Quantifying posterior corneal astigmatism’s contribution to ocular residual astigmatism: implications for personalized refractive surgery

Abstract

Purpose

To determine the main sources of ocular residual astigmatism (ORA) and to quantify the contribution of posterior corneal astigmatism (PCA) to ORA.

Methods

This cross-sectional study was conducted on 177 right eyes of 177 patients. Anterior corneal astigmatism (ACA) and PCA were obtained from Pentacam. ORA was calculated by the vector difference between refractive astigmatism (RA) and ACA. The Pearson correlation coefficient was employed to assess relationships between variables, and the contribution of PCA to ORA was assessed using a physical method.

Results

The mean magnitude of PCA was 0.40 ± 0.15 D (range 0.10–0.70 D), while the mean ORA was 0.69 ± 0.33 D (range 0.05–1.63 D). There was a statistically significant correlation between the magnitudes of PCA and ORA (r = 0.38, P < 0.001). PCA had a positive effect on ORA in 171 eyes (96.61%), with a mean contribution value (CV) of 0.37 ± 0.16 D (range 0.01–0.70 D). The against-the-rule PCA exhibited both positive and negative effects on with-the-rule and oblique ORA, whereas with-the-rule, against-the-rule, and oblique PCA all positively influenced against-the-rule ORA. In 85 eyes (49.71%), the CV exceeded half of the ORA magnitude. In a small subset of eyes, PCA had a negative effect on ORA, potentially exceeding its magnitude.

Conclusions

Approximately half of the magnitude of ORA comes from PCA. The CV/ORA ratio may be used to personalize the surgical method proposed by Alpins, resulting in better outcomes.

Introduction

Astigmatism, one of the most prevalent refractive defects, refracted each point on an object into two focal lines with specific directions [1]. Clinically significant astigmatism (≥ 1.0 D) can impair contrast sensitivity and vision acuity, interfere with the development of vision, and cause a variety of visual symptoms [2]. Astigmatism is broadly classified into anterior corneal (ACA), ocular residual (ORA) and refractive astigmatism (RA). RA represents the total ocular astigmatism and is derived from the combined effects of ACA and ORA. ACA is the differential refractive power between the steepest and flattest meridians on the anterior corneal surface. ORA is referred to the component of ocular astigmatism that cannot be attributed to the anterior corneal surface [3]. In addition to encompassing astigmatism from the crystalline lens and posterior corneal surface, ORA also incorporates influences from retinal astigmatism and perceptual factors [4–6]. Due to its composite nature, ORA cannot be directly measured and is typically determined by calculating the vector difference between RA and ACA [4]. ORA is mainly against-the-rule (ATR) astigmatism; it can be significant and lead to poor outcomes after corneal refractive surgery or cataract surgery.

Posterior corneal astigmatism (PCA) is the power difference between the weakest and the strongest power meridians on the posterior corneal surface. Similar to ORA, PCA is predominantly ATR in orientation. ACA and PCA conduce to total corneal astigmatism. PCA plays an important role in cataract surgery. Generally speaking, ignoring the contribution of PCA, total corneal astigmatism was underestimated by 0.22 @180. This could cause unanticipated outcomes in eyes having toric IOL [7].

PCA is a significant contributor to ORA. As previously noted, both PCA and ORA are predominantly ATR astigmatism, suggesting that PCA exerts a positive influence on ORA. However, research exploring this relationship remains limited. Furthermore, the primary source of ORA is still a topic of debate. While it is widely accepted that lenticular astigmatism constitutes the largest component of ORA [8], some researchers argue that PCA plays a more dominant role [9].

The aim of the current study was to determine the main source of ORA and to quantify the specific contribution of PCA to ORA.

Materials and methods

In this study, the Declaration of Helsinki was followed and the Lianyungang Maternal and Child Health Hospital review board approved it. Following an explanation of the study’ nature, informed consent was obtained from each subject.

Participants selection

177 right eyes of 177 subjects aged 18–40 were evaluated in this cross-sectional study. Patients were included in this study if they had myopia and regular astigmatism, had good quality of Pentacam examination images, and best corrected monocular visual acuity 20/20 or better. Exclusion criteria included cataracts, glaucoma, keratoconus, previous ocular surgery, uveitis, irregular astigmatism, and nystagmus. A person who wore soft contact lenses within 2 weeks or rigid gas-permeable lenses within 1 month before examination, as well as anyone younger than 18 years or older than 40 years, was also excluded from the examination [10].

