Accuracy of 10 IOL power calculation formulas in 100 short eyes (≤ 22 mm)

Background To assess and compare the accuracy of 10 intraocular lens (IOL) power calculation formulas after cataract surgery in eyes with an axial length (AL) shorter than or equal to 22.00 mm. Methods A retrospective case series included 100 eyes with an AL ≤ 22.00 mm that underwent uneventful cataract surgery. The refractive prediction error (PE) was calculated using 10 different IOL power calculation formulas: Barrett Universal II, EVO 2.0, Haigis, Hill RBF 2.0, Hoffer Q, Holladay 1 and 2, Kane, SRK/T and SuperLadas. The median absolute prediction error (MedAE ± SD) and mean absolute prediction error (MAE ± SD) were calculated after adjusting the mean prediction error (ME) to 0. Results Hoffer Q obtained the lowest MedAE (0.292 D) after adjusting the ME to 0, followed very closely by EVO 2.0 (0.298 D) and Kane (0.300 D). EVO 2.0 and Kane obtained both the lowest MAE after adjusting the ME to 0 (0.386). Differences in MAE among the different formulas were not statistically significant ( p > 0.05). Conclusions Our study reflects a tendency of the EVO 2.0 formula and the Kane formula along with the older Hoffer Q formula, to predict more accurately the refractive outcomes in short eyes that undergo cataract phacoemulsification surgery compared to the other formulas, despite this difference could not be statistically proved.

In 2008 and 2011, Gavin et al. [7] and Aristodemou et al. [8], respectively, demonstrated that the Hoffer Q [9] was more accurate for IOL power calculation in short eyes compared to the existing formulas at that time (Holladay 1 [10] and SRK/T [11]). Over the last years, new formulas have appeared, which mainly attempt to improve the accuracy of their refractive outcomes by better predicting the ELP. Others, such as Hill RBF 2.0 [12] (hereafter "RBF 2.0"), are based on artificial intelligence and huge datasets to calculate the most appropriate IOL power for a given eye with particular biometric values [13].
There are multiple previous reports [4,6,[13][14][15][16][17][18] that have tried to elucidate which is the most accurate IOL formula for short eyes (defined as ≤ 22.00 mm of AL by almost all authors). However, even with the new artificial intelligence method, the ray tracing method and the last generation theoretical formulas, it seems difficult to arrive to a consensus about the matter. Nevertheless, two formulas seem to have recently obtained promising results: the EVO [19] and the Kane [20] formula.
The Kane formula was developed in 2017 and combines theoretical optics, regression and artificial intelligence based on huge database to make its predictions [13]. The Kane formula has been recently demonstrated to be significantly more accurate in calculating IOL power in short eyes (≤ 22.00 mm), compared to other formulas [21].
The emmetropia verifying optical (EVO) 2.0 formula is a recently created formula, based on the theory of emmetropization and the thick lens formula. Although it has not been widely studied, it has already shown promising results together with Kane in calculating IOL power in short eyes [22], and overall axial length [23].
The aim of this study is to assess and compare the accuracy of 10 old and new IOL power calculation formulas after standard cataract surgery in eyes with an axial length shorter than or equal to 22.00 mm.

Subjects and methods
All cataract surgeries conducted at Hospital Universitari Arnau de Vilanova de Lleida (HUAV, Spain) between April 2014 and March 2020 were retrospectively reviewed. Relevant patients' data were obtained from the electronical medical record of our hospital (SAP software; Systems, Applications, Products in Data Processing).

