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Validation of the SToP formula for calculating intraocular lens power in eyes with previous myopic excimer laser surgery

Savini, Giacomo MD1,*; Hoffer, Kenneth MD, FACS4,5; Barboni, Piero MD2,3; Balducci, Nicole MD1; Schiano-Lomoriello, Domenico MD1

Author Information
Journal of Cataract & Refractive Surgery: November 2019 - Volume 45 - Issue 11 - p 1562-1567
doi: 10.1016/j.jcrs.2019.06.011
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Rotating Scheimpflug cameras have been available for nearly 15 years. Their ability to measure the curvature of both anterior and posterior corneal surfaces makes them an excellent tool to accurately calculate the intraocular lens (IOL) power in eyes with previous myopic laser in situ keratomileusis (LASIK) or photorefractive keratectomy (PRK). However, this interesting feature has never had a significant impact from a clinical point of view because we cannot take advantage of it when using traditional thin-lens IOL power formulas, such as the Barrett Universal II (unpublished),A Haigis,1 Hoffer Q,2 Holladay 1,3 Holladay 2 (unpublished),B and SRK/T.4 These formulas, in fact, were developed to be used with keratometric (K) values of corneal power that are derived solely from the anterior corneal curvature.

The Pentacam (OCULUS Optikgeräte GmbH) was the first rotating Scheimpflug camera, and several authors investigated its measurements when looking for a method to improve IOL power calculation in eyes with previous corneal refractive surgery. In 2006, Borasio et al.5 developed the BESSt formula, which modifies the corneal power (as calculated with the Gaussian optics formula) to be used with thin-lens formulas. In 2009, Holladay et al.6 described the equivalent K-reading (EKR), which takes the posterior corneal curvature measurements into account. In the same year, Kim et al.7 used the “true net power” (ie, a corneal power value obtained with a modified Gaussian optics formula) to calculate the IOL power with the standard SRK/T formula. More recently, Schuster et al.8 used the ratio of posterior-to-anterior corneal radius to adjust the Holladay 1 and the SRK/T formulas by means of regression formulas, which they called the “Schuster/Schanzlin-Thomas-Purcell (SToP)” IOL calculator.8 This no-history method was developed in a sample of 61 eyes, and it offered promising results in a separate group of 14 eyes. However, it has not yet been validated by an independent study. The present investigation was designed to compare the SToP formula to the other methods that rely on the corneal power measurements provided by this rotating Scheimpflug camera and have been described to calculate the IOL power after myopic laser refractive surgery.

Patients and methods

Patients and Surgery

This was a retrospective study involving all patients with previous myopic laser refractive surgery who underwent cataract surgery and had undergone anterior segment imaging with the rotating Scheimpflug camera. Informed consent was obtained from each of them, and the study was approved by the IRCCS - Bietti Foundation ethics committee. The study methods adhered to the tenets of the Declaration of Helsinki for the use of human participants in biomedical research. Exclusion criteria included vitreoretinal or corneal disease; a history of other ocular surgery, uveitis, trauma, or systemic disease affecting vision; intraoperative complications (during either refractive or cataract surgery); and postoperative corrected distance visual acuity worse than 0.8 (20/25) for any reason.

Phacoemulsification was performed through a 2.75 mm temporal incision under topical anesthesia. Different IOL models were implanted and IOL power calculation was performed using the optimized constants available on the User Group for Laser Interference Biometry (ULIB) website.C

Preoperative Measurements

Before surgery, all patients underwent anterior segment imaging with the Pentacam HR (software version from 1.17r89 to 1.20r10) between February 2011 and April 2017, and with the Pentacam AXL (software version 1.21r43) between May 2017 and November 2018 (both OCULUS Optikgeräte GmbH). At the end of this period, all measurements were analyzed with the latest software (1.21r43). The Pentacam HR measures corneal curvature and thickness by means of a rotating Scheimpflug camera. Scans were taken in the automatic release mode. The 25-picture scan was used. Only scans that had an examination quality specification graded by the instrument as “OK” were saved. The Pentacam AXL adds the measurement of axial length (AL) by means of partial coherence interferometry.

Among the corneal power measurements provided by the rotating Scheimpflug camera, two were evaluated: the average K power (defined as the mean of the power of the steep and flat corneal meridians, and calculated using the standard 1.3375 K index) and the EKR was measured at 4.5 mm. The 4.5 mm diameter was selected because Holladay et al.6 had previously found that this was the optimal diameter.

