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Visual Psychophysics and Physiological Optics  |   May 2012
Methods to Estimate the Size and Shape of the Unaccommodated Crystalline Lens In Vivo
Author Affiliations & Notes
  • Jos J. Rozema
    Department of Ophthalmology, Antwerp University Hospital, Edegem, Belgium;
    Department of Medicine and Health Science, University of Antwerp, Wilrijk, Belgium;
  • David A. Atchison
    School of Optometry and Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, Queensland, Australia;
  • Sanjeev Kasthurirangan
    Abbott Medical Optics, Milpitas, California; and
  • James M. Pope
    Faculty of Science and Engineering and Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, Queensland, Australia.
  • Marie-José Tassignon
    Department of Ophthalmology, Antwerp University Hospital, Edegem, Belgium;
    Department of Medicine and Health Science, University of Antwerp, Wilrijk, Belgium;
  • Corresponding author: Jos Rozema, Department of Ophthalmology, Antwerp University Hospital, Wilrijkstraat 10, 2650 Edegem, Belgium; Jos.Rozema@uza.be
Investigative Ophthalmology & Visual Science May 2012, Vol.53, 2533-2540. doi:10.1167/iovs.11-8645
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      Jos J. Rozema, David A. Atchison, Sanjeev Kasthurirangan, James M. Pope, Marie-José Tassignon; Methods to Estimate the Size and Shape of the Unaccommodated Crystalline Lens In Vivo. Invest. Ophthalmol. Vis. Sci. 2012;53(6):2533-2540. doi: 10.1167/iovs.11-8645.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose.: The purpose of this article was to present methods capable of estimating the size and shape of the human eye lens without resorting to phakometry or magnetic resonance imaging (MRI).

Methods.: Previously published biometry and phakometry data of 66 emmetropic eyes of 66 subjects (age range [18, 63] years, spherical equivalent range [−0.75, +0.75] D) were used to define multiple linear regressions for the radii of curvature and thickness of the lens, from which the lens refractive index could be derived. MRI biometry was also available for a subset of 30 subjects, from which regressions could be determined for the vertex radii of curvature, conic constants, equatorial diameter, volume, and surface area. All regressions were compared with the phakometry and MRI data; the radii of curvature regressions were also compared with a method proposed by Bennett and Royston et al.

Results.: The regressions were in good agreement with the original measurements. This was especially the case for the regressions of lens thickness, volume, and surface area, which each had an R 2 > 0.6. The regression for the posterior radius of curvature had an R 2 < 0.2, making this regression unreliable. For all other regressions we found 0.25 < R 2 < 0.6. The Bennett-Royston method also produced a good estimation of the radii of curvature, provided its parameters were adjusted appropriately.

Conclusions.: The regressions presented in this article offer a valuable alternative in case no measured lens biometry values are available; however care must be taken for possible outliers.

