Comparison of the formula accuracy for calculating multifocal intraocular lens power: a single center retrospective study in Korean patients

This study evaluated the accuracy of newer formulas (Barrett Universal II, EVO 2.0, Kane, Hoffer QST, and PEARL-DGS) and the Haigis formula in Korean patients with the Alcon TFNT multifocal intraocular lens. In total, 3100 randomly selected eyes of 3100 patients were retrospectively reviewed. After constant optimization, the standard deviation (SD) of the prediction error was assessed for the entire group, and the root mean square error was compared for short and long axial length (AL) subgroup analysis. The Cooke-modified AL (CMAL) was experimentally applied to the Haigis formula. All the newer formulas performed well, but they did not significantly outperform the Haigis formula. In addition, all the newer formulas exhibited significant myopic outcomes (− 0.23 to − 0.29 diopters) in long eyes. Application of the CMAL to the Haigis formula with single constant optimization produced similar behavior and higher correlation with the newer formulas. The CMAL-applied triple-optimized Haigis formula yielded a substantially smaller SD, even superior to the Barrett and Hoffer QST formulas. The AL modification algorithms such as the CMAL used in newer formulas to cope with optical biometry’s overestimation of the AL in long eyes seemed to overcompensate, particularly in the long eyes of the East Asian population.


Results
A total of 3100 eyes from 3100 patients were included.
Table 1 summarizes the patients' demographic characteristics.Overall, the newer formulas and Haigis formula showed a substantially lower standard deviation (SD) than the Hoffer Q, Holladay 1, and SRK/T formulas (Table 2).As this result was well anticipated 2,15,18,19,21 , these older two-variable formulas, legacies of the pre-optical biometry era, were excluded from further analysis to avoid less-engaging comparisons.
The Kane, EVO 2.0, and PEARL-DGS formulas exhibited similar levels of accuracy.The Haigis formula with both single and triple optimizations ranked next.The discrepancy between the three best-performing newer formulas and the conventional AL-applied Haigis formula (both single-and triple-optimized) was not statistically significant (P > 0.05, Table 3).The Barrett formula did not outperform the Haigis formula and significantly underperformed compared to the more recently developed formulas, except for the Hoffer QST formula, which utilizes a smaller number of variables 25 (Tables 2, 3).All novel formulas performed excellently in the short eye subgroup (see Supplementary Table S1).
While they exhibited statistically significant (P < 0.05, see Supplementary Table S1) hyperopic shifts, except for the Hoffer QST formula which resulted in myopia, their overall performances were generally better than those of the conventional axial length (AL)-applied Haigis formula (both single-and triple-optimized).However, it was not statistically significant by the root mean square error (RMSE) comparison (see Supplementary Table S2).
The newer formulas performed well in the medium eye subgroup.As this subgroup comprised > 90% of the entire population, accuracy in this subgroup was the main factor affecting overall performance (see Supplementary Table S3).
In the long eye subgroup, a marked difference was observed between the novel and Haigis formulas (Fig. 1, Table 4).All newer formulas showed a significant (P < 0.001, Table 4) myopic mean numerical prediction error (ME) ranging between − 0.2 D and − 0.3 D (Fig. 1a).In contrast, the Haigis formula (both single-and triple-optimized) had an ME close to zero (Fig. 1b).The myopic ME of the newer formulas increased the RMSE and absolute errors in this subgroup more than the older two-variable formulas in most cases, despite the SDs of the newer formulas being smaller.The RMSE comparison confirmed that the refractive errors of the new formulas were significantly higher than those of the Haigis (conventional AL-applied) formula (Table 5).The systemic deviation, not the random error, determined the performance and eventually affected the SD of the entire population.
The experimentally Cooke-modified AL (CMAL)-applied, single-optimized Haigis formula showed a similar myopic shift in long eyes (Fig. 1c).The correlation between the newer and Haigis formulas' prediction error increased with CMAL replacement (Table 6).Additionally, with CMAL application and triple optimization, the SD of the Haigis formula markedly decreased, even significantly smaller than that of the Barrett and Hoffer QST formulas (Tables 2, 3).
Another experiment with the PEARL-DGS formula, replacing the AL with the reversed CMAL (AL + 0.05467 × lens thickness [LT] − 1.23853)/0.95855),calculated using the conventional AL instead of the CMAL in its inner algorithm 15 , showed that the myopic shift substantially decreased in the long AL subgroup (Fig. 1d), and the overall SD decreased (see Supplementary Table S4) significantly (P < 0.001).The correlations between the prediction errors of the PEARL and EVO, as well as between PEARL and Kane formulas decreased with the reversed CMAL (Table 6).The comparison between the average values of conventional AL, CMAL, and reversed CMAL in AL subgroups are described (see Supplementary Table S5).
The Cochran's Q test revealed significant differences in the percentage of cases within 0.25, 0.50, and 0.75 D, but not within the 1.0 D range of absolute errors, as all formulas recorded > 99.4% within the 1.0 D range.All the newer formulas showed > 86.6% within the 0.50 D range (Fig. 2).However, they did not outperform the Haigis formula (both single-and triple-optimized) (see Supplementary Table S5).The separate results of the non-toric and toric IOL subgroups exhibited no significant differences compared to the overall analysis, including the newer formulas' myopic tendencies in long eyes (see Supplementary Tables S6-S8).

