Article Text

## Abstract

**Background/aims** To compare the accuracy of 13 formulas for intraocular lens (IOL) power calculation in cataract surgery.

**Methods** In this retrospective interventional case series, optical biometry measurements were entered into these formulas: Barrett Universal II (BUII) with and without anterior chamber depth (ACD) as a predictor, EVO 2.0 with and without ACD as a predictor, Haigis, Hoffer Q, Holladay 1, Holladay 2AL, Kane, Næser 2, Pearl-DGS, RBF 2.0, SRK/T, T2 and VRF. The mean prediction error (PE), median absolute error (MedAE), mean absolute error and percentage of eyes with a PE within ±0.25, ±0.50, ±0.75 and ±1.00 diopters (D) were calculated.

**Results** Two hundred consecutive eyes were enrolled. With all formulas, the mean PE was zero. The BUII with no ACD had the lowest standard deviation (±0.343 D), followed by the T2 (0.347 D), Kane (0.348 D), EVO 2.0 with no ACD (0.348 D) and BUII with ACD (0.353 D) formulas. The difference among the MedAEs of all formulas was statistically significant (p<0.0001); the lowest values were achieved with the Kane (0.214 D), RBF 2.0 (0.215 D), BUII with and without ACD (0.218 D) and SRK/T (0.223 D). A percentage ranging from 80% to 88.5% of eyes showed a PE within ±0.50 D and all formulas achieved more than 50% of eyes with a PE within ±0.25 D.

**Conclusion** All investigated formulas achieved good results; there was a tendency towards better outcomes with newer formulas. Traditional formulas can still be considered an accurate option.

- lens and zonules
- optics and refraction

## Statistics from Altmetric.com

## Introduction

The refractive accuracy of intraocular lens (IOL) power calculation is still far from being perfect, notwithstanding the increasing expectations of patients. Recent studies have reported that only 70%–80% of eyes achieved a postoperative refraction within ±0.50 diopters (D) of the predicted value.1–3 Higher percentages were reported for smaller series,4 5 but they do not yet reach those achieved with corneal refractive surgery by excimer laser, where more than 90% of eyes are within ±0.50 D of the intended refraction.6 As a consequence, several (mostly unpublished) formulas have been introduced with the ultimate goal of improving these numbers.7–10 The aim of this study was to compare the results of newer and older formulas when using measurements of optical biometry.

## Materials and methods

In a retrospective interventional study, we analysed consecutive patients undergoing cataract surgery with the same monofocal IOL between June 2018 and May 2019 at University Eye Clinic, San Martino-IST Hospital (Genoa, Italy). Patients were excluded for the following conditions: prior corneal or intraocular surgery, any corneal disease, contact lens wearing during the previous month and postoperative distance corrected visual acuity <0.8 (20/25). Patients were excluded also when optical biometry was not possible because of lens opacities. All patients gave their written consent.

Phacoemulsification was performed by one surgeon through a 2.75 mm temporal incision. All patients received the same IOL model (Si 255, Hoya), so that formula constant optimisation could be carried out.11

Patients underwent optical biometry with the Nidek AL-Scan (software V.1.03), which is based on partial coherence interferometry.

The IOL power was calculated according to the following formulas:

Barrett Universal II (hereafter referred to as the ‘BUII’): this is a modified unpublished version of the original formula,12 13 available at www.apacrs.org (accessed on 26 February 2020). Constant optimisation and data analysis were performed for us by the author, professor Barrett, MD. Since he found that slightly better results were achieved without using anterior chamber depth (ACD) measurements, we also included in our analysis the outcomes of the BUII with no ACD (BUII

_{noACD}).Emmetropia Verifying Optical (EVO) formula (V.2.0): this unpublished formula, developed by Tun Kuan Yeo, MD, is available at www.evoiolcalculator.com (accessed on 26 February 2020). Optimisation and data analysis were carried out by Dr Yeo. Since he found that slightly better results were achieved without using ACD measurements, we also included in our analysis the outcomes of the EVO 2.0 with no ACD (EVO 2.0

