October 2011
Volume 52, Issue 11
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Visual Psychophysics and Physiological Optics  |   October 2011
Comparing Methods to Estimate the Human Lens Power
Author Affiliations & Notes
  • Jos J. Rozema
    From the Department of Ophthalmology, Antwerp University Hospital, Edegem, Belgium;
    the Department of Medicine, Antwerp University, Wilrijk, Belgium; and
  • David A. Atchison
    the School of Optometry and Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, Queensland, Australia.
  • Marie-José Tassignon
    From the Department of Ophthalmology, Antwerp University Hospital, Edegem, Belgium;
    the Department of Medicine, Antwerp University, Wilrijk, Belgium; and
  • Corresponding author: Jos J. Rozema, Department of Ophthalmology, Antwerp University Hospital, Wilrijkstraat 10, 2650 Edegem, Belgium; jos.rozema@uza.be
Investigative Ophthalmology & Visual Science October 2011, Vol.52, 7937-7942. doi:10.1167/iovs.11-7899
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      Jos J. Rozema, David A. Atchison, Marie-José Tassignon; Comparing Methods to Estimate the Human Lens Power. Invest. Ophthalmol. Vis. Sci. 2011;52(11):7937-7942. doi: 10.1167/iovs.11-7899.

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Abstract

Purpose.: To compare the accuracy of different methods of calculating human lens power when lens thickness is not available.

Methods.: Lens power was calculated by four methods. Three methods were used with previously published biometry and refraction data of 184 emmetropic and myopic eyes of 184 subjects (age range, 18–63 years; spherical equivalent range, −12.38 to +0.75 D). These three methods consist of the Bennett method, which uses lens thickness, a modification of the Stenström method and the Bennett-Rabbetts method, both of which do not require knowledge of lens thickness. These methods include c constants, which represent distances from lens surfaces to principal planes. Lens powers calculated with these methods were compared with those calculated using phakometry data available for a subgroup of 66 emmetropic eyes (66 subjects).

Results.: Lens powers obtained from the Bennett method corresponded well with those obtained by phakometry for emmetropic eyes, although individual differences up to 3.5 D occurred. Lens powers obtained from the modified-Stenström and Bennett-Rabbetts methods deviated significantly from those obtained with either the Bennett method or phakometry. Customizing the c constants improved this agreement, but applying these constants to the entire group gave mean lens power differences of 0.71 ± 0.56 D compared with the Bennett method. By further optimizing the c constants, the agreement with the Bennett method was within ±1 D for 95% of the eyes.

Conclusions.: With appropriate constants, the modified Stenström and Bennett-Rabbetts methods provide a good approximation of the Bennett lens power in emmetropic and myopic eyes.

