The customized lens power calculation is crucial for ensuring satisfactory visual outcomes, thereby achieving successful refractive surgery. Our optimal stacking ML models achieved numerically lower MAE and SD than the MVF and demonstrated strong agreement with MVF. The optimal ML model yielded 48.46% and 40.77% of eyes with a predictive error within 0.25 D, which was approximately 4% and 5% more than the OCOS calculator of STAAR company. Our results demonstrated that surgeons could use stacking ML models to combine various ocular parameters for each case, achieving comparable accuracy to MVF in predicting postoperative sphere and SE. Furthermore, in low-to-moderate myopia (preoperative SE >  − 6.00 D), the MAE difference between our ML models and MVF was more obvious, indicating that stacking ML models may have a potential advantage in low-to-moderate myopia. For clinical application, we developed an ICL power calculator displaying the predicted ICL powers for ICL lens power selection.

According to the Van der Heijde vergence formula, phakic IOL lens power is theoretically calculated based on preoperative manifest refraction, keratometry, and expected lens position [27]. The present study involved preoperative manifest refraction, age, and ocular dimensional parameters (keratometry, AL, IOP, ACV, ACA, ACD, scotopic PD, and Pentacam PD) as predictors of postoperative refractive errors. As expected, preoperative refraction errors (SE and sphere) were the most important factor in predicting postoperative refractive outcomes. The crystalline lens in phakic eyes after ICL implantation retains accommodation function to compensate for the postoperative refractive error caused by the change of ICL position. Thus, we considered that the recovery of accommodation function affects postoperative refractive errors in phakic eyes, especially in the early postoperative period [28]. Previous studies reported that postoperative intended refractive errors varied with the severity of myopia [17,18,19], and higher myopia (with an SE ≤  − 6.00 D) tends to have worse accommodation function[17]. Since there are far more patients with high myopia (preoperative SE − 6.00 to − 10.00 D), the MAEs of MVF and ML models were lower in the high myopia group in NT-ICL cases. Contrarily, the MAEs were smaller in the low-to-moderate myopia group(previous SE >  − 6.00 D) in TICL cases.

Age is another important variable affecting accommodation function [29] and showed a certain correlation with the postoperative outcomes in our study. Luo et al. found that vision and accommodative functions improved significantly after ICL implantation in younger patients than in patients aged over 35 years [30]. Here, we observed that older patients usually have less accommodation during manifest refraction than younger patients, which may cause more stable preoperative and postoperative refraction evaluations. Nevertheless, our findings showed that the preoperative manifest refraction should be accurately determined.

The AL seems to have a significant impact on predicting postoperative refraction. Additional file 6 shows that the MAE and SD are smaller in AL, between 26 and 30 mm, which the larger sample size can explain in the moderately long AL. Previous studies with existing IOL formulas demonstrated strong correlations between AL and PE. In our study, only the PE of postoperative SE for NT-ICL by MVF was statistically correlated with AL (r =  − 0.222, P = 0.011). The fact that ICL is implanted in the sulcus but not the capsular bag explains this. The difference of MAE between MVF and ML models was larger in extreme AL (less than 26 mm and over 30 mm), which shows that ML-based methods have the potential to better capture the nonlinearity of the relationship between biometric variables, ICL power, and the postoperative refractive refraction, resulting in substantially smaller AL bias.

Herein, various ocular dimensional parameters (IOP, ACV, ACA, scotopic PD, and Pentacam PD) were involved in our models because of their potential association with the postoperative lens position [31, 32]. Previous studies have observed a tendency toward a hyperopic shift with a higher vault and vice versa [14,15,16, 33]. With the vault higher, the distance of ICL and ocular posterior polar increases so that ICL lens power increases and vice versa. In our study, we analyzed the correlation between PE and vault (Additional file 7). The result showed that PE of the MVF was significantly hyperopic (PE > 0.00 D) with a higher vault and myopic (PE < 0.00 D) with a lower vault (postoperative ICL SE: r = 0.245, P = 0.007; postoperative ICL sphere: r = 0.237, P = 0.009; postoperative TICL SE: r = 0.153, P = 0.03; postoperative TICL sphere: r = 0.142, P = 0.004). Our ML models demonstrated a flatter slope than MVF. However, the correlation did not reach statistical significance (SVR for ICL SE: r = 0.128, P = 0.161; XGBoost for TICL SE: r = 0.124, P = 0.079; XGBoost for TICL sphere: r = 0.114, P = 0.107), except for that of postoperative SE prediction after ICL by stacking RF (r = 0.206, P = 0.023). The result showed that our ML models might correct the vault bias by incorporating the ocular dimensional parameters related to the vault. Considering that we must enter preoperative parameters to predict postoperative refraction error in clinical applications and the limited accuracy of vault prediction, we did not include vault into our ML model.