The AL was always measured by partial coherence interferometry in patients who had preoperative examinations before May 2017, either an IOLMaster 500 (Carl Zeiss Meditec AG) or an Aladdin (Topcon Europe Medical B.V.) was used.

Intraocular Lens Power Calculation

The SToP Formula is a no-history method used to calculate the IOL power with both Holladay 1 formula3 and the SRK/T formula.4 The IOL power is then adjusted according to the posterior/anterior corneal radius ratio and (for the Holladay 1 only) the AL.8 More specifically, the following equations are used to calculate the IOL adjustments with the Holladay 1:

and the SRK/T:

where rpost/ant is the ratio between the posterior and the anterior corneal radius and AL is the axial length.

The SToP formula was compared with two categories of methods: those requiring historical data and those using no previous data.

Methods Requiring Historical Data

  • Masket: the IOL power is calculated by entering the average K into the SRK/T formula.9 The IOL power is then modified by adding an amount of corneal optical power equal to the myopic laser correction multiplied by −0.326 with the addition of +1.01.
  • Savini: the corneal power is obtained from the average K by means of a modified K index (npost),10 calculated according to the equation:

The IOL power is determined by entering the adjusted corneal power into the double-K SRK/T formula, which requires the pre-LASIK corneal power.4,11

Methods Using No Previous Data

  • Shammas no-history: corneal power is calculated according to the formula:

where Kc. is the corrected K value and Sim-K is simulated K.12 This Kc can be entered into the Shammas-post-LASIK (PL) formula.13

  • Barrett True-K no-history: the IOL power is calculated according to the unpublished formulaD using the internal software of the Scheimpflug camera, which provides users with the expected refraction for the selected IOL power. In this sample, the outcome does not take the pre-LASIK refraction into account. The lens factor is derived from the ULIB A-constant of the SRK/T formula.
  • Seitz/Speicher/Savini (Triple-S) method: the K value given by the Scheimpflug camera is changed by separately considering the anterior and posterior corneal curvatures, as advocated by Seitz and Langenbucher14 in 2000 and later reviewed by Speicher.15 Savini et al.16 further modified it using a mean value of −4.98 diopters (D) for the posterior corneal power, so that the formula is:

The corneal power calculated with this method has to be entered into the double-K SRK/T formula.

  • BESSt formula: the corneal power is calculated according to the formula published by Borasio et al.,5 and it is then entered into the double-K SRK/T formula.
  • EKR: this value is calculated by the software of the Scheimpflug camera (according to the article by Holladay et al.6) and then entered into the double-K Holladay 1 formula.

In all cases requiring a double-K formula (Savini method, Triple-S method, BESSt formula, EKR), the pre-LASIK corneal power was required. If this could be retrieved from clinical charts, the available value was used. If it was not possible, it was calculated using the method described by Savini et al.,17 according to the equation:

Calculation of the Prediction Error in Refraction

The prediction error (PE) in refraction was calculated as the difference between the measured and predicted postoperative refractive spherical equivalent (measured refraction − predicted refraction) with the actual IOL power implanted. A negative PE value indicates a more myopic result than the predicted refraction and a positive PE represents a more hyperopic result. The predicted postoperative refraction was obtained with different procedures depending on the methods for IOL power calculation. For methods based on the double-K SRK/T (BESSt, Savini, Triple-S), it was calculated using the equations in the original papers.4,11 The value shown on the IOL calculation display of the Scheimpflug camera was used for the Barrett formula. For the remaining methods, the IOL power was back-calculated first for emmetropia in 4 steps, as previously suggested by Aramberri11: (1) the expected refraction (according to the double-K SRK/T formula) was calculated with the implanted IOL and with an IOL power 1.0 D higher; (2) the former was subtracted from the latter and the refraction difference was obtained for a change of 1.0 D in IOL power (Rx1DIOL); (3) 1.0 was divided by Rx1DIOL, resulting in the magnitude in IOL power that should produce 1.0 D of refractive change (IOL1DRx); (4) this value was multiplied by the refractive error (Rx) and added to the implanted IOL power. The difference between the implanted IOL power and the back-calculated IOL power for emmetropia is defined as the PE in IOL power. To obtain the PE in refraction, the PE in IOL power has to be divided by the Rx1DIOL.