Ocular refraction is determined by axial length, anterior chamber depth, corneal power, and lens power. Although axial length and keratometry measurements have become routine clinically, determining lens power is problematic, as the lens radii of curvature and refractive index distribution are usually not available. Although techniques have been proposed in the literature to estimate the radii in vivo, 1 5 they are currently too complicated to be used in large-scale studies or clinical practice. 
Because of this impracticality, various methods have been proposed that use ocular biometry, such as keratometry, ocular axial length, anterior chamber depth, lens thickness, and ocular refraction, to estimate the power of an equivalent lens at a location near that of the lens. Since these biometric parameters are easily determined, such methods can provide a quick estimate of the equivalent lens power. 
The most well known of these methods was proposed by Bennett, 6 who used a thick-lens description that makes assumptions about the shape and refractive index distribution of the lens based on the Gullstrand-Emsley schematic eye. 7 From this, he could calculate the equivalent lens power in a way that has been shown to be accurate in comparison with phakometry. 8 However, his method requires knowledge of the lens thickness, which is sometimes not available. 
Other methods do not require this knowledge of the lens thickness, such as the approaches proposed by Stenström 9,10 and by Bennett and Rabbetts. 11 These approaches might be useful in a clinical practice using biometry devices that do not provide lens thickness (e.g., IOL Master; Carl Zeiss Meditec, Dublin, CA), or in analysis of historical biometry data. 
The purposes of this study are (1) to verify the agreement that Dunne et al. 8 found between the Bennett method and phakometry; to (2) compare lens powers obtained with the Bennett method, our modification of the Stenström method, and the Bennett-Rabbetts method for previously published data of emmetropic and myopic eyes, and (3) to provide customized constants to optimize the performance of these three methods. The results allow improvement of our statistical eye model 12 by including a more reliable method to estimate lens power when lens thickness is not available. 
Methods
Subjects
To estimate the accuracy of the lens power calculations with respect to phakometry, we need the biometry and phakometry data of a population of normal subjects. For this purpose, we used previously published data by Atchison et al. 13 for a group of 66 eyes of 66 emmetropic subjects (32 men, 34 women; 62 Caucasian, 4 non-Caucasian). The average spherical equivalent refraction of this group was +0.01 ± 0.38 D (range, –0.88 to +0.75), and the mean age was 42.4 ± 14.4 years (range 19–69). 
To compare the results of the three power calculation methods for a wider range of refractions, the first dataset was supplemented by a second set from the same research group. 14 This dataset contained 118 eyes of 118 emmetropic and myopic subjects (43 men, 75 women; 74 Caucasian, 44 non-Caucasian) with a mean spherical equivalent refraction of −2.69 ± 2.79 D (range, −12.3 to +0.75 D) and an average subject age of 25.4 ± 5.1 years (range, 18–36 years). No phakometry data were available for this second dataset. 
Inclusion criteria were stringent, to ensure that only healthy eyes were included. These entailed, among others, corrected visual acuity better than 6/6 on an ETDRS chart, an intraocular pressure below 21 mm Hg, and a Pelli-Robson contrast sensitivity of 1.65 or better for subjects of 40 years of age and younger and a contrast sensitivity of 1.50 or better for subjects older than 40 years of age. In the myopic dataset, eyes with astigmatism larger than 0.5 D were also excluded. 
The subjects' eyes were not dilated or cyclopleged before testing, which might have caused some degree of accommodation in some of the younger subjects, resulting in slightly more hyperopic refraction, increased lens thickness, and decreased anterior chamber depth. 
The data collection complied with the Declaration of Helsinki and received ethical committee approval from the QUT University Human Research Ethics Committee and the Prince Charles Hospital Human Research Ethics Committee. All subjects gave written informed consent before participation. 
Biometry
Subjects' refractions were determined monocularly using Jackson crossed cylinders in a phoropter. Keratometry was performed with a corneal topographer (E300; Medmont, Vermont, VIC, Australia), and axial length, anterior chamber depth, lens thickness, and vitreous depth were measured by A-scan ultrasonography (Quantel Medical AXIS-II). For the emmetropic group, the radii of curvature of the anterior and posterior lens surfaces, as well as the lens equivalent refractive index, were obtained by analyzing Purkinje images, refraction, and biometry, by using a setup and calculations 13 similar to that described by Rosales and Marcos. 3 Note that phakometry data were not available for the second dataset. 
Bennett's Method: Lens Power Calculation Using Known Lens Thickness
Bennett's method 6 calculates lens power P L when lens thickness T is available by keeping the distances from the surfaces to the principal planes of the lens in the same proportion as in the lens of the Gullstrand-Emsley eye model. 7 Using the parameters defined in Table 1, the steps in his method can be combined as the single equation:   with n = 4/3 the aqueous and vitreous index, c 1 T = 1000n(nn L)T/(n L P L r La) the distance between the anterior lens surface and first lenticular principal plane, and c 2 T = 1000n(nn L)T/(n L P L r La) the distance between the posterior lens surface and second lenticular principal plane. The latter is negative because the principal plane is in front of the back surface. Bennett estimated the c 1 and c 2 constants using the Gullstrand-Emsley eye model, for which the lens refractive index n L = 1.416. 
Table 1.
 
Overview of the Parameters Used
Table 1.
 
