Development and Clinical Accuracy of a New Intraocular Lens Power Formula (VRF) Compared to Other Formulas OLEKSIY V. VOYTSEKHIVSKYY

PURPOSE:

To develop and compare the accuracy and reproducibility of the VRF intraocular lens (IOL) power calculation formula with well-known methods. DESIGN: Development and validation study. METHODS: This analysis comprised 823 eyes of 823 patients at Kiev Clinical Ophthalmology Hospital Eye Microsurgery Center, Kiev, Ukraine, operated on by 1 surgeon with 3 different types of hydrophobic lenses: IQ SN60WF (494 eyes) and ReSTOR SN6AD1 (169 eyes) (Alcon Labs, Fort Worth, Texas, USA) and AMO Tecnis MF ZMB00 (160 eyes) (J&J Vision, Santa Ana, California, USA). The full data set was divided into 2 subsets, the first to develop the new formula and the second to evaluate their performance with other most commonly used modern methods of IOL power calculation (Haigis, Hoffer Q, Holladay 1, Holladay 2, SRK/T, and T2). The VRF algorithm is empirical; it uses 4 predictors for estimation of postoperative lens position, including axial length, corneal power (K), preoperative anterior chamber depth (corneal epithelium to lens), and horizontal corneal diameter. The results are also stratified into groups of short (£22 mm), medium (>22 to 0.05), but not for MedAE and MAE (each P < .05). Homogeneity of variance was checked with the F test for each sample pair. Comparisons of the mean values of normally distributed data were performed using Student t test for paired samples with 2 tails (ME). For non-normally distributed data (MedAE and MAE), the Wilcoxon rank sum test (W test) was applied. The differences between all formulas were assessed using the

AMERICAN JOURNAL OF OPHTHALMOLOGY

JANUARY 2018

TABLE 3. Mean Refractive Error, Median Absolute Error, Mean Absolute Error, Standard Deviation of Errors, Maximum Refractive Error, and Percentage of Eyes With Refractive Prediction Errors Within 60.25, 60.50, 61.0, and 62.0 Diopters for Each Formula in the Short Axial Length Groupa Mean (D) Formula

Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2 VRF

Mean Absolute (D)

Max Error (6) (D)

Eyes Within PE (%)

ME (D)

SD (D)

MedAE (D)

MAE (D)

SD (D)

Minus

Plus

Range

60.25 D

60.50 D

61.0 D

62.0 D

0.011 0.012 0.024 0.018 0.015 0.068 0.009

0.546 0.538 0.521 0.523 0.565 0.542 0.548

0.386 0.350 0.376 0.361 0.426 0.418 0.345

0.447 0.434 0.422 0.419 0.461 0.446 0.428

0.306 0.311 0.300 0.309 0.320 0.305 0.336

1.359 1.169 1.212 1.308 1.204 1.257 1.314

1.057 0.947 0.918 0.911 1.083 0.956 1.076

2.416 2.116 2.130 2.219 2.287 2.213 2.390

26.4 37.7 33.9 33.9 30.2 30.2 35.8

56.6 62.3 60.4 67.9 60.4 62.3 64.1

96.2 96.2 96.2 98.1 92.4 94.4 92.5

100.0 100.0 100.0 100.0 100.0 100.0 100.0

D ¼ diopters; MAE ¼ mean absolute error; Max Error ¼ maximum error; ME ¼ mean refractive error; MedAE ¼ median absolute error; PE ¼ prediction error; SD ¼ standard deviation. AcrySof IQ SN60WF intraocular lens was used for the evaluation. a _ 22 mm. Short axial length group (n ¼ 53 cases) defined as axial length

(Supplemental Appendix; Supplemental Material available at AJO.com). The second factor is associated with the conversion of the refractive power of the cornea in true optical power. Recently, many authors3,12,17 have shown the irrationality of using the classic 1.3375 index refraction and the error in the refractive power of the cornea from VOL. 185