Ethical Approval
Institutional ethics approval from the ethics committee of HUAV was obtained for this study. This work was designed following the recommendations on intraocular lens power formula accuracy study from Wang et al. [24] and Hoffer et al. [25] All uneventful sutureless phacoemulsification cataract surgeries in eyes with AL equal to or lower than 22.00 mm were included. In all cases, preoperative biometry was measured using the IOL Master 500 (IOL Master V 3.1, Zeiss), and manifest refraction was performed by an hospital optometrist from 1 to 3 months after the surgical procedure [25].
Exclusion criteria were corneal astigmatism higher than 3.50 dioptres, additional procedures during cataract surgery, previous intraocular surgery (including refractive surgery), previous corneal pathology, complicated cataract surgery (posterior capsule rupture, sulcus IOL placement), presence of post-operative complications, toric lens implantation, best post-operative corrected visual acuity worse than 20/40, and incomplete documentation.
Radius of curvature of the cornea in the flattest meridian (K1), radius of curvature of the cornea in the steepest meridian (K2), corneal cylinder power, anterior chamber depth (ACD) and AL were the only biometric measurements recorded. No details of lens thickness (LT), corneal central thickness (CCT) or corneal diameter (CD) measurements were registered.
Ten IOL power calculation formulas were evaluated: Barrett Universal II [26] (hereafter "BU II"), EVO 2.0, Haigis [27], Hoffer Q, Holladay 1 and 2 [28], Kane, RBF 2.0, SRK/T and SuperLadas [29]. The VERION™ Reference Unit (Vision Planner 3.1 software, Alcon Laboratories, Inc.) was used for calculating the Haigis, Hoffer Q, Holladay 1, Holladay 2, and SRK/T formulas. The BU II, EVO 2.0, Kane, RBF 2.0 and SuperLadas formulas were calculated online on their respective websites [12,19,20,26,29]. The A constant for SRK/T, Surgeon Factor (SF) for Holladay 1, pACD for Hoffer Q and the three constants (a 0 , a 1 , a 2 ) for Haigis were obtained from the ULIB (User Group for Laser Interference Biometry) online database [30], which contains optimized constants for the Zeiss IOL Master for each one of the lenses included in this study, except for EyeCee®ONE. The manufacturer provided constants were used for the latter. The Holladay 2 ACD constant was obtained for each lens by entering the A constant in the Verion™ Vision Planner constant converter, as it is not available in the ULIB database nor provided by the manufacturer. The same A constant used for SRK/T was also used to calculate the BU II, EVO 2.0, Kane, RBF 2.0 and SuperLadas formulas, as recommended in the instructions of these calculators in absence of a specifically optimized A constant [12,19,20,26,29]. We could have used the A constant available in the Kane and BU II online calculators for the SN60WF and SN60AT lenses; however, they were very similar to the A constant obtained from the ULIB database. The VERION system was used instead of the IOL Master for calculating some formulas in order to obtain the Holladay 2 formula, as it is not available in the IOL Master 500.
The refractive prediction error (PE) for each eye and formula was calculated as follows: actual postoperative refraction expressed as spherical equivalent (SE) minus predicted post-operative refraction for the actual implanted IOL power.
To eliminate the systematic error derived from using non-specifically optimized IOL constants, the mean prediction error (ME) of each formula was zeroed out by adjusting the prediction error of each eye up or down by an amount equal to the arithmetic mean error [6,24]. After this correction was made, the median absolute prediction error (MedAE ± SD), the mean absolute prediction error (MAE ± SD), and the percentage of eyes within different prediction errors (± 0.25 D, ± 0.50 D, ± 1.00 D) were calculated for each formula. Zeroing out the arithmetic mean prediction error is an alternative method to lens constant optimization, described by Wang et al. [24], that has been previously employed in other studies [6,21].

Statistical analysis
For data collection and statistical analysis, Excel software (Microsoft Corp.) and SPSS software (International Business Machines, corp.) were used, respectively. A probability of less than 5% (P < 0.05) was considered statistically significant. The Kolmogorov-Smirnov test was used to check for normal distribution of data. To compare the MAE and the percentage of eyes within different prediction errors (± 0.25 D, ± 0.50 D, ± 1.00 D) between formulas, the Friedman test and Cochran's Q test were used, respectively [24]. Post hoc analysis with the Wilcoxon test was conducted if significant differences were found, applying the Bonferroni correction [21]. Statistical analysis for comparison between formulas was carried out after zeroing out the ME of each formula to 0 [6]. To account for correlation between pairs of eyes, a multiple regression analysis with GEE (generalized estimating equations) was performed to evaluate different parameters that contribute to the refractive prediction error for each formula: mean K (keratometry) and AL for Hoffer Q, Holladay 1 and SRK/T; mean K, AL and ACD for BU II, EVO 2.0, Haigis, Holladay 2, RBF 2.0 and SuperLadas; and mean K, AL, ACD and age for Kane [6].
We calculated the sample size for a repeated-measures ANOVA in the open-source statistical power application, G*Power (G*Power 3.1.9.7 software version for Windows). With a significance level of 0.05 (5%) and a test power of 0.8 (80%), 100 eyes were required, which matches with the number of eyes used in our study.

