Highlights
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Evaluated 8 IOL formulas in 330 Asian eyes with extreme axial length (≥32.00 mm).
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Zhu-Lu and Cooke K6 achieved the lowest root mean square absolute errors.
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Cooke K6 achieved the lowest median absolute error across groups.
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Pearl-DGS, EVO, Cooke K6, and Zhu-Lu showed robust accuracy within ±0.50 D.
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Hoffer QST showed the greatest hyperopic drift with increasing axial length.
Objective
To compare the accuracy of 8 modern intraocular lens (IOL) power calculation formulas in Asian eyes with axial length (AL) ≥32.00 mm.
Design
Retrospective observational study to compare IOL power calculation formulas.
Subjects
A total of 330 eyes from 330 Asian cataract patients with AL ≥ 32.00 mm were included. Eyes with previous ocular surgery, vision-threatening corneal disease, or surgical complications were excluded.
Methods
The accuracy of eight formulas was evaluated: Barrett Universal II, Cooke K6, EVO 2.0, Hill-RBF 3.0, Kane, Pearl-DGS, Hoffer QST, and Zhu-Lu. To minimize systematic refractive bias, prediction error (PE) values were arithmetically adjusted to zero the mean PE for each formula and each IOL model. Due to the 35.00 mm AL input limit of Hill-RBF and Kane, refractive outcomes were analyzed separately in the Main Group (32.00 mm ≤ AL < 35.00 mm, n = 307) and Ultra-Long Group (AL ≥ 35.00 mm, n = 23). The correlation between PE and AL was also assessed.
Main Outcome Measures
Root mean square absolute error (RMSAE), median absolute error (MedAE), and percentages of eyes with prediction error within ±0.50 D.
Results
In the Main Group, Zhu-Lu showed the lowest RMSAE (0.562 D), with Cooke K6 ranking closely behind (0.563 D). Cooke K6 achieved the lowest MedAE (0.271 D), followed by EVO and Pearl-DGS (both 0.272 D). Pearl-DGS had the highest proportion of eyes within ±0.50 D (75.24%), whereas Zhu-Lu had the highest proportion within ±1.00 D (93.49%). Hoffer QST had the highest RMSAE (0.682 D), the highest MedAE (0.369 D), and the lowest proportion within ±0.50 D (61.89%). In the small Ultra-Long Group, Zhu-Lu yielded the lowest RMSAE (0.438 D) and the highest proportion within ±0.50 D (78.26%), whereas Cooke K6 had the lowest MedAE (0.281 D). Hoffer QST again showed the greatest hyperopic drift with increasing AL.
Conclusions
In eyes with AL ≥ 32.00 mm, modern formulas showed distinct differences in predictive accuracy. Based on an integrated assessment across multiple accuracy metrics, Cooke K6, EVO, Pearl-DGS, and Zhu-Lu are clinically preferable options for this specific population, although their relative strengths differ across endpoints.
INTRODUCTION
H igh myopia, commonly defined as an axial length (AL) exceeding 26 mm, is a major cause of visual impairment worldwide. , The burden is particularly heavy in East Asia, where myopia is highly prevalent. ,
Cataract develops much earlier in highly myopic eyes, making surgical intervention necessary at a relatively younger age. Although modern cataract surgery can substantially improve visual function, achieving accurate refractive prediction remains challenging in highly myopic eyes, especially those with extremely long axial lengths. Long eyes are prone to postoperative hyperopic error because traditional formulas tend to generate insufficient IOL power. This inaccuracy primarily arises from two major sources. First, AL measurement may be biased in eyes with posterior staphyloma, as the measured signal may correspond to the deepest point of the staphyloma rather than the fovea, thereby overestimating the true AL. , Second, conventional formulas may be less reliable in extremely long eyes, given that effective lens position (ELP) estimation remains an important source of error and formula performance may decline at ALs beyond the range in which these formulas were originally developed. ,
To address these limitations, surgeons now have access to advanced formulas based on diverse principles. The European Society of Cataract and Refractive Surgeons (ESCRS) calculator integrates seven modern options. Barrett Universal II (BUII) based on paraxial ray tracing treats the IOL as a thick lens, making it perform well across diverse patient populations. , Cooke K6 and EVO 2.0 are thick-lens vergence formulas, with EVO rooted in emmetropization theory. , The Kane formula and Hoffer QST both enhance thin-lens concepts with machine learning support. , Hill-RBF 3.0 takes a different approach as a purely data-driven artificial intelligence formula. , Pearl-DGS combines a thick-lens model with AI. , Beyond these, the Zhu-Lu formula is a newly developed ensemble machine learning model specifically trained on highly myopic eyes.