Overview of the Parameters Used
Parameter Unit Calculation Description
S D Spherical refraction at spectacle back vertex plane
S CV D S/(1 − 0.014 S) Spherical refraction at corneal vertex
S PP D S/(1 − 0.0155 S) Spherical refraction at first principal plane of the eye
K D Corneal power
ACD mm Anterior chamber depth (corneal epithelium to anterior lens)
T mm Lens thickness
L mm Axial length
V mm L-ACD-T Vitreous depth
P L D Lens power
n Refractive index of aqueous and vitreous humors
n L Refractive index of crystalline lens
P L,Bennett D Equation 1 Lens power using Bennett method
r La mm Anterior radius of curvature of lens
r Lp mm Posterior radius of curvature of lens
P La D (n Ln)/r La Power of anterior lens surface
P Lp D (nn L)/r Lp Power of posterior lens surface
c 1 T mm 1000 n(nn L)T/(n L P L r Lp) Distance between anterior lens surface and first principal plane of lens
c 2 T mm 1000 n(nn L)T/(n L P L r La) Distance between posterior lens surface and second principal plane of lens
P L,Sten D Equation 2 Lens power using modified-Stenström method
P eye D Equation 3 Equivalent power of combination of eye and a thin correcting lens placed at the cornea
c Sten mm Equation 2 + 3 solved for c Sten Distance between anterior lens surface and first principal plane of lens
P L,BR D Equation 4 Lens power using Bennett-Rabbetts method
c BR mm Equation 4 solved for c BR Distance between thin lens position and anterior lens surface
Modified-Stenström Method: Lens Power Calculation if Lens Thickness Is Not Known
If lens thickness T is not available, one can estimate the lens power PL using Stenström's method,9,10 which provides the lens power referenced to its anterior vertex rather than to the principal planes. We modified the method by including the parameter cSten, which is the estimated distance between the anterior lens surface and the first lenticular principal plane. The modified-Stenström method is given by:   using the parameters in Table 1, with n = 1.336. This equation contains the equivalent power of the eye Peye. Based on Stenström's derivation, we calculated this as:   Here the ocular refraction at the first principal plane of the eye SPP is used. Lens power PL can be found by substituting the value for Peye derived from equation 3 into the right side of equation 2
A simplification of equation 3 was proposed by van Alphen, 15 who used the approximation P eye = 1392/LS PP. However, since this simplification deviates considerably from values obtained from equation 3 for c Sten > 0 mm, we did not include it in our analysis. 
Bennett-Rabbetts Method: Lens Power Calculation if Lens Thickness Is Not Known
Another approach to calculating P L without knowing T is to modify an equation proposed by Bennett and Rabbetts 11 for the purpose of calculating the spherical refraction of an eye when its biometry is known. They replaced the lens with an equivalent thin lens located at the midpoint between the lenticular principal planes, using the Bennett-Rabbetts eye model. 11 If the ocular refraction at the corneal vertex S CV is known, their equation can be rewritten to give P L:   with n = 1.336 and c BR the distance between the anterior lens surface and the thin lens position. This parameter can be found by solving equation 4 for c BR when P L is known. 
Phakometry
Using the lens surface radii of curvature and lens refractive index determined using phakometry, along with the lens thickness, the lens equivalent power was calculated using the thick lens formula 16 :   with P La and P Lp as defined in Table 1
Comparing Lens Powers with the Different Methods
To compare lens powers obtained with the methods detailed above, we determined the c constants c 1, c 2, c Sten, and c BR for both Gullstrand-Emsley and Bennett-Rabbetts eye models. As both eye models will differ from actual ocular biometry, we determined the optimal c constants also for each eye individually. For the Bennett method, these constants were easily determined by filling in the available phakometry of the emmetropic dataset into the formulas for c 1 and c 2 in Table 1, using n = 1.336. The optimal c constants of the modified-Stenström and Bennett-Rabbetts methods were found by using the phakometry lens powers of the emmetropic dataset for P L and solving equations 2 and 3 and equation 4 for the c constants, also using n = 1.336. The analytical solution for c Sten in the modified-Stenström method was mathematically complicated and could not be used in Excel (Microsoft, Redmond, WA); Mathematica (Wolfram Research, Champaign, IL) was used instead, to estimate values numerically. Means and standard deviations of these optimal c constants were called the customized c constants and are given in Table 2
Table 2.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry and Phakometry of the Emmetropic Data
Table 2.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry and Phakometry of the Emmetropic Data
Method Symbol Eye Model c Constants Average Within ±1 D from P L (%) Pearson Correlation Coefficients with Phakometry
Phakometry P L 22.87 ± 2.42 D
Bennett P L,Bennett Gullstrand-Emsley c 1 = 0.596; c 2 = −0.358 22.50 ± 2.02 D 45.5 0.778 (P < 0.001)
Bennett-Rabbetts c 1 = 0.599; c 2 = −0.353 22.74 ± 2.03 D 50.0 0.779 (P < 0.001)
Customized c 1 = 0.571 ± 0.028 22.54 ± 2.00 D 45.5 0.778 (P < 0.001)
c 2 = −0.378 ± 0.029
Modified Stenström P L,Sten Gullstrand-Emsley cSten = 2.145 mm 21.04 ± 1.94 D 19.7 0.720 (P < 0.001)
Bennett-Rabbetts cSten = 2.221 mm 21.36 ± 1.97 D 27.3 0.720 (P < 0.001)
Customized c Sten = 2.875 ± 0.763 mm 22.78 ± 2.12 D 42.4 0.721 (P < 0.001)
Bennett-Rabbetts P L,BR Gullstrand-Emsley c BR = 2.230 mm 21.21 ± 1.96 D 24.2 0.720 (P < 0.001)
Bennett-Rabbetts c BR = 2.306 mm 21.54 ± 1.99 D 36.4 0.720 (P < 0.001)
Customized c BR = 2.891 ± 0.778 mm 22.81 ± 2.13 D 40.9 0.721 (P < 0.001)
Statistics
The significance level was P < 0.05 (analyses by Excel; Microsoft; SPSS ver. 12; Chicago, IL). 
Results
Agreement between Calculated and Phakometry Derived Lens Powers for Emmetropes
The mean lens power determined with phakometry was P L = 22.87 ± 2.42 D, which may be considered the target value that the calculation methods must approximate (Table 2). Using both the Gullstrand-Emsley and Bennett-Rabbetts eye models, the lens powers with the Bennett method were not significantly different from the phakometry powers. Using the customized c constants did not improve the agreement. A Bland-Altman plot shows that the differences between Bennett and phakometry lens power remained between ±3 D (Fig. 1a) and for 45% and 50% of the eyes were less than ±1 D (Table 2). These differences did not correlate with subject age (Pearson < 0.01, P > 0.05), which excludes accommodation as a possible source of these differences. 
Figure 1.
 
