Several intraocular lens (IOL) power calculation formulas are in use today. Many have been compared with each other. In a previous paper,^{1} we compared 6 formulas to determine which formulas were better “out of the box” on 2 different optical biometers. This study broadens the topic to include formulas that were not available on these biometers at the time the first study was performed. The discussion section of this paper is organized by each of these new formulas.

The following 6 formulas were previously evaluated on our dataset of 1079 eyes: Haigis,^{2} Holladay 1,^{3} Hoffer Q,^{4–6} Sanders-Retzlaff-Kraff/Theoretical (SRK/T),^{7} SRK II,^{8} and Olsen.^{9} These formulas came preinstalled on the IOLMaster (Carl Zeiss Meditec AG) or the Lenstar LS 900 (Haag-Streit AG), the only optical biometers available in the United States when this study was initiated. The IOLMaster is based on partial coherence interferometry (PCI) and the Lenstar on optical low-coherence interferometry (OLCR).

The SRK II has been shown to be markedly worse than other formulas.^{1} It was thus excluded from analysis in this study. The additional formulas used were the purchased (standalone) version of the Olsen formula, Holladay 2,^{A} T2,^{10} Super Formula,^{11} and the Barrett Universal II.^{B}

We present predictions from 9 formulas in this study. However, some tables present up to 12 formulas. This is because we present 2 distinct forms of 3 formulas. We listed them as though they were separate formulas because each gave distinctly different results. We were surprised that Olsen varied depending on whether the formula came preinstalled on the Lenstar LS 900 biometer (Olsen_{OLCR} ) or whether we used software we purchased (Olsen_{Standalone} ). Because prediction accuracy was improved when we omitted the preoperative refraction from Holladay 2, we elected to include this formula with a preoperative refraction (Holladay 2_{PreSurgRef} ), and without it (Holladay 2_{NoRef} ). In addition to the Holladay 1 formula, we included the Holladay 1 formula with the Wang-Koch adjustment for long eyes (H1_{Long (adj)} ).^{12}

The formulas evaluated in our previous study are identified in the tables with an asterisk. This study emphasizes the newer formulas, which did not come preinstalled on any machine at the time this study was initiated.

Patients and methods
This study conformed to ethics codes based on the tenets of the Declaration of Helsinki. An institutional review board (IRB) (Lakeland Hospitals Niles and Saint Joseph, IRB number 1) exempted the study from review.

Inclusion Criteria
Our database has been described elsewhere.^{1} The medical records of all patients from our office who had phacoemulsification cataract surgery between March 2010 and December 2012 were evaluated. Eyes were included if they had uneventful in-the-bag placement of an Acrysof SN60WF IOL (Alcon Laboratories, Inc.). Ocular pathology was excluded. These patients were all measured preoperatively with both the Lenstar LS 900 and the IOLMaster (versions 3.02 and 5.4). Inclusion criteria required full data capture by both biometers. No eyes were excluded for unexpected refractions. Long eyes and short eyes were evaluated separately.

Postoperative Assessment
The postoperative assessment included a subjective manifest refraction obtained between 3 weeks and 3 months postoperatively. Those refractions were performed by qualified technicians who had passed standardized in-office accuracy training.

Availability of Formulas
The Haigis, Hoffer Q, Holladay 1, and SRK/T formulas are available on many devices in the U.S. The Super Formula was released recently and is not currently available preinstalled on any biometer.

The Olsen_{Standalone} formula was accessed via PhacoOptics software^{C} (version 1.10.100.2020, IOL Innovations ApS). The Olsen_{OLCR} was only available on the Lenstar biometer (EyeSuite i8.0.0.0, Haag-Streit AG). The Holladay 2 formula was used within Holladay IOL Consultant (version 2014.0607, Holladay Consulting).^{A} The Barrett Universal II formula is available online.^{B} This formula was provided by its author, Graham Barrett, in an Excel (Microsoft Corp.) spreadsheet^{D} because it was not otherwise available at the time of the initial writing of this paper. The formula has recently become available preinstalled on the Lenstar biometer. The T2 formula calculator is available online as a free Excel download.^{E} The Barrett Universal II and the Holladay 2 formulas have not been formally published. The Olsen formula was published in a patent^{9} and has been partially described in the medical literature.^{13}

Preoperative Parameters (Biometry Measurements)
Four formulas use 2 measurement parameters. The Hoffer Q, Holladay 1, SRK/T, and T2 require only keratometry (K) values and axial length (AL). The Haigis formula uses 3 parameters: K values, AL, and anterior chamber depth (ACD). Three IOL formulas are multiple-parameter formulas designed for 5 to 7 parameters. The Olsen uses 6 inputs (K values, AL, ACD, lens thickness, patient age, and central corneal thickness). The Barrett Universal II uses 5 inputs (AL, K values, ACD, lens thickness, and horizontal white-to-white [WTW] distance). The Holladay 2 uses 7 inputs (K values, AL, ACD, lens thickness, horizontal WTW, preoperative refraction, and age).