Despite these advancements, a critical gap still exists in the literature regarding the optimal formula selection for extreme cases (AL ≥ 32.00 mm). High myopia affects 10% to 20% of the East Asian population, and epidemiological studies from China, Singapore, and Japan indicate a substantial regional burden of high myopia and associated axial elongation. ,,, However, these patients are typically aggregated into broader high myopia cohorts (AL > 26 mm), resulting in a scarcity of independent research. ,,, Furthermore, except for Zhu-Lu, most formulas were derived from general Caucasian populations where high myopia is less prevalent. This results in a sparse representation of extremely long eyes in training datasets, which may compromise prediction accuracy for this specific subgroup. Moreover, given that the ocular anatomy of Asian populations may differ from that of Caucasians, the direct application of these formulas to Asian eyes with extreme axial elongation requires rigorous validation. To address this critical need, we leveraged the clinical database of the Shanghai High Myopia Study Group (ClinicalTrials.gov identifier NCT03062085). Established in 2015 at the Eye & ENT Hospital of Fudan University, this hospital based prospective study has continuously enrolled highly myopic patients undergoing cataract surgery, thereby providing a valuable clinical resource for the evaluation of formula performance in eyes with extreme axial elongation.
Therefore, this study aims to evaluate the refractive prediction accuracy of the 7 ESCRS formulas and the Zhu-Lu formula in Asian cataract patients with extremely long axial lengths (AL ≥ 32.00 mm). By identifying the most robust formulas for this rare but clinically challenging population, we seek to provide evidence-based guidance to optimize refractive outcomes for these patients.
METHODS
This retrospective observational study was approved by the Institutional Review Board of the Eye & ENT Hospital of Fudan University, and was conducted under the Shanghai High Myopia Study, which was registered at www.clinicaltrials.gov (NCT03062085). All procedures adhered to the tenets of the Declaration of Helsinki. Written informed consent for the use of clinical data was obtained from all patients prior to cataract surgery.
SUBJECTS
The dataset was collected from the Eye & ENT Hospital of Fudan University from Nov 2018 to Oct 2024. We reviewed medical records of patients aged 18 years or older with high myopia who underwent uneventful phacoemulsification and posterior chamber IOL implantation.
The inclusion criteria were: (1) AL ≥ 32.00 mm; (2) implantation of a monofocal IOL; (3) availability of complete preoperative biometric data; and (4) reliable postoperative manifest refraction measured between one and 3 months after surgery. Postoperative subjective refraction was performed by a licensed optometrist during routine follow up, and the spherical equivalent was calculated as the sphere plus half of the cylinder. Postoperative corrected distance visual acuity (CDVA) was measured at 5 m. To ensure the reliability of subjective refraction, eyes with postoperative CDVA worse than 20/60 were excluded. Patients were also excluded if they had a history of previous ocular surgery, vision-threatening corneal diseases, or severe intraoperative or postoperative complications. If both eyes met the eligibility criteria, one eye was randomly selected using the sample function in R software. In total, 330 highly myopic eyes of 330 patients were included in the final dataset.
Preoperative biometric parameters, including AL, anterior chamber depth (ACD), lens thickness (LT), white-to-white (WTW) and keratometry values, were measured by IOLMaster 700. All cataract surgeries were performed by experienced eye surgeons using standard phacoemulsification techniques. Two types of monofocal IOLs were implanted: the HumanOptics MC X11 ASP in 190 eyes and the Rayner 920H in 140 eyes.
