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Comparison of formula accuracy for intraocular lens power calculation based on measurements by a swept-source optical coherence tomography optical biometer

Savini, Giacomo MD; Hoffer, Kenneth J. MD, FACS; Balducci, Nicole MD; Barboni, Piero MD; Schiano-Lomoriello, Domenico MD

Author Information
Journal of Cataract & Refractive Surgery: January 2020 - Volume 46 - Issue 1 - p 27-33
doi: 10.1016/j.jcrs.2019.08.044
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The OA-2000 (Tomey Corp.) is an optical biometer based on swept-source optical coherence tomography (SS-OCT). Previous studies have shown that it provides repeatable and reproducible measurements that are similar to those of the main benchmark for comparison, the IOLMaster 500 (Carl Zeiss Meditec).1 We have shown that such measurements lead to accurate intraocular lens (IOL) power calculation when entered into standard theoretical thin-lens vergence formulas,2 such as the Hoffer Q, Holladay 1, and SRK/T.3–5 Recently, however, several new formulas have been introduced, and authors using either the IOLMaster 500 or the Lenstar (Haag-Streit AG), have found that some of them are more accurate than the older standard formulas.6–8 The aim of this study was to investigate the refractive outcomes of older and new formulas using the measurements provided by the OA-2000.


This was a prospective interventional study. Consecutive patients having cataract surgery and implanted with nontoric, nonmultifocal IOLs were enrolled between March 2016 and December 2018 at Fondazione G.B. Bietti. Exclusion criteria were previous corneal or intraocular surgery, keratoconus and other corneal disease, contact lens usage during the previous month, and postoperative corrected distance visual acuity lower than 0.8 (20/25). Patients were also excluded when optical biometry measurements were not possible because of lens opacities (graded using Lens Opacities Classification System III).9 Before being included in the study, all patients were informed of its purpose and gave their written consent. The study protocol was approved by the Ethics Committee of the G.B. Bietti Foundation, and the study complied with the tenets of the Declaration of Helsinki.

Phacoemulsification was performed through a 2.75 mm temporal incision under topical anesthesia. All patients received the same IOL (AcrySof SN60WF; Alcon Laboratories, Inc.) so that formula constant optimization could be performed according to the protocols recommended by Hoffer et al.10

Before surgery, all patients underwent optical biometry with the OA-2000. This instrument combines an optical biometer, based on SS-OCT, and a Placido ring topographer. It can measure axial length (AL), keratometry (K), anterior chamber depth (ACD, measured from the epithelium to the lens), lens thickness (LT), corneal diameter, central corneal thickness, and pupil diameter. The SS-OCT uses a wavelength of 1060 nm. Placido disk corneal topography can simultaneously measure the radius of curvature of the cornea (r) at diameters of 2.5 mm and 3.0 mm, and K is calculated using the keratometric index of n = 1.3375. For the purposes of this study, the 2.5 mm diameter was selected, and the mean (Km) of the flattest (Kf) and steepest (Ks) meridian was recorded.

Intraocular lens power was calculated according to the following formulas:

  • Barrett Universal II (hereafter simply referred to as the “Barrett”): this formula is included in the current software of the optical biometer and its constant, known as the Lens Factor, is calculated from the A-constant. Because the formula is not published and cannot be entered into an Excel (Microsoft Corporation) file, it is difficult to optimize its constant. Optimization and data analysis, therefore, were performed for us by Dr. Barrett himself.11
  • Emmetropia Verifying Optical (EVO) formula: this unpublished formula, developed by Tun Kuan Yeo, MD, is available at (accessed on May 3, 2019). Because the formula is not published and cannot be entered into an Excel file, it is difficult to optimize its constant. Optimization and data analysis, therefore, were performed for us by Dr. Yeo himself.
  • Haigis: the original formula was computed in Excel, and triple optimization was performed according to the method described by Haigis, that is, using multiple linear regression to correlate d (dependent variable) with preoperative ACD and AL (independent variables).12,13
  • Hoffer Q: the formula was computed on Excel, and constant optimization was performed using the Excel “goal/seek” tool.3,10
  • Holladay 1: the original formula was computed on Excel, and constant optimization was performed using the Excel goal/seek tool.4–10
  • Holladay 2: this formula, which is unpublished, was accessed on Holladay software (version 2019.0302; Holladay Consultant Software & Surgical Outcomes Assessment), whose latest version contains, as an option, a nonlinear AL adjustment for eyes longer than 24.0 mm. The following parameters were entered into the software: age, AL, Kf, Ks, ACD, corneal diameter, and LT. Calculations were performed with and without AL adjustment, and the constant was optimized by the software itself.14
  • Kane: this unpublished formula, developed by Jack Kane, MD, is available at (accessed on May 3, 2019). According to the author, it “is based on theoretical optics and also incorporates regression and artificial intelligence components.”15 It uses AL, K, ACD, and patient sex along with optional variables of LT and central corneal thickness to predict the refractive outcome. Because the formula is not published and cannot be entered into an Excel file, it is difficult to optimize its constant. Optimization and data analysis, therefore, were performed for us by Dr. Kane himself.
  • Olsen: the version of this formula included in the optical biometer software (OlsenSS-OCT) is the one based on the C-constant,16 where the IOL position is predicted from the ACD and LT. We analyzed this version and the one available in PhacoOptics software (version; IOL innovations), which can predict the IOL position from 4 parameters: AL, K, ACD, and LT (Olsenstandalone).
  • Panacea: this unpublished formula, developed by David Flikier, MD, is available for free at (accessed on May 3, 2019). It is a thin-lens vergence formula that features the unique possibility of including the anterior-to-posterior corneal curvature ratio and the asphericity (Q value) of the anterior corneal surface. These values were obtained from a rotating Scheimpflug camera combined with Placido disk topography (Sirius, Schwind eye-tech-solutions GmbH & Co. KG). Because the formula is not published and cannot be entered into an Excel file, the value of the optimized A-constant had to be empirically derived by reiteration with multiple attempts until a zero mean prediction error (PE) was obtained.
  • Radial Basis Function (RBF): version 2.0, available at (accessed on May 3, 2019), was used. The optimized A-constant calculated from the SRK/T formula was entered into the online calculator.
  • SRK/T: the original formula was programmed on Excel, and constant optimization was performed using the goal/seek tool.5,10
  • T2: this formula was published as an improvement over the SRK/T.17 The formula was computed on Excel, and constant optimization was performed using the Excel goal/seek tool.10
  • VRF-IOL: the results of this formula, developed by Oleksiy V. Voytsekhivskyy, MD,18 were obtained after entering the biometric measurements in the specific software (VIOL Commander V. Optimization and data analysis were performed for us by Dr. Voytsekhivskyy himself.

A final evaluation was performed by assessing the postoperative subjective refractive outcomes 1 month postoperatively, which is when refractive stability can be expected with small-incision clear corneal surgery and this type of IOL.19 Postoperative subjective refraction was measured at 4 m and then adjusted to infinity by subtracting 0.25 D, as recommended by Simpson and Charman.20 The PE was calculated as the difference between the measured and predicted postoperative refractive spherical equivalent for the IOL power implanted (measured refraction − predicted refraction). A negative PE value indicates that the result achieved was more myopic than the predicted refraction, whereas a positive refractive prediction error represents a more hyperopic result. A calculation was made of the mean error, the median absolute error (MedAE), and the mean absolute error, as well as the rate of eyes with a PE ≤ ±0.50 D.10,21

Predictions made using each formula were optimized in retrospect by adjusting the respective constants to give an arithmetic PE of zero in the average case. As a result of constant optimization, it was possible to evaluate the statistical error as representing the optimum prediction error rather than offset errors related to incorrect lens constants or systematic errors in the measuring environment.

Statistical Analysis

Normality of data distribution was assessed by means of the Kolmogorov–Smirnov test. Comparison of the arithmetic PEs was performed by repeated-measures analysis of variance (ANOVA). Comparison of the absolute errors was performed by means of the Friedman test (nonparametric ANOVA) with the Dunn post-test. The Cochran Q test was used to compare the percentage of eyes within <±0.50 D of the predicted refraction. A P value less than .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 performed using GraphPad software (version 3.1; Instat) and MedCalc (version 12.3.0; MedCalc Software Inc.).