Finally, the Super Formula uses no parameters of its own; it merely uses other formulas. For negatively powered IOLs, the Super Formula uses the Haigis formula. It also uses the Hoffer Q for eyes with ALs that are less than 21.50 mm, the Holladay 1 formula with the Wang-Koch adjustment for long eyes (H1_{Long (adj)} ) for eyes with ALs greater than 25.00 mm, and the Holladay 1 for all other eyes.

Computations and Access to Formulas
Most formula calculations were performed with Excel 2010 (version 14.0) and Microsoft SQL Server (version 12.0.2000.8, Microsoft Corp.). Standard deviations (SDs) were calculated using the STDEV.P statistical function within Excel. F tests were used to analyze significant differences between formulas.

Prediction Error
The prediction error is defined as the measured postoperative spherical equivalent (SE) refraction minus the predicted SE that was calculated by each IOL formula. Thus, a positive prediction error indicates a refractive outcome that was more hyperopic than predicted.

Lens Constants
Group-optimized constants were derived using computer software developed by one of the authors (T.L.C.). This software automatically entered patient measurements directly into PhacoOptics, Holladay IOL Consultant, and Eyesuite software. Data from 10 eyes were manually entered into PhacoOptics, Eyesuite, and Holladay IOL Consultant software to verify the accuracy of this method.

The measurements from each patient’s eye were entered multiple times into these programs, each time with a different lens constant. The predictions were stored. Different lens constants were tested using a trial-and-error method until the mean prediction error for the entire dataset was as close to zero as practical. This value was considered the optimized lens constant for that particular formula. This same process was repeated for each of the other formulas. Every optimized lens constant was rounded to 3 digits except for the Olsen formula because its software rounded lens constants to only 2 digits. The Haigis lens constants were obtained using linear regression as described by Haigis et al.^{2}

Results
Entrance criteria were met by 1454 eyes. When both eyes of the same patient met the entrance criteria, the second eye to have surgery was excluded to avoid having any 2 genetically identical eyes in the data pool. There remained 1079 eyes that were analyzed.

Tables 1 through 6 show the prediction results. The lens constants for PCI data and OLCR data were developed separately. Only lens constants for datasets containing all eyes were optimized because 1 goal was to determine how formulas perform at different ALs without adjusting lens constants.

Table 1: Formula performance for all eyes using optimized lens constants with PCI measurements (mean AL = 23.81 mm; range = 20.87 to 29.44 mm; N = 1079).

Table 2: Formula performance for all eyes using optimized lens constants with OLCR measurements (mean AL = 23.81 mm; range = 20.84 to 29.51 mm; N = 1079).

Table 3: Formula performance for long eyes using optimized lens constants with PCI measurements (mean AL = 26.84 mm; range = 25.97 to 29.44 mm; n = 54).

Table 4: Formula performance for long eyes using optimized lens constants with OLCR measurements (mean AL = 26.85 mm; range = 26.02 to 29.51 mm; n = 54).

Table 5: Formula performance for short eyes using optimized lens constants with PCI measurements (mean AL = 21.71 mm; range = 20.87 to 22.01 mm; n = 41).

Table 6: Formula performance for short eyes when using optimized lens constants with OLCR measurements (mean AL = 21.69 mm; range = 20.84 to 22 mm; n = 41).

Table 1 shows the results of the 9 formulas for the entire dataset (all eyes) using PCI-optimized lens constants with PCI measurements. Table 2 is similar, except OLCR measurements and OLCR-optimized constants are used. In the same manner, Tables 3 and 4 show long-eye results (AL ≥ 26 mm) and Tables 5 and 6 show short-eye results (AL ≤ 22 mm), again using lens constants that were optimized for all eyes. Table 7 lists the optimized lens constants that have not been published by the authors previously.

Table 7: Lens constants for Holladay 2, Barrett Universal II, T2, and Olsen formulas.*

Formula Ranking
When a formula’s mean prediction error is moved to zero by optimizing the formula’s lens constants, as in Tables 1 and 2 , any of the last 5 of these statistics will give almost the same ranking as this mean of 6 ranks. The formulas in Tables 1 and 2 were ranked by the SD.