Discussion
To our knowledge, this report comprises the largest study population for the TFNT IOL and adheres to the generally accepted guidelines [22][23][24] .The use of high-quality data from uniform clinical settings, including a single IOL model and the same biometer throughout the study, is another strength of this study.Additionally, we compared the accuracy of the novel IOL formulas.The superiority of the newer formulas to the older two-variable formulas, as previously demonstrated with monofocal IOLs 2,18,21 is well reproduced, herein.
The performance of the minimally structured Haigis formula was comparable to that of the more sophisticated newer formulas; none of them significantly outperformed the Haigis formula.Interestingly, although there was no notable difference between the single-optimized, triple-optimized AL-applied, and single-optimized CMALapplied Haigis formulas, the accuracy of the triple-optimized CMAL-applied version improved significantly (P < 0.001, Table 3).The triple-optimized CMAL-applied Haigis formula was significantly superior to the Barrett and Hoffer QST formulas (P < 0.001, Table 3).A noticeable change was also observed in the optimized constant triplets with the CMAL (3.526/0.523/0)compared with those with AL (1.304/0.442/0.104),which demonstrated little difference from the default values (a1: 0.4, a2: 0.1).The nullification of the a2 constant is noteworthy, indicating that the CMAL had no influence on the ELP prediction.While our results are divergent from previous large population studies that demonstrated the superiority of the newer formulas over the Haigis formula 2,18,20,21 , they align well with the fundamental principle that formula accuracy comparison results can vary significantly depending on factors such as the test population characteristics, constant optimization, and clinical settings 17,22 .In our population, the alleged exceptional adjustability of the triplet-constant Haigis formula 15,17 enabled it to demonstrate comparability to the more complex but less flexible single-constant novel formulas.Depending on the clinical setting, investing extra time and effort in the undisclosed working of the online calculators may not guarantee more advantages than actually optimizing the Haigis formula to its own population and relying on it.
The most distinct outcome of this study was that all novel formulas showed statistically significant hyperopic results in short eyes (except for the Hoffer QST formula), and more pronounced myopic outcomes in long eyes (Fig. 1a), which contradicts the findings of previous reports [26][27][28] .Contrarily, the Haigis formula that used the conventional AL exhibited consistent performance throughout the entire AL (Fig. 1b).The IOLMaster 700 follows the AL calculation of its predecessors 13 .It tends to underestimate the AL in shorter eyes and overestimate it in longer eyes 13 , consequently resulting in myopic errors in shorter eyes and hyperopic errors in longer eyes 13,[26][27][28] .To compensate for this, the CMAL was developed and is considered to better resemble the actual AL 26 .The PEARL-DGS formula is known to incorporate the CMAL in its algorithm 15 ; this was ensured after a recent update (confirmed by the author through personal communication).Moreover, the Hoffer QST formula incorporated a customized AL adjustment algorithm developed using machine learning methods 25 .
As there was no way to further investigate the underlying causes of the unpublished newer formulas' myopic shifts in the long eyes, we experimentally applied the CMAL to the Haigis formula and single optimized, expecting that this experimental approach would provide insights into the workings of the newer formulas (both share the same optical principles, single constant, and AL modification).The single-optimized CMAL-applied Haigis formula revealed similar hyperopic shifts in short eyes and myopic shifts in long eyes (Fig. 1c), and the overall SD remained nearly unchanged (Table 2).The correlation between the errors of the Haigis and newer formulas increased after applying the CMAL to the Haigis formula with single optimization (Table 6).We also experimented with the online PEARL-DGS formula calculator, entering a reversed CMAL, calculating with the conventional AL instead of the CMAL in its inner algorithm 15 .The results showed that the myopic shift substantially decreased in the long AL subgroup (Fig. 1d), and the overall SD decreased significantly (P < 0.001, see Supplementary Table S4).The correlation between the errors of the PEARL and EVO, as well as between the PEARL and Kane formulas decreased with the reversed CMAL (Table 6).
Based on our results, we speculate that other three novel formulas (Barrett, EVO, and Kane) also use some AL modification algorithms more or less similar to the CMAL in their algorithms, which seem to overcorrect, yielding statistically significant hyperopic errors in short eyes and more pronounced myopic errors in the long eyes of our population.Future studies should further investigate the myopic results in the long AL range.
Another remarkable strength of this study was the clinical relevance of the results.Without awareness of this myopic deviation of the newer formulas in the long AL range, surgeons might deliberately aim for myopia in selecting IOL power, as in a monofocal IOL implantation, or in an attempt to compensate for the well-known hyperopic shifts [26][27][28] .For instance, in a normal distribution of postoperative refraction error when targeting emmetropia with an ME of − 0.28 D and an SD of 0.35 D, aiming for a mean refraction target of − 0.21 D would result in a proportion of 7.2% exceeding − 1.0 D, which corresponds to an approximately three-step difference in IOL power in long eyes.This extent of residual myopia, considering the low tolerance of multifocal IOL designs for myopia 10 , may potentially compromise spectacle independence and necessitate secondary interventions, such as IOL exchange or laser corneal surgeries.In case of a more myopic target or bigger SD, these suboptimal proportions would escalate.Moreover, as the population size increases, the clinical relevance of these findings would become more pronounced 19 .The less apparent scale of this pattern, compared to the well-known hyperopia in older formulas [26][27][28] , might have allowed them to be left undetected, thereby insidiously affecting the refractive outcomes.
A limitation of this study is the inclusion of eyes that underwent IOL exchange.To relieve patients' early preoperative discomfort, IOL exchange was performed within 3 months postoperatively before strong adhesion between the IOL and the lens capsule was established.Despite these eyes having no chance of being evaluated during the same period as others, excluding them would have resulted in a positive bias.Therefore, we included them, and the refractive results obtained at least 3 months after IOL exchange were used instead of the early post-cataract surgery outcomes to best meet the standards of this study.
In conclusion, although the new formulas performed well in Korean patients with multifocal TFNT IOL implantation, they did not show significant superiority over the Haigis formula using conventional AL, mainly because of substantial myopic errors in the long AL subgroup.The Haigis formula with CMAL application and single optimization showed similar myopic shifts in long eyes, while triple optimization yielded a significantly smaller SD than some of the newer formulas.Considering the potential clinical consequences when combined with conventional myopic targeting, surgeons implanting the same IOL in East Asian patients need to be aware of the newer formulas' potential myopic shifts in long eyes and take them into consideration when selecting IOL power.

