**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.

^{ 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.

^{ 2–5 }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).

^{ 11–14 }

^{ 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.

^{ 8 }may be used to estimate lens volume, nonoptical methods, such as ultrasound and magnetic resonance imaging (MRI),

^{ 20–22 }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.

^{ 23 }with a number of lenticular parameters.

^{ 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).

^{ 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 as

^{1}: where

*n*= 1.336 is the refractive index of the humors,

*c*

_{1}

*T*= 0.571

*T*the distance between the anterior lens surface and the first lenticular principal plane, and

*c*

_{2}

*T*= −0.378

*T*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.**

**Table 1.**

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 | mm^{2} | Appendix A | 1.11 | Surface area of lens |

Vol | mm^{3} | Appendix A | 2.49 | Volume of lens |

^{ 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.

*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 }

*k*is related to the surface coordinates (

*x*,

*y*) and vertex radius of curvature

*r*by $ y = x 2 / ( r + r 2 \u2212 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.

^{ 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.

*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).

*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.

*r*

_{La}[S] and

*r*

_{Lp}[S]

*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.**

**Table 2.**

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] |

*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*].

**Figure 1**

**.**

**Figure 1**

**.**

*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).

*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.

*R*

^{2}correlation with phakometry was not influenced by the use of regression (5).

*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.

^{ 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.**

**Table 3.**

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] |

*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.

*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 mm

^{3}and 1.11 mm

^{2}. The 95% confidence intervals of the differences between regression (8) and MRI were [−21.34, 22.32] mm

^{3}for the volume and [−15.6, 15.6] mm

^{2}for the surface area.

*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).

*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.

*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.

*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.

*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).

*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.

^{ 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.

*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 mm

^{3}and

*Surf =*173.7 ± 12.0 mm

^{2}) correspond well with the values recently published by Hermans et al.

^{ 21 }(

*Vol =*160.1 ± 2.5 mm

^{3}and

*Surf*= 175.5 ± 2.8 mm

^{2}).

^{ 16,33 }

*Invest Ophthalmol Vis Sci*. 2011; 52:7937–7942. [CrossRef] [PubMed]

*Am J Optom Arch Am Acad Optom*. 1968; 45:743–759. [CrossRef] [PubMed]

*Ophthalmic Physiol Opt*. 1992; 12:370–375. [CrossRef] [PubMed]

*Ophthalmic Physiol Opt*. 1997; 17:75–80. [CrossRef] [PubMed]

*J Vis*. 2008; 818; 1–12.

*Vision Res*. 2001; 41:1867–1877. [CrossRef] [PubMed]

*J Vis*. 2006; 6:1057–1067. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 1998; 39:120–133. [PubMed]

*J Opt Soc Am A Opt Image Sci Vis*. 2006; 23:509–520. [CrossRef] [PubMed]

*Ophthalmic Physiol Opt*. 2004; 24:65–73. [CrossRef] [PubMed]

*J Cat Refract Surg*. 2007; 33:418–429. [CrossRef]

*J Cat Refract Surg*. 2010; 36:1529–1535. [CrossRef]

*Clin Exp Optom*. 2010; 6:400–408. [CrossRef]

*Ophthalmic Physiol Opt*. 2010; 30:43–54. [CrossRef] [PubMed]

*Ophthalmic Physiol Opt*. 1989; 9:412–414. [CrossRef] [PubMed]

*Ophthalmic Physiol Opt*. 1988; 8:53–59. [CrossRef] [PubMed]

*Visual Optics*. Vol. 1. 5th ed. London, UK: Butterworth; 1952: 40–42, 344, 360–361.

*Trans Ophthalmol Soc*. 1956; 54:675–708.

*Vision Res*. 2006; 46:1002–1009. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 1999; 40:1162–1169. [PubMed]

*Invest Ophthalmol Vis Sci*. 2009; 50:281–289. [CrossRef] [PubMed]

*J Opt Soc Am A Opt Image Sci Vis*. 2001; 18:265–272. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 2011; 52:4525–4533. [CrossRef] [PubMed]

*J Vis*. 2008; 8:29.1–29.20. [CrossRef]

*Invest Ophthalmol Vis Sci*. 2008; 49:2531–2540. [CrossRef] [PubMed]

*J Vis*. 2011; 11:19.1–19.16. [CrossRef]

*J Res Nat Bur Stand Sect C*. 1966; 70:263–273. [CrossRef]

*Optics and Vision*. 1st ed. Upper Saddle River, NJ: Prentice Hall; 1998: 77.

*Optics Express*. 2010; 18:21905–21917. [CrossRef] [PubMed]

*J Opt Soc Am A Opt Image Sci Vis*. 2007; 24:2175–2185. [CrossRef] [PubMed]

*J Opt Soc Am A Opt Image Sci Vis*. 2007; 24:2911–2920. [CrossRef] [PubMed]

*Graefes Arch Clin Exp Ophthalmol*. 1986; 224:165–173. [CrossRef] [PubMed]

*Ophthalmic Physiol Opt*. 1989; 9:69–71. [CrossRef] [PubMed]

^{ 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.

*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**

**.**

**Figure A1**

**.**

_{I}and CO

_{III}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:

*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 2

*x*(

_{n}*n*= 1, … ,

*N*), lower diameter 2

*x*

_{ n+1}(

*n*= 1, … ,

*N*− 1), and height

*δ*can be defined for which the volume can easily be calculated using the following equation:

*D*

_{e}/2 and height

*CO*

_{III}−

*CO*

_{I}, for which the volume can be found directly. Adding the volumes of all three Sections, leads to the total lens volume.

*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*

_{III}−

*CO*

_{I}is used. Adding the surface areas of all three Sections, leads to the total surface area.

^{3}and 0.1 mm

^{2}for the lens volume and surface area, respectively. From this, we concluded that approximations (A3) and (A4) were adequate for our purposes.