Introduction
To perform accurate ray tracing inside the human eye, it is important to know the in vivo size and shape of the eye lens. Although the equivalent power of the lens can be easily calculated, 1 obtaining a reliable estimate of its dimensions is more difficult because lens biometry, except for lens thickness, requires dedicated equipment that is available only as experimental devices. 
In ray tracing, the lens radii of curvature are of particular importance. These are traditionally determined by either phakometry, that is, the analysis of the location and relative sizes of the Purkinje reflections, 25 or a Scheimpflug camera corrected for refractive distortions of the images. 6 These techniques give similar results for the anterior lens radius of curvature and slightly differing results for the posterior radius of curvature, 7 and have been valuable in studies on ocular development, 8 accommodation, 9,10 and alignment of intraocular lenses (IOLs). 1114  
As there is no commercial device available to determine the lens radii of curvature in a clinical setting, Royston et al. 15 developed a method to estimate them by extending Bennett's method for the calculation of lens power. 16 This method uses the Gullstrand-Emsley eye model, 17 supplemented by ocular refraction, corneal power, and intraocular distances. 
Other parameters of interest are the lens equatorial diameter, volume, and surface area, which could be used in cataract surgery for the determination of a suitable IOL haptic diameter or predicting the amount of postoperative lens epithelial cell proliferation based on the size of the capsular bag. Although optical methods 8 may be used to estimate lens volume, nonoptical methods, such as ultrasound and magnetic resonance imaging (MRI), 2022 are preferred, as these are able to image the peripheral regions of the lens that are covered by the iris. In vitro studies 18,19 are not suitable, as the lens loses its physiological shape. 
This work proposes multiple linear regressions using common biometry parameters to estimate parameters that are difficult to measure in vivo. For the regressions of the radii of curvature of spherical lens surface fits, a comparison with the Bennett-Royston method is made, and all regressions are evaluated for their quality of fit to measured data, both when lens thickness is known and when it is not known. The latter may be useful in a clinical practice that uses a biometry device that does not provide lens thickness (e.g., Zeiss IOLMaster, Jena, Germany) or in analysis of historical biometry data. Combined with calculations of lens power or equivalent refractive index, these methods expand our statistical eye model 23 with a number of lenticular parameters. 
Methods
Subjects
This work uses previously published biometry and phakometry data 24 of 66 eyes of 66 emmetropes (32 male, 34 female; 62 Caucasian, 4 non-Caucasian) with a mean age of 42.4 ± 14.4 years, range [19.0, 69.3] years, and mean spherical equivalent of 0.01 ± 0.38 D with range [−0.88, 0.75] D. MRI data 25,26 (Kasthurirangan S, et al. IOVS 2007;48:ARVO E-Abstract 6008), containing estimates of the lens size and shape in the axial (horizontal) plane of the unaccommodated eye, were available for two subpopulations of 15 young and 15 older subjects (mean ages 22.3 ± 3.4 years and 63.6 ± 3.1 years, respectively). 
Subject inclusion criteria were stringent so as to ensure that only healthy eyes were included. These included corrected visual acuity better than 6/6 on an Early Treatment Diabetic Retinopathy Study chart, Pelli-Robson contrast sensitivity higher than 1.65 for subjects younger than 40 years and higher than 1.50 for subjects older than 40 years, and intraocular pressures below 21 mm Hg. 
Subjects' eyes were not dilated nor cyclopleged before testing. This might have caused some degree of accommodation in the younger subjects, resulting in slightly more myopic refraction, increased lens thickness, and decreased anterior chamber depth. 
The data collection followed the tenets of the Declaration of Helsinki and received ethical committee approval from the Queensland University of Technology Human Research Ethics Committee and the Prince Charles Hospital Human Research Ethics Committee. All subjects gave written informed consent before participation. 
Bennett-Royston Method for Estimating Lens Power and Radii of Curvature
Bennett 16 published a method to calculate the lens power P L using ocular refraction and biometry. Using the parameters defined in Table 1, Bennett's method can be written as1:  where n = 1.336 is the refractive index of the humors, c 1 T = 0.571T the distance between the anterior lens surface and the first lenticular principal plane, and c 2 T = −0.378T the distance between the posterior lens surface and the second lenticular principal plane. These c 1 and c 2 constants were obtained by optimization as reported in a previous article. 1  
Table 1.
 
Parameters
Table 1.
 