Bland-Altman plots showing the differences between the phakometry lens powers and the lens powers calculated using (a) the Bennett method and (b) the Bennett-Rabbetts method. Powers for the calculation methods are shown for both eye model c constants and customized c constants. As the modified-Stenström and the Bennett-Rabbetts methods gave similar lens powers, results are shown only for the latter.
Figure 1.
 
Bland-Altman plots showing the differences between the phakometry lens powers and the lens powers calculated using (a) the Bennett method and (b) the Bennett-Rabbetts method. Powers for the calculation methods are shown for both eye model c constants and customized c constants. As the modified-Stenström and the Bennett-Rabbetts methods gave similar lens powers, results are shown only for the latter.
Using the Gullstrand-Emsley and Bennett-Rabbetts eye models, the modified-Stenström and Bennett-Rabbetts methods gave lens powers that were approximately 1.5 D lower and were significantly different from phakometry lens powers (paired t-tests, P < 0.01). By customizing the c constants, the differences with phakometry reduced remarkably to nonsignificance (paired t tests, P > 0.05), and for ∼40% of the eyes the differences were less than ±1 D (Table 2). 
The Pearson correlation coefficients between the calculated and phakometry lens powers were high (Table 2) and independent of the eye model used. The correlation coefficients were higher for the Bennett method than for the modified-Stenström and the Bennett-Rabbetts methods. 
Comparison of the Three Lens Power Calculation Methods for the Whole Population
Figure 2a shows lens powers calculated using the Bennett method with customized c constants calculated for the combination of the two datasets (184 eyes) as a function of axial length L. The lens power has a negative correlation with axial length for L < 24 mm (r = –0.624; P < 0.001), with a slope that matches that of the measured lens power data. Above L = 24 mm, approximately corresponding with the onset of myopia, the lens power plateaus to become constant (r = –0.036; P > 0.05). Because phakometry was not available for the second dataset, this plateauing could not be confirmed experimentally. However, a similar trend was found in the raw data published by Sorsby et al. 17 Thus, in the absence of phakometry data for the entire dataset, the Bennett power with customized c constants was used as a benchmark. This choice is based on the observation of Dunne et al. 8 that the Bennett power corresponds well with phakometry in myopic refractions up to –9.37 D, including the long eyes for which the plateauing is shown in Figure 2a. 
Figure 2.
 
(a) Phakometry lens power and powers calculated using the Bennett method with customized c constants, plotted as a function of axial length L. (b) Difference between the lens power values calculated using the modified-Stenström or Bennett-Rabbetts methods with the customized 2 constants and the Bennett method with customized c constants, plotted as a function of axial length L. The calculated powers use data from both datasets (184 eyes), whereas the phakometry powers contain data only from the first dataset (66 eyes).
Figure 2.
 