0.5 to 1 diopter.13,17 The standardized keratometric index of refraction was chosen many years ago, so that an anterior radius of curvature of the cornea of 7.5 mm would yield a power of 45.0 D. The cornea is a thick lens with 2 surfaces and thicknesses. Using the index of refraction of the corneal stroma of 1.376, a posterior corneal radius that is 1.2 mm steeper, and a corneal thickness of 0.55 mm, results in a net corneal power of 44.4 D. This value is approximately 0.56 D less than the standardized keratometric power. As described in detail by Holladay, the value of 4/3 for the net corneal index of refraction is an appropriate value and would have the minimum impact, and thus was recommended for use in modern formulas. Olsen recommend using an even lower value of 1.3315 that yielded an appropriate corneal power of 44.20 diopters.17 Holladay’s value of the refraction index was chosen (1.3333) because a more appropriate result was achieved with it than using Olsen value (1.3315) that overestimated the resulting IOL power.13,17 Therefore, we used the following correction factor: Ktrue ¼

ð4=3 1Þ ¼ 0:987654313K ð1:3375 1Þ

(1)

Thus, we used a classical stigmatic, paraxial optical formula with an adjusted axial length and a correction of the true optical power of the cornea: P¼

1336 AL0 C

1336 1336 1000 þKtrue 1000 tgRef

C

;

(2)

Vd

_26:5 mm; ALo ALo ¼ AL þ 0:20; if AL> ¼ AL þ ð 0:1593AL þ 4:401Þ:

(3)

P is the optical power of the implanted IOL for emmetropia (D), n is the refraction index of aqueous humor and

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TABLE 4. Mean Refractive Error, Median Absolute Error, Mean Absolute Error, Standard Deviation of Errors, Maximum Refractive Error, and Percentage of Eyes With Refractive Prediction Errors Within 60.25, 60.50, 61.0, and 62.0 Diopters for Each Formula in the Medium Axial Length Groupa Mean (D)

Mean Absolute (D)

Max Error (6) (D)

Eyes Within PE (%)

Formula

ME (D)

SD (D)

MedAE (D)

MAE (D)

SD (D)

Minus

Plus

Range

60.25 D

60.50 D

61.0 D

62.0 D

Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2 VRF

0.077 0.079 0.083 0.081 0.084 0.084 0.080

0.475 0.478 0.473 0.498 0.508 0.481 0.473

0.320 0.330 0.314 0.338 0.336 0.317 0.302

0.370 0.377 0.371 0.393 0.402 0.377 0.367

0.297 0.290 0.290 0.305 0.312 0.297 0.296

1.829 1.877 1.725 1.620 1.565 1.678 1.719

1.479 1.344 1.304 1.365 1.326 1.289 1.459

3.308 3.221 3.029 2.985 2.891 2.967 3.178

40.0 38.1 37.8 39.4 34.4 37.5 41.8

70.9 71.8 70.9 68.4 66.2 69.0 71.8

96.6 95.6 96.6 95.9 94.0 96.6 95.6

100.0 100.0 100.0 100.0 100.0 100.0 100.0

D ¼ diopters; MAE ¼ mean absolute error; Max Error ¼ maximum error; ME ¼ mean refractive error; MedAE ¼ median absolute error; PE ¼ prediction error; SD ¼ standard deviation. AcrySof IQ SN60WF intraocular lens was used for the evaluation. a Medium axial length group (n ¼ 320 cases) defined as axial length >22 to

TABLE 7. Mean Refractive Error, Median Absolute Error, Mean Absolute Error, Standard Deviation of Errors, Maximum Refractive Error, and Percentage of Eyes With Refractive Prediction Errors Within 60.25, 60.50, 61.0, and 62.0 Diopters for Each Formula Over the Entire Axial Length Range Mean (D)

Mean Absolute (D)

Max Error(6) (D)

Eyes Within PE (%)

Formula

ME (D)

SD (D)

MedAE (D)

MAE (D)

SD (D)