The distribution of the absolute PEs was graphically shown by means of box-and-whisker plots, where the central box represents the values from the lower to the upper quartile (25th to 75th percentile), the middle line represents the median value, and the horizontal lines represent the minimum and maximum values, excluding outliers and far out values, which are displayed as separate points. An outlier is defined as a value that is smaller than the lower quartile minus 1.5 times the interquartile range or larger than the upper quartile plus 1.5 times the interquartile range. A far out value is defined as a value that is smaller than the lower quartile minus three times the interquartile range or larger than the upper quartile plus three times the interquartile range.

Based on power and sample size calculations performed using the PS program (version 3.0.12; Dupont WD, Plummer WD Jr. PS: Power and Sample Size Calculation, version 3.0. Nashville, TN, Department of Biostatistics, Vanderbilt University, 2012. Available at:, it was estimated that a sample size of 21 eyes would be necessary to detect a difference in median absolute error of 0.05 D with a power of 95% at a significance level of 5%, given a within-subject SD for simulated keratometry equal to 0.06 D.1


We enrolled 155 eyes of 155 patients; 4 cases subsequently had to be excluded because the target refraction was too myopic for the RBF formula (between −3.25 D and −5.00 D) and 1 eye because of impossibility to achieve a correct AL measurement (in this case, the eye was classified as NO5, NC5, C5, and P5 according to the Lens Opacities Classification System III). Thus, the final analysis was performed on 150 eyes of 150 patients [mean age: 77.2 ± 10.0 years; men: 88 (58.7%)]. Table 1 contains the mean values of the measured parameters. Based on AL, 3 (2.00%) eyes were classified as short (<22.00 mm), 100 (66.67%) as medium (22.00 to 24.50 mm), 29 (19.33%) as medium long (24.51 to 26.00 mm), and 19 (12.67%) as long (>26.00 mm).

Table 1
Table 1:
Mean values of the parameters measured by the optical biometer.

Table 2 shows the optimized constants and the refractive outcomes of all formulas for the 150 eyes investigated in the present study. The optimized constants with the OA-2000 are slightly higher than those available on the User Group for Laser Interference Biometry website (, accessed on May 3, 2019), where the pACD of the Hoffer Q is 5.64, the Surgeon Factor of the Holladay 1 is 1.84, and the A-constant of the SRK/T is 119.0.

Table 2
Table 2:
Refractive outcomes obtained with the formulas investigated and the biometric measurements obtained with the SS-OCT optical biometer.

Repeated-measures ANOVA did not reveal any statistically significant difference for the mean PE (P = .2728), which was close to zero with all formulas due to constant optimization. Comparison of the absolute prediction errors, on the other hand, revealed a statistically significant difference (P = .0004). The lowest MedAE values were achieved with the following formulas: Kane (0.200 D), T2 (0.200 D), Barrett (0.202 D), EVO (0.205 D), RBF (0.205 D), Olsenstandalone (0.209 D), and VRF (0.215 D). Dunn post-test analysis showed that only the following paired comparison had statistically significant differences (P < .005): EVO vs Haigis, EVO vs Hoffer Q, and RBF vs Haigis.

Figure 1 shows the box-and-whisker plots and the distribution around the MedAE for the investigated formulas. The most interesting finding is the lack of far outliers for most of the recent formulas: Barrett, EVO, Kane, Olsenstandalone, T2, and VRF. The Holladay 2 (with and without AL adjustment) and the SRK/T, although they do not belong to the last generation formulas, similarly did not show far outliers. In contrast, a few far outliers were produced by older formulas such as the Haigis, Hoffer Q, and Holladay 1 and newer formulas such as the RBF and Panacea.

Figure 1
Figure 1:
Distribution of the absolute prediction errors. Formulas are ranked according to the median absolute error, increasing from left to right (EVO = Emmetropia Verifying Optical; RBF = Radial Basis Function).