Constant Optimization

As previously stated, IOL power calculations were based on the ULIB optimized values. However, constant optimization was carried out on a subgroup of cases, that is, on the largest sample with the same IOL model, to obtain a mean zero PE. For all formulas using the A-constant, optimization was performed using the Goal/Seek tool in Excel software (Microsoft Corp.), as suggested by Hoffer et al.18

Statistical Analysis

Statistical analyses were carried out using InStat software (version 3.10, GraphPad Software, Inc.) and MedCalc software (version, MedCalc Software bvba). Normality of data distribution was assessed by means of the Kolmogorov-Smirnov test. Data with a normal distribution (eg, the PE) were compared using repeated-measures analysis of variance (ANOVA), whereas data without a normal distribution (eg, the absolute PE) were compared using the Friedman test. The percentages of eyes within a given PE (±0.50 D, ±0.75 D, and ±1.00 D) were compared using a Cochran Q test. A P value less than 0.05 was considered statistically significant. For each method, the mean PE in refraction, the median absolute error (MedAE), and the interquartile range of the absolute PE in refraction, as well as the percentage of eyes with an absolute PE in refraction within ±0.50 D, ±0.75 D, and ±1.00 D were recorded.


Whole Group Analysis

Fifty-seven eyes of 57 patients were identified. Seven patients whose laser myopic correction was unknown were eliminated, and then the following analysis was performed in the remaining 50 eyes of 50 patients, so that the historical and no-history methods could both be investigated. AL was measured with the Pentacam AXL in 13 eyes. In 27 out of 50 eyes, the original corneal power could be retrieved; the mean corneal power was higher (P = .0393) in eyes with estimated pre-excimer laser keratometry than in eyes with measured pre-excimer laser keratometry. Table 1 shows the patients’ demographic data. An AL longer than 26.0 mm was noted in 41 (82%) of 50 eyes. Eight different IOL models were implanted, and the most common one was the AcrySof SN60WF (Alcon Laboratories, Inc.), which was chosen for 24 eyes.

Table 1
Table 1:
Demographic data on the 50 patients whose myopic laser correction was known.

Table 2 shows the results of IOL power calculation ranked on the basis of the MedAE (from lowest to highest). The SToP formula based on the SRK/T and Triple-S methods gave the lowest MedAE, and a very close value was obtained with the Masket formula. A low MedAE was observed with another method specifically developed for the Scheimpflug camera, that is, the BESSt formula, and the Savini method. The remaining methods, including the SToP formula based on the Holladay 1, yielded worse outcomes, with MedAEs higher than 0.4 D.

Table 2
Table 2:
Results of IOL power calculation ranked on the basis of the MedAE in eyes with historical data (N = 50).

The highest percentage of eyes with a PE in refraction within ±0.50 D was achieved by the Masket method, followed by the Triple-S method, BESSt formula, Savini method, and the two SToP formulas. With all these methods, the percentage was ≥60%. According to the Cochran Q test, the proportion of eyes with a PE within ±0.50 D showed a statistically significant difference (P = .01) among the investigated methods; paired multiple comparisons revealed a significant difference (P < .05) between the Masket formula and the EKR method. The difference was also significant when comparing the PE within ±0.75 D (P = .015) and ±1.00 D (P = .003); multiple comparisons detected only one significant difference (P < .05), that is, a higher proportion of eyes within ±1.00 D with the Triple-S method than with Shammas-PL formula.

With all methods (except for the Shammas formula), the lack of back-calculated constant optimization was not a relevant clinical issue because the arithmetic PE ranged between zero and slightly more than a quarter of a diopter. The PE was on the hyperopic side with the two SToP formulas and on the myopic side with the Shammas-PL formula. Both differences between the arithmetic and the absolute PEs were statistically significant (P < .0001 and P = .0204, respectively).

With most formulas, the arithmetic PE was negatively correlated to the amount of myopic correction induced by the excimer laser. This means that eyes with previous higher myopic correction were more likely to have a hyperopic error after cataract surgery (the opposite being true for eyes with previous lower myopic correction). Such a negative correlation was observed with the SToP formula based on the SRK/T (r = −0.5624, P < .0001), the Holladay 1 (r = −0.5626, P < .0001), the Shammas formula (r = −0.5616, P < .0001), Triple-S method (r = −0.3985, P = .0042), BESSt formula (r = −0.3245, P = 0.0215), Barrett True-K method (r = −0.5124, P < .0001), and EKR (r = −.3801, P = .0065). In contrast, it was not detected with the Masket and Savini methods.

