In our study, Holladay 1modified W-K yielded the best results in terms of MedAE, followed closely by Kane; and subsequently by Barrett, Pearl and EVO. Statistically significant differences were observed when comparing the MedAE of three new-generation formulas (Barrett, Kane and EVO) with the older SRK/T. Additionally, we found significant differences in the MedAE between Holladay 1modified W-K and SRK/Tmodified W-K.

In terms of MAE, the Holladay 1modified W-K and the Barrett formulas produced the most accurate results, followed again by other new-generation formulas (Pearl, Kane and EVO). There were no significant differences in the percentage of eyes within specific prediction error ranges among formulas. However, it is important to highlight that the Holladay 1modified W-K had the highest percentage of refractive errors within the range below 0.25 DP, followed by Kane and Barrett. Additionally, Holladay 2NP W-K had the highest percentage of refractive errors within the range below 0.50 DP, followed by Holladay 1modified W-K.

Our study is consistent with previous literature, which indicates a greater accuracy of the new-generation formulas and the Wang-Koch AL adjustment formulas for IOL power calculation in eyes with axial myopia. However, the question of which new-generation formula or WK AL adjustment formula performs better varies across studies. In a study involving 370 long eyes, Cheng et al. [26] reported the superiority of the Holladay 1-WK (both original and modified versions) and Kane formulas over the EVO and SRK/T-WK (original and modified) formulas, with statistically significant differences in the MedAE. In that study [26], the Kane and Holladay 1-WK formulas also achieved a lower MedAE compared to Hill RBF 2.0 and BU II. In another investigation including 164 myopic eyes, Zhang et al. [8] observed that the Holladay 1 formula, with both the first linear and nonlinear versions of the WK AL adjustment, and the EVO formula showed the lowest MAE and MedAE across all formulas; while Barrett and SRK/T with original AL adjustment displayed a slightly but not significantly lower accuracy. However, the modified WK AL adjustment for the Holladay 1 formula resulted in a significantly higher MedAE compared to the original version. In a study involving 136 long eyes, Liu et al. [24] observed that the Barrett formula produced the lowest MAE and MedAE values, followed by Hill RBF 2.0, Holladayoriginal W-K and Holladay 1modified W-K. In the mentioned study [24], Barrett obtained significantly lower MAEs than did the Holladay 1, SRK/T and both the original and modified SRK/T WK AL adjustment formulas; and the Hill RBF 2.0 and the Holladayoriginal W-K obtained significantly lower MAEs than did the SRK/T formula. Moreover, Barrett and Hill produced a significantly higher percentage of eyes within a 0.5 DP prediction error compared to Holladay 1, SRK/T and SRK/Tmodified W-K. In a subgroup analysis of eyes with an AL greater than or equal to 26.00 mm, Savini et al. [19] reported that the EVO formula yielded the lowest MAE and MedAE, followed by Kane and Barrett. The only WK AL adjustment formula employed was for the Holladay 2 formula, which produced substantially worse outcomes compared to EVO and Kane. Darcy et al. [14], in a subgroup analysis of 637 long eyes, found that the Kane formula had a significantly lower MAE compared to the Hill 2.0, Barrett, Holladay 2, Olsen, Haigis, and third-generation formulas. Additionally, Barrett also had a significantly lower MAE compared to all the formulas mentioned, except for Kane. A retrospective study comparing the Kane, Barrett, EVO, Haigis and SRK/T formulas in 175 eyes with high axial myopia reported that, in the subgroup of eyes with an AL between 26 and 28 mm, the EVO formula demonstrated significantly higher accuracy [12]. The study concluded that, overall, the Kane and EVO formulas achieved better results compared to the other formulas [12]. Finally, a recent investigation comparing seven AI-based formulas in 48 eyes with extremely long axial length (AL > 30 mm) found that the Hill-RBF 3.0 formula yielded the lowest root mean square absolute error (RMSAE), with statistically significant differences compared to the Karmona formula [27]. Meanwhile, the lowest MedAE was achieved by the Hoffer QST formula, followed by Kane and Pearl-DGS [27].