Because many power formulas are not calibrated for AL variation, several rows in Tables 3 to 6 have mean prediction errors far from zero. For such situations, our previously described method of ranking formula performance was used. This method averages the relative ranks of the following 6 performance indicators: the mean prediction error, the mean absolute error (MAE), the SD, the maximum prediction error, the percentage of final refractions within ±0.5 diopter (D) of the predicted value, and the percentage of final refractions within ±1.0 D of the predicted value. Each of these statistics has special clinical relevance. The last column in Tables 3 to 6 lists the mean ranking score. Ties were broken based on rankings of MAE. The median absolute error (Med AE) was also included in each table, but not in the ranking system. The current study used the original ranking system based on the MAE to allow for better comparison with previous studies.

Preoperative Refraction
The Holladay 2 is the only formula that uses preoperative refraction as an input. When this retrospective analysis was initially designed, the refraction at the visit immediately before cataract extraction was recorded. If the refraction was not available, the prescription of the patient’s spectacles or the most recently recorded refraction in the patient’s chart was used. This was called the presurgical refraction, and predictions using this measurement were labeled Holladay 2_{PreSurgRef} .

In addition, the most hyperopic preoperative refraction was recorded. After the data were collected, the pre-cataract refraction was defined as the recorded refraction that occurred between 2 years and 6 years before cataract extraction. Predictions that used this measurement were called Holladay 2_{PreCatRef} . This pre-cataract refraction was only obtained in 557 of the initial 1079 eyes. Only OLCR measurements were used for this subsection analysis (Table 8 ). That table shows predictions for preoperative refractions and pre-cataract refractions. The mean time between the preoperative refraction (used in Holladay 2_{PreSurgRef} ) and cataract extraction was 1.5 months. The mean time between the pre-cataract refraction (used in Holladay 2_{PreCatRef} ) and cataract extraction was 43.8 months.

Table 8: Holladay 2 formula optimized refraction results (557 eyes).

Discussion
When OLCR values were used (Tables 2, 4, and 6 ), the Olsen formula outperformed the other formulas in every category. The Olsen_{Standalone} and the Olsen_{OLCR} had different predictions even though the same lens constants were used in both formulas. The difference was more apparent in short eyes (Table 6 ) than in long eyes (Table 4 ). The Olsen_{Standalone} outperformed the Olsen_{OLCR} in all categories except in long eyes, for which the difference between them was minimal (difference in MAE 0.001 D).

We did not anticipate there would be 2 versions for the Olsen formula and contacted Haag-Streit.^{F} We were informed that Olsen_{OLCR} is preset to use only preoperative lens thickness and preoperative ACD to predict the postoperative ACD. The Olsen_{Standalone} default settings include 2 more variables to predict postoperative ACD–preoperative AL and preoperative K value readings. When these 2 fields are deselected in a settings popup window, the Olsen_{Standalone} predictions become equivalent to the Olsen_{OLCR} predictions.

Given its superb ranking with OLCR data, it was surprising to see the Olsen_{Standalone} was the worst of all 9 formulas with PCI measurements (Table 1 ). This might be because OLCR measures lens thickness, but PCI does not. We would expect the Olsen to perform well with other optical biometers that measure lens thickness, such as the IOLMaster 700 series (Carl Zeiss Meditec AG) or the Argos (Movu Inc.) biometers. The Olsen formula uses the C constant as 1 input in determining the IOL position for its calculations, and the C constant requires lens thickness.^{13} The Olsen_{Standalone} from the OLCR biometer was better than the best formula on the PCI biometer, which was the Barrett Universal II (P < .05). The Olsen_{OLCR} outperformed all other formulas for eyes longer than 26.0 mm.

The Haigis and both Olsen formulas seem to be best tuned to AL. For extreme ALs (Tables 3 to 6 ), their mean prediction errors were consistently the closest to zero of all the formulas.

Without special computing skills, the Olsen formula is difficult to study because it is proprietary. It is also more cumbersome to use than the other formulas because it requires 6 lens constants to be filled in by the user. Five of these constants are related to IOL manufacturing specifications. The constants we use for this formula were previously published for this IOL.^{1} In our experience, the most difficult constants to obtain from IOL manufacturers are the front and back curves of an average IOL.

The Barrett Universal II formula achieved the best results across the board when PCI measurements were used. When OLCR values were used, the Barrett Universal II outperformed the preinstalled version of the Olsen_{OLCR} except in eyes longer than 26.0 mm. It was only slightly worse than Olsen_{Standalone} using OLCR measurements (P = .77). This makes the Barrett formula an attractive formula for both biometers.