With all formulas, the PE was ±0.50 D or less in at least 80% of eyes (Table 2 and Figure 2). The highest percentages were achieved with the EVO and RBF (90.67%), followed by the Kane (90.00%), Holladay 2 with AL adjustment and Olsenstandalone (89.33%), T2 (88.67%), and Barrett (88%) formulas. Interestingly, good outcomes were also obtained with more traditional vergence formulas (Haigis, Hoffer Q, Holladay 1, and SRK/T), which generated a PE within 0.50 D in a percentage of eyes between 84.67% and 85.33%. According to the Cochran Q test, the proportion of eyes with a PE within 0.50 D was statistically significantly different (P < .0001) among the investigated formulas. Table 3 shows the results of post-test multiple comparisons. Interestingly, we found that nine formulas yielded a percentage of eyes with a PE within 0.25 D higher than 55%.

Figure 2
Figure 2:
Stacked histogram comparing the percentage of cases with a given prediction error. Formulas are ranked according to the higher percentage for the prediction error within 0.50 D (EVO = Emmetropia Verifying Optical; RBF = Radial Basis Function; SS-OCT = swept-source optical coherence tomography).
Table 3
Table 3:
Multiple comparisons according to the Cochran Q test.

We also investigated the subgroup of long eyes (AL > 26.0 mm). Their outcomes are reported in Table 4, which shows that many formulas were able to achieve more than 80% of eyes with a PE of ±0.50 D or less. These include the Barrett (84.21%), EVO (89.47%), Haigis (84.21%), Holladay 2 with AL adjustment (84.21%), Kane (94.74%), OlsenSS-OCT (84.21%), Olsenstandalone (89.47%), RBF (94.74%), SRK/T (84.21%), and T2 (89.47%).

Table 4
Table 4:
Refractive outcomes obtained in eyes with AL >26.0 mm.


The present investigation was designed to assess the refractive outcomes of IOL power calculation using the measurements provided by the SS-OCT optical biometer and the newest and older formulas. The outcomes we obtained exceeded our expectations because all formulas, including the older ones (ie, the Haigis, Hoffer Q, Holladay 1, and SRK/T), achieved a PE of ±0.50 D or less in at least 80% of eyes. With 11 formulas out of 15, the percentage was even higher than 85%, and with two formulas, it was higher than 90%. To our knowledge, only rarely have similar results been reported before.22,23 Even more interestingly, with 9 formulas, more than 55% of eyes had a PE within 0.25 D: we should remember that according to the benchmark established by the National Health Service of the United Kingdom, this percentage should be reached for eyes with a PE within 0.50 and not 0.25 D.24 This finding demonstrates the improved accuracy of IOL power calculation because the standards were published 10 years ago.

In a previous multicenter study with the same optical biometer and the same IOL model, we obtained lower percentages with the older formulas, as the PE of ±0.50 D or less was reached in a percentage of eyes ranging between 67.1% (SRK/T) and 71.5% (Hoffer Q).2 The improvement between the previous and the current study can be related to at least two factors: first, because in this study, all eyes were operated on by the same surgeon, constant optimization did not have to compensate for different surgical techniques, such as different capsulorhexis size, which influence the postoperative IOL position; second, the postoperative refraction was assessed by the same surgeon, who measured it with the maximum attention.

Our results are considerably better also when compared with those previously reported by several authors, who investigated the accuracy of IOL power calculation with different formulas, different biometers, and the same IOL model as in our study.6–8Table 5 clearly reveals the better outcomes obtained with the SS-OCT optical biometer compared with those previously reported. Possible explanations for such a difference include the fact that both surgery and the assessment of postoperative refraction were performed in all cases by the same surgeon, as noted above. Moreover, our sample included only a small fraction of eyes with an AL shorter than 22.0 mm: short eyes are known to have poorer refractive outcomes,6 which can influence the results in the whole sample. It is likely that a larger proportion of short eyes would lead to worse results.

Table 5
Table 5:
Comparison between refractive outcomes achieved with different formulas in the current and previous studies.

To our knowledge, the studies with the results most similar to the ones reported here were published in 2017 and 2018 by our own group, when we analyzed the results of IOL power calculation with two different optical biometers: the Aladdin (Topcon Europe) and the Galilei G6 (Ziemer Ophthalmology GmbH). In both studies, all patients received the same type of surgery and IOL model as in this investigation. In the first study, the percentage of eyes with a PE of ±0.50 D or less, as calculated by the Hoffer Q, Holladay 1, and SRK/T formulas, was even higher, as it ranged between 87.67% and 89.04%. The MedAE was 0.25 D for all formulas.22 In the second study, we found that the MedAE was 0.19 D for the Barrett formula and ranged between 0.23 D and 0.26 D for older formulas. Similarly, the percentages of eyes with a PE of ±0.50 D or less were also close to the ones found in the current study, as they ranged between 80.88% and 83.82%.23 The good results reported by our group are likely to be due to the factors explained above.