Parameters
Parameter Unit Calculation Uncertainty Description
S D 0.25 Spherical refraction at spectacle back vertex plane
S CV D S/(1 − 0.014 S) 0.25 Spherical refraction at corneal vertex
K D 0.25 Corneal power
ACD mm 0.05 Anterior chamber depth (corneal epithelium to anterior lens)
T mm 0.05 Lens thickness
D e mm 0.06 Equatorial lens diameter
L mm 0.05 Axial length
V mm 0.05 Vitreous depth
n 1.336 Refractive index of aqueous and vitreous humors
n L equation (3) 0.003 Equivalent refractive index of lens
P L D equation (1) 0.61 Lens power
r La[S] mm 0.18 Anterior lens radius of curvature (spherical fit)
r Lp[S] mm 0.11 Posterior lens radius of curvature (spherical fit)
r La[A] mm 0.14 Anterior lens radius of curvature (aspherical fit)
r Lp[A] mm 0.01 Posterior lens radius of curvature (aspherical fit)
k La 0.13 Conic constant of anterior lens surface
k Lp 0.03 Conic constant of posterior lens surface
c 1 T mm 0.571T 0.03 Distance between anterior lens surface and first principal plane of the lens
c 2 T mm −0.378T 0.02 Distance between posterior lens surface and second principal plane of the lens
Surf mm2 Appendix A 1.11 Surface area of lens
Vol mm3 Appendix A 2.49 Volume of lens
Royston et al. 15 later expanded Bennett's method to determine the lens radii of curvature using:  where Q = P La/P L is the ratio of the power of the anterior lens surface P La to the lens equivalent power. Equation (2) provides estimates of a spherical fit to the lens surfaces, as indicated by the “[S]” in r La[S] and r Lp[S]. This is in contrast with the “[A]” to be used later with the MRI data to which aspherical surfaces are fitted.  
As will be shown in the results section, using n L = 1.416 and Q = 0.380, as used by Royston et al., 15 gives radii of curvature that are significantly smaller than found using phakometry. As this would lead to a considerable bias in our comparison, we used the Bennett-Royston method with customized values n L = 1.431 ± 0.011 and Q = 0.405 ± 0.031, which were the mean values determined by phakometry for all 66 subjects. This follows the same reasoning as the customized c 1 and c 2 constants for the Bennett method mentioned previously. 1  
Multiple Linear Regression Modeling of Lens Parameters
As lens power is determined by thickness, radii of curvature, and equivalent refractive index, one can make a first-order approximation of the radii of curvature by a multiple linear regression of lens thickness and power. These radii of curvature can then be used to derive the equivalent refractive index, as is shown in the next section. Using the MRI data, regressions of the lens radii of curvature, conic constants, equatorial diameter, volume, and surface areas can be derived. The lens surface conic constant k is related to the surface coordinates (x, y) and vertex radius of curvature r by y = x 2 / ( r + r 2 k x 2 ) . The method to derive lens volume and surface area from the MRI data is described in Appendix A. All regressions are evaluated both assuming lens thickness is known and that it is not known.  
Using the uncertainties (or margins of error) on the biometry measurements given in Table 1, error propagation analysis 27 was used to estimate the uncertainty of the regressions. This involved calculating the partial derivatives of the regression with respect to each of the included variables, resulting in long and complicated equations that go beyond the scope of this article. Hence, only the results of the calculations are given and the error propagation formulas can be found in an annotated Mathematica notebook (Supplementary material). Note that, as knowledge of the covariances between the uncertainty estimates was not available, the covariance terms were discarded from the analysis. As this may lead to minor underestimation of the uncertainty, uncertainties presented in the following should be considered as indicative. 
Calculating Equivalent Refractive Index of the Lens
Using the radii of curvature and calculated lens power P L, it is possible to derive the equivalent refractive index of the lens n L from the thick lens formula 28  
P L = P La + P Lp − 0.001 TP La P Lp/n L, where P La and P Lp are the anterior and posterior lens surface powers. From this, the equivalent refractive index can be derived as:  where A = T − r La[S] + r Lp[S]. The equivalent refractive index may then be found by using the Bennett power from equation (1) as P L in equation (3).  
Statistics
All statistical calculations were performed using Excel 2003 (Microsoft Corp., Redmond, WA) and SPSS 12 (SPSS Inc., Chicago, IL). As in this work a large number of statistical tests are performed (about 50), a Šidák correction was applied to reduce the effect of α inflation. Hence, a significance level of P < 0.0044 was used to indicate statistically significant differences to ensure that the probability of obtaining a false test by chance is less than 20%. The multiple linear regressions were optimized using the linear regression function in SPSS, in which a significance level of P < 0.01 was chosen to identify significant terms. 
Results
Estimating Radii of Curvature rLa[S] and rLp[S]
The radii of curvature determined with phakometry were r La[S] = +10.38 ± 1.37 mm and r Lp[S] = −6.85 ± 0.86 mm, which are the target values to compare with the other methods (Table 2). The Bennett-Royston method (as published) gave the significantly lower values of r La[S] = +9.73 ± 0.88 mm and r Lp[S] = −5.85 ± 0.53 mm (paired t-test, P = 0.000 for both parameters). Replacing the original lens refractive index n L and Q in the Bennett-Royston method by mean n L and Q determined from phakometry (n L = 1.431 and Q = 0.405), the radii of curvature increased to r La[S] = +10.49 ± 0.95 mm and r Lp[S] = −6.94 ± 0.63 mm, which were much closer to phakometry. In the following, the Bennett-Royston radii of curvature is reported only using these mean n L and Q values. 
Table 2.
 
Comparison of Methods to Determine Lens Parameters (66 Eyes)
Table 2.
 