Subgroup Analysis with Constant Optimization

In the 24 eyes with the same IOL model (AcrySof SN60WF), constant optimization could be performed, and the mean PE was zeroed out. Table 3 shows the results in this subgroup, which did not show any relevant improvement with respect to the results for the whole group. With most formulas, neither the MedAE nor the percentage of eyes with a PE within ±0.5 D improved; the only exceptions were the SToP formulas based on the SRK/T (percentage of eyes within ±0.50 D from 62.0% to 62.5%), the Holladay 1 (percentage of eyes within ±0.50 D from 60.0 to 62.5%), and the EKR (percentage of eyes within 0.50 D from 44.0 to 54.2% and MedAE from 0.59 to 0.45 D).

Table 3
Table 3:
Results of IOL power calculation in the subgroup with the same IOL model (n = 24) and with optimized constants.


Our data suggest that corneal power measurements from the rotating Scheimpflug camera can be successfully used to calculate the IOL power in eyes with previous myopic excimer laser surgery. Several formulas and methods can be relied on for this purpose.

In this study, we specifically aimed to investigate the accuracy of a new formula termed SToP,8 which was published in 2016 and has not yet been validated. The variant based on the SRK/T formula generated excellent outcomes because it yielded the lowest MedAE (0.31 D) and a relatively high percentage of eyes with a PE within ±0.50 D (62%). These results allow it to be ranked among the best options to calculate the IOL power in eyes with previous myopic excimer laser surgery, with the advantage of being a fully no-history method. The SToP formula based on the Holladay 1 had less accurate (but still acceptable) outcomes, probably because the original Holladay 1 formula is known to be less accurate than the SRK/T in long eyes.19

Other formulas yielded good results with the biometric measurements by the rotating Scheimpflug camera. The Masket method, a classical “historical” method of fudging the IOL power calculated by the SRK/T formula,9 achieved a low MedAE (0.32 D) and the highest percentage (76%) of eyes with a PE within ±0.50 D. Several previous studies9,20–23 found it to be one of the most accurate options (with more than 60% of cases showing a PE within ±0.5 D), and the present study confirms their findings. This percentage of eyes (76%) with a PE within ±0.50 D is impressive because more than three quarters of eyes had an AL longer than 26.0 mm; and in this category of eyes, the PE within ±0.50 D is known to range between 53% and 64%, even when no excimer laser surgery has been previously performed.24 The other historical method, that is, the one described by Savini et al.,10 yielded less accurate outcomes, with 64% of eyes showing a PE ≤±0.50 D. This percentage is lower than previously reported,20–23 probably because in this study, the original corneal power had to be estimated in about half of cases, which might have influenced the effective lens position estimation by the double-K SRK/T formula. Interestingly, the two historical methods were the only ones that did not show any correlation between the laser-induced myopic correction and the post-cataract refractive outcome. This result is likely dependent upon the fact that both methods take the myopic correction into account and adjust their calculations accordingly. No-history methods, by contrast, treat all eyes in the same way, disregarding the amount of treatment induced by the excimer laser. In other words, as discussed by Shammas et al.12 in 2003, with no-history formulas, “a −10 D and a −1 D treatment with the same measured postoperative K readings would have the same adjustment.” We believe this is one reason why no-history methods showed a trend toward hyperopic refraction in cases with higher laser-induced myopic correction and a trend toward myopic refraction in cases with lower laser-induced myopic correction. An additional reason might be the well-known risk for postoperative hyperopia in highly myopic eyes, which can be reduced by using specifically optimized formula constants,25 optimization of AL measurements,26 or new methods to calculate the AL from the optical path length.27

Among the no-history options, the best results were achieved using the BESSt formula5 and Triple-S method.16 The Triple-S method yielded one of the highest percentages of eyes with a PE within ±0.50 D (35 [70%] of 50 eyes), a value close to that previously found in another study by our group,20 and the lowest MedAE (0.31 D). Actually, this method is not a completely no-history one because it requires a double-K formula and thus, the original corneal power must be known. However, the rotating Scheimpflug camera used in the present study enables us to estimate the pre-LASIK/PRK corneal power by means of a regression formula published in 2016,17 and therefore makes it possible to use the Triple-S method as if it were a no-history method. The same consideration holds true for the corneal power calculated according to the BESSt formula because it also requires a double-K approach to obtain the IOL power. Interestingly, this is the first study confirming the accuracy of the BESSt formula since it was published in 2006.

