The study was approved by the Institutional Review Board of the Eye & Ear, Nose, and Throat (ENT) Hospital of Fudan University (ID: 2020005). Written informed consent was obtained from all participants for the use of their clinical data. All procedures adhered to the tenets of the Declaration of Helsinki.

High myopia was defined as AL ≥ 26.00 mm. The internal dataset was collected from the Eye & ENT Hospital of Fudan University from Jan 2019 to Aug 2021, and the external test datasets were collected from the Shanghai Aier Eye Hospital and the Eye Hospital of Wenzhou Medical University. Highly myopic eyes undergoing uneventful cataract surgeries with IOL implantation were reviewed. Eyes with complete preoperative biometric data and credible postoperative (one to two months after surgery) manifest refraction outcomes were included. Postoperative subjective refractive outcomes were assessed by a licensed optometrist and best corrected vision acuity was assessed at 5 m. Then, the refractions were standardized to a 6 m distance by adding − 0.03 D to the spherical equivalent. One eye was randomly selected if both eyes met the criteria. The exclusion criteria were eyes with postoperative best-corrected distance visual acuity (BCVA) less than 20/40, severe corneal opacity or other ocular diseases that may influence the accuracy of manifest refraction, and history of ocular trauma or surgery. In total, 1828 highly myopic eyes of 1828 patients were included in the internal dataset, and 151 highly myopic eyes of 151 patients were included in the external test dataset.

The complete preoperative biometric data included AL, flattest and steepest keratometry (K) values, anterior chamber depth (ACD, as measured from corneal epithelium to the lens), lens thickness (LT), and horizontal corneal diameter (CD), which were all measured by IOLMaster 700 (version 1.50, Carl Zeiss Meditec AG, Jena, Germany). In the internal dataset, the implanted IOL types included data obtained from the Tecnis ZCB00 (33 cases), Tecnis ZA9003 (29 cases), Zeiss CT ASPHINA 409MP (353 cases), Alcon SN60WF (34 cases), HumanOptics MC X11 ASP (658 cases), HumanOptics ASPIRA-aAY (39 cases), Rayner 920H (646 cases), and Ophtec B.V. 52501TW/TY (36 cases). In the external test dataset, the implanted IOL types included data from the Alcon SN6CWS (33 cases), Alcon SN60WF IQ (63 cases), Bausch & Lomb Akreos AO MI60 (28 cases), HumanOptics ASPIRA-aAY (7 cases), and Zeiss CT ASPHINA 509 M (20 cases). “A constants” were obtained from the IOL Con website (www.iolcon.org) [46] for each IOL type, as advised by Professors Hoffer and Savini [17].

The internal dataset was randomly split into training dataset and internal test dataset with a fixed ratio (8:2, 1462 eyes in the training dataset and 366 eyes). The machine learning features used can be classified into three categories: (1) ocular biometrics: AL, flattest and steepest K values, ACD, LT, and CD; (2) IOL information: implanted IOL power and A constants suggested by the IOL Con website for each IOL type; (3) parameters after transformations: predicted refractions calculated by the Haigis and the SRK/T theoretical formulas [18, 19].

The training dataset consisted of 1462 highly myopic eyes with complete information that was used for modeling. After a series of attempts at feature selection and combination, we constructed two sets of learning features. In addition to all ocular biometric features and all IOL information features, feature set 1 incorporates results from the Haigis and SRK/T formulas, while feature set 2 only contains results from the Haigis formula. Two supervised learning models, i.e., the XGBoost and the SVR, were trained with each of the two sets of learning features. The actual postoperative manifest refraction (presented as spherical equivalent [SE]) was set as the training target. Therefore, we obtained four sub-models, each of which can function independently for IOL power calculation in highly myopic eyes. To increase the model robustness, we adopted the weighted average of the calculation results from four sub-models and generated an assembled prediction model. Figure 1 shows the flow diagram of the model construction. Based on this assembled model, we built a novel IOL power calculation formula, named the Zhu-Lu formula (IOL power calculation formula for highly myopic eyes developed by Zhu and Lu). The software used in model construction was Python 3.7 with the scikit-learn package.

The flow diagram of model construction. In addition to all ocular biometric features and all IOL information features, feature set 1 incorporates results from the Haigis and SRK/T formulas, while feature set 2 only shows results from the Haigis formula. AL, axial length; CD, horizontal corneal diameter; LT, lens thickness; ACD, anterior chamber depth; IOL, intraocular lens; XGBoost, eXtreme Gradient Boosting; SVR, support vector regression; BUII, Barrett Universal II; EVO, Emmetropia Verifying Optical; RBF, Radial Basis Function; SD, standard deviation; PE, prediction error; D, diopter; MAE, mean absolute error; MedAE, median absolute error

The internal and external test datasets consisted of 366 and 151 highly myopic eyes, respectively. Due to the restriction of AL input (up to 35.00 mm) of the Kane and RBF 3.0 formulas, five cases in the internal test dataset and one case in the external test dataset were further excluded from analysis. The remaining 361 and 150 eyes in the internal and external test datasets were used. The prediction error (PE) was calculated as the actual postoperative refraction minus the predicted refraction back-calculated with the implanted IOL power using the BUII, EVO 2.0, Kane, Pearl-DGS, RBF 3.0 and Zhu-Lu formulas (URLs: additional sources [40,41,42,43,44,45,46]). The mean absolute errors (MAEs), median absolute errors (MedAEs) and percentages of eyes within ± 0.25 D, ± 0.50 D, ± 0.75 D, and ± 1.00 D of the PE were calculated and compared, as well as the cumulative percentages of eyes within different absolute errors. The formula performance index for each formula was calculated following recommendations by Hoffer et al. [17] which is based on four parameters: standard deviation (SD) of the PE, MedAE, the correlation between PE and AL (evaluated by the Pearson’s correlation test), and the inverse value of the percentage of eyes within ± 0.50 D of the PE. The higher the formula performance index, the more accurate the formula. Furthermore, accuracies were compared in subgroups according to different AL ranges (26.00–28.00 mm, 28.00–30.00 mm, and ≥ 30.00 mm).

The website for the Zhu-Lu formula is currently available (URL: https://HM-ZLF.com/). Surgeons could easily calculate the optimal IOL power and predicted refraction by entering highly myopic eye’s ocular biometry data, target refraction, and IOL constants including A constant for the SRK/T formula and a0, a1, a2 constants for the Haigis formula.

Quantitative data were expressed as means ± SD and categorical data were displayed as proportions in demographics. After data normality was assessed with the Kolmogorov-Smirnov test, the one-way ANOVA test or the Kruskal-Wallis test was performed to analyze the normally or non-normally distributed continuous variables. Categorical data from demographics were compared using the χ2 test. The Friedman test with Bonferroni correction was used to assess the differences of MedAEs among formulas. The Cochran’s Q test with Bonferroni correction was conducted for comparisons of percentages of eyes within ± 0.25 D, ± 0.50 D, ± 0.75 D, and ± 1.00 D of the PE among formulas. The cumulative percentages of eyes within different absolute errors were compared using the log-rank test. Correlations between AL and PEs were assessed using the Pearson’s correlation analysis. A P value of less than 0.05 was considered statistically significant. All statistical analyses were performed using SPSS (version 26.0, IBM Corp., New York, US).