Results
Three-hundred eighty-eight cataract phacoemulsification surgeries in eyes with an AL equal to or shorter than 22.00 mm were evaluated. From these, after applying all inclusion and exclusion criteria listed before, 100 eyes of 84 patients were finally included in the study. The demographics, preoperative and post-operative data are shown in Table 1. IOL constants used are shown in Table 2. Generalized estimating equations with the parameters contributing to the refractive prediction error for each formula are exhibited in Table 3. We only found statistically significant parameters (p value < 0.05) contributing to the refractive prediction error in five formulas. Results of ME before and after adjusting the ME to 0 are shown in Table 4.
Final outcomes of the 10 formulas after adjusting the ME to 0 are shown in Table 5. Hoffer Q had the lowest median absolute prediction error (0.292 D), followed very closely by EVO 2.0 (0.298 D) and Kane (0.300 D). Figure 1 shows the distribution around the MedAE of the different formulas after adjusting the ME to 0. EVO 2.0 and Kane had the lowest mean absolute prediction error compared to the other formulas (0.386 D). However, no statistically significant differences in the MAE between formulas were found (p > 0.05).
Kane obtained the highest number of eyes with a PE equal to or lower than 0.25 D (48%), whereas Holladay 2 and EVO 2.0 achieved the highest percentage of eyes with a PE equal to or lower than ± 0.50 D (73%) and ± 0.75 D (89%), respectively. There were no statistically significant differences in the percentages of eyes within different PE between formulas (p > 0.05). In 64 to 71% of eyes, the PE was within ± 0.50 D, while, in 91% to 94% of eyes, the PE was within ± 1.00 D.