The outcomes achieved using the Shammas no-history formula and the Barrett True-K formula were satisfactory; however, they were not as good as those obtained with the best methods because the MedAE was greater than 0.45 D and the percentage of eyes with a PE within ±0.50 D was between 52% and 54%. It is difficult to comment on the Barrett formula because it has never been published and nothing is known about its nature. However, it should be noted that the percentage of eyes with a PE within ±0.50 D in this study (52%) was lower than that previously reported by Wang et al.21 (58.7%), Abulafia et al.22 (67.2%), and Savini et al.23 (63.6%). Differences in the technique used to measure corneal curvature might play a role in the lower accuracy of the Barrett formula observed in this study. As regards the Shammas formula, our data were almost the same as those previously reported by Wang et al.21 and Abulafia et al.22

On the other hand, the EKR provided us only with moderately good outcomes. The EKR, which has been available since 2006 on the Scheimpflug camera, is the “K-reading that should be used for IOL calculations, which takes into account the front and back surface power and thickness of the cornea.”6 Theoretically it should not require any adjustment, and it should be entered in a double-K formula, as we did by entering it into the double-K Holladay 1 formula. We have already expressed our concerns about this method some years ago.28

In this study, we did not evaluate all the corneal powers provided by the rotating Scheimpflug camera: we intentionally discarded the values calculated by ray tracing (so called “total corneal refractive power”) and Gaussian optics formula (so called “true net power”) because they require specific formula constants and are therefore unlikely to be adopted by clinicians.29 For similar reasons, we did not include the method described by Kim et al.7 because they relied on the true net power,7 with no specifically optimized constants and no double-K formulas.

This study has some limitations. First, because of its retrospective nature, we enrolled eyes implanted with different IOL models, which prevented back-calculation of optimized constants for the whole sample. However, the lack of constant optimization and the adoption of the ULIB values did not affect the clinical results, since the mean arithmetic PE was very low with all methods except the Shammas method. Moreover, we optimized in retrospect the constants in a subgroup of eyes with the same IOL model and did not observe any improvement of the results, so that we can conclude that ULIB constants can be used for IOL power calculation with the formulas investigated in this study. Second, we did not have the ability to test IOL power calculation by means of specific ray-tracing software30 because this option was not available on our instrument, or using other formulas, such as the formula by Olsen and Hoffman,31 because lens thickness was not available for all eyes.

In conclusion, our study demonstrates that based on the measurements of the rotating Scheimpflug camera, several methods and formulas can be used to accurately calculate IOL power in eyes with previous myopic LASIK or PRK. The SRK/T variant of the SToP formula is one of the best formulas for this purpose. Other formulas that should be taken into consideration are the Triple-S, Masket, BESSt, and Savini.

What Was Known

  • Intraocular lens (IOL) power calculation in eyes with previous excimer laser surgery can lead to refractive surprises.
  • Several methods can be used to achieve accurate results in these eyes.

What This Paper Adds

  • A recently described no-history formula (SToP) can help surgeons obtain reliable IOL power calculations in eyes with previous myopic correction by excimer laser because it ranked among the most accurate methods.
  • The version of the SToP formula based on the SRK/T led to better outcomes with respect to the version based on the Holladay 1.


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Dr. Hoffer licenses the registered trademark name Hoffer to ensure accurate programming of his formulas to Carl Zeiss Meditec AG (IOLMasters), Haag-Streit AG (LenStar/EyeStar), Heidelberg Engineering GmbH (Anterion), OCULUS Optikgeräte GmbH (Pentacam AXL), Movu, Inc. (Argos), NIDEK Co., Ltd. (AL-Scan), Tomey GmbH (OA-2000), Topcon Europe Medical B.V./Visia Imaging S.r.l. (Aladdin), Ziemer Ophthalmic Systems AG (Galilei G6), and all A-scan biometer manufacturers. None of the other authors has a financial or proprietary interest in any material or methods mentioned.


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C. User Group for Laser Interference Biometry. Available at: Accessed 20-1-2019
D. Barrett GD. Barrett True K formula for prior myopic or hyperopic LASIK/PRK. Available at:
© 2019 by Lippincott Williams & Wilkins, Inc.