Minus

Plus

Range

60.25 D

60.50 D

61.0 D

62.0 D

Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2 VRF

0.044 0.043 0.045 0.045 0.045 0.041 0.045

0.505 0.509 0.494 0.523 0.522 0.501 0.500

0.329 0.338 0.326 0.351 0.336 0.321 0.305

0.393 0.402 0.391 0.413 0.412 0.393 0.386

0.317 0.312 0.303 0.321 0.320 0.310 0.316

1.960 1.877 1.725 1.911 1.832 2.013 1.881

1.479 1.559 1.511 1.365 1.326 1.289 1.459

3.439 3.436 3.236 3.276 3.158 3.302 3.340

39.0 38.0 37.8 37.7 35.0 38.0 41.3

68.0 69.4 69.6 67.0 66.4 68.0 69.4

94.9 94.5 95.7 95.1 93.3 95.4 94.4

100.0 100.0 100.0 100.0 100.0 99.8 100.0

D ¼ diopters; MAE ¼ mean absolute error; Max Error ¼ maximum error; ME ¼ mean refractive error; MedAE ¼ median absolute error; PE ¼ prediction error; SD ¼ standard deviation. AcrySof IQ SN60WF intraocular lens was used for the evaluation. Total n ¼ 494 cases.

INVESTIGATION OF ESTIMATED LENS POSITION:

To obtain the regression algorithm of ELP prediction, we used the secondary data group of patients with 2 different types of lenses, Alcon ReSTOR SN6AD1 (169 eyes) and AMO Tecnis MF ZMB00 (160 eyes). In total, there were 329 eyes. Preoperative assessment included examination using the IOLMaster 500 (Sarl Zeiss Meditec AG, Jena, Germany) software version 7.3. All eyes meeting the following inclusion criteria were included: complete preoperative data, postoperative visual acuity of at least 6/12, no astigmatism more than 1 D, no prior refractive surgery, and no intraoperative complications. The mean numeric error of each formula was set to 0 by adjusting the IOL constant using Microsoft Excel (the IF function). The absolute error was the absolute value of the numeric error. Postoperative assessment included

VOL. 185

subjective manifest refraction obtained 3 months postoperatively. Based on the data of the preoperative parameters of the eye (AL, K, ACDpre, and CD), the values of the optical power of the 2 different types of implanted IOLs and the received postoperative manifest refraction empirically based on the multiple regression analysis (SPSS 22.0; IBM) obtained the equation describing the postoperative position of the IOL in the eye, namely the postoperative ELP. To develop the regression formula, multiple linear regression was performed using the ELP as the dependent variable and the axial length (AL), corneal power (K), preoperative anterior chamber depth (epithelium to lens) (ACDpre), and horizontal corneal diameter (CD) as independent variables. For each value of the predicted postoperative ACD, the corresponding regression equation was obtained. More

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61

than 700 iterations were performed to obtain the averaged regression equation model. Accordingly, for 2 different types of lenses (Alcon ReSTOR SN6AD1 and AMO Tecnis MF ZMB00), 2 regression models were derived as follows: AL3ðCACD30:051 0:006Þ þ K3ðCACD30:019 0:008Þ þ ACDpre 3ðCACD30:053 þ 0:005Þ CD3ðCACD30:013 0:003Þ ðCACD 30:959 0:013Þ; AL3ðCACD30:050 0:007Þ þ K3ðCACD30:018 0:001Þ þ ACDpre 3ðCACD30:056 þ 0:004Þ CD3ðCACD30:012 0:003Þ ðCACD 30:974 0:005Þ;

constant from the manufacturer, D constants 1–4 and E constants 1–4 are the regression constants obtained empirically by the study, and the offset is the regression equation obtained empirically. The regression constants are: D1 ¼ 0:051; D2 ¼ 0:019; D3 ¼ 0:053; D4 ¼ 0:013; (8) E1 ¼ 0:006; E2 ¼ 0:008; E3 ¼ 0:005; E4 ¼ 0:003: The offset is:

(4)

(5)

where CACD is an ACD constant from the manufacturer, AL is the axial length of the eye (optical method) (mm), K is the refractive power of the cornea (D), K ¼ (nc-1)/r (D), r is the radius of curvature of the anterior corneal surface (mm), nc is the refractive index of 1.3375, ACDpre is the preoperative anterior chamber depth (epithelium to lens) (mm), and CD is the horizontal cornea white-towhite diameter (mm). Equation 4 had a higher correlation coefficient (R2 ¼ 0.922 vs R2 ¼ 0.895) and a lower standard error (0.316 vs 0.334) than equation 5 and was therefore selected

Offset ¼ CACD30:959 0:013:

for further evaluation. A new formula was programmed using equation 4. The proposed method was called VRF. Thus, in the new formula, the ELP is a function of 5 variables, as follows: ELP ¼ f CACD; AL; K; ACDpre ; CD ; (6)