_{noACD}).Haigis: this formula was entered into an Excel file and triple optimisation was carried.14 15

Hoffer Q: this formula was entered into an Excel file and constant optimisation was carried out using the Excel ‘goal/seek’ tool.11 16

Holladay 1: this formula was entered into an Excel file and constant optimisation was carried out using the Excel ‘goal/seek’ tool.11 17

Holladay 2: this unpublished formula was accessed on Holladay’s software (V.2019.0302, Holladay Consultant Software & Surgical Outcomes Assessment), which features an optional non-linear AL adjustment for eyes longer than 24.0 mm.18 Since we selected this option, we will refer to this formula as the Holladay 2AL.

Kane: this unpublished formula, developed by Jack Kane, MD, and available at www.iolformula.com (accessed on 26 February 2020), is based on theoretical optics and incorporates regression and artificial intelligence components.19

Næser 2: this formula is based on paraxial ray tracing, predicts the anatomical ACD (not the ELP) and considers the IOL as a ‘thick’ lens.7 For this study, we assumed an IOL edge thickness of 0.20 mm and an equal distribution of power on the anterior and posterior IOL surface, although no information was provided by the manufacturer.

Pearl-DGS: this recently introduced formula (whose abbreviation stands for Precision Enhancement using Artificial Intelligence and Output Linearization) is available at www.iolsolver.com (accessed on 26 May 2020). Data analysis and constant optimisation were performed by one of the authors, Guillaume Debellemanière, MD.

Radial basis function (RBF): this method is based on artificial intelligence. We used V.2.0, available at www.rbfcalculator.com (accessed on 26 February 2020). To make the formula comparison fair, we did not exclude ‘out of bounds’ cases.

SRK/T: this formula was computed on Excel and constant optimisation was carried out using the ‘goal/seek’ tool.12 20

T2: this formula was developed as an improvement over the SRK/T.9 It was programmed on Excel; data analysis and constant optimisation were carried out by one of the coauthors, David Cooke, MD.

VRF-IOL: this is a thin-lens formula, developed by Oleksiy V. Voytsekhivskyy, MD10; the results can be achieved after entering the biometric measurements into the specific software (VIOL Commander V.2.0.0.0). Data analysis and constant optimisation were carried out by the author of the formula.

Optimisation and data analysis of the Holladay 2AL, Kane and RBF formulas were done for us by Jack Kane, MD. LT was not entered (because not provided by the optical biometer) in several formulas in which it can be used, such as the BUII, Holladay 2AL, Kane, Pearl-DGS and RBF. Due to the lack of LT measurements, we did not investigate the Olsen formula.8

One month after surgery, when refraction is stable,21 we assessed the postoperative subjective refractive outcomes. We calculated the prediction error (PE) as the difference between the measured and the predicted postoperative refractive spherical equivalent for the power of the implanted IOL. In addition to the SD of the PE, we calculated the median absolute error (MedAE) and the mean absolute error (MAE), as well as the percentage of eyes with a PE≤±0.25, 0.50, 0.75 and 1.00 D.16 22 Predictions made by each formula were optimised by adjusting their constants to give an arithmetic PE of zero in the average case.

### Statistical analysis

Normality of data distribution was assessed by the Kolmogorov-Smirnov test, which revealed a non-Gaussian distribution of both the arithmetic and the absolute PEs. The Wilcoxon signed-rank test was used to assess whether the median of the PE was significantly different from zero. Comparison of both arithmetic PEs and absolute errors was performed by means of Friedman’s test with Dunn’s post hoc test. Cochran’s Q test was used to compare the percentage of eyes within ±0.25 and ±0.50 D of the predicted refraction. A p value less than 0.05 was considered statistically significant. For patients who had bilateral surgery, only the first eye operated on was considered for statistical analysis. All statistical analyses were carried out using GraphPad software (V.3.1, Instat) and MedCalc (V.12.3.0, MedCalc Software).

As previously reported,5 a sample size of at least 21 eyes was considered necessary to detect a difference in MedAE of 0.05 D with a power of 95% at a significance level of 5%.