(a) Phakometry lens power and powers calculated using the Bennett method with customized c constants, plotted as a function of axial length L. (b) Difference between the lens power values calculated using the modified-Stenström or Bennett-Rabbetts methods with the customized 2 constants and the Bennett method with customized c constants, plotted as a function of axial length L. The calculated powers use data from both datasets (184 eyes), whereas the phakometry powers contain data only from the first dataset (66 eyes).
The mean powers with the modified-Stenström and Bennett-Rabbetts methods, using the Gullstrand-Emsley and Bennett-Rabbetts eye models, were 0.5 to 1.0 D less than the mean powers obtained with the Bennett method and its customized c constants (Table 2). These differences were statistically significant (paired t-test, P < 0.001). Using the customized c constants, the modified-Stenström and Bennett-Rabbetts methods each yielded lens powers that were 0.71 ± 0.56 D greater than those with the Bennett method (Table 3), and this difference was also statistically significant (P < 0.001). 
Table 3.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry of Both the Emmetropic and Myopic Datasets
Table 3.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry of Both the Emmetropic and Myopic Datasets
Method Symbol Eye Model c Constants Average Within ±1 D from P L,Bennett (%) Pearson Correlation Coefficient with Phakometry
Bennett P L,Bennett Customized c 1 = 0.571 ± 0.028 22.31 ± 1.72 D
c 2 = −0.378 ± 0.029
Modified Stenström P L,Sten Gullstrand-Emsley c Sten = 2.145 mm 21.30 ± 1.61 D 61.4 0.942 (P < 0.001)
Bennett-Rabbetts c Sten = 2.221 mm 21.62 ± 1.63 D 71.7 0.943 (P < 0.001)
Customized* c Sten = 2.875 ± 0.763 mm 23.01 ± 1.76 D 64.7 0.947 (P < 0.001)
Customized 2 c Sten = 2.550 mm 22.30 ± 1.69 D 95.1 0.945 (P < 0.001)
Bennett-Rabbetts P L,BR Gullstrand-Emsley c BR = 2.230 mm 21.45 ± 1.62 D 67.9 0.946 (P < 0.001)
Bennett-Rabbetts c BR = 2.306 mm 21.77 ± 1.64 D 78.8 0.947 (P < 0.001)
Customized* c BR = 2.891 ± 0.778 mm 23.02 ± 1.76 D 66.8 0.950 (P < 0.001)
Customized 2 cBR = 2.564 mm 22.31 ± 1.69 D 95.1 0.948 (P < 0.001)
To improve the matches of the modified-Stenström and Bennett-Rabbetts methods with the Bennett method, a second c constant (customized 2) was determined for the modified-Stenström and Bennett-Rabbetts methods that minimized the mean lens power difference with the Bennett method over the entire population. When these customized 2 constants were used, the lens power differences with the Bennett method were no longer statistically significant (P > 0.05) and were within ±1 D for about 95% of eyes (Table 3). For both methods, the power differences with the Bennett method were correlated significantly with axial length L (r = 0.390, P < 0.001 and r = 0.329, P < 0.001 for the modified-Stenström and the Bennett-Rabbetts methods, respectively; Fig. 2b). 
Discussion
Using the customized c constants all three lens power calculation methods are in reasonable agreement with the phakometry lens power for emmetropic eyes. This answers the first purpose of this article, which was to confirm the agreement that Dunne et al. 8 found between the Bennett method and phakometry. However, for individual eyes, differences between calculated and phakometric power of up to 3.5 D occurred (Fig. 1, Table 2), which is considerably larger than the differences of up to 0.77 D that Dunne et al. reported for the Bennett method. These differences could result from biometric errors and Bennett's assumption that the lens shapes of the eye models are representative for all eyes (the ratio P Lp/P La of posterior to anterior lens powers was 1.52 ± 0.19 for phakometry, but 1.67 and 1.70 for the Gullstrand-Emsley and Bennett-Rabbetts eye models, respectively). Using the argument of Bennett 6 and Dunne et al. 8 that lens power provided by the Bennett method is likely to be more accurate than phacometry, because of the inherent difficulties in performing the latter accurately, we considered that the Bennett method derived power as a reasonable approximation of the real equivalent lens power and used it as a reference to compare the modified-Stenström and Bennett-Rabbetts methods. 
For the Bennett method, the choice of eye model did not influence the calculated lens power significantly, which may be a consequence of the fact that the method is based on ray tracing of a thick lens model rather than a thin lens approximation. It can be used accurately for emmetropic eyes with either the c 1 and c 2 constants of the two eye models or the customized constants derived in this work. 