Methods
This study conformed to the tenets of the Declaration of Helsinki and was approved by the Korean Public Institutional Review Board.Due to the anonymized data extraction and analysis, the requirement for informed consent was waived by the Korean Public Institutional Review Board.
We performed a retrospective chart review of consecutive patients who underwent cataract surgery between January 2020 and April 2022 at the Miracle eye clinic, Seoul, Korea.Eyes with uneventful in-the-bag implantation of the AcrySof TFNT IOL (including toric versions; Alcon Laboratories, Fort Worth, TX) were selected.The specifications of the studied IOL have been described in the literature 29 .
All patients were evaluated using IOLMaster 700 (software versions 1.88 to 1.90; Carl Zeiss Meditec AG, Jena, Germany) to obtain the following parameters: AL, conventional keratometry, total keratometry (TK), central corneal thickness (CCT), anterior chamber depth (measured from the corneal epithelium to the lens), LT, and horizontal corneal diameter (also known as white-to-white) 22 .The TK value was used to represent the corneal power in practice and in this study.The selection of IOL power in practice was based on the biometer printout, which provided results of four formulas using default constants: SRK/T (A-constant: 119.1),Hoffer Q (pACD: 5.61), Haigis (a0: 1.390, a1: 0.4, a2: 0.1), and Barrett (LF: 1.94, equivalent to A-constant 119.1 on the online calculator).Previous large population studies 2,18,21 have consistently shown that the Barrett formula is the most accurate among these.Therefore, the recommendation of the Barrett formula was primarily followed, especially in situations where there were discrepancies among the recommendations.
Four experienced surgeons performed all surgeries using the standard technique.Candidates for multifocal IOL implantation were strictly screened 3,7,8 .Surgeons strongly discouraged performing implantation on patients with any significant visual impairment due to ocular pathology other than cataracts.Patients with these unfavorable conditions, in whom the procedure was not indicated and who underwent implantation at their own request, were eventually excluded from the analysis.Eyes that had undergone additional previous or postoperative ocular surgery that may affect refractive status and eyes with intra-or postoperative complications were excluded.As the studied IOL has a power range of 6.0-34.0dioptres (D) and a toric range of 1.0-3.75D, extremely long (> 30 mm) or short eyes (< 20 mm) and eyes with severe corneal astigmatism (> 3.5 D) were automatically excluded.
Final postoperative refraction was evaluated by in-house optometrists using an automated refractometer (RK-F2, Canon, Tokyo, Japan) and confirmed subjectively by non-cycloplegic manifest refraction, using a 4 m lane and a − 0.25 D adjustment, at least 3 months after surgery 22,23 .Twenty-six eyes (0.8%) underwent IOL exchange with the same IOL model at different powers or toricities to reduce postoperative refractive errors, mostly within 1 month postoperatively.For these eyes, the final refractive results were obtained at least 3 months after the IOL exchange.Predicted refractions were calculated from the final implanted IOL power and biometric data before cataract surgery.Of the 5973 eyes that met the study criteria, by using the RAND() function in Excel spreadsheets (Microsoft, Redmond, WA), one eye from each patient was randomly selected [22][23][24] .
The Holladay 1 30 , Hoffer Q 31 , SRK/T 32 , and Haigis 33 formulas were programmed into Excel spreadsheets.The online calculators of the Barrett Universal II 34 , EVO 2.0 35 , Kane 36 , and PEARL-DGS 37 formulas were accessed using a robotic process automation software (UIPath Studio, Uipath, New York, NY).If the implanted IOL power was not within the range suggested by each formula, as in cases of IOL exchange, the predicted refraction was obtained by entering a different target refraction other than 0. The Hoffer QST formula's 25 dedicated research section 38 was used for optimized calculation (optimized pACD: 5.664).
The Hill-RBF formula 3.0 39 was not included because its online calculator refused automated access, and manual data transcription was not an alternative option because of the large data size.Instead, to obtain results from pure artificial intelligence (AI) based formula, the Nallasamy formula 40 was included for the initial calculation.

