{19–22} None of the previous studies reported an emmetropic group, so no comparison was possible for this group. Erdem et al^{19} and Yang et al^{22} had also reported the X and Ycomponents for pupil offset, which mirrored the findings of the current study of a temporal skew for the Xcomponent, but a symmetrical distribution for the Ycomponent.
Beyond the gross comparison of ametropic groups, we found the correlation for pupil offset extended to SEQ, demonstrating a linear relationship of pupil offset magnitude increasing from high myopia through emmetropia to high hyperopia. The same relationship was found for the pupil offset Xcomponent. Although axial length was not measured in the current study, these results indirectly support the concept that the pupil offset is larger for shorter (ie, hyperopic) eyes than longer (ie, myopic) eyes, as has been described, for example, by Artal et al^{24} and Berrio et al.^{25} The reason for this is related to the geometry of the eye. The visual axis is the line connecting the fovea with the fixation point passing through the nodal point. Because the eye is not a rotationally symmetric optical system, the visual axis does not pass through the entrance pupil center. If the distance from the optic nerve to the fovea is similar between eyes, it would be expected for the pupil offset to be larger in a short (likely hyperopic) eye. This is also supported by the correlation found for larger pupil offset in flatter keratometry, because flat keratometry is more commonly found in hyperopia.
The pupil offset was found to be in the temporal hemisphere in the majority of eyes (85.2% for the myopic group, 96.0% for the emmetropic group, and 98.8% for the hyperopic group), and the majority of these in the temporal quadrant. This was similar to previous reports, with 81% to 90% temporally in the study by Mabed et al^{20} and 88% temporally in the study by Erdem et al.^{19} This was also similar to the study of Orbscan II angle kappa by Hashemi et al,^{17} which found the angle kappa intercept was temporal in 99.5% of eyes (inferotemporal in 89.7% and superotemporal in 9.8%).
A correlation was also found between the pupil offset and scotopic pupil diameter. Other studies have found an increase in pupil offset when increasing the pupil diameter by changing the lighting conditions from photopic to mesopic.^{20,26} However, this may not represent the same finding as the current study, where all measurements were obtained with the same lighting conditions using the Orbscan II in a dark room. The correlation in the current study shows that pupil offset increased for larger pupil diameter between subjects, whereas the previous studies reported an increase within subjects. The correlation with pupil diameter may also be affected by the presence of pupil centroid shift for some patients.^{27} However, although this correlation was statistically significant, the R^{2} values were low, indicating a large degree of variation in the population.
There are many changes in the eye that are known to occur over time, so it was interesting to evaluate the pupil offset with age. There was a trend for decreasing pupil offset magnitude with age consistent among the three groups (0.08 mm over a 50year period), but there was significant scatter with R^{2} values below 0.02 and the correlation was only statistically significant for the hyperopic group. Age was also not found to be statistically significant in the multivariate regression analysis. This lack of correlation agreed with previous findings.^{26}
Orbscan II angle kappa value has been reported for large populations (> 300 eyes) in three previous studies, as set out in Table A (available in the online version of this article).^{16–18} The results were similar for comparison between groups, although Hashemi et al^{17} found the largest angle kappa in the emmetropic group. Within our population, we found a larger angle kappa in every refractive group compared to the previous studies also using the Orbscan II. One difference between study populations was the mean age, being 55 years in the current study, 40 years in the study by Hashemi et al,^{17} and 28 years in the study by Basmak et al.^{16} Hashemi et al^{17} reported a decrease in angle kappa with increasing age, but this would predict a smaller angle kappa in the current study. Combining this with the lack of correlation between pupil offset and age, as described above, it is unlikely that the difference between studies can be due to the age. There may be other factors contributing to the difference in findings that have not been explored. In our data, even the parameter that showed the strongest trend—the pupil offset Xcomponent—showed a large variance for the myopic (R^{2} = 0.1484) and hyperopic (R^{2} = 0.0725) groups when analyzing linear regression equations for pupil offset magnitude plotted against SEQ (Figure 3). The difference in findings demonstrates the need for large population studies to be performed in this area, as well as appropriate standardization of testing protocols and measurement techniques.
Table A: Literature Review for Orbscan II Angle Kappa for the Myopic, Emmetropic, and Hyperopic Groups 
In the current study, the pupil offset was slightly larger in left eyes compared to right eyes, although this was only statistically significant for the hyperopic group (a difference of 0.03 mm between right and left eyes). The regression line for all three groups showed a trend for the pupil offset Xcomponent to be larger in left eyes when there was a small pupil offset, and for the pupil offset Ycomponent to be larger in right eyes when there was a large pupil offset. Although there is no obvious clinical explanation for such a relationship, one possibility may be a systematic measurement tilt within the Orbscan device. A degree of scatter in the data (R^{2} between 0.33 and 0.56) and the presence of some outliers may also partly explain this. Comparing these results to previous studies, there appears to be no consensus yet. Erdem et al^{19} found no difference in pupil offset between right and left eyes using the Topolyzer Vario (Alcon Laboratories, Inc), whereas Basmak et al^{16} did find a larger Orbscan II angle kappa value in left eyes compared to right eyes in all groups. They postulate this could be related to eye dominance, or even to variables that will affect the distance of the eye to the instrument, such as head posture or facial asymmetry.