## Results

We enrolled 205 eyes of 205 consecutive patients; five cases had to be excluded because the AL could not be measured. Therefore, we analysed 200 eyes of 200 subjects (mean age: 75.2±7.1 years; female: 118 (59%)). Table 1 reports the mean values of the measured parameters. Based on the AL, 10 eyes (5%) were classified as short (<22.00 mm), 154 (77%) as medium (22.00–24.50 mm), 26 (13%) as medium-long (24.51–26.00 mm) and 10 (5%) as long (>26.00 mm).

The optimised constants and the refractive outcomes of all formulas for the 200 eyes investigated in the present study are shown in table 2. The optimised constants with the AL-Scan are similar to those provided by the manufacturer (https://hoyasurgicaloptics.com, accessed on 26 February 2019), where the Hoffer Q pACD is 5.30, the Holladay 1 Surgeon Factor (SF) is 1.52, the SRK/T A-constant is 118.5, and the Haigis a0, a1 and a2 are −0.542, 0.161 and 0.204, respectively.

### Comparison of the arithmetic PE

As expected, all arithmetic PEs were close to zero due to constant optimisation. The SD ranged from 0.343 to 0.409 D: the lowest values were achieved by the BUII_{noACD} (0.343 D), T2 (0.347 D), Kane (0.348 D) and EVO 2.0_{noACD} (0.348 D) formulas. The median PE was not significantly different from zero for any formula. When all formulas were compared, Friedman’s test revealed a statistically significant difference for the PE (p<0.0001). Dunn’s post test showed a statistically significant difference for the BUII PE and BUII_{noACD} compared with all the other formulas (p<0.001); no difference was detected between the two versions of the BUII. A statistically significant difference was observed when the SRK/T was compared with the Haigis (p<0.05).

### Comparison of MedAE

The lowest MedAE values were obtained with the following formulas: Kane (0.214 D), RBF 2.0 (0.215 D), and BUII and BUII_{noACD} (0.218 D). The MedAEs showed a statistically significant difference from one another (p<0.0001). However, Dunn’s post test revealed only a few significant differences: the MedAE was lower with Kane’s formula than with the Holladay 2AL, Naeser and SRK/T formulas (p<0.05).

Figure 1 shows the box-and-whisker plots and the distribution around the MedAEs for the analysed formulas. The most interesting finding is the similar distribution of data among all formulas, with little differences between Kane’s (lowest MedAE, 0.214 D) and Næser’s (highest MedAE, 0.256 D).

### Percentage of eyes with a PE within 0.25, 0.50, 0.75 and 1.00 D

With all formulas, a range from 80% to 88.5% of eyes showed a PE within ±0.50 D (table 2). The T2 and the Holladay 1 formulas showed the highest percentage (88.5%), followed by the BUII_{noACD} (88.0%) and EVO 2.0_{noACD} (87.0%) formulas. All formulas achieved more than 50% of eyes with a PE of ±0.25 D or less, and the BUII was the only one to yield as many as 60% of eyes within this limit. Figure 2 shows the percentages of eyes within each PE interval, ranked according to the highest percentage of eyes within 0.5 D.

According to Cochran’s Q test, the proportion of eyes with a PE within ±0.50 D was significantly different (p=0.002) among the investigated formulas. Post-test multiple comparisons revealed, as the only statistically significant difference, that Naeser’s formula had a lower percentage than the BUII_{noACD}, T2 and Holladay 1 formulas.

## Discussion

Our data show that good refractive outcomes can be achieved with both newer and older formulas. With all of them, at least 51.5% and 80% of eyes had a PE within ±0.25 and ±0.50 D, respectively. Some of the newer formulas (eg, BUII_{noACD}, EVO 2.0_{noACD}, RBF 2.0 and Kane) achieved very good results, as they ranked first, for example, for the lowest MedAE. On the other hand, in some cases, older formulas reached better results: the Holladay 1 formula obtained the highest percentage (88.5%) of eyes with a PE within ±0.50 D and the SRK/T one of the lowest MedAEs (0.223 D). A more detailed discussion of each formula may help us to better understand the outcomes of this study.