Comparison of Methods to Determine Lens Parameters (66 Eyes)
Phakometry T Available, Bennett-Royston (2) T Available, Regression (4) T Not Available, Regression (4) + (5)
Lens power P L
 Mean ± SD (D) 22.87 ± 2.42 22.54 ± 2.01 22.54 ± 2.01 22.55 ± 1.93
 Coeff. of determination R 2 (P) 0.605 (0.000) 0.605 (0.000) 0.574 (0.000)
 95% CI of difference [−2.72, +3.37] [−2.72, +3.37] [−2.83, +3.48]
Anterior radius of curvature r La[S]
 Mean ± SD (mm) 10.38 ± 1.37 10.49 ± 0.95 10.40 ± 1.20 10.36 ± 1.01
 Coeff. of determination R 2 (P) 0.099 (0.010) 0.527 (0.000) 0.541 (0.000)
 95% CI of difference [−2.92, +2.69] [−1.96, +1.90] [−1.85, +1.87]
Posterior radius of curvature r Lp[S]
 Mean ± SD (mm) −6.85 ± 0.86 −6.94 ± 0.63 −6.86 ± 0.76 −6.83 ± 0.64
 Coeff. of determination R 2 (P) 0.096 (0.011) 0.134 (0.002) 0.109 (0.015)
 95% CI of difference [−1.70, +1.88] [−1.83,+ 1.83] [−1.80, +1.76]
Thickness T
 Mean ± SD (mm) 4.11 ± 0.41 4.11 ± 0.41 4.11 ± 0.41 4.13 ± 0.36
 Coeff. of determination R 2 (P) 0.751 (0.000)
 95% CI of difference [−0.43, +0.40]
Refractive index n L
 Mean ± SD 1.431 ± 0.011 1.431 ± 0.000 1.430 ± 0.011 1.430 ± 0.011
 Coeff. of determination R 2 (P) 0.011 (0.402) 0.244 (0.000) 0.231 (0.000)
 95% CI of difference [−0.022, +0.022] [−0.021, +0.022] [−0.021, +0.023]
As an alternative to the Bennett-Royston method, r La[S] and r Lp[S] can be estimated using a multiple linear regression of lens thickness T and estimated lens power P L as follows:    
The third and fourth columns of Table 2 compare the radii of curvature obtained from phakometry with those determined from the Bennett-Royston method with adjusted parameters and from regression (4). The radii of curvature calculated with the Bennett-Royston method did not differ significantly from phakometry (paired t-test: P = 0.516 and 0.427 for the anterior and posterior radii, respectively) and the same was found for the radii of curvature given by regression (4) (P = 0.807 and 0.979, respectively). The R 2 correlations between the Bennett-Royston and phakometry methods were low and not statistically significant for both r La[S] and r Lp[S]. The R 2 correlation of the regression (4) was high for r La[S], but low for r Lp[S]. 
Figures 1a and 1b show the differences between radii of curvature obtained with the Bennett-Royston method and phakometry, which had 95% confidence intervals of [−2.92, +2.69] mm and [−1.70, + 1.88] mm for the anterior and posterior lens surfaces, respectively. The confidence intervals for differences between regression (4) and phakometry were similar for the anterior surface and posterior surfaces ([−1.96, +1.90] mm and [−1.83, +1.83] mm), respectively. 
Figure 1 .
 
Bland-Altman plots showing differences between biometric parameters determined by phakometry and calculated using the Bennett-Royston method, regression equation (4) and regression (5) for (a) lens anterior radius of curvature rLa[S], (b) the lens posterior radius of curvature rLp[S], (c) lens thickness T, and (d) lens refractive index nL.
Figure 1 .
 
Bland-Altman plots showing differences between biometric parameters determined by phakometry and calculated using the Bennett-Royston method, regression equation (4) and regression (5) for (a) lens anterior radius of curvature rLa[S], (b) the lens posterior radius of curvature rLp[S], (c) lens thickness T, and (d) lens refractive index nL.
Estimating Lens Thickness
If no measurement for lens thickness T is available, it can be approximated by a regression of subject age (in years) and anterior chamber depth ACD:  where R 2 = 0.751 (P = 0.000) and an uncertainty of ±0.05 mm. The coefficient of determination R 2 between the measured thickness and equation (5) was high (Table 2) and the 95% confidence interval of the difference with measured lens thickness was [−0.43, +0.40] mm (Fig. 1c).  
Using regression (5) and vitreous depth V = L − ACD − T in conjunction with Bennett's equation (1) gives a mean lens power of P L = 22.55 ± 1.93 D, which is not significantly different from the phakometry lens power of 22.87 ± 2.42 D (paired t-test, P = 0.927; Table 2). The R 2 between the lens powers calculated for regression (5) and phakometry was high (0.574) and the 95% confidence interval for the differences between them was [−2.83, +3.48] D. 
Similarly, the radii of curvature can be estimated using lens thickness regression (5) in conjunction with regression equations (4). The radii of curvature are not significantly different from those obtained with phakometry or when measured lens thickness was used (see Table 2). The R 2 correlation with phakometry was not influenced by the use of regression (5). 
Estimating Equivalent Refractive Index of the Lens
The lens refractive index n L calculated using equation (3) for the Bennett-Royston method and regression (4) (Table 2) did not differ significantly from the phakometry values (paired t-test, respectively P = 0.553 and P = 0.894). The uncertainties on these calculations were both ±0.003. 
The refractive index of the Bennett-Royston method was a constant, resulting in a linear Bland-Altman plot (Fig. 1d), and a 95% confidence interval of [−0.022, +0.022]. Regression (4) was reasonably correlated with phakometry and their difference had a 95% confidence interval of [−0.021, +0.022] (Fig. 1d). 
If no measurement for the lens thickness is available, the refractive index may be calculated using regression (5). This gives results that are not significantly different from either phakometry or from using equation (3) with a measured lens thickness (Table 2). 
Estimating Radii of Curvature and Conic Constants of Aspherical Fits to the Lens Surfaces, Equatorial Diameter, Volume, and Surface Area
Using MRI data 25,26 from a subpopulation of 30 eyes, the radii of curvature and conic constants can be estimated by the following multiple linear regressions:  with uncertainties of ±0.14, ±0.01, ±0.13 and ±0.03, respectively. The R 2 values and 95% confidence intervals of the differences with MRI are given in Table 3. Note that as the MRI subgroup is considerably smaller than the entire group, these R 2 values may be more sensitive to individual variations than the correlations reported in Table 2.  
Table 3.
 