The second purpose of the paper was to compare lens powers obtained with the Bennett method, the modified-Stenström method, and the Bennett-Rabbetts method for emmetropic and myopic eyes. The modified-Stenström and Bennett-Rabbetts methods gave lens powers that were significantly lower than those given by phakometry (mean, 1.6 D) and the Bennett method (mean, 1.3D) for emmetropic eyes and for the Bennett method in combined emmetropic and myopic eyes (mean, 0.8 D). 
The third purpose of the paper was to provide customized constants to optimize the performances of the Bennett, modified-Stenström, and Bennett-Rabbetts methods. For the Bennett method using customized c constants made little difference in the results, but in emmetropic eyes the customized c of 2.875 and 2.891 mm for modified-Stenström and Bennett Rabbetts methods, respectively, gave nonsignificant lens power differences with phakometry and produced more accurate results than the constants of the eye models. When comparing lens powers for combined emmetropic and myopic eyes, the customized c constants for emmetropic eyes produced systematic lens power differences between the Bennett method and the modified-Stenström and Bennett-Rabbetts methods. This was improved by new customized 2 constants for the latter two methods (c = 2.550 and 2.564 mm for modified-Stenström and Bennett Rabbetts methods, respectively), which brought the lens power differences to within ±1 D for ∼95% of the eyes. If lens thickness is not available, both methods with the customized 2 constants may be considered to be good approximations of the Bennett method. 
Although the three calculation methods now match well with one another for a wide range of refractions, there are still theoretical issues to consider. The first is that the modified-Stenström and Bennett methods produce the same results when both lenticular principal planes coincide (i.e., c 1 T = c Sten and c2 T = T − c Sten). This can be confirmed mathematically by comparing equation 1 with equations 2 and 3 for the special case when S pp = S CV = 0. The more general case, when S pp and S CV are different from 0, could only be confirmed numerically because of the mathematically complicated equation 3. Although this seems to point at some common origin of both formulas, the meaning of this observation remains unclear. 
A second relationship was found between the modified-Stenström and Bennett-Rabbetts methods, which, despite being mathematically very different, produced very similar lens powers. Again, a possible relationship between both methods could not be investigated further due to the complexity of equation 3
As lens power depends on lens refractive index, one could expect a correlation between the c constants and lens refractive index values n L determined from phakometry. For this reason, the results of the lens power calculations were given for each eye model separately. However, a significant correlation with n L was seen only for c 1 of the Bennett method; the other c constants were either constant or randomly distributed. 
Finally, we would like to point out that one could also use IOL calculation formulas, such as the Hoffer Q 18 or the SRK/T formula 19,20 to calculate the lens power, provided appropriate values for the IOL constants are used. Here, one has to deal with the added difficulty of estimating the final postoperative position of the lens, 21,22 which may explain the large variety in IOL calculation formulas in the literature. 
In conclusion, if lens thickness is known the equivalent lens power is best calculated using the Bennett method with either the published or the customized c constants. The modified-Stenström and Bennett-Rabbetts methods, with appropriate c constants, provide reasonable approximations of equivalent lens power when lens thickness is not known. These methods allow the application of the concept of our statistical eye model 12 to datasets without lens thickness or can be included in the software of a biometry device alongside IOL calculation formulas, thus providing physicians with access to the important parameter of lens power. 
Footnotes
 Disclosure: J.J. Rozema, None; D.A. Atchison, None; M.-J. Tassignon, None
The authors thank Kristien Wouters for statistical advice and Rafael Irribaren and Wolfgang Haigis for discussions on the topic of this work. 
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Figure 1.
 