The BUII_{noACD} formula ranked as one of the best formulas, as it achieved the highest rate of eyes with a PE within 0.25 D (60.0%), the second highest rate of eyes with a PE within 0.50 D (88.0%), the lowest SD (0.343 D) and the third lowest MedAE (0.218 D). The original BUII formula, including ACD as a predictor of the IOL position, performed slightly worse.

Several articles previously reported the high accuracy of this unpublished formula,1–3 5 23–25 so our study simply confirms their findings. Interestingly, Barrett (who analysed our data) found that in this sample his formula worked better without using the ACD measurements (the rate of eyes with a PE within 0.50 D increased from 85.5% to 88.0%).

The EVO 2.0 formula also performed very well, as 87.0% of eyes ended up with a PE within 0.50 D. Our data confirm those recently reported for another IOL model and another optical biometer, where as many as 90.7% of eyes reached that target.5 The EVO formula was also investigated in a larger study by Melles *et al*
19, who found a lower percentage of eyes with a PE within 0.50 D (~80%) and a higher MedAE (0.236).18 It is interesting to note that, as in the case of the BUII formula, the results (analysed by the author) proved to be better when the ACD measurements were not entered.

The results of the Haigis formula were not as good as expected. The percentage of eyes with a PE within 0.50 D was relatively low (82.0%). Together with the results of the BUII and EVO 2.0 formulas, this finding raises some suspicions about the accuracy of ACD measurements.

The Hoffer Q performed quite well, as 84% of eyes had a PE within 0.50 D, a result close to those recently reported by Savini *et al* for the OA-2000 biometer (85.3%),5 the Galilei G6 (Ziemer) (83.8%)23 and the Aladdin (Topcon EU, Visia Imaging) (89.0%).4

The Holladay 1 was one of the most relevant surprises of this study, due to an outstanding 88.5% of patients with a PE within 0.50 D, the highest of this study (together with the T2). The good performance is in good agreement with the data recently reported by Savini *et al*.5

The Holladay 2AL obtained fair results; probably the lack of LT measurements and the influence of ACD played a role, as the outcomes were slightly worse than those recently reported by Savini *et al*, who used a different optical biometer based on swept-source optical coherence tomography.5

Kane’s formula yielded some of the best outcomes, such as the lowest MedAE, one of the lowest SDs and one of the highest percentages of eyes with a PE within 0.50 D. The results are close to those previously reported.5 19 It therefore stands out as one of the most promising options among the new formulas.

The Næser 2 formula had worse results than those reported in a recent study, where the BUII, Næser 2 and Haigis formula were the most accurate.7 It should be noted that this is the only thick-lens formula among those evaluated in this study and therefore requires the physical properties of the IOL. Such properties were not available for the IOL of the present study, whereas they were known in the previous one. It is likely that this difference played a major role in the accuracy of the formula here. It should also be noted that the previous study used the Aladdin optical biometer (Tomey Europe, Visia Imaging Srl, Italy), so differences in biometry principles and quality may also play a role in the results.

The Pearl-DGS has not yet been investigated in any previous study of which we are aware. Our findings show good results, although it did not rank among the best formulas. Since it has just been released, it is likely that future versions of this formula will yield better outcomes. Moreover, the IOL investigated in this study was not in the database of the authors and this might have negatively affected the results.

The RBF 2.0 provided us with the second lowest MedAE (0.215 D) and a fairly high percentage of eyes with a PE within 0.50 D (85.0%). These results show that artificial intelligence is not yet able to obtain better results than optical models, although it is close to the best formulas. These findings are in good agreement with those previously reported using the RBF 2.0.5 19 24

The SRK/T was another huge surprise in this sample, as it achieved better results than many formulas recently introduced, notwithstanding its 29-year lifespan. Like other formulas, the outcomes of the SRK/T in the present study were better than in previous larger studies.1–3

The T2 performed very well, as shown by the highest percentage of eyes with a PE within 0.50 D and the second lowest SD. The results of the T2 are in good agreement with the data reported by other authors, who have always found it to rank among the best formulas.1 2 5