Postoperative refraction was measured between one and 3 months after cataract surgery and compared with the predicted refraction obtained from 8 IOL power calculation formulas, including the Barrett Universal II (BUII), Cooke K6, and EVO 2.0, as well as five AI-based formulas, ie, Hill-RBF 3.0, Kane, Pearl-DGS, Hoffer QST, and Zhu-Lu. Calculations were performed using their corresponding web resources (eg, https://iolcalculator.escrs.org and https://HM-ZLF.com/ ).
For formula calculations, the original constants entered for the two implanted IOL models were as follows. For the HumanOptics MC X11 ASP, A constant = 119.4, pACD = 5.85, and Haigis constants a0 = 1.59, a1 = 0.40, and a2 = 0.10. For the Rayner 920H, A constant = 118.3, pACD = 5.21, and Haigis constants a0 = 1.02, a1 = 0.40, and a2 = 0.10. A keratometric index of 1.3375 was used for all formula calculations.
The prediction error (PE) was defined as the actual postoperative spherical equivalent (SE) minus the predicted SE. To assess formula accuracy while minimizing systematic refractive bias, we performed an arithmetic adjustment to zero the mean PE on the entire dataset (N = 330 for most formulas, and N = 307 for Hill-RBF and Kane because of their AL input limitations), stratified by IOL model, as recommended by Hoffer et al and Wang et al. Specifically, for each formula and each IOL model, the mean raw PE was subtracted from the individual PE values. This arithmetic adjustment was applied to the PE values rather than by iteratively modifying formula-specific constants in the online calculators. Unless otherwise specified, all subsequent accuracy and correlation analyses were based on the PE values after arithmetic adjustment.
Following this arithmetic adjustment, the absolute error (AE) was calculated as the absolute value of the PE. The accuracy of each formula was evaluated and compared through root mean square absolute error (RMSAE), median absolute error (MedAE), and the percentages of eyes with a PE within ±0.25 D, ±0.50 D, ±0.75 D, and ±1.00 D. In addition, the standard deviation (SD) of PE and the mean absolute error (MAE) were also provided.
Subsequently, given that the Kane and Hill-RBF 3.0 formulas restrict AL input to 35.00 mm, eyes with an AL ≥ 35.00 mm (n = 23) were classified as the Ultra-Long Group, while the remaining eyes (n = 307) constituted the Main Group. Analyses were performed separately for the Main Group and the Ultra-Long Group. Furthermore, to investigate potential lens-specific performance differences, we compared refractive predictions for the two IOL models respectively within the Main Group. Finally, the trends of PE with increasing AL were evaluated to investigate the refractive drift of each formula. As a supplementary analysis, refractive outcomes before arithmetic adjustment were summarized separately for the Main Group and Ultra-Long Group using the same accuracy metrics.
STATISTICAL ANALYSIS
All statistical analyses were performed using R software (version 4.5.2; R Foundation for Statistical Computing). Descriptive statistics were calculated, with quantitative data expressed as means ± standard deviation (SD) and categorical data displayed as proportions. Demographic characteristics were compared using the independent samples t-test or Pearson’s Chi-squared test. The normality of the PE distribution was assessed using the Kolmogorov–Smirnov test. Differences in MedAE between formulas were analyzed using the Friedman analysis of variance with Dunn’s post hoc test and Holm correction for multiple comparisons. The RMSAEs of PEs were compared using a bootstrap-t method with Holm correction, utilizing the specific R package provided by Holladay et al. A nonparametric Cochran Q test followed by the McNemar post hoc test was used to compare the percentage of eyes with a PE within ±0.50 D. Additionally, the relationship between AL and PE was evaluated using Pearson’s correlation analysis. A P value of less than 0.05 was considered statistically significant.
RESULTS
CHARACTERISTICS
The demographics and biometric characteristics of the whole cohort, stratified by the Main Group (32.00 mm ≤ AL < 35.00 mm) and the Ultra-Long Group (AL ≥ 35.00 mm), are summarized in Table 1 . The Main Group comprised 307 eyes with the AL ranging from 32.00 to 34.86 mm, while the Ultra-Long Group included 23 eyes with the AL ranging from 35.01 to 36.46 mm (independent samples t-test, P <.001). There were no statistically significant differences between the two groups regarding age, eye laterality, or the distribution of implanted IOL models (independent samples t-test and Pearson’s Chi-squared test, P >.05). However, a significant difference in sex distribution was observed (Pearson’s Chi-squared test, P =.018), with a higher proportion of male patients in the Ultra-Long Group.