Bland-Altman plots showing the differences between the phakometry lens powers and the lens powers calculated using (a) the Bennett method and (b) the Bennett-Rabbetts method. Powers for the calculation methods are shown for both eye model c constants and customized c constants. As the modified-Stenström and the Bennett-Rabbetts methods gave similar lens powers, results are shown only for the latter.
Figure 1.
 
Bland-Altman plots showing the differences between the phakometry lens powers and the lens powers calculated using (a) the Bennett method and (b) the Bennett-Rabbetts method. Powers for the calculation methods are shown for both eye model c constants and customized c constants. As the modified-Stenström and the Bennett-Rabbetts methods gave similar lens powers, results are shown only for the latter.
Figure 2.
 
(a) Phakometry lens power and powers calculated using the Bennett method with customized c constants, plotted as a function of axial length L. (b) Difference between the lens power values calculated using the modified-Stenström or Bennett-Rabbetts methods with the customized 2 constants and the Bennett method with customized c constants, plotted as a function of axial length L. The calculated powers use data from both datasets (184 eyes), whereas the phakometry powers contain data only from the first dataset (66 eyes).
Figure 2.
 
(a) Phakometry lens power and powers calculated using the Bennett method with customized c constants, plotted as a function of axial length L. (b) Difference between the lens power values calculated using the modified-Stenström or Bennett-Rabbetts methods with the customized 2 constants and the Bennett method with customized c constants, plotted as a function of axial length L. The calculated powers use data from both datasets (184 eyes), whereas the phakometry powers contain data only from the first dataset (66 eyes).
Table 1.
 
Overview of the Parameters Used
Table 1.
 
