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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 5, Iss. 5 — May. 1, 2014
  • pp: 1649–1663
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An analytical method for predicting the geometrical and optical properties of the human lens under accommodation

Conor J. Sheil, Mehdi Bahrami, and Alexander V. Goncharov  »View Author Affiliations


Biomedical Optics Express, Vol. 5, Issue 5, pp. 1649-1663 (2014)
http://dx.doi.org/10.1364/BOE.5.001649


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Abstract

We present an analytical method to describe the accommodative changes in the human crystalline lens. The method is based on the geometry-invariant lens model, in which the gradient-index (GRIN) iso-indicial contours are coupled to the external shape. This feature ensures that any given number of iso-indicial contours does not change with accommodation, which preserves the optical integrity of the GRIN structure. The coupling also enables us to define the GRIN structure if the radii and asphericities of the external lens surfaces are known. As an example, the accommodative changes in lenticular radii and central thickness were taken from the literature, while the asphericities of the external surfaces were derived analytically by adhering to the basic physical conditions of constant lens volume and its axial position. The resulting changes in lens geometry are consistent with experimental data, and the optical properties are in line with expected values for optical power and spherical aberration. The aim of the paper is to provide an anatomically and optically accurate lens model that is valid for 3 mm pupils and can be used as a new tool for better understanding of accommodation.

© 2014 Optical Society of America

1. Introduction

The mechanism of accommodation in the human eye is still an area of on-going research. The most widely accepted theory is that proposed by Helmholtz in his Handbuch der Physiologischen Optik in 1867 [1

1. H. von Helmholtz, Handbuch der Physiologischen Optik (Leopold Voss, 1867).

]. This Helmholtzian theory of accommodation states that contraction of the ciliary muscle body reduces zonular and capsular tension, allowing the lens to form a steeper, more convex shape. This corresponds to the accommodated state, with an associated increase in lenticular refractive power; the accommodated state provides clear vision at near distances. For distant vision, the ciliary muscle relaxes (dilates), and as a result the lens is stretched into a thinner shape with reduced refractive power. This is the unaccommodated state, in which the lens is kept under constant tension. Experimental support for this most commonly accepted Helmholtzian theory includes, amongst many others, an interesting paper by Shao et al. (2013) showing real-time imaging of accommodative changes in the anterior segment of the eye [2

2. Y. Shao, A. Tao, H. Jiang, M. Shen, J. Zhong, F. Lu, and J. Wang, “Simultaneous real-time imaging of the ocular anterior segment including the ciliary muscle during accommodation,” Biomed. Opt. Express 4, 466–480 (2013). [CrossRef] [PubMed]

]. While experimental efforts show that lenticular spherical aberration (SA) becomes more negative with accomodation, it is clear that increasing surface curvatures cause more positive SA; thus the accommodative decrease in lenticular radii should produce more positive SA. We aim to investigate this apparent contradiction by looking at the lens’ surface and gradient index contributions to lenticular SA.

The field of ocular biometry is fast improving, with new and enhanced imaging methods capable of taking more accurate data with better sampling. While wavefront sensing continues to provide high-quality measurement of ocular wavefront aberration, the improving temporal and spatial resolution of magnetic resonance imaging (MRI) [3

3. K. Richdale, L. T. Sinnott, M. A. Bullimore, P. A. Wassenaar, P. Schmalbrock, C.-Y. Kao, S. Patz, D. O. Mutti, A. Glasser, and K. Zadnik, “Quantification of age-related and per diopter accommodative changes of the lens and ciliary muscle in the emmetropic human eye,” Invest. Ophthalmol. Visual Sci. 54, 1095–1105 (2013). [CrossRef]

] and optical coherence tomography (OCT) [4

4. A. de Castro, S. Ortiz, E. Gambra, D. Siedlecki, and S. Marcos, “Three-dimensional reconstruction of the crystalline lens gradient index distribution from OCT imaging,” Opt. Express 18, 21905–21917 (2010). [CrossRef] [PubMed]

, 5

5. A. de Castro, J. Birkenfeld, B. Maceo, F. Manns, E. Arrieta, J.-M. Parel, and S. Marcos, “Influence of shape and gradient refractive index in the accommodative changes of spherical aberration in nonhuman primate crystalline lenses,” Invest. Ophthalmol. Visual Sci. 54, 6197–6207 (2013). [CrossRef]

] may help with obtaining additional biometric information. The availability of these more accurate instruments for ocular biometry allows one to develop more advanced optical and physical models of the human eye; in particular, the crystalline lens.

