Author Correction: The contribution of ocular residual astigmatism to anterior corneal astigmatism in refractive astigmatism eyes

To determine the distribution of ocular residual astigmatism (ORA) in astigmatic eyes and the influence on the anterior corneal (ACA) and refractive astigmatism (RA). A total of 165 children met the inclusion criteria. Right eyes’ data were analyzed. Using Thibos vector analysis to calculate ORA. Spearman correlation analysis was used to obtain the correlation between the magnitude of ORA, ACA and RA. The median magnitude of ORA in astigmatic eyes was 0.57 D, with interquartile range was 0.42 D. And they were main against-the-rule (57.6–75.8%) and oblique astigmatism (13.9–34.5%) ORA. The ORA in 140 eyes (84.8%) acted as an offset to ACA, meanwhile, 25 eyes (15.2%) superimposed it. About 98% (97.9–98.4%) against-the-rule and 75% (73.9–82.5%) oblique ORA counteracted ACA, nevertheless, all with-the-rule ORA had a superimposed effect on ACA. For with-the-rule ACA, about 86% (85.4–85.9%) ORA worked to offset it. There was statistically correlations between ORA and ACA (r = 0.17, P = 0.03), ORA and RA (r = − 0.27, P = 0.001). The magnitude of ocular residual astigmatism was relatively small in children’s astigmatic eyes. Both against-the-rule and oblique ORA can counteract with-the-rule ACA.

The orientation of astigmatism was grouped for with-the-rule (ie, correcting axis of positive cylinders at or near 90°), against-the-rule (ie, correcting axis of positive cylinders at or near 180°) and oblique astigmatism. There were different classification standards, including ± 15°, ± 20° and ± 30°. Statistical methods. SPSS statistics software package version 17.0 for Windows (IBM, Armonk, NY, USA) was used for the statistical analysis and calculations. Normality of all data samples was checked by means of the Kolmogorov-Smirnov test. The magnitude of ACA, RA and ORA all were non-normally distributed. The nonnormality measurement data were expressed as median value and interquartile distance. Correlation coefficients (Pearson or Spearman depending if normality condition could be assumed) were used to assess the correlation of ORA and ACA, RA. Correlations were considered to be statistically significant when the associated p-value was < 0.05.
ORA outcomes. Figure    www.nature.com/scientificreports/ The influence of ORA on ACA . In mathematics and physics fields, it was considered that the angles between two vectors cancel out each other when they were greater than 90° and overlap each other when they were less than 90°. Since the vector angle was twice as large as the axis, therefore, when the axial difference between ORA and ACA was greater than 45°, ORA will canceled out ACA. When the difference was less than 45°, ORA had the superposition effect on ACA. In this group of data, the ORA of 140 eyes (84.8%) counteracted the ACA, 25 eyes (15.2%) shown superimposed effect on ACA. The influence of ORA on ACA was analyzed and found that, with different classification standards, about 98% (from 93/95 to 123/125) against-the-rule and 75% (from 17/23 to 47/57) oblique ORA counteracted ACA, nevertheless, all with-the-rule ORA had a superimposed effect on ACA (Table 2). For with-the-rule ACA, about 86% (from 134/157 to 140/163) ORA worked to offset it. All with-the-rule ORA played a superimposed role on with-the-rule ACA, and all against-the-rule ORA canceled with-the-rule ACA, with approximately 76% (17/23-44/54) of oblique ORA cancelling with-the-rule ACA (Table 3).

Discussion
The ORA was defined as that cannot be attributed to the anterior corneal surface astigmatism, including posterior corneal astigmatism, lenticular astigmatism and retinal astigmatism 12,17 . It is also designated with other terminologies, such as lenticular astigmatism 18 . The ORA were main against-the-rule astigmatism in human eyes. So most clinicians suggested that the ORA provided a compensatory effect for ACA, which was normally with-therule in the normal population 14,15 . In this group of data, 95 (57.6%) to 125 (75.8%) eyes shown against-the-rule ORA with different limits of astigmatism axes classification. ORA of 140 eyes (84.8%) counteracted the ACA, 25 eyes (15.2%) shown superimposed effect on ACA. However, the specific impact of ORA on ACA remains unclear. Duke-Elder 19 considered that lenticular astigmatism can counteract with-the-rule corneal astigmatism and superimpose against-the-rule corneal astigmatism, however, the rule of oblique astigmatism is not clear. This paper describes the contribution of ORA to ACA, we found that, with different classification standards, about 98% (from 93/95 to 123/125) against-the-rule and 75% (from 17/23 to 47/57) oblique ORA counteracted ACA, nevertheless, all with-the-rule ORA had a superimposed effect on ACA (Table 2). For with-the-rule ACA, about 86% (from 134/157 to 140/163) ORA worked to offset it. All with-the-rule ORA played a superimposed   Table 3). The magnitude of ORA in children refractive astigmatism eyes range from 0.01D to 1.87D (median value: 0.57D, IQR : 0.42 D). This value is less than the results in normal adult eyes from other studies. Piñero et al. 20 found the magnitude of ORA in normal adult eyes ranged between 0.07 and 2.58 D, with a mean value of 0.79 D (SD: 0.43). Plech et al. 21 found it ranged from 0.13 to 2.88 D , with 0.78 ± 0.61 D. The difference of RA may be the main reason for this result. Spearman correlation analysis showed that there was statistically correlations between the magnitude of ORA and ACA (r = 0.17, P = 0.03), ORA and RA (r = -0.27, P = 0.001). This implies that ORA is more strongly but negatively correlated with RA than ACA, i.e., the greater the RA are, the smaller the ORA are. In addition, the magnitude of ORA also determines the ability of its influence on ACA. A study had shown that the prevalence of ACA is significantly higher than RA 16 . This findings suggested that low amounts of ORA counteracts too little ACA and eventually leads to too much RA. One recent study with comparison on the ORA between myopes and non-myopes found that ORA may decrease the contribution of ACA to RA in eyes with small amounts of ACA 22 .
To investigate the contribution of ORA to ACA is also crucially in laser surgery. The ORA had been shown to be ubiquitous after keratorefractive procedures and to be in relation with postoperative astigmatism 2,13,17 . Therefore, the contribution of ORA to ACA should not be ignored in corneal laser surgery. This suggests that before corneal refractive surgery, the magnitude and axis of ORA should be calculated first. For patients with high ORA / RA, the purpose of laser ablation is to retain corneal astigmatism with the same magnitude as ORA and the axial difference of 90°, which may be more beneficial. When large ORA existed, Alpins 23 suggested to leave 60% of ORA on the cornea (rather than the customary 100%) and 40% in the wavefront refraction 2ndorder component (instead of the usual 0%). This treatment produced a greater reduction in ACA and better visual outcomes than the conventional treatment. It is generally known that ablation on the steepest meridian of ACA can reduce the depth of ablation. If the ORA is decomposed to the steepest meridian of ACA and its vertical meridian, the amount of surgery can be redesigned, which may benefit the patients with thin cornea.
In conclusion, in this study we found the magnitude of ORA was relatively small in children's astigmatic eyes. Most of ORA (84.8% in this study) counteracted the anterior corneal astigmatism. There was a significant correlation between ACA, RA and ORA.
Ethics approval and consent to participate. The study protocol was approved by the Lianyungang Maternal and Child Health Hospital, adhered to the tenets of the Declaration of Helsinki.