Examination protocol and collect parameters

All patients were examined in detail, including standard subjective refraction tests, intraocular pressure (IOP), slit-lamp examination of the anterior segment, corneal topography, and Pentacam measurements. The RA was received by subjective manifest refraction. Anterior segment data were exported from the Pentacam machine. ACA and PCA were calculated using corneal curvature radii in the central 3-mm ring mode.

Data analysis and calculations

A vector difference between RA and ACA was used to calculate ORA. Before calculating ORA, RA was transformed into the corneal plane, and both RA and ACA were converted into positive-cylinder notation. Please refer to our previous article for the specific calculation process of ORA [11].

The distribution of astigmatic axes was described by categorizing with-the-rule (WTR) astigmatism as positive-cylinder axes between 60° and 120°, and against-the-rule (ATR) as positive-cylinder axes between 1° to 30° or 150° to 180°. Oblique astigmatism was characterized by positive-cylinder axes falling within 31–59°or 121–149° [12].

Analysis process of the contribution of PCA to ORA

An astigmatism is composed of magnitude and direction and is analyzed as a vector.

ORA is a collective term encompassing PCA, lenticular astigmatism, retinal astigmatism, and other components. PCA can be accurately measured through instruments. To study the contribution of PCA to ORA, ORA is tentatively divided into two parts: PCA and other internal astigmatism. ORA is the vector sum of PCA and other internal astigmatism (Fig. 1). PCA was obtained from Pentacam, while ORA was calculated as the vector difference between RA and ACA. If the direction of ORA was defined as positive, when the angle difference between ORA and PCA vectors was smaller than 90° on the double angle vector diagram, PCA had a positive effect to ORA. At this point, PCA had a positive contribution value (PCV) to ORA. Conversely, if the angular difference exceeded 90°, PCA exerted a negative effect on ORA, yielding a negative contribution value (NCV).

Fig. 1.

Calculation process of the positive contribution values and negative contribution values. ORA,ocular residual astigmatism; PCA, posterior corneal astigmatism; The α was the included angle between the vectors of ORA and PCA

PCA was decomposed into two orthogonal vector components: one was in the direction of ORA and the other perpendicular to it. The perpendicular component completely canceled out with the component of other internal astigmatism. The vector component retained in the direction of ORA represented the part of ORA originating from PCA (Fig. 1). Through analysis of this vector component, we were able to quantify the specific contribution of PCA to the total ORA.

PCV were calculated by multiplying PCA by cosine (α), where α (range 0–90°) was the included angle between the vectors of ORA and PCA on the double angle vector diagram (Fig. 1A). The NCV were calculated by multiplying PCA by cosine (180°- α), where α ranged 90–180° (Fig. 1B). Both PCV and NCV represented the portion of ORA that originated from PCA.

Statistical methods

SPSS statistics software package version 25.0 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, PCA, RA, ORA and spherical refraction were normal or approximate normal distribution. They were expressed as the mean ± standard deviation (SD). Pearson correlation analysis was used to assess the correlation between ORA and PCA. Correlations were considered to be statistically significant when the associated p-value was < 0.05.

Results

Characteristics of the study population

Of the 177 patients (177 right eyes), 155 (87.57%) were male. The mean age of the patients was 20.3 ± 3.6 years old (range: 18–37 years). The mean sphere refraction was − 4.01 ± 1.65 D (range − 9.00 D to − 1.00 D). At the corneal plane, the mean RA was 0.77 ± 0.56 D (range 0.0–2.41 D). The prevalence of RA (≥ 1.00 D) was 28.25% (50 eyes). The RA was zero in 13 eyes (7.34%). The WTR RA was observed in 138 eyes (77.97%), ATR RA was seen in 14 eyes (7.91%), and oblique RA was discovered in 12 eyes (6.78%). Mean magnitude of ACA was 1.36 ± 0.65 D (range 0.10–3.00 D). The prevalence of ACA (≥ 1.00 D) was 71.75% (127 eyes). The WTR ACA was observed in 164 eyes (92.66%), the ATR ACA was seen in 6 eyes (3.39%), and oblique ACA was discovered in 7 eyes (3.95%). The mean magnitude of PCA was 0.40 ± 0.15 D (range 0.10–0.70 D). 32 eyes (18.08%) eyes had PCA of more than 0.50 D. 1 eyes (0.56%) had WTR PCA, 175 eyes (98.88%) shown ATR PCA, 1 eyes (0.56%) was oblique PCA. Mean magnitude of ORA was 0.69 ± 0.33 D (range 0.05–1.63 D). The prevalence of ORA (≥ 1.00 D) was 18.64% (33 eyes). 6 eyes (3.39%) were WTR ORA.159 eyes (89.83%) shown ATR ORA. 12 eyes (6.78%) had oblique ORA. The patient’s characteristics are listed in Table 1. Figure 2 indicates the distributions of different types of astigmatism.