Overview of the Parameters Used
Parameter Unit Calculation Description
S D Spherical refraction at spectacle back vertex plane
S CV D S/(1 − 0.014 S) Spherical refraction at corneal vertex
S PP D S/(1 − 0.0155 S) Spherical refraction at first principal plane of the eye
K D Corneal power
ACD mm Anterior chamber depth (corneal epithelium to anterior lens)
T mm Lens thickness
L mm Axial length
V mm L-ACD-T Vitreous depth
P L D Lens power
n Refractive index of aqueous and vitreous humors
n L Refractive index of crystalline lens
P L,Bennett D Equation 1 Lens power using Bennett method
r La mm Anterior radius of curvature of lens
r Lp mm Posterior radius of curvature of lens
P La D (n Ln)/r La Power of anterior lens surface
P Lp D (nn L)/r Lp Power of posterior lens surface
c 1 T mm 1000 n(nn L)T/(n L P L r Lp) Distance between anterior lens surface and first principal plane of lens
c 2 T mm 1000 n(nn L)T/(n L P L r La) Distance between posterior lens surface and second principal plane of lens
P L,Sten D Equation 2 Lens power using modified-Stenström method
P eye D Equation 3 Equivalent power of combination of eye and a thin correcting lens placed at the cornea
c Sten mm Equation 2 + 3 solved for c Sten Distance between anterior lens surface and first principal plane of lens
P L,BR D Equation 4 Lens power using Bennett-Rabbetts method
c BR mm Equation 4 solved for c BR Distance between thin lens position and anterior lens surface
Table 2.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry and Phakometry of the Emmetropic Data
Table 2.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry and Phakometry of the Emmetropic Data
Method Symbol Eye Model c Constants Average Within ±1 D from P L (%) Pearson Correlation Coefficients with Phakometry
Phakometry P L 22.87 ± 2.42 D
Bennett P L,Bennett Gullstrand-Emsley c 1 = 0.596; c 2 = −0.358 22.50 ± 2.02 D 45.5 0.778 (P < 0.001)
Bennett-Rabbetts c 1 = 0.599; c 2 = −0.353 22.74 ± 2.03 D 50.0 0.779 (P < 0.001)
Customized c 1 = 0.571 ± 0.028 22.54 ± 2.00 D 45.5 0.778 (P < 0.001)
c 2 = −0.378 ± 0.029
Modified Stenström P L,Sten Gullstrand-Emsley cSten = 2.145 mm 21.04 ± 1.94 D 19.7 0.720 (P < 0.001)
Bennett-Rabbetts cSten = 2.221 mm 21.36 ± 1.97 D 27.3 0.720 (P < 0.001)
Customized c Sten = 2.875 ± 0.763 mm 22.78 ± 2.12 D 42.4 0.721 (P < 0.001)
Bennett-Rabbetts P L,BR Gullstrand-Emsley c BR = 2.230 mm 21.21 ± 1.96 D 24.2 0.720 (P < 0.001)
Bennett-Rabbetts c BR = 2.306 mm 21.54 ± 1.99 D 36.4 0.720 (P < 0.001)
Customized c BR = 2.891 ± 0.778 mm 22.81 ± 2.13 D 40.9 0.721 (P < 0.001)
Table 3.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry of Both the Emmetropic and Myopic Datasets
Table 3.
 
Comparison of the Measured and Calculated Lens Powers Using the Biometry of Both the Emmetropic and Myopic Datasets
Method Symbol Eye Model c Constants Average Within ±1 D from P L,Bennett (%) Pearson Correlation Coefficient with Phakometry
Bennett P L,Bennett Customized c 1 = 0.571 ± 0.028 22.31 ± 1.72 D
c 2 = −0.378 ± 0.029
Modified Stenström P L,Sten Gullstrand-Emsley c Sten = 2.145 mm 21.30 ± 1.61 D 61.4 0.942 (P < 0.001)
Bennett-Rabbetts c Sten = 2.221 mm 21.62 ± 1.63 D 71.7 0.943 (P < 0.001)
Customized* c Sten = 2.875 ± 0.763 mm 23.01 ± 1.76 D 64.7 0.947 (P < 0.001)
Customized 2 c Sten = 2.550 mm 22.30 ± 1.69 D 95.1 0.945 (P < 0.001)
Bennett-Rabbetts P L,BR Gullstrand-Emsley c BR = 2.230 mm 21.45 ± 1.62 D 67.9 0.946 (P < 0.001)
Bennett-Rabbetts c BR = 2.306 mm 21.77 ± 1.64 D 78.8 0.947 (P < 0.001)
Customized* c BR = 2.891 ± 0.778 mm 23.02 ± 1.76 D 66.8 0.950 (P < 0.001)
Customized 2 cBR = 2.564 mm 22.31 ± 1.69 D 95.1 0.948 (P < 0.001)
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