Comparison of MRI Data and Regressions (6), (7), and (8) (30 Eyes)
Table 3.
 
Comparison of MRI Data and Regressions (6), (7), and (8) (30 Eyes)
Regression Using Measured T Regression Using Equation (5)
Anterior radius of curvature r La[A]
 Coeff. of determination R 2 (P) 0.610 (0.000) 0.494 0.000)
 95% CI of difference* [−2.83, +3.07] [−3.19, +3.48]
Posterior radius of curvature r Lp[A]
 Coeff. of determination R 2 (P) 0.367 (0.000) 0.367 0.000)
 95% CI of difference* [−1.45, +1.43] [−1.45, +1.43]
Anterior lens conic constant k La
 Coeff. of determination R 2 (P) 0.335 (0.000) 0.267 (0.000)
 95% CI of difference* [−5.25, +5.29] [−5.52, +5.61]
Posterior lens conic constant k Lp
 Coeff. of determination R 2 (P) 0.304 (0.002) 0.358 (0.000)
 95% CI of difference* [−1.19, +1.17] [−1.15, +1.12]
Equatorial diameter D e
 Coeff. of determination R 2 (P) 0.402 (0.000) 0.403 (0.000)
 95% CI of difference* [−0.50, +0.51] [−0.50, +0.51]
Lens volume Vol
 Coeff. of determination R 2 (P) 0.876 (0.000) 0.806 (0.000)
 95% CI of difference* [−21.3, +22.3] [−26.1, +26.3]
Lens surface area Surf
 Coeff. of determination R 2 (P) 0.669 (0.000) 0.632 (0.000)
 95% CI of difference [−15.6, +15.6] [−16.2, +16.5]
The equatorial diameter of the lens can be estimated using:  for which R 2 = 0.402 (Table 3) and with an uncertainty ±0.06 mm. The difference with the MRI data had a 95% confidence interval of [−0.50, +0.51] mm.  
Lens volume and surface area were determined from MRI biometry using the method described in Appendix A. These may be given by:    
The R 2 values were 0.876 and 0.669 for the lens volume and surface area, respectively (Table 3), and the uncertainties were 2.49 mm3 and 1.11 mm2. The 95% confidence intervals of the differences between regression (8) and MRI were [−21.34, 22.32] mm3 for the volume and [−15.6, 15.6] mm2 for the surface area. 
If lens thickness T is not known, regressions (6), (7), and (8) can be used with the lens thickness regression (5). This did not affect statistical significance (Table 3). 
Discussion
Using previously published phakometry and MRI data, we obtained estimates of crystalline lens parameters that are difficult to measure in vivo, using multiple linear regressions from parameters easily determined in clinical practice. Most regressions were in good agreement with the original measurements (Tables 2 and 3), with the regressions for thickness, volume, and surface area of the lens having R 2 values above 0.6 (i.e., explaining more than 60% of the variance). Therefore, these may be used confidently if no measurements are available. The regressions of r La[S], r La[A], r Lp[A], k La, k Lp, and D e had R 2 values between 0.25 and 0.6 (explaining about 25%–60% of the variance), meaning that these regressions may be cautiously used instead of measured values, bearing in mind that there may be outliers. 
The R 2 value for the posterior radius of curvature in the spherical fit r Lp[S] was below 0.2, thus making this particular regression unreliable when used in wavefront calculations, as it may incorrectly estimate the lenticular aberrations. This may be because of the calculations involved in phakometry, which assume a constant (equivalent) refractive index for the lens and do not take the gradient index into account. As the fourth Purkinje reflection (associated with the posterior surface of the lens) has to pass this gradient index twice, this may introduce an error in the calculation, as it has been recently shown 29 that spherical aberrations differ when they are calculated using a gradient index lens or using an equivalent refractive index. Another reason may be the uncertainty in determining the locations of the dim fourth Purkinje reflection, used in phakometry to estimate the posterior lens curvature. The estimates obtained using MRI were more accurate, as reflected by the better R 2 values. Furthermore, there was a difference in age dependency of both lens radii, with r La[S] decreasing significantly with age and almost no systematic variation in r Lp[S]. 24 Because regression (4) uses age-dependent variables T and P L, and a wide range of ages was considered, this particular model may not be able to predict the variation in r Lp[S] adequately. Therefore, a single mean value for posterior lens curvature (i.e., 6.85 mm) may be sufficient for most purposes, keeping in mind that wavefront calculations performed using this value may not always reliably represent the physiological reality. 
Provided the customized n L and Q values are used, the Bennett-Royston method produced similar radii of curvature (r La[S], r Lp[S]) as phakometry, but the coefficients of determination R 2 between them were low at 0.099 and 0.096, respectively. For r La[S], regression (4) produced a much higher R 2 with phakometry (0.527) than did the Bennett-Royston method, and the 95% confidence interval of the differences with phakometry was much smaller for the former. For r Lp[S], the differences in R 2 and size of the confidence interval were less pronounced. Note that the uncertainty for the Bennett-Royston method is 2.5 times higher than that of regression (4), making an estimate of both radii of curvature using the former method less reliable. 
In the MRI subgroup, the radii of curvature of the spherical fit to the lens surfaces (r La[S] = +10.77 ± 1.03 mm and r Lp[S] = −6.99 ± 1.03 mm) were different from those of the aspherical fit (r La[A] = +11.84 ± 2.40 mm and r Lp[A] = −5.87 ± 0.89 mm). This can be attributed to the differences in techniques and the attempt in MRI analysis to give a more sophisticated fit than simply obtaining a best-fit radius of curvature to a proportion of each surface. Note that the anterior lens radius of curvature r La[A] could not always be determined reliably for the entire horizontal meridian because of contact between the lens and the iris 25,26 (Kasthurirangan S, et al. IOVS 2007;48:ARVO E-Abstract 6008). 