It can be desirable to analyse accommodation-dependant optical properties of the human lens featuring gradient index in ray-tracing software such as Zemax. When starting with a set of data for the unaccommodated lens, currently one must predict what values the accommodated lens parameters will take. If, for example, the lenticular radii, conic constants and central thickness of the accommodated lens are inserted into the ray-tracing software, these accommodated parameters will not necessarily be optically or physically appropriate. That is, the accommodated lens will not display the correct power or SA, and it may have changed substantially in volume. The current study aims to develop an analytical method of describing the geometrical and optical changes in the lens under accommodation. In this paper, we describe an anatomically realistic, adjustable lens, which can be used as a tool for ocular modelling and understanding of accommodation.

2. Analytical method

The GRIN structure of this lens model is coupled to the shape of the external surface of the lens. This is an attractive feature for modelling the accommodative changes since it secures the optical integrity of the GRIN structure. The general outline of the GRIN medium can be visualised by looking at its iso-indicial contours, which are surfaces of constant refractive index; for example, Fig. 1 contains five iso-indicial contours with a refractive index step of 0.008. Preserving the optical integrity of the GRIN structure requires that the number of contours does not increase or decrease with accommodation. Few recent lens models [13

13. M. Bahrami and A. V. Goncharov, “Geometry-invariant gradient refractive index lens: analytical ray tracing,” J. Biomed. Opt. 17, 055001 (2012). [CrossRef] [PubMed]

15

15. R. Navarro, F. Palos, and L. González, “Adaptive model of the gradient index of the human lens. i. formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A 24, 2175–2185 (2007). [CrossRef]

] feature this coupling of GRIN structure and external lens shape. Of these recent models, we choose the geometry-invariant model [13

13. M. Bahrami and A. V. Goncharov, “Geometry-invariant gradient refractive index lens: analytical ray tracing,” J. Biomed. Opt. 17, 055001 (2012). [CrossRef] [PubMed]

] since it offers both a realistic GRIN distribution with the age-dependent parameter P and continuous iso-indicial surfaces. Note also that n(ζ) is a continuous function.

Fig. 1: Visualisation of Ta, Tp and Zc in a GRIN lens; age-related parameter P = 2.94.

Since Ta and Tp are positive quantities, and Zc is a negative quantity [13

13. M. Bahrami and A. V. Goncharov, “Geometry-invariant gradient refractive index lens: analytical ray tracing,” J. Biomed. Opt. 17, 055001 (2012). [CrossRef] [PubMed]

], the anterior and posterior axial thicknesses (sags) of the lens are given by Za = Ta + Zc and Zp = TpZc. The anterior and posterior sags are defined from their respective poles (vertices) to the lens equator, while distances Ta and Tp are defined from their respective poles to the centre of the nucleus. The centre of the nucleus is the position of peak refractive index and is defined as the origin; it is not necessarily at the same axial position as the lens’ equator [15

15. R. Navarro, F. Palos, and L. González, “Adaptive model of the gradient index of the human lens. i. formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A 24, 2175–2185 (2007). [CrossRef]

, 17

17. C. Jones, D. Atchison, R. Meder, and J. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vis. Res. 45, 2352–2366 (2005). [CrossRef] [PubMed]

]. See Fig. 1 for a visualisation of the two definitions.

Next, an analytical formula for the lens’ volume can be derived using the disc method of integral calculus for the volume of a solid of revolution:
V=πTaZcρa2(z)dz+πZcTpρp2(z)dz,
(5)
taking note again that Ta and Tp are positive, while Zc is negative. Since the lenticular epithelial stroma constitutes a soft condensed medium and is essentially incompressible, it seemed reasonable to assume that the volume does not change on accommodation. A recent study by Hermans et al. came to the same conclusion experimentally [18

18. E. A. Hermans, P. J. W. Pouwels, M. Dubbelman, J. P. A. Kuijer, R. G. L. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: An MRI and scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50, 281–289 (2009). [CrossRef]

]. Furthermore, the relatively short timescale over which the accommodation process occurs is rather short to facilitate any volume change through passage of water/aqueous into or out of the lens material [19

19. C. E. Jones, D. A. Atchison, and J. M. Pope, “Changes in lens dimensions and refractive index with age and accommodation,” Opt. Vis. Sci. 84, 990–995 (2007). [CrossRef]

]. A study by Gerometta et al. [20

20. R. Gerometta, A. C. Zamudio, D. P. Escobar, and O. A. Candia, “Volume change of the ocular lens during accommodation,” Am. J. Physiol.: Cell Physiol. 293, C797–C804 (2007). [CrossRef]