Discussion
When analysing the results after adjusting the ME to 0, Hoffer Q showed the lowest median absolute prediction error (0.292), followed very closely by EVO 2.0 (0.298) and Kane (0.300) in second and third  According to Aristodemou et al. [8], the predictability of theoretical post-operative refractive outcomes after cataract surgery substantially improves when optimizing IOL constants for AL and keratometry measurements with the IOL Master PCI biometer compared with using the manufacturer's A constants. Lens constant optimization is performed to reduce the arithmetic ME to 0, thereby eliminating the systematic myopic or hyperopic prediction error [24]. However, some online formula calculators limit the entry of IOL constants to only 2 decimal places, which makes it impossible to achieve a ME of exactly 0 [21]. For this reason, we chose another way of eliminating the systematic error, which consists in zeroing out the arithmetic ME by adjusting the PE for each eye up or down by an amount equal to the arithmetic ME in that group [6,24]. Eliminating the systematic error derived from using a non-optimized constant is necessary to evaluate more accurately the precision of the formula in predicting the post-operative refraction. Therefore, the refractive outcomes we most focus on are those obtained after adjusting the ME to 0.  There are some previous reports in literature comparing BU II, Haigis, Hoffer Q, Holladay 1, Holladay 2, RBF 2.0, SRK/T and other formulas not included in this study for eyes with an AL of less than 22.00 mm. The results of these studies are often dissimilar between them, and there is not a formula which clearly stands out over the others for its precision in estimating the refractive outcomes. Aristodemou et al. [8] proved the Hoffer Q formula was better than Holladay 1 and SRK/T in eyes shorter than 21 mm, that Hoffer Q and Holladay 1 were equal for eyes between 21.00 and 21.49 mm, and that Hoffer Q, Holladay 1 and SRK/T were equal for AL's between 21.50 and 21.99 mm. Gavin et al. [7] proved the Hoffer Q formula was significantly more accurate than SRK/T in eyes with an AL shorter than 22.00 mm. Kane et al. [16] found no statistically significant differences in the MAE between BU II, Haigis, Hoffer Q, Holladay 1, Holladay 2, SRK/T and T2 for short eyes. However, Holladay 1 performed the best in terms of MAE [16]. Göcke et al. [6] did not find significant differences between BU II, Hoffer Q, Holladay 1 and 2, Olsen, RBF 2.0 and SRK/T when comparing the MAE in a series of 86 eyes with an AL equal to or less than 22.00 mm. However, in this work [6], Holladay 2 performed the best after adjusting the ME to zero. In another study of IOL calculation formula accuracy [31], Barrett Universal II obtained the lowest ME in the short eyes subgroup, while Hoffer Q obtained the highest, compared to Haigis, Holladay 1, Holladay 2, Olsen Haag-Streit, SRK/T and 4 Wang-Koch modified formulas (Haigis, Hoffer Q, Holladay 1 and SRK/T). In the mentioned work [31], however, the short eyes subgroup included those eyes with an AL ≤ 22.50 mm. A meta-analysis of accuracy of IOL power calculation formulas in short eyes [18] published in 2017, in which 6 formulas were compared, concluded that Haigis was superior to Hoffer Q, SRK/T and SRK II with statistical significance. However, no statistical difference between Haigis, Holladay 1 and Holladay 2 was identified [18]. In another study [17] involving 163 short eyes, there was no statistically significant difference in the MAE between the Haigis, Hoffer Q, Holladay 1, and SRK/T formula. However, Haigis obtained the lowest MAE [17]. Finally, a review article published by Hoffer et al. [4] on 2017 concluded that one can reliably depend on the Haigis, Hoffer Q and Holladay 2 formulas for IOL prediction in short eyes. However, this review [4] did not include EVO, Kane, RBF 2.0 and Super Ladas.
The Ladas Super Formula predicts the IOL power by using 1 of 5 existing formulas depending on what the literature has shown to be the most accurate formula for that particular combination of AL and keratometry [15]. To our knowledge, there is only one study [15] which analyses the precision of SuperLadas in predicting the refractive outcomes after cataract surgery. In such study [15], SuperLadas performed worse than Hill RBF and Holladay 1 in predicting the refractive outcomes for short eyes, but significantly better than BU II. Some late reports [21,22] present the EVO 2.0 and the Kane formulas as the most accurate for predicting the IOL power in short eyes that undergo cataract surgery. The Kane formula was developed in 2017 and combines theoretical optics, regression and artificial intelligence based on huge database to make its predictions [13]. The Kane formula has been recently demonstrated to be significantly more accurate in calculating IOL power in short eyes (≤ 22.00 mm), compared to the BU II, Haigis, Hoffer Q, Holladay 2, Olsen (2007 version), RBF 2.0 and SRK/T in a study carried out by Darcy et al. [21] that included 766 short eyes. Another study [13] comparing the Kane formula with the BU II, Haigis, Hoffer Q, Holladay 1, Holladay 2, Olsen, RBF 2.0 and SRK/T did not find statistically significant differences between them for the ≤ 22.00 mm AL subgroup, probably due to the low number of eyes included in this group (n = 46). In the mentioned study [13], Holladay 1 obtained the lowest ME, and Kane the third lowest, in the ≤ 22.00 mm AL subgroup.
The EVO 2.0 formula is a recently created formula, based on the theory of emmetropization and the thick lens formula. In a study [22] including 182 eyes with high axial hyperopia which underwent cataract phacoemulsification surgery, the Kane formula had a statistically significant lower MAE compared with all other formulas (BU II, Haigis, Hoffer Q, Holladay 1, Holladay 2, Olsen, RBF 2.0 and SRK/T) except the EVO 2.0. To our knowledge, this is the only study [22] of IOL formula accuracy in short eyes involving the EVO 2.0 formula. Other works [23,32] which analyse IOL formula accuracy overall axial length report promising results with regard to this formula. A summary of studies on formula accuracy in short eyes is exhibited in Table 6.
We explored different parameters contributing to the refractive prediction errors with each formula. We found that mean K is a factor significantly contributing to the refractive prediction errors in RBF 2.0, Haigis, SRK/T and BU II formulas. We also found that AL is a parameter significantly contributing to the refractive prediction errors in SuperLadas and BU II formulas.
There were some limitations in our study. The absence of lens thickness, corneal central thickness and corneal diameter measurements probably limits the capacity of some formulas to generate more accurate results [21]. However, none of these parameters is mandatory to calculate any of the formulas included in this work. Both CCT and LT are optional parameters for the EVO 2.0, Kane and RBF 2.0 formulas, while LT is an optional data to calculate the BU II and Holladay 2 formulas. The IOL Master 500 biometer is a device based on partial coherence interferometry (PCI) not able to estimate the LT nor the CCT. Many surgeons around the world still do not have the possibility to measure these parameters because they use older biometers.
Another limitation of our study is that, because lens constant optimization is difficult for some newer formulas, we decided to eliminate the systematic error by zeroing out the ME. The refractive outcomes derived from this method may be not exactly the same as those that would be obtained from lens constant optimization [6]. Also, we included different IOL types, which may be a source of error [25].
The relative low number of eyes included in this series was also a limitation, especially to obtain statistically significant results. However, this derives from a relative low incidence of short eyes in general population and from strict inclusion and exclusion criteria that make our results more reliable and reproducible. Nevertheless, this is one of the largest studies of IOL formula accuracy for short eyes until today. Another possible limitation may be the long recruitment period, which can introduce variations in surgical technique, surgeons, and refraction tests.
One of the strengths of our study is that it includes 10 popular IOL calculation formulas, including the new EVO 2.0 and Kane.

Conclusions
In conclusion, our study reflects a tendency of the EVO 2.0 formula and the Kane formula along with the older Hoffer Q formula, to predict more accurately the refractive outcomes in short eyes (AL < 22.00 mm) that undergo cataract phacoemulsification surgery compared to the other formulas, despite this difference could not be statistically proved.