(7)

where AL is the axial length of the eye (optical method) (mm), K is the refractive power of the cornea (D) and K ¼ (nc-1)/r (D), r is the radius of curvature of the anterior corneal surface (mm), nc is the refractive index of 1.3375, ACDpre is the preoperative anterior chamber depth (epithelium to lens) (mm), CD is the horizontal cornea white-to-white diameter (mm), CACD is an ACD 62

(10)

The main feature of this algorithm is the use of a single IOL constant that is repeated several times, and not the use of a number of different constants.11,18 Each of the 4 preoperative parameters of the eye affects a constant and gives a final value corresponding to the postoperative position of the IOL in the eye. The socalled optical constant of the anterior chamber depth (optical CACD) was used as a constant. The CACD constant was used exclusively as the optical constant owing primarily to the fact that the sample was taken from patients whose AL was measured using an optical method (PCI, IOLMaster 500, software version 7.3, Sarl Zeiss Meditec AG). There is a method to determine the appropriate optical CACD. The option is to use the regression equation proposed by Haigis18,19 for optimized constants to obtain the values of the optical CACD constant from the optical A-constants given by the manufacturer

Optical CACD constant ¼ ðOptical A constant 30:62467Þ 68:82:

ELP ¼ AL3ðCACD3D1 E1Þ þ K3ðCACD3D2 E2Þ þ ACDpre 3ðCACD3D3 þ E3Þ CD3ðCACD3D4 E4Þ offset;

(9)

(11)

where the optical CACD is a constant depth of the anterior chamber for optical measurement techniques, and the optical A constant is a constant for optical measurement techniques by the manufacturer of the intraocular lens. EVALUATION OF THE VRF FORMULA:

The aim of this study was to develop and compare a new method for predicting the postoperative IOL position and further calculating the optical power of the implanted lens using 4 parameters: the axial length of the eye (AL), the optical refractive power of the cornea (K), the preoperative anterior chamber depth (epithelium to lens) (ACDpre), and the horizontal corneal diameter (CD). The clinical performance of the VRF formula was compared to that of the other formulas by calculating the spectacle prediction error of each formula in the evaluation subset of eyes using separate IOL-specific constants optimized for each formula. AcrySof IQ SN60WF IOL was used for the evaluation of the second subgroup of patients (494 eyes,

AMERICAN JOURNAL OF OPHTHALMOLOGY

JANUARY 2018

FIGURE 1. Median absolute error (MedAE) plotted against axial length groups for the Haigis, Hoffer Q, Holladay 1, Holladay 2, SRK/T, T2, and VRF formulas.

FIGURE 2. Mean absolute error (MAE) plotted against axial length groups for the Haigis, Hoffer Q, Holladay 1, Holladay 2, SRK/T, T2, and VRF formulas.

Alcon Labs, Fort Worth, Texas, USA). Overall, there was good correlation between the prediction errors of the 7 formulas (best, r2 ¼ 0.905 Haigis; worst, r2 ¼ VOL. 185

0.844 Holladay 2). In general, the VRF formula produced a prediction error similar to that of the Hoffer Q on short eyes, Holladay 1 on medium eyes, T2 on medium-long

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63

.274 .051 .865 .007 .029 .842 .939 .845 .986 .953 .984 .583 .263 .124 .209 .031 .978 .680 .379 .943 .921 .975 .334 .311

DISCUSSION

a

ME ¼ mean refractive error; MedAE ¼ median absolute error; t test ¼ Student t test; W test ¼ Wilcoxon rank sum test. P value less than .05 was considered statistically significant.