Interestingly, the importance of these ocular dimensional parameters weighed heavily in the TICL dataset, which means that the postoperative refractive errors after TICL implantation was affected by more factors, including the preoperative lens power calculation and postoperative rotation [34, 35]. We found that the MAEs in the low and moderate myopia subgroups were smaller than those in the high and super high myopia. The difference in ocular structure in low and high myopia can explain this. The higher levels of myopia have a weaker ocular structure, such as a larger sulcus diameter, which may lead to less vault and easier rotation. The correlation of vault change and rotation was also observed in previous studies [36, 37]. Park et al. found that the absolute value of rotation was correlated with the spherical power of TICL [38]. The spherical power of TICL may increase its thickness and height, and the cylinder power of TICL may increase its asymmetry, which may contribute to its postoperative rotation. In addition, a large pupil size (over 4 mm) may influence the prolate or aspheric shape of the cornea, which may cause the overcorrection of astigmatism [39]. Further studies with larger samples are needed to explore the relationship between ocular dimensional parameters and postoperative astigmatism in the TICL model.

Regarding the application of our model, we evaluated the most appropriate lens sphere for targeting emmetropia by enumerating the lens power 0.25 D step. The divergence between the NT-ICL and implanted lens power did not differ significantly from zero, showing that our ML models and the MVF are alternatives for the calculation of NT-ICL lens sphere. However, the recommended TICL sphere was lower by − 0.729 D than the implanted TICL sphere, which would cause a myopic shift, which may be explained by the larger interval of the TICL sphere provided by the manufacturer (0.50 D) than our interval (0.25 D). However, the result of divergence between recommended and implanted power reflects differences in clinical selection. However, it does not mean that our formula is not sufficiently accurate. Age and preoperative SE may affect the clinical selection. When designing the lens power, younger patients (under 30 years) tended to leave hyperopia. Contrarily, elder patients (over 40 years) tended to leave myopia for monovision design [40]. A hyperopic shift was more likely to occur in individuals with more severe myopia (> − 16.00 D), which may be because of the upper limit of the lens sphere (− 18.00 D) provided by the manufacturer. In clinical application, the predicted postoperative SE and corresponding ICL or NT-ICL spheres can be demonstrated for surgeons, making it convenient to adjust the lens power according to the needs and targets of each patient. Future studies are needed to explore a more personalized lens power design for different age groups and preoperative power groups.

The accuracy of our ML models can be attributed to the stacking ML technique, which is one of the strengths of our study. In addition, we trained ML models based on the merging data and the separate NT-ICL or TICL datasets to capture similar and different factors influencing postoperative refractive error. Thus, we provided a reliable result and a novel perspective on factors of postoperative refraction by exploring the interpretability of the ML model.

This study had some limitations. First, our study used data from a single center. Our data remains to be validated with an external multicenter dataset. Second, we did not predict the postoperative cylinder or calculate the TICL cylinder because more complex models are needed to perform vector analysis. Our ML models for TICL showed a relatively weaker correlation with MVF and a larger disparity of lens sphere. Third, we excluded the patients with a myopic target (patients with presbyopia or monovision surgeries) when analyzing the result of recommended lens power (Table 5). Since we did not involve the target refractive error when calculating our model’s lens power, the target refractive error varies with individuals. Recommended lens power has to be adjusted manually according to the predictive result in patients with monovision design. Finally, we used data obtained over a short period postoperatively. ICL implantation through a 3-mm corneal incision has a negligible effect on the refractive outcome and is less subject to the wound-healing response of the cornea. We included the cases of two experienced surgeons (XYW and XTZ) performed with over 1,000 eyes of ICL implantation to ensure the surgical quality and minimize the other causes of refractive instability. In our clinical experience, the perioperative medication stopped one week after surgery, and the effect of viscosurgical device disappeared after one day postoperatively. Our previous study found that patients with high myopia showed continuous myopic progression and axial elongation at an adult age one year after ICL surgery [41]. We did not use long-term visual outcomes. In the future, we will introduce external data, combine more parameters, and explore models for predicting astigmatism with wider applications and higher accuracy. The longer period of follow-up data will also be included to predict the refractive stability after ICL surgery.