One weakness of the study is that the groups were not matched for age, refractive cylinder, scotopic pupil diameter, and keratometry. Age was highest in the hyperopic group and lowest in the myopic group. However, age was found to have only a very weak influence on pupil offset, if at all, as discussed earlier. Therefore, it is unlikely that age has contributed to the difference in pupil offset between groups. The difference in average keratometry was only apparent for the emmetropic group, only 0.60 D in magnitude, and only statistically significant due to the large population size. Pupil diameter was higher in the myopic group, but the trend was for a larger pupil offset with larger pupil diameter, so the pupil offset may be slightly overestimated relative to the other groups. Finally, refractive cylinder was highest in the myopic group and lowest in the emmetropic group. Multivariate regression found a correlation with cylinder with increasing pupil offset for higher cylinder. This would act to increase the pupil offset in the myopic group, which means that the pupil offset may be slightly overestimated relative to the other groups. The differences between the groups would be acting to reduce the difference between the groups, but the differences are significant despite these factors.
The pupil offset magnitude and Orbscan angle kappa demonstrated a reasonable correlation, although perhaps not as strong a correlation as might be expected. This indicates that although associated, the pupil offset and angle kappa may be two separate entities. In particular, the intercept was nonzero for each group, which implies that an eye may present with a pupil offset despite having no angle kappa. However, a zero angle kappa is clinically extremely unlikely due to the anatomy of the eye, as demonstrated by there being only one eye with an angle kappa value of less than 1°. Similarly, the pupil offset magnitude was less than 0.05 mm in only 4 eyes (0.05%). Therefore, it is difficult to draw conclusions about this range because this is extrapolating outside the range of data in the population.
Another potential weakness of the study is that both eyes from each patient have been used, which may introduce some bias into the statistical analysis. This had been done to perform a contralateral comparison analysis. However, the possible bias from including nonindependent data is significantly reduced for larger populations, and it would be unlikely to have an effect for a 250 eye population. To rule out this possibility, we reanalyzed the data using one eye only and did not find any difference in the statistical results as presented for both eyes. Therefore, we have chosen to retain the data for both eyes in the current study.
Measurement of pupil offset using the Orbscan II has also not been validated for use in guiding treatment centration for corneal laser refractive surgery, and these data may not be as reliable as those obtained by synoptophore. Another potential source of error is the repeatability and reproducibility of the Orbscan measurements. Although the large population size mitigates this to a degree, as demonstrated by the confidence intervals, an average of multiple scans would ideally have been used for this study.
There is a need for further studies in this area, including comparisons between treating on the center of the pupil versus treating on the CSCLR, especially in patients where these points are significantly different.^{28} This is particularly important because, although some excimer laser platforms provide an option as to where to lock the eyetracker and center treatment ablation zone, there are still laser platforms that force users to perform a treatment on the entrance pupil (ie, wholeeye aberrometer wavefrontguided ablation).^{28}
The assessment of angle kappa and its influence in treatment centration is also complicated by the difficulty in measuring the true visual axis. The CSCLR and corneal vertex are two of the fiducial points that most closely match the visual axis,^{4} but there is variability in the alignment of these points, which can result in decentration even in an apparently wellcentered treatment. It is also important to consider that the method used in the current study to measure the pupil offset is specific to the Orbscan and that the results may not be interchangeable with pupil offset derived from other devices.
The pupil offset was found to be largest in the hyperopic group, but the scatter within all groups demonstrates the importance and need for consideration of how angle kappa may affect centration, efficacy, and safety in corneal laser refractive surgery and intraocular lens surgery when using lenses with multifocal optics.
$j(window).load(function(){ !function(f,b,e,v,n,t,s) {if(f.fbq)return;n=f.fbq=function(){n.callMethod? n.callMethod.apply(n,arguments):n.queue.push(arguments)}; if(!f._fbq)f._fbq=n;n.push=n;n.loaded=!0;n.version='2.0'; n.queue=[];t=b.createElement(e);t.async=!0; t.src=v;s=b.getElementsByTagName(e)[0]; s.parentNode.insertBefore(t,s)}(window,document,'script', 'https://connect.facebook.net/en_US/fbevents.js'); fbq('init', '472649976441888'); fbq('track', 'PageView'); });