In addition, the Barrett Universal II requires only 1 constant and has been tuned to the SRK/T A constant; therefore, it is much easier to find lens constants for it than for the Olsen formula. It is available online, where patient measurements can be entered for free.^{B}

The Holladay 2 performed better when the preoperative refraction was excluded (Tables 1, 2, and 4 ). We initially used the refraction immediately before cataract surgery (Holladay 2_{PreSurgRef} ). We then turned our attention to the refraction of the eye before it developed a cataract (Holladay 2_{PreCatRef} ). Table 8 shows predictions from 557 eyes for which refractions were available from immediately before surgery (Holladay 2_{PreSurgRef} ) and before cataract development (Holladay 2_{PreCatRef} ). Analysis of Table 8 confirms that results were better when neither refraction was included. Specifically, 4 of the 5 statistical measures in Table 8 were best when refractions were excluded (Holladay 2_{NoRef} ). There might be a situation in which refractions improve results in extremely long or short eyes; however, our data do not support the general use of preoperative refraction in the Holladay 2 formula.

One peculiarity of the Super Formula is that it could not be optimized (the mean prediction error could not be brought to zero) (Tables 1 and 2 ). The formula has no lens constant of its own; it uses the other formulas’ lens constants. The Super Formula generally performed better than the traditional formulas; however, it never performed as well as the T2 or the Haigis.

The T2 is identical to the SRK/T except it includes a patch for an error in the SRK/T effective lens position (ELP) calculation.^{10} It performed remarkably well, despite using only 2 preoperative variables. It almost always predicted better than the Holladay 2, which uses up to 7 variables.

Its simplicity makes it ideal for outreach in a low-resource setting where testing equipment and access to IOL formulas can be limited. This is helpful in locations where internet access is limited because the formula can be used on any device that can download Excel.^{E} Because all values likely have to be manually entered in outreach scenarios, it is valuable that the T2 does not require the ACD measurement. With fewer variables to enter, there is a lower chance for data-entry error.

Short eyes (Tables 5 and 6 ) generally yielded myopic predictions. This finding has been reported by others.^{14} None of the formulas performed as well for short eyes as they did for other eyes. Most had mean prediction errors far from zero. Long eyes (Tables 3 and 4 ) yielded hyperopic mean prediction errors in all traditional formulas except the Haigis. This has also been reported previously.^{14} The Haigis equation ELP is tuned to AL by a2, one of its 3 lens constants. This is most likely the reason it had good results across the AL spectrum.

This study has noteworthy limitations. Most important, we restricted our study to 1 IOL platform. Because the range of available Acrysof SN60WF IOLs is somewhat limited (6.0 to 30.0 D), this dataset does not have many extreme-length eyes. We look forward to results in similar studies using different IOL models.

Our practice contributed data from eyes with Acrysof SN60WF IOLs to the development of the T2 formula. Our contribution represented 13% of the 11 189 total eyes used in developing the T2. The eyes with an Acrysof SN60WF IOL that contributed toward the T2 formula development were not included in this study. Although unlikely, the contribution of data derived from our surgical and refractive techniques could have resulted in some favorable bias for the T2 formula.

Two formulas were developed primarily from Acrysof SN60WF IOLs, giving them a potential advantage in this study. The Barrett Universal II formula was reformulated exclusively based on data from these IOLs,^{G} and 62% of the data used in developing the T2 formula were from these IOLs.^{10} We are unaware of what IOL types were used in developing the other formulas.

In conclusion, the formulas gave different results depending on which optical biometry measurements were used. The Olsen_{Standalone} formula was the most accurate if OLCR measurements were available, regardless of AL. If only PCI measurements were available, the Barrett Universal II performed the best and the Olsen_{Standalone} formula performed the worst. The Barrett Universal II performed nearly as well as the Olsen_{Standalone} with OLCR data, and it is easier to use and access. The preinstalled version of the Olsen was not as good as the standalone version. The Holladay 2 formula generally performed better when the preoperative refraction was intentionally excluded.

What Was Known
Intraocular lens formulas differ in their predictive ability.
Using more preoperative biometry variables has been assumed to improve formula predictions.
Various IOL formulas have been compared, but typically only a few formulas or only a subset, such as long eyes, have been presented in 1 study.
Biometer measurements have often been compared. There is general agreement of individual parameters. However, formula predictions have rarely been compared.
What This Paper Adds
The standalone Olsen IOL formula ranked higher than 8 other formulas in prediction accuracy for all eyes, regardless of AL, as long as OLCR measurements were available. This formula was significantly better than the best formula using PCI values.
The Olsen ranked worst of all formulas when only PCI measurements were used.
The Olsen that comes preinstalled on the OLCR biometer is a different formula than the standalone version. It generally underperformed compared with the standalone version.
The Barrett Universal II formula was the best when PCI measurements were used. It predicted nearly as well as the Olsen when OLCR measurements were used.
The Holladay 2 performed better when the preoperative refraction was intentionally excluded.
The T2, a 2-variable formula, generally predicted better than the Holladay 2, a 7-variable formula.
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http://www.phacooptics.net/ . Accessed July 5, 2016

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