Pearl-DGS is a thick lens and AI-based formula, launched in 2017 by Debellemanière, Gatinel and Saad [21]. The performance of this formula has not been extensively studied in long eyes; therefore, we decided to include it in our study. We found that, in terms of MAE and MedAE, the Pearl-DGS formula achieved an accuracy comparable to other new-generation formulas (Kane, EVO, Barrett); however, no statistically significant differences were observed between Pearl-DGS and any of the other formulas included in our work. In a subgroup analysis of long eyes within a study comparing the formulas included in the ESCRS calculator [22], the authors observed that the Pearl-DGS formula achieved similar results to the other formulas in terms of MAE and MedAE. In the same study [22], the Kane formula demonstrated a significantly lower MAE compared to Barrett, Hoffer QST and EVO; and the Kane and Hill 3.0 formulas demonstrated a significantly lower MedAE compared to Hoffer QST, in the long-eye subgroup analysis.

Although the differences in refractive prediction error observed between formulas in our study were modest (approximately 0.10 D), they are consistent with those commonly reported in comparative studies of IOL power calculation formulas. We believe these differences are clinically relevant, as formulas that consistently perform better in controlled studies often yield noticeably superior outcomes in real-world clinical settings. Even small improvements in prediction accuracy can contribute to higher rates of emmetropia and patient satisfaction when applied across larger and more diverse patient populations. Therefore, we believe these findings have practical implications for surgical planning and formula selection.

As shown in the multiple regression analysis evaluating the association between biometric variables and refractive prediction error for each formula in our study (Table 4), certain formulas appear to be influenced by variability in specific biometric parameters. Consequently, differences in the biometric profile of patient populations across studies may contribute to the discrepancies in formula performance reported in the literature. When comparing the mean values of keratometry (K), anterior chamber depth (ACD), and axial length (AL) in our sample to those reported in other studies involving eyes with AL > 26.00 mm, we observe that our mean K and ACD values are very similar to those in previous reports. However, with respect to mean AL, while some studies report comparable values [8, 28], others include populations with considerably longer eyes (mean AL ranging from 28.78 to 29.63 mm) [9, 24, 26, 29,30,31]. This variation should be considered when interpreting the generalizability of our findings.

We decided to include several earlier IOL power calculation formulas in our investigation. SRK/T, Holladay 1, Haigis, and Holladay 2 are vergence formulas based on thin-lens optics [21]. All of them were included in the study due to their widespread use over the past decades, their demonstrated accuracy in IOL power calculation, and their continued use in clinical practice worldwide. We believe it is important to compare them with newer formulas to determine whether recent developments truly offer improved refractive prediction.

A limitation of our study is the relatively small number of eyes included (72), especially in relation to the high number of formulas analysed (12), which restricts the ability to detect statistically significant differences. In addition to its impact on statistical power, this also limits the generalizability of the results, which would likely have been greater with a more diverse and representative sample. The difficulty of retrospectively identifying long eyes that meet the inclusion criteria makes it challenging to achieve a large sample size. Nevertheless, the strict inclusion and exclusion criteria applied enhance the reliability and reproducibility of our study. Moreover, the primary aim of this study is not solely to identify statistical differences between formulas, but also to serve an exploratory purpose by generating hypotheses that can be tested in future studies.

Similarly, both eyes of some patients were included in the analysis, which may have influenced the statistical results due to the potential correlation between data from bilateral eyes [32]. To account for the correlation between pairs of eyes, we performed regression analysis with generalized estimating equations to explore the association between biometric variables and the RPE for each formula, as reported in previous studies [24, 25]. Also, we employed three different IOL models, which may introduce variability and serve as a potential source of error [32].

Recently, a new predictive error metric known as the root mean square absolute error (RMSAE) has been proposed for evaluating the performance of IOL power calculation formulas [33]. We did not include this measure in our analysis, as we employed more established metrics such as the MAE, MedAE, and the percentage of eyes within specific prediction error thresholds. However, the exclusion of RMSAE may be considered a limitation, as it has been reported to be a reliable indicator and has been utilized in recent studies [27, 34, 35].

The absence of LT and CCT represents another limitation, as it may affect the accuracy of some formulas employed in our investigation. However, these parameters are not mandatory for calculating IOL power in any of the formulas used. The biometry device utilized to obtain the biometric variables in our study, the IOL Master 500, is based on Partial-Coherence Interferometry (PCI), and is unable to measure CCT or LT values. Many surgeons worldwide still lack the opportunity to measure these parameters, as they use older biometers. Additionally, in certain cases, depending on the eye’s characteristics, newer biometers may not provide reliable LT values.

Among the strengths of our study, in addition to the strict inclusion and exclusion criteria, is the large number of formulas compared, including both widely used traditional formulas and newer generation formulas.