As there may be circumstances in which the lens thickness of an eye is not known, such as in the analysis of historical or incomplete data or in a clinical practice that uses a biometry device that does not record lens thickness (e.g., Zeiss IOLMaster), the lens thickness regression (5) was introduced. It incorporates the thickness increase with age, and the resulting decrease in anterior chamber depth. The last column in Tables 2 and 3 shows that, when the lens thickness regression is used in conjunction with Bennett lens power (1), lens refractive index (3), or regressions (4), (6), (7), and (8), similar results were found as when the measured lens thickness was used. Using this regression in such a manner will slightly increase uncertainty. 
Once power, radii of curvature, and thickness of the lens are available, equation (3) can be used to calculate an equivalent refractive index n L that does not include the refractive index gradient of the natural lens. For regression (4), this yielded refractive indices that were not significantly different from the phakometric refractive index, whereas the Bennett-Royston method gave an unrealistic constant refractive index. This constant value is a result of using the thick lens approach in the Bennett-Royston method in combination with equation (3), which is the thick lens formula solved for n L. The Bennett-Royston method is therefore unsuitable for studies of the lens refractive index. Using the approximated lens shape, through equations (6) and (7), its equivalent refractive index (3) and the subject age, it is possible to make an estimate of the gradient index distribution using the formulas recently published by Navarro et al. 30,31 or de Castro et al. 29 We would not recommend this, however, as each consecutive model (combination of four regressions followed by the gradient index model) would add uncertainty to the results. 
Some of the proposed regressions, such as the lens equatorial diameter, volume, and surface area, may have applications in cataract surgery. A priori knowledge of the lens diameter could help cataract surgeons in choosing the right haptic size of an IOL. Similarly, lens volume and surface area, corresponding with the size of the capsular bag, may play a role in the capsular healing process after IOL implantation. Adapting the choice of IOL and surgical procedure to the dimensions of the natural lens could potentially reduce postoperative complications. Lens volume could also play a role in optimizing the phako-ersatz technique, 32 in which the capsular bag of a cataractous lens is emptied and subsequently filled with a clear gel in an attempt to preserve or restore accommodation. Until now, this technique has not been very successful, among other reasons, because of insufficient knowledge of the lens volume, resulting in an under- or overfilling of the capsular bag. 
As the regressions for lens volume and surface area use common biometry parameters, and the estimates were highly correlated with the original MRI data, these parameters can be easily and reliably used in a clinical practice (e.g., by incorporating the calculation into a biometry device). These strong correlations may be because of the lens thickness T, which is used in the calculation of the lens volume and surface area in Appendix A, as well as in regression (8). The dimensions of the lens found in this work (Vol = 172.8 ± 30.5 mm3 and Surf = 173.7 ± 12.0 mm2) correspond well with the values recently published by Hermans et al. 21 (Vol = 160.1 ± 2.5 mm3 and Surf = 175.5 ± 2.8 mm2). 
A shortcoming of this study is that all subjects were emmetropes, and it is not known how well these regressions will work in ametropic eyes. Because none of the regressions include refraction as a parameter, and this emmetropic group contained an axial length range of [21.47, 25.25] mm, we feel confident that the regressions will also work for an ametropic range of about [−6, +6] D. Another shortcoming is that the MRI subgroup was divided into a young and an older age group, which might have an influence on the results as well. 
Although the differences between the regressions presented in this work and phakometry may at times seem large, using the regression estimates in calculations is more appropriate than using the lens parameters of a mathematical eye model. As eye models use constant radii and refractive indices to describe individual eyes, they have no correlation with phakometry or MRI measurements and are unable to account for variations in biometry and the influence of ageing. The large 95% confidence intervals on the differences with phakometry given in Tables 2 and 3 may be attributable to some of the younger subjects accommodating slightly, as cycloplegia was not used. Other factors that may increase the confidence intervals include compounding of measurement errors and the errors of the phakometry technique. 16,33  
Supplementary Materials
Acknowledgments
We thank Kristien Wouters for statistical advice and Michiel Dubbelman for providing the numerical data of reference 21. 
References
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Footnotes
 Disclosure: J.J. Rozema, None; D.A. Atchison, None; S. Kasthurirangan, None; J.M. Pope, None; M.-J. Tassignon, None
Appendix A: Surface and Volume of a Solid of Revolution
Assuming that the lens is rotationally symmetric around its optical axis, its shape can be approximated by a solid of revolution, as was done previously by Koretz et al. 22 This solid can either be defined exactly using integration or by approximation using a stack of infinitesimally thin cylinders. For reasons of simplicity, we opted to use the latter method. 
The surface profile of the lens can be divided into (I) an aspherical anterior section, (II) a cylindrical midsection, and (III) an aspherical posterior section, each of which is calculated separately (Fig. A1a). Each of the two aspherical Sections I and III are given by:  which is derived from the general equation for an aspherical surface, with x as the radial component, y as position along the optical axis, y 0 as the offset, r L as the radius of curvature and k as the conic constant.  
Figure A1 .
 