.715 .331 .495 .094 .886 .494 .765 .865 .938 .771 .678 .402 .573 .276 .916 .046 .020 .689 .706 .887 .788 .940 .784 .685 .957 .927 .603 .780 .878 .060 Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2

.444 .705 .870 .533 .300 .493

MedAE

W Test t Test

ME MedAE

W Test t Test

ME MedAE

W Test t Test

ME MedAE

W Test t Test

ME MedAE

W Test t Test

ME

_26 mm) (N ¼ 51 Cases) Long (> P Value (a ¼ .05)a

_24.5 to Medium (>22 to

PURPOSE:

To develop and compare the accuracy and reproducibility of the VRF intraocular lens (IOL) power calculation formula with well-known methods. DESIGN: Development and validation study. METHODS: This analysis comprised 823 eyes of 823 patients at Kiev Clinical Ophthalmology Hospital Eye Microsurgery Center, Kiev, Ukraine, operated on by 1 surgeon with 3 different types of hydrophobic lenses: IQ SN60WF (494 eyes) and ReSTOR SN6AD1 (169 eyes) (Alcon Labs, Fort Worth, Texas, USA) and AMO Tecnis MF ZMB00 (160 eyes) (J&J Vision, Santa Ana, California, USA). The full data set was divided into 2 subsets, the first to develop the new formula and the second to evaluate their performance with other most commonly used modern methods of IOL power calculation (Haigis, Hoffer Q, Holladay 1, Holladay 2, SRK/T, and T2). The VRF algorithm is empirical; it uses 4 predictors for estimation of postoperative lens position, including axial length, corneal power (K), preoperative anterior chamber depth (corneal epithelium to lens), and horizontal corneal diameter. The results are also stratified into groups of short (£22 mm), medium (>22 to 0.05), but not for MedAE and MAE (each P < .05). Homogeneity of variance was checked with the F test for each sample pair. Comparisons of the mean values of normally distributed data were performed using Student t test for paired samples with 2 tails (ME). For non-normally distributed data (MedAE and MAE), the Wilcoxon rank sum test (W test) was applied. The differences between all formulas were assessed using the

AMERICAN JOURNAL OF OPHTHALMOLOGY

JANUARY 2018

TABLE 3. Mean Refractive Error, Median Absolute Error, Mean Absolute Error, Standard Deviation of Errors, Maximum Refractive Error, and Percentage of Eyes With Refractive Prediction Errors Within 60.25, 60.50, 61.0, and 62.0 Diopters for Each Formula in the Short Axial Length Groupa Mean (D) Formula

Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2 VRF

Mean Absolute (D)

Max Error (6) (D)

Eyes Within PE (%)

ME (D)

SD (D)

MedAE (D)

MAE (D)

SD (D)

Minus

Plus

Range

60.25 D

60.50 D

61.0 D

62.0 D

0.011 0.012 0.024 0.018 0.015 0.068 0.009

0.546 0.538 0.521 0.523 0.565 0.542 0.548

0.386 0.350 0.376 0.361 0.426 0.418 0.345

0.447 0.434 0.422 0.419 0.461 0.446 0.428

0.306 0.311 0.300 0.309 0.320 0.305 0.336

1.359 1.169 1.212 1.308 1.204 1.257 1.314

1.057 0.947 0.918 0.911 1.083 0.956 1.076

2.416 2.116 2.130 2.219 2.287 2.213 2.390

26.4 37.7 33.9 33.9 30.2 30.2 35.8

56.6 62.3 60.4 67.9 60.4 62.3 64.1

96.2 96.2 96.2 98.1 92.4 94.4 92.5

100.0 100.0 100.0 100.0 100.0 100.0 100.0

D ¼ diopters; MAE ¼ mean absolute error; Max Error ¼ maximum error; ME ¼ mean refractive error; MedAE ¼ median absolute error; PE ¼ prediction error; SD ¼ standard deviation. AcrySof IQ SN60WF intraocular lens was used for the evaluation. a _ 22 mm. Short axial length group (n ¼ 53 cases) defined as axial length

(Supplemental Appendix; Supplemental Material available at AJO.com). The second factor is associated with the conversion of the refractive power of the cornea in true optical power. Recently, many authors3,12,17 have shown the irrationality of using the classic 1.3375 index refraction and the error in the refractive power of the cornea from VOL. 185