TABLE 1
Demographics and Biometric Characteristics.
| Characteristic | Total (N = 330) |
Main Group (AL < 35.00 mm)
N = 307 |
Ultra-Long Group (AL ≥ 35.00 mm)
N = 23 |
P -Value |
|---|---|---|---|---|
| Age (years) | 58.7 ± 8.8 | 58.7 ± 8.9 | 59.0 ± 7.6 | .849 |
| Sex | .018 | |||
| Female | 192 (58%) | 184 (60%) | 8 (35%) | |
| Male | 138 (42%) | 123 (40%) | 15 (65%) | |
| Eye | .164 | |||
| Left | 161 (49%) | 153 (50%) | 8 (35%) | |
| Right | 169 (51%) | 154 (50%) | 15 (65%) | |
| IOL Model | .740 | |||
| MC X11 ASP | 190 (58%) | 176 (57%) | 14 (61%) | |
| Rayner 920H | 140 (42%) | 131 (43%) | 9 (39%) | |
| Axial length (mm) | 33.22 ± 1.02 | 33.03 ± 0.76 | 35.77 ± 0.48 | <.001 |
| Anterior chamber depth (mm) | 3.46 ± 0.38 | 3.47 ± 0.38 | 3.38 ± 0.36 | .234 |
| Keratometry K1 (D) | 43.25 ± 1.50 | 43.28 ± 1.50 | 42.84 ± 1.48 | .176 |
| Keratometry K2 (D) | 44.45 ± 1.61 | 44.46 ± 1.64 | 44.27 ± 1.10 | .459 |
| Corneal astigmatism (D) | 1.20 ± 0.75 | 1.18 ± 0.74 | 1.44 ± 0.87 | .177 |
| Postoperative SE (D) | −2.86 ± 1.18 | −2.84 ± 1.18 | −3.06 ± 1.28 | .441 |
The Main Group included eyes with 32.00 mm ≤ AL < 35.00 mm; the Ultra-Long Group included eyes with AL ≥ 35.00 mm.
MAIN GROUP
Following arithmetic adjustment to zero the mean PE on the full available dataset (N = 330 for most formulas, and N = 307 for Hill-RBF and Kane), the mean PE in the Main Group (n = 307) remained close to zero for all formulas, ranging from −0.031 D to 0.000 D. The detailed refractive accuracy outcomes are presented in Table 2 .
TABLE 2
Refractive Outcomes of Each Formula (Main Group, n = 307).
| Formula | Mean PE | SD | RMSAE | MAE | MedAE | ±0.25 D (%) | ±0.50 D (%) | ±0.75 D (%) | ±1.00 D (%) |
|---|---|---|---|---|---|---|---|---|---|
| Barrett | −0.005 | 0.595 | 0.594 | 0.409 | 0.288 | 42.35 | 73.62 | 85.34 | 93.16 |
| Cooke K6 | −0.005 | 0.564 | 0.563 | 0.387 | 0.271 | 46.58 | 74.27 | 87.95 | 93.16 |
| EVO | −0.017 | 0.595 | 0.594 | 0.401 | 0.272 | 46.58 | 74.92 | 85.67 | 92.83 |
| Hill-RBF | 0.000 | 0.585 | 0.584 | 0.427 | 0.329 | 37.13 | 68.73 | 87.62 | 92.83 |
| Hoffer QST | −0.031 | 0.682 | 0.682 a | 0.495 | 0.369 | 36.48 | 61.89 b | 78.50 | 87.30 |
| Kane | −0.000 | 0.581 | 0.580 | 0.418 | 0.315 | 43.65 | 70.36 | 84.69 | 92.51 |
| Pearl-DGS | −0.003 | 0.587 | 0.586 | 0.396 | 0.272 | 45.93 | 75.24 | 87.62 | 92.51 |
| Zhu-Lu | −0.013 | 0.563 | 0.562 | 0.398 | 0.299 | 42.02 | 74.27 | 87.62 | 93.49 |
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