(a) Division of the lens profile into the three sections. (b) Detail of a, showing the subdivision into N parts with height δ to approximate the lens profile using a series of infinitesimally thin cone slices.
Figure A1 .
 
(a) Division of the lens profile into the three sections. (b) Detail of a, showing the subdivision into N parts with height δ to approximate the lens profile using a series of infinitesimally thin cone slices.
The three Sections are separated by cutoff points COI and COIII for the anterior and posterior aspherical surfaces respectively. These are found by determining the x for which y = D e/2 (i.e., the point where the aspherical section is equal to the equatorial semidiameter of the lens). These cutoff points are given by:    
From this, the solid of revolution of Section I can be approximated by dividing the section [0 CO I] into N parts with a thickness of δ = CO I/N (Fig. A1b). Choosing a large N yields a more accurate approximation. For each of these N parts, a corresponding cone slice with upper diameter 2xn (n = 1, … , N), lower diameter 2x n+1 (n = 1, … , N − 1), and height δ can be defined for which the volume can easily be calculated using the following equation:    
By adding up all cylinder volumes, the total volume of Section I can be found. The volume of Section III is found in a similar way and Section II is a cylinder with a diameter D e/2 and height CO IIICO I, for which the volume can be found directly. Adding the volumes of all three Sections, leads to the total lens volume. 
For the lens surface area, a similar approach is used by calculating the mantle surface area of the cone slices:    
Adding all the mantle surfaces of all N cone slices gives the surface area of Sections I and III. For Section II, mantle surface of a cylinder with diameter D e/2 and height CO IIICO I is used. Adding the surface areas of all three Sections, leads to the total surface area. 
The results obtained by approximations (A3) and (A4) were verified with numerical integration using Mathematica (Wolfram Research, Champaign, IL), for which we found differences of 0.1 mm3 and 0.1 mm2 for the lens volume and surface area, respectively. From this, we concluded that approximations (A3) and (A4) were adequate for our purposes. 
Figure 1 .
 
Bland-Altman plots showing differences between biometric parameters determined by phakometry and calculated using the Bennett-Royston method, regression equation (4) and regression (5) for (a) lens anterior radius of curvature rLa[S], (b) the lens posterior radius of curvature rLp[S], (c) lens thickness T, and (d) lens refractive index nL.
Figure 1 .
 
Bland-Altman plots showing differences between biometric parameters determined by phakometry and calculated using the Bennett-Royston method, regression equation (4) and regression (5) for (a) lens anterior radius of curvature rLa[S], (b) the lens posterior radius of curvature rLp[S], (c) lens thickness T, and (d) lens refractive index nL.
Figure A1 .
 
(a) Division of the lens profile into the three sections. (b) Detail of a, showing the subdivision into N parts with height δ to approximate the lens profile using a series of infinitesimally thin cone slices.
Figure A1 .
 
(a) Division of the lens profile into the three sections. (b) Detail of a, showing the subdivision into N parts with height δ to approximate the lens profile using a series of infinitesimally thin cone slices.
Table 1.
 
Parameters
Table 1.
 