0.5 to 1 diopter.13,17 The standardized keratometric index of refraction was chosen many years ago, so that an anterior radius of curvature of the cornea of 7.5 mm would yield a power of 45.0 D. The cornea is a thick lens with 2 surfaces and thicknesses. Using the index of refraction of the corneal stroma of 1.376, a posterior corneal radius that is 1.2 mm steeper, and a corneal thickness of 0.55 mm, results in a net corneal power of 44.4 D. This value is approximately 0.56 D less than the standardized keratometric power. As described in detail by Holladay, the value of 4/3 for the net corneal index of refraction is an appropriate value and would have the minimum impact, and thus was recommended for use in modern formulas. Olsen recommend using an even lower value of 1.3315 that yielded an appropriate corneal power of 44.20 diopters.17 Holladay’s value of the refraction index was chosen (1.3333) because a more appropriate result was achieved with it than using Olsen value (1.3315) that overestimated the resulting IOL power.13,17 Therefore, we used the following correction factor: Ktrue ¼

ð4=3 1Þ ¼ 0:987654313K ð1:3375 1Þ

(1)

Thus, we used a classical stigmatic, paraxial optical formula with an adjusted axial length and a correction of the true optical power of the cornea: P¼

1336 AL0 C

1336 1336 1000 þKtrue 1000 tgRef

C

;

(2)

Vd

_26:5 mm; ALo ALo ¼ AL þ 0:20; if AL> ¼ AL þ ð 0:1593AL þ 4:401Þ:

(3)

P is the optical power of the implanted IOL for emmetropia (D), n is the refraction index of aqueous humor and

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TABLE 4. Mean Refractive Error, Median Absolute Error, Mean Absolute Error, Standard Deviation of Errors, Maximum Refractive Error, and Percentage of Eyes With Refractive Prediction Errors Within 60.25, 60.50, 61.0, and 62.0 Diopters for Each Formula in the Medium Axial Length Groupa Mean (D)

Mean Absolute (D)

Max Error (6) (D)

Eyes Within PE (%)

Formula

ME (D)

SD (D)

MedAE (D)

MAE (D)

SD (D)

Minus

Plus

Range

60.25 D

60.50 D

61.0 D

62.0 D

Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2 VRF

0.077 0.079 0.083 0.081 0.084 0.084 0.080

0.475 0.478 0.473 0.498 0.508 0.481 0.473

0.320 0.330 0.314 0.338 0.336 0.317 0.302

0.370 0.377 0.371 0.393 0.402 0.377 0.367

0.297 0.290 0.290 0.305 0.312 0.297 0.296

1.829 1.877 1.725 1.620 1.565 1.678 1.719

1.479 1.344 1.304 1.365 1.326 1.289 1.459

3.308 3.221 3.029 2.985 2.891 2.967 3.178

40.0 38.1 37.8 39.4 34.4 37.5 41.8

70.9 71.8 70.9 68.4 66.2 69.0 71.8

96.6 95.6 96.6 95.9 94.0 96.6 95.6

100.0 100.0 100.0 100.0 100.0 100.0 100.0

D ¼ diopters; MAE ¼ mean absolute error; Max Error ¼ maximum error; ME ¼ mean refractive error; MedAE ¼ median absolute error; PE ¼ prediction error; SD ¼ standard deviation. AcrySof IQ SN60WF intraocular lens was used for the evaluation. a Medium axial length group (n ¼ 320 cases) defined as axial length >22 to

TABLE 7. Mean Refractive Error, Median Absolute Error, Mean Absolute Error, Standard Deviation of Errors, Maximum Refractive Error, and Percentage of Eyes With Refractive Prediction Errors Within 60.25, 60.50, 61.0, and 62.0 Diopters for Each Formula Over the Entire Axial Length Range Mean (D)

Mean Absolute (D)

Max Error(6) (D)

Eyes Within PE (%)

Formula

ME (D)

SD (D)

MedAE (D)

MAE (D)

SD (D)

Minus

Plus

Range

60.25 D

60.50 D

61.0 D

62.0 D

Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2 VRF

0.044 0.043 0.045 0.045 0.045 0.041 0.045

0.505 0.509 0.494 0.523 0.522 0.501 0.500

0.329 0.338 0.326 0.351 0.336 0.321 0.305

0.393 0.402 0.391 0.413 0.412 0.393 0.386

0.317 0.312 0.303 0.321 0.320 0.310 0.316

1.960 1.877 1.725 1.911 1.832 2.013 1.881

1.479 1.559 1.511 1.365 1.326 1.289 1.459

3.439 3.436 3.236 3.276 3.158 3.302 3.340

39.0 38.0 37.8 37.7 35.0 38.0 41.3

68.0 69.4 69.6 67.0 66.4 68.0 69.4

94.9 94.5 95.7 95.1 93.3 95.4 94.4

100.0 100.0 100.0 100.0 100.0 99.8 100.0

D ¼ diopters; MAE ¼ mean absolute error; Max Error ¼ maximum error; ME ¼ mean refractive error; MedAE ¼ median absolute error; PE ¼ prediction error; SD ¼ standard deviation. AcrySof IQ SN60WF intraocular lens was used for the evaluation. Total n ¼ 494 cases.