The VRF yielded good results, although not at the same level as the best performing formulas. The percentage of eyes with a PE within 0.50 D (84.5%) was close to that reported by Savini *et al* (86%)5 and higher than that reported by the author of the formula in the original paper (72.5%).10

The results of IOL power calculation with the AL-Scan have been assessed by two papers. Kaswin *et al* found that the Haigis and SRK/T formulas achieved worse results than in our series, but their study was flawed by the lack of constant optimisation and the small sample size (n=50).26 Suto *et al* reported higher MAEs for the Haigis (0.40 D) and SRK/T (0.37 D) formulas.27 In this paper as well, constant optimisation was flawed, as it was carried out in only 81 out of 262 eyes and just the a0 constant of the Haigis formula was optimised. Overall, our data show that the measurements with this optical biometer lead to a more accurate IOL power calculation if the formula constants are optimised.

As previously stated, some concerns were raised about ACD measurements provided by the AL-Scan for two reasons: (1) the results of the BUII and EVO2 formulas were better when this parameter was not used to estimate the IOL position and (2) the results of the Haigis and Næser 2 formulas, which heavily depend on ACD, were among the poorest ones. Our group previously found that the AL-Scan provided higher mean ACD values compared with the IOLMaster 500 (probably because of a different technology used to measure this parameter).28 Such a difference may play a role in the accuracy of some formulas that rely on ACD measurements to estimate the IOL position and were developed based on the ACD values provided by the IOLMaster 500. Further investigations, however, are required to assess the influence of ACD measurements by different devices on IOL power calculation.

In addition to its retrospective nature, this study has two limitations. First, the sample size is considerably smaller than in other recent studies which relied on very large samples. Second, formulas based on LT measurements, such as the Olsen formula,8 were not investigated. The lack of LT measurements, in addition, does not show the full capabilities of formulas that can use this parameter, such as the BUII, EVO 2.0, Holladay 2AL, Kane, Pearl-DGS and RBF 2.0.

In conclusion, our series demonstrates that both older and newer formulas can achieve good results when the IOL power is calculated based on the measurement of PCI.

## References

## Footnotes

Contributors GS: concept and design, data acquisition, data analysis, drafting manuscript, statistical analysis and final approval. MDM: concept and design, data acquisition, data analysis, drafting manuscript, statistical analysis and final approval. KJH: concept and design, data analysis, drafting manuscript, clinical revision of the manuscript, supervision and final approval. KN: concept and design, drafting manuscript, clinical revision of the manuscript, supervision and final approval. DS-L: concept and design, data analysis, drafting manuscript, clinical revision of the manuscript, securing funding and final approval. AV: concept and design, data acquisition, clinical revision of the manuscript and final approval. LDC: concept and design, data acquisition, clinical revision of the manuscript and final approval. CET: concept and design, drafting manuscript, clinical revision of the manuscript, supervision and final approval.

Funding The contribution of IRCCS - G.B. Bietti Foundation was supported by the Italian Ministry of Health and Fondazione Roma.

Competing interests KJH licenses the registered trademark name Hoffer to ensure accurate programming of his formulas to Carl Zeiss-Meditec (IOLMasters), Haag-Streit (LenStar/EyeStar), Heidelberg Engineering (Anterion), Oculus (Pentacam AXL), Movu (Argos), Nidek (AL-Scan), Tomey (OA-2000), Topcon EU/VisiaImaging (Aladdin), Ziemer (Galilei G6) (except Alcon (Verion)) and all A-scan biometer manufacturers. Dr Savini is a consultant to CSO, and has received speaker honoraria from Alcon, Oculus and Zeiss.The remaining authors have no financial interests.

Patient consent for publication Not required.

Ethics approval The study protocol was approved by the local ethics committee and the study complied with the tenets of the Declaration of Helsinki.

Provenance and peer review Not commissioned; externally peer reviewed.

Data availability statement Data are available on reasonable request. Deidentified participants' data (biometric measurements, implanted IOL power, postoperative refraction, prediction error by each formula) are available from Giacomo Savini, MD (giacomo.savini@fondazionebietti.it).

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