Parameters
Parameter Unit Calculation Uncertainty Description
S D 0.25 Spherical refraction at spectacle back vertex plane
S CV D S/(1 − 0.014 S) 0.25 Spherical refraction at corneal vertex
K D 0.25 Corneal power
ACD mm 0.05 Anterior chamber depth (corneal epithelium to anterior lens)
T mm 0.05 Lens thickness
D e mm 0.06 Equatorial lens diameter
L mm 0.05 Axial length
V mm 0.05 Vitreous depth
n 1.336 Refractive index of aqueous and vitreous humors
n L equation (3) 0.003 Equivalent refractive index of lens
P L D equation (1) 0.61 Lens power
r La[S] mm 0.18 Anterior lens radius of curvature (spherical fit)
r Lp[S] mm 0.11 Posterior lens radius of curvature (spherical fit)
r La[A] mm 0.14 Anterior lens radius of curvature (aspherical fit)
r Lp[A] mm 0.01 Posterior lens radius of curvature (aspherical fit)
k La 0.13 Conic constant of anterior lens surface
k Lp 0.03 Conic constant of posterior lens surface
c 1 T mm 0.571T 0.03 Distance between anterior lens surface and first principal plane of the lens
c 2 T mm −0.378T 0.02 Distance between posterior lens surface and second principal plane of the lens
Surf mm2 Appendix A 1.11 Surface area of lens
Vol mm3 Appendix A 2.49 Volume of lens
Table 2.
 
Comparison of Methods to Determine Lens Parameters (66 Eyes)
Table 2.
 
Comparison of Methods to Determine Lens Parameters (66 Eyes)
Phakometry T Available, Bennett-Royston (2) T Available, Regression (4) T Not Available, Regression (4) + (5)
Lens power P L
 Mean ± SD (D) 22.87 ± 2.42 22.54 ± 2.01 22.54 ± 2.01 22.55 ± 1.93
 Coeff. of determination R 2 (P) 0.605 (0.000) 0.605 (0.000) 0.574 (0.000)
 95% CI of difference [−2.72, +3.37] [−2.72, +3.37] [−2.83, +3.48]
Anterior radius of curvature r La[S]
 Mean ± SD (mm) 10.38 ± 1.37 10.49 ± 0.95 10.40 ± 1.20 10.36 ± 1.01
 Coeff. of determination R 2 (P) 0.099 (0.010) 0.527 (0.000) 0.541 (0.000)
 95% CI of difference [−2.92, +2.69] [−1.96, +1.90] [−1.85, +1.87]
Posterior radius of curvature r Lp[S]
 Mean ± SD (mm) −6.85 ± 0.86 −6.94 ± 0.63 −6.86 ± 0.76 −6.83 ± 0.64
 Coeff. of determination R 2 (P) 0.096 (0.011) 0.134 (0.002) 0.109 (0.015)
 95% CI of difference [−1.70, +1.88] [−1.83,+ 1.83] [−1.80, +1.76]
Thickness T
 Mean ± SD (mm) 4.11 ± 0.41 4.11 ± 0.41 4.11 ± 0.41 4.13 ± 0.36
 Coeff. of determination R 2 (P) 0.751 (0.000)
 95% CI of difference [−0.43, +0.40]
Refractive index n L
 Mean ± SD 1.431 ± 0.011 1.431 ± 0.000 1.430 ± 0.011 1.430 ± 0.011
 Coeff. of determination R 2 (P) 0.011 (0.402) 0.244 (0.000) 0.231 (0.000)
 95% CI of difference [−0.022, +0.022] [−0.021, +0.022] [−0.021, +0.023]
Table 3.
 
Comparison of MRI Data and Regressions (6), (7), and (8) (30 Eyes)
Table 3.
 
Comparison of MRI Data and Regressions (6), (7), and (8) (30 Eyes)
Regression Using Measured T Regression Using Equation (5)
Anterior radius of curvature r La[A]
 Coeff. of determination R 2 (P) 0.610 (0.000) 0.494 0.000)
 95% CI of difference* [−2.83, +3.07] [−3.19, +3.48]
Posterior radius of curvature r Lp[A]
 Coeff. of determination R 2 (P) 0.367 (0.000) 0.367 0.000)
 95% CI of difference* [−1.45, +1.43] [−1.45, +1.43]
Anterior lens conic constant k La
 Coeff. of determination R 2 (P) 0.335 (0.000) 0.267 (0.000)
 95% CI of difference* [−5.25, +5.29] [−5.52, +5.61]
Posterior lens conic constant k Lp
 Coeff. of determination R 2 (P) 0.304 (0.002) 0.358 (0.000)
 95% CI of difference* [−1.19, +1.17] [−1.15, +1.12]
Equatorial diameter D e
 Coeff. of determination R 2 (P) 0.402 (0.000) 0.403 (0.000)
 95% CI of difference* [−0.50, +0.51] [−0.50, +0.51]
Lens volume Vol
 Coeff. of determination R 2 (P) 0.876 (0.000) 0.806 (0.000)
 95% CI of difference* [−21.3, +22.3] [−26.1, +26.3]
Lens surface area Surf
 Coeff. of determination R 2 (P) 0.669 (0.000) 0.632 (0.000)
 95% CI of difference [−15.6, +15.6] [−16.2, +16.5]
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