Table 1.

Patient’s characteristics

Variable	Mean ± SD	Range
Age (y)	20.3 ± 3.6	18–37
Spherical refraction (D)	− 4.01 ± 1.65	− 9.00 to − 1.00
RA (D)	0.77 ± 0.56	0.0–2.41
ACA (D)	1.36 ± 0.65	0.10–3.00
PCA (D)	0.40 ± 0.15	0.10–0.70
ORA (D)	0.69 ± 0.33	0.05–1.63

RA, refractive astigmatism; ACA, anterior corneal astigmatism; PCA, posterior corneal astigmatism; ORA, ocular residual astigmatism

Fig. 2.

Distributions of different types of astigmatism

The relationship of PCA and ORA

A significant but weak linear correlation was found between the magnitude of PCA and ORA (r = 0.38, P < 0.001). The prediction equation for predicting ORA based on the amount of PCA is as follows (Fig. 3):

Fig. 3.

Correlation between the magnitude of ORA and PCA

ORA = 0.36 + 0.82 × PCA (R² = 0.14)

The contribution of PCA to ORA

The contribution of PCA to ORA was analyzed and found: the PCA of 6 eyes (3.39%) had a negative impact for ORA. The mean NCV was—0.18 ± 0.13 D (range − 0.40 D to − 0.06 D). Of the 6 eyes, 5 eyes had WTR ORA and 1 eye had oblique ORA, meanwhile, they were all ATR PCA. In contrast, 171 eyes (96.61%) shown a positive effect on ORA. The mean PCV was 0.37 ± 0.16 D (range 0.01–0.70 D). Of these, the ATR PCA had a positive effect on ATR ORA of 157 eyes (91.81%), oblique ORA of 11 eyes (6.43%) and WTR ORA of 1 eye (0.58%). One eye (0.58%) with WTR PCA and one eye (0.58%) with oblique PCA played a positive effect on ATR ORA (Table 2). Of the 171 eyes, the PCV of 86 eyes (50.29%) were no more than half of the magnitude of the ORA. Among them, the PCV of 6 eyes (3.51%) were equal to half of the magnitude of the ORA. The PCV were larger than ORA in 23 eyes (13.45%). The maximum of the PCV/ORA was 6.53. (Table 3).

Table 2.

The effects of PCA on ORA (N = 177)

	PCA
	Positive effects (n = 171)			Negative effects (n = 6)
	WTR	ATR	Oblique	WTR	ATR	Oblique
WTR ORA (n = 6)	0	1	0	0	5	0
ATR ORA (n = 159)	1	157	1	0	0	0
Oblique ORA (n = 12)	0	11	0	0	1	0

ORA, ocular residual astigmatism; PCA, posterior corneal astigmatism; WTR, with-the-rule; ATR, against-the-rule

Table 3.

The ratio of PCV/ORA (N = 171)

Eyes (Percentage of Eyes)
≤ 0.25	≤ 0.50	≤ 0.75	≤ 1.00	1.00 < , ≤ 6.53
18(10.53%)	86(50.29%)	130(76.02%)	148(86.55%)	23(13.45%)

PCV, positive contribution values; ORA, ocular residual astigmatism

Discussion

In this study, the mean PCA was 0.40 ± 0.15 D and 18.08% of eyes had PCA of more than 0.50 D. These findings align with Shao et al.’s investigation of 3769 right eyes (mean age 38.17 ± 20.22 years), which reported comparable PCA magnitudes (0.33 ± 0.16 D) and a 14.27% prevalence of PCA > 0.50 D [13]. Notably, Koch et al. [7] observed lower PCA values (0.30 ± 0.15 D) with only 9% of eyes surpassing 0.50 D in an older population (mean age 55 ± 20 years), suggesting an inverse relationship between PCA magnitude and advancing age. This age-dependent decline in PCA had been further substantiated by subsequent investigations [7, 13, 14]. The mean magnitude of ORA was 0.69 ± 0.33 D (range 0.05–1.63 D) and the prevalence of ORA (≥ 1.00 D) was 18.64% (33 eyes). These results were consistent with Wallerstein et al.’s large-scale analysis of 21,580 myopic eyes, which documented a similar ORA magnitude (0.73 ± 0.36 D) and 24.6% prevalence of ORA ≥ 1.00 D [15]. However, our observed ORA prevalence was substantially lower than the 46% reported by Frings et al. [16] in 2991 myopic patients (mean ORA 0.75 ± 0.39 D). This discrepancy likely stems from variations in age distribution across study populations. Specifically, the mean participant ages differed markedly between studies: 20.3 ± 3.6 years (current study), 29.0 ± 7.0 years (Wallerstein et al.), and 35.0 ± 10.0 years (Frings et al.). Our analysis reveals a distinct age-related pattern—both ORA magnitude and the prevalence of ORA ≥ 1.00 D demonstrate progressive increases with higher mean population age, potentially reflecting crystalline lens modifications during aging. Table 4 summarized the results of several studies mentioned above.