] found that bovine lenticular volume increases by 8% with accommodation, and expected the human lenticular volume to increase by 3%; however, one cannot be certain of the applicability of the bovine study to the human lens. Another study by Strenk et al. [21

21. S. A. Strenk, L. M. Strenk, J. L. Semmlow, and J. K. DeMarco, “Magnetic resonance imaging study of the effects of age and accommodation on the human lens cross-sectional area,” Invest. Ophthalmol. Visual Sci. 45, 539–545 (2004). [CrossRef]

] concluded that the lenticular volume increases with accommodation, but there has been considerable discussion regarding the accuracy and validity of their method [22

22. S. J. Judge and H. J. Burd, “The MRI data of strenk et al. do not suggest lens compression in the unaccommodated state (e-letter),” Invest. Ophthalmol. Visual Sci. 45, 539 (2004).

, 23

23. R. A. Schachar, “The change in intralenticular pressure during human accommodation (e-letter),” Invest. Ophthalmol. Visual Sci. 45, 539 (2004).

]. In any instance, we anticipate a small change in lens volume which, although non-negligible, will not drastically affect the results of our model. Any future definitive answer for the accommodative change in lenticular volume can be incorporated into our analytical method.

In our present study, the constant volume condition is used as a control for the accommodated lens. The volume calculated using Eq. (5) is given below:
V=16π[5RpZp2+5RaZa2QpZp3QaZa3].
(6)

3. Finding the conic constants of the lens

To demonstrate application of this analytical method, we use sample experimental data for the accommodative changes in lenticular radii and central thickness. We present an initial example using population-averaged Scheimpflug data from the work of Dubbelman et al. (2005) [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

], which have consistently smaller error compared to data obtained with other imaging instruments, for example MRI [3

3. K. Richdale, L. T. Sinnott, M. A. Bullimore, P. A. Wassenaar, P. Schmalbrock, C.-Y. Kao, S. Patz, D. O. Mutti, A. Glasser, and K. Zadnik, “Quantification of age-related and per diopter accommodative changes of the lens and ciliary muscle in the emmetropic human eye,” Invest. Ophthalmol. Visual Sci. 54, 1095–1105 (2013). [CrossRef]

,18

18. E. A. Hermans, P. J. W. Pouwels, M. Dubbelman, J. P. A. Kuijer, R. G. L. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: An MRI and scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50, 281–289 (2009). [CrossRef]

]. The application of these Scheimpflug data helps to ensure that the accommodative changes in radii and central thickness are anatomically realistic. The model is defined for a 30 year old lens, covering a range of accommodative amplitudes A =0–8 D. In this example, the anterior and posterior radii of curvature and central lens thickness were calculated for each accommodative amplitude (in steps of 2 D, for clarity) using empirical linear equations given in Dubbelman et al. [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

]. The change per dioptre in the anterior radius of curvature Ra, posterior radius of curvature Rp and central thickness T are:
Ra(A)=R0a+ΔRa=R0a+(0.350.084R0a)A,Rp(A)=R0p+ΔRp=R0p+(0.370.082R0p)A,T(A)=T0+ΔT=T0+(0.0436)A,
(7)
where R0a, R0p and T0 are the unaccommodated radii and thickness, and A is accommodative amplitude in D. Note that these equations represent the change in lenticular parameters per dioptre of stimulus. Objective accommodative response was not measured, and so these formulae may include accommodative lag.

Conic constants can only be accurately determined by imaging a relatively large diameter of the lens [16

16. G. Smith, “The optical properties of the crystalline lens and their significance.” Clin. Experimental Opt. 86, 3–18 (2003). [CrossRef]

]. The accuracy in measuring the conic constants of the human lens’ surfaces by Scheimpflug imaging is limited by lenticular occlusion by the finite maximum size of the iris and ocular refractive distortion. Consequently, the main motivation in our study was to find a way of predicting the conic constants of the human lens at different accommodative amplitudes, while the data for accommodative changes in the lenticular radii and central thickness were taken from the literature.

In order to develop an analytical method of finding the conic constants, we also use additional information for the accommodative changes in the anterior chamber depth (ACD) of the eye. As the lens accommodates, its anterior pole moves forward into the anterior chamber and this can be seen as a reduction in the ACD. In the same study by Dubbelman et al. [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

], partial coherence interferometry was used to measure a change per dioptre in the anterior chamber of ΔACD(A) = −0.036A mm for the 30 year old eye.