INVESTIGATION OF ESTIMATED LENS POSITION:

To obtain the regression algorithm of ELP prediction, we used the secondary data group of patients with 2 different types of lenses, Alcon ReSTOR SN6AD1 (169 eyes) and AMO Tecnis MF ZMB00 (160 eyes). In total, there were 329 eyes. Preoperative assessment included examination using the IOLMaster 500 (Sarl Zeiss Meditec AG, Jena, Germany) software version 7.3. All eyes meeting the following inclusion criteria were included: complete preoperative data, postoperative visual acuity of at least 6/12, no astigmatism more than 1 D, no prior refractive surgery, and no intraoperative complications. The mean numeric error of each formula was set to 0 by adjusting the IOL constant using Microsoft Excel (the IF function). The absolute error was the absolute value of the numeric error. Postoperative assessment included

VOL. 185

subjective manifest refraction obtained 3 months postoperatively. Based on the data of the preoperative parameters of the eye (AL, K, ACDpre, and CD), the values of the optical power of the 2 different types of implanted IOLs and the received postoperative manifest refraction empirically based on the multiple regression analysis (SPSS 22.0; IBM) obtained the equation describing the postoperative position of the IOL in the eye, namely the postoperative ELP. To develop the regression formula, multiple linear regression was performed using the ELP as the dependent variable and the axial length (AL), corneal power (K), preoperative anterior chamber depth (epithelium to lens) (ACDpre), and horizontal corneal diameter (CD) as independent variables. For each value of the predicted postoperative ACD, the corresponding regression equation was obtained. More

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than 700 iterations were performed to obtain the averaged regression equation model. Accordingly, for 2 different types of lenses (Alcon ReSTOR SN6AD1 and AMO Tecnis MF ZMB00), 2 regression models were derived as follows: AL3ðCACD30:051 0:006Þ þ K3ðCACD30:019 0:008Þ þ ACDpre 3ðCACD30:053 þ 0:005Þ CD3ðCACD30:013 0:003Þ ðCACD 30:959 0:013Þ; AL3ðCACD30:050 0:007Þ þ K3ðCACD30:018 0:001Þ þ ACDpre 3ðCACD30:056 þ 0:004Þ CD3ðCACD30:012 0:003Þ ðCACD 30:974 0:005Þ;

constant from the manufacturer, D constants 1–4 and E constants 1–4 are the regression constants obtained empirically by the study, and the offset is the regression equation obtained empirically. The regression constants are: D1 ¼ 0:051; D2 ¼ 0:019; D3 ¼ 0:053; D4 ¼ 0:013; (8) E1 ¼ 0:006; E2 ¼ 0:008; E3 ¼ 0:005; E4 ¼ 0:003: The offset is:

(4)

(5)

where CACD is an ACD constant from the manufacturer, AL is the axial length of the eye (optical method) (mm), K is the refractive power of the cornea (D), K ¼ (nc-1)/r (D), r is the radius of curvature of the anterior corneal surface (mm), nc is the refractive index of 1.3375, ACDpre is the preoperative anterior chamber depth (epithelium to lens) (mm), and CD is the horizontal cornea white-towhite diameter (mm). Equation 4 had a higher correlation coefficient (R2 ¼ 0.922 vs R2 ¼ 0.895) and a lower standard error (0.316 vs 0.334) than equation 5 and was therefore selected

Offset ¼ CACD30:959 0:013:

for further evaluation. A new formula was programmed using equation 4. The proposed method was called VRF. Thus, in the new formula, the ELP is a function of 5 variables, as follows: ELP ¼ f CACD; AL; K; ACDpre ; CD ; (6)