The present study shows some interesting findings about the latest formulas. The Kane and EVO formulas provided outstanding outcomes, as their MedAE was ≤0.205 D and both yielded at least 90% of eyes with a PE of ±0.50 D or less. Our data for the Kane formula mirror those recently reported by its author.15,25 Because neither the Kane nor the EVO formulas have been published and little is known about their structure, it is not possible to discuss the reasons for their excellent performance. However, they look promising and deserve our attention.

Barrett formula confirmed its reputation as being one of the most accurate, as previously found by many researchers.6–8 Good results are also achieved with the RBF version 2.0, as recently reported by Connell and Kane,25 the T2 formula, as previously reported by Cooke and Cooke and Kane et al.,6,7 the VRF formula, as previously reported by its author,18 and the Holladay 2 formula with AL adjustment (for eyes with AL > 24.0 mm). With 89.33% of eyes showing a PE of ±0.50 D or less, the Holladay 2 can now be considered one of the most accurate formulas, with a clear improvement over previous versions and studies.6,7,21 Such improvement was confirmed by a direct comparison with the standard Holladay 2, as we can see in Table 2.

As regards the Olsen formula, we observed more accurate results with the standalone version, which includes K and AL among the IOL position predictors, compared with the version installed on the SS-OCT, which adopts the C-constant and thus relies only on LT and ACD to predict the IOL position. This finding is in good agreement with the data reported by Cooke and Cooke.7

One of the most interesting findings of this study was the good performance of older formulas, which enabled us to have more than 84% of eyes with a PE of ±0.50 D or less. This percentage is considerably higher than the corresponding values of previous studies with large samples6–8; the good outcomes suggest that caution should be used before abandoning these formulas, which are still accurate and have one big advantage: the possibility of optimizing their constants independently, as they can be entered into Excel. With unpublished formulas, this is not possible.

On the other hand, Panacea performed slightly worse than the other formulas, although its results still have to be considered good. In this case, a role is played by the Scheimpflug camera measuring the Q value and the A/P ratio. We used a rotating Scheimpflug camera combined with Placido corneal topography, but other devices may offer different measurements and improve the results of Panacea.

As regards myopic eyes with AL >26.0 mm, this subgroup revealed interesting results, as many formulas achieved excellent results (ie, more than 80% of eyes with a PE of ±0.50 D or less). These include some traditional formulas (Haigis and SRK/T) and most of the newer formulas. A larger sample is needed to confirm our preliminary data.

As a potential limitation, a smaller number of eyes were analyzed in this study than in other recent investigations, which included more than 1000 eyes.6–8,15 Although our sample size was sufficient to detect a statistically significant difference in the MedAE (on the basis of sample size calculation), we acknowledge that big data can provide us with additional information and therefore will go on enrolling patients. Because of the relatively small sample size, we did not investigate the influence of each biometric parameter on the outcomes of all formulas, as larger subgroups (eg, short eyes) would have been required.

In conclusion, our data suggest that all formulas can be successfully used to calculate the IOL power using the measurements provided by the SS-OCT. In most cases, newer formulas show higher accuracy because of the lack of far outliers. Older formulas, however, are still a valid option.


  • Optical biometry leads to accurate intraocular lens (IOL) power calculation, with 70% to 80% of eyes showing a prediction error within 0.50 D.
  • The swept-source optical coherence tomography (SS-OCT) optical biometer investigated in this study provides repeatable measurements that, according to a previous multicenter study, lead to a prediction error of ±0.50 D or less in about 70% of eyes, using older formulas.


  • Using data from a single surgeon and a large variety of newer and older formulas, measurements by the SS-OCT optical biometer enabled us to improve the refractive outcomes of IOL power calculation compared with previous studies because we were able to achieve a prediction error within 0.50 D in at least 80% of eyes with all formulas.


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