Table 4.

Summary of results from several studies on PCA and ORA

Study (Years)	Sample size(n)	Mean PCA(D)	PCA > 0.50D (%)	Mean ORA(D)	ORA ≥ 1.00D (%)	Mean age (Years)
Shao et al. (2017) [13]	3769	0.33 ± 0.16	14.27	Not reported	Not reported	38.17 ± 20.22
Koch et al. (2012) [7]	715	0.30 ± 0.15	9.00	Not reported	Not reported	55.0 ± 20.0
Wallerstein et al. (2020) [15]	21,580	Not reported	Not reported	0.73 ± 0.36	24.6	29.0 ± 7.0
Frings et al. (2014) [16]	2991	Not reported	Not reported	0.75 ± 0.39	46.0	35.0 ± 10.0
Current Study	177	0.40 ± 0.15	18.08	0.69 ± 0.33	18.64	20.3 ± 3.6

ORA, ocular residual astigmatism; PCA, posterior corneal astigmatism

Refractive surgery, including techniques like laser-assisted in situ keratomileusis (LASIK) and small incision lenticule extraction (SMILE), has emerged as a mainstream approach for correcting refractive errors such as myopia, hyperopia, and astigmatism. These procedures reshape the cornea to enhance visual acuity and reduced dependence on corrective lenses. However, despite the advancements in corneal laser surgery, some patients may still experience severe residual astigmatism after the surgery. Multiple studies have found that lower efficiency in astigmatic correction is related to higher ORA. A series of studies by Qian and her collaborators demonstrated that ORA influenced the efficacy of correcting myopic astigmatism in LASIK, LASEK, and SMILE [8, 17, 18]. This finding was corroborated by Chao et al. [19], who found that a higher preoperative ORA was related to a weaker efficacy of astigmatic correction after SMILE. These collective findings underscore the imperative for comprehensive preoperative ORA evaluation in refractive surgery planning. In addition, ORA is one of the most significant factors leading to poor postoperative visual quality in cataract surgery [20]. The impact of ORA on the effectiveness of cataract surgery is mainly achieved through PCA. The refractive power of a toric intraocular lens (IOL) was routinely calculated basing on anterior keratometry measurements. Therefore, the influence of PCA is ignored in astigmatism correction in traditional cataract surgery. Patients with significant PCA may experience relatively lower visual quality after implantation of intraocular lenses (IOLs) [21, 22]. It is possible to reduce residual astigmatism after toric IOL implantation by taking the posterior corneal astigmatism into account [22]. Jin et al. found that even using total corneal astigmatism (measured by Pentacam HR) to calculate the toric IOL in the case of high PCA may lead to potential over-correction in ATR and WTR eyes [23]. However, they still recommend using measured total corneal astigmatism for toric IOL calculation in ATR eyes. In conclusion, PCA plays a crucial role in IOL surgery and can significantly impact refractive outcomes. Surgeons must be aware of its presence and accurately measure its magnitude to ensure optimal visual results for their patients.