(7)

where AL is the axial length of the eye (optical method) (mm), K is the refractive power of the cornea (D) and K ¼ (nc-1)/r (D), r is the radius of curvature of the anterior corneal surface (mm), nc is the refractive index of 1.3375, ACDpre is the preoperative anterior chamber depth (epithelium to lens) (mm), CD is the horizontal cornea white-to-white diameter (mm), CACD is an ACD 62

(10)

The main feature of this algorithm is the use of a single IOL constant that is repeated several times, and not the use of a number of different constants.11,18 Each of the 4 preoperative parameters of the eye affects a constant and gives a final value corresponding to the postoperative position of the IOL in the eye. The socalled optical constant of the anterior chamber depth (optical CACD) was used as a constant. The CACD constant was used exclusively as the optical constant owing primarily to the fact that the sample was taken from patients whose AL was measured using an optical method (PCI, IOLMaster 500, software version 7.3, Sarl Zeiss Meditec AG). There is a method to determine the appropriate optical CACD. The option is to use the regression equation proposed by Haigis18,19 for optimized constants to obtain the values of the optical CACD constant from the optical A-constants given by the manufacturer

Optical CACD constant ¼ ðOptical A constant 30:62467Þ 68:82:

ELP ¼ AL3ðCACD3D1 E1Þ þ K3ðCACD3D2 E2Þ þ ACDpre 3ðCACD3D3 þ E3Þ CD3ðCACD3D4 E4Þ offset;

(9)

(11)

where the optical CACD is a constant depth of the anterior chamber for optical measurement techniques, and the optical A constant is a constant for optical measurement techniques by the manufacturer of the intraocular lens. EVALUATION OF THE VRF FORMULA:

The aim of this study was to develop and compare a new method for predicting the postoperative IOL position and further calculating the optical power of the implanted lens using 4 parameters: the axial length of the eye (AL), the optical refractive power of the cornea (K), the preoperative anterior chamber depth (epithelium to lens) (ACDpre), and the horizontal corneal diameter (CD). The clinical performance of the VRF formula was compared to that of the other formulas by calculating the spectacle prediction error of each formula in the evaluation subset of eyes using separate IOL-specific constants optimized for each formula. AcrySof IQ SN60WF IOL was used for the evaluation of the second subgroup of patients (494 eyes,

AMERICAN JOURNAL OF OPHTHALMOLOGY

JANUARY 2018

FIGURE 1. Median absolute error (MedAE) plotted against axial length groups for the Haigis, Hoffer Q, Holladay 1, Holladay 2, SRK/T, T2, and VRF formulas.

FIGURE 2. Mean absolute error (MAE) plotted against axial length groups for the Haigis, Hoffer Q, Holladay 1, Holladay 2, SRK/T, T2, and VRF formulas.

Alcon Labs, Fort Worth, Texas, USA). Overall, there was good correlation between the prediction errors of the 7 formulas (best, r2 ¼ 0.905 Haigis; worst, r2 ¼ VOL. 185

0.844 Holladay 2). In general, the VRF formula produced a prediction error similar to that of the Hoffer Q on short eyes, Holladay 1 on medium eyes, T2 on medium-long

VRF INTRAOCULAR LENS POWER CALCULATION FORMULA

63

.274 .051 .865 .007 .029 .842 .939 .845 .986 .953 .984 .583 .263 .124 .209 .031 .978 .680 .379 .943 .921 .975 .334 .311

DISCUSSION

a

ME ¼ mean refractive error; MedAE ¼ median absolute error; t test ¼ Student t test; W test ¼ Wilcoxon rank sum test. P value less than .05 was considered statistically significant.

.715 .331 .495 .094 .886 .494 .765 .865 .938 .771 .678 .402 .573 .276 .916 .046 .020 .689 .706 .887 .788 .940 .784 .685 .957 .927 .603 .780 .878 .060 Haigis Hoffer Q Holladay 1 Holladay 2 SRK/T T2

.444 .705 .870 .533 .300 .493

MedAE

W Test t Test

ME MedAE

W Test t Test

ME MedAE

W Test t Test

ME MedAE

W Test t Test

ME MedAE

W Test t Test

ME

_26 mm) (N ¼ 51 Cases) Long (> P Value (a ¼ .05)a

_24.5 to Medium (>22 to