Table 1 .
Demographics of the patient population (N = 3100).Values are shown as numbers (percentage) or mean ± standard deviation.AL axial length, BCDVA best corrected distance visual acuity, D diopter, IOL intraocular lens, LogMAR Logarithm of the minimum angle of resolution, SE spherical equivalent, UCDVA Uncorrected distance visual acuity, UCIVA Uncorrected intermediate visual acuity, UCNVA Uncorrected near visual acuity.

Table 2 .
Prediction errors of each formula in the whole group.The optimized constants for the formulas are: AL axial length, CMAL Cooke-modified axial length, HSAL Haigis formula, single-optimized and AL-applied, HSCL Haigis formula, single-optimized and CMAL-applied, HTAL Haigis formula, triple-optimized and AL-applied, HTCL Haigis formula, triple-optimized and CMALapplied, MAE mean absolute error, ME mean numerical prediction error, MedAE median absolute error, RMSE root mean square numerical error, SD standard deviation, EVO Emmetropia Verifying Optical formula, Hoffer QST Hoffer Q/Savini/Taroni formula, PEARL-DGS Prediction Enhanced by Artificial Intelligence and output Linearization-Debellemanière Gatinel, and Saad.

Table 3 .
Statistical comparison of the standard deviation of the formulas for the whole group with adjusted P-values (heteroscedastic test and Holm correction).AL axial length, CMAL Cooke-modified axial length, HSAL Haigis formula, single-optimized and AL-applied, HSCL Haigis formula, single-optimized and CMALapplied, HTAL Haigis formula, triple-optimized and AL-applied, HTCL Haigis formula, triple-optimized and CMAL-applied, EVO Emmetropia Verifying Optical formula, Hoffer QST Hoffer Q/Savini/Taroni formula, PEARL-DGS Prediction Enhanced by Artificial Intelligence and output Linearization-Debellemanière Gatinel, and Saad.The bolded values with asterisks (*) represent significant differences between the formulas compared.

Table 4 .
Prediction errors of each formula in the long axial length subgroup.

Table 5 .
Statistical comparison of the root mean square error of the formulas for the long axial length subgroup with adjusted P-values (heteroscedastic test and Holm correction).AL axial length, CMAL Cookemodified axial length, HSAL Haigis formula, single-optimized and AL-applied, HSCL Haigis formula, singleoptimized and CMAL-applied, HTAL Haigis formula, triple-optimized and AL-applied, HTCL Haigis formula, triple-optimized and CMAL-applied, EVO Emmetropia Verifying Optical formula, Hoffer QST Hoffer Q/ Savini/Taroni formula, PEARL-DGS Prediction Enhanced by Artificial Intelligence and output Linearization-Debellemanière, Gatinel, and Saad.The bolded values with asterisks (*) represent significant differences between the formulas compared.