There was a significant but weak correlation between the magnitude of PCA and ORA (r = 0.38, P < 0.001). Linear regression modeling demonstrated limited predictive capability of PCA for ORA estimation (ORA = 0.82 × PCA + 0.36; R²: 0.14), consistent with previous findings. Piñero et al. [24] found a significant positive correlation between PCA and ORA (r = 0.34, P < 00.1) and obtained a linear equation (ORA = 0.64 × PCA + 0.55; R²: 0.08). Astigmatism on the retina, lens, and posterior corneal surface are all contributing factors to ORA [25]. At present, there is no consensus about which is the main sources of ORA. Additionally, there is limited research on the specific contribution of PCA to ORA. Our investigation provides novel insights into PCA’s contribution to ORA, demonstrating positive effects in 96.61% of cases (171 eyes) and negative effects in 3.39%. Specifically, ATR PCA had a positive effect on ATR ORA in 157 eyes (91.81%), oblique ORA in 11 eyes (6.43%) and WTR ORA in 1 eye (0.58%). One eye (0.58%) with WTR PCA and one eye (0.58%) with oblique PCA played positive effects on ATR ORA. As for the negative effect, the ATR PCA had a negative effect on WTR ORA of 5 eyes and oblique ORA of one eye (Table 2). All in all, ATR PCA had both positive and negative effects on WTR and oblique ORA. The WTR, ATR and oblique PCA can all play positive effects on ATR ORA. Of the 171 eyes, the PCV of 85 eyes (49.71%) were larger than half of the magnitude of ORA. The PCV were larger than ORA in 13.45% of eyes (Table 3). The PCV represented the portion of ORA that originates from PCA. Therefore, approximately half of the magnitude of ORA comes from PCA in myopic patients aged 18–40 years. Based on this, it can be inferred that both PCA and lenticular astigmatism may be the main sources of ORA. Before a certain age, the primary source of ORA is PCA. However, after the certain age, due to changes in the crystalline lens, the primary source of ORA shifts to the crystalline lens. The age at which this transition occurs is still under further investigation. Not all PCA have a positive impact on ORA. In a minority of eyes, PCA can have a negative effect on ORA and may be greater than ORA. Therefore, the specific contribution of PCA to ORA should be analyzed on a case-by-case basis in order to develop personalized treatment plans based on each patient’s unique corneal characteristics.

When ORA is large, Alpins [26] recommends applying 60% of the ORA correction on the cornea (rather than the usual 100%), while leaving 40% in the second-order component of wavefront refraction (instead of the customary 0%). Compared to conventional treatment, this design achieved better visual outcomes and greater reduction in ACA. But why was surgery performed according to this percentage? Alpins explained that this percentage was based on the average optimized distribution calculated from a previous study [27] on myopic astigmatism treatment in patients with forme fruste or mild keratoconus. This can be extended to normal eyes as well, as the amount of ORA is smaller in these patients, making it less challenging in terms of corneal and refractive astigmatism outcomes. However, he emphasized that individualizing this percentage would provide more scope for achieving better results. The PCV/ORA ratio may be used to personalize the surgical method proposed by Alpins, resulting in better outcomes. When significant ORA is present, we recommend calculating the PCV/ORA ratio first and then designing a refractive surgery plan based on the PCV/ORA value. For example, if PCV/ORA = 0.65, it indicates that 65% of the ORA is derived from PCA. In this case, it is advised to correct 65% of the ORA on the cornea and leave 35% in the wavefront refraction second-order component. This approach could also be applied to the correction of hyperopic astigmatism and simple astigmatism, though further research is necessary to confirm this.

Some limitations of our study should be mentioned. Firstly, our participants were myopic patients aged 18–37 years, which may limit the applicability of our conclusions to teenagers or elderly individuals. With increasing age, a significant change of ORA was induced by crystalline lens. Further research is needed to determine whether the contribution of PCA to ORA is different in people over 40 years. In elderly cataract patients, due to uncorrected manifest astigmatism measurements, the calculations of ORA may be inaccurate. Secondly, our findings may not be generalizable to emmetropic or hyperopic individuals. Finally, Patel and Tutchenko emphasize the need for standardized calibration protocols to minimize device-to-device variations [28]. We did not use other measuring devices to prove the differences between devices in calculating ORA. Our future research will pay attention to the differences in calculating ORA between different devices.

In conclusion, we found the PCA of 96.61% of total cases (171 eyes) shown positive effect on ORA. In a minority of eyes, PCA can have a negative effect on ORA and may be greater than ORA. Furthermore, ATR PCA had not only positive but also negative effects on WTR and oblique ORA. The WTR, ATR and oblique PCA can all play positive effects on ATR ORA. Notably, approximately half of the magnitude of ORA comes from PCA in myopic patients aged 18–37 years. The PCV/ORA ratio may be used to personalize the surgical method proposed by Alpins in 2008. Our results may help surgeons identify the main sources of ORA in individual eye to develop personalized correction programs and to optimize the outcome of astigmatism correction in the refractive surgery or cataract surgery.

Author contributions

JL the main author, DM Y and BW the corresponding authors designed the study, collected, analyzed, interpreted data, wrote the manuscript, approved the final version of the manuscript, and agreed to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. DX A, HQ W, YL collected, analyzed, and interpreted the manuscript data. JL wrote the main manuscript text. All authors read and approved the final manuscript.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors declare no competing interests.

Ethical approval and consent to participate

Institutional Review Board (IRB)/Ethics Committee approval was obtained by the Human Medical Ethics Committee of Lianyungang Maternal and Child Health Hospital. The authors confirm that the research followed the tenets of the Declaration of Helsinki and that informed consent was obtained from each subject after explanation of the nature and possible consequences of the study.