We use the relation above together with Eq. (7) as a basis for finding Ra(A), Rp(A) and T(A), and ACD(A) for a given accommodative amplitude. To find the anterior and posterior conic constants, we apply the following two conditions to the lens: constant volume (Eq. (6)) and fixed equator position. The latter assumes that the equator of the lens does not move axially within the eye during accommodation; thus, the reduction in ACD can be seen as an increase in the anterior sag Za of the lens and can be expressed as:
Za(A)=Z0aΔACD(A).
(8)

Finally, using Eqs. 4, 6 and 8, we can obtain the anterior and posterior conic constants as a function of accommodative amplitude:
Ka=π(TZa)(Rp(TZa)RaZa)6V0πTZa2+5RaZa1,and
(9)
Kp=πZa(RaZaRp(TZa))6V0πT(TZa)2+5RpTZa1.
(10)
Note that calculation of the initial volume V0 requires sample data for R0a, R0p, K0a, K0p and T0. Alternatively, if we have data for the accommodative change in Ka, we can solve Eq. (9) for Za, and subsequently find Kp from Eq. (10). This has physical significance since it allows us to predict, using data for Ka, the accommodative change in Kp, which is difficult to measure in vivo.

4. Analysis of lenticular geometry

We define the initial lens geometry based on the recent work of Ortiz et al. [25

25. S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3, 2471–2488 (2012). [CrossRef] [PubMed]

]. We used the following parameters for the unaccommodated lens: R0a = 12.48 mm, R0p = 7.25 mm, T0 = 3.18 mm, K0a = −2.57 and K0p = −1.64. Assuming the centre of the lens’ nucleus is located more posteriorly [15

15. R. Navarro, F. Palos, and L. González, “Adaptive model of the gradient index of the human lens. i. formulation and model of aging ex vivo lenses,” J. Opt. Soc. Am. A 24, 2175–2185 (2007). [CrossRef]

, 17

17. C. Jones, D. Atchison, R. Meder, and J. Pope, “Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI),” Vis. Res. 45, 2352–2366 (2005). [CrossRef] [PubMed]

, 26

26. R. Navarro, F. Palos, and L. M. González, “Adaptive model of the gradient index of the human lens. ii. optics of the accommodating aging lens,” J. Opt. Soc. Am. A. 24, 2911–2920 (2007). [CrossRef]

], we set Ta = 0.6T and Tp = 0.4T, so that T0a = 1.91 mm and T0p = 1.27 mm.

These initial parameters allow us to calculate the geometrical features of the lens, which were derived analytically. First, the position of the lens’ equator with respect to the centre of the nucleus is given by Eq. (4): Z0c = −0.74 mm. Hence, we can calculate the anterior sag Z0a = T0a + Z0c = 1.17 mm, the posterior sag Z0p = T0pZ0c = 2.01 mm and the two B coefficients: B0a = −6.93 mm−1 and B0p = −1.41 mm−1. Finally, the initial volume V0 is calculated using Eq. (6) to give V0 = 125.45 mm3. Using Eqs. 7, 3 and 8, we can calculate the parameters of the lens at different accommodative amplitudes, as shown in Tables 1 and 2 below for A = 0, 2, 4, 6 and 8 D. Figure 2 shows the transverse cross-section of the lens, with the anterior and posterior surfaces to the left and right, respectively—note that the lens is axisymmetric.

Table 1:. Predicted changes in geometrical parameters of the lens under accommodation. All distances are in mm, with area in mm2, and power is given in D.

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Table 2:. Figuring and approximate conic constants of the lens’ surfaces, and their contribution to SA. The change in SA per dioptre is calculated as a linear fit of the SA versus accommodation for the ranges 0–2 D, 0–4 D, 0–6 D and 0–8 D.

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Fig. 2: Lens profile for ocular accommodative amplitudes from 0D (orange) to 8D (blue).

The negative value of the conic constants, seen in Table 2, can be compared with the literature [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

, 25

25. S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3, 2471–2488 (2012). [CrossRef] [PubMed]

, 27

27. C. E. Campbell, “Nested shell optical model of the lens of the human eye,” J. Opt. Soc. Am. A 27, 2432–2441 (2010). [CrossRef]

, 28

28. E. Lanchares, R. Navarro, and B. Calvo, “Hyperelastic modelling of the crystalline lens: Accommodation and presbyopia,” J. Optometry 5, 110–120 (2012). [CrossRef]

]. The in vivo study of Dubbelman et al. [29

29. M. Dubbelman and G. van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vis. Res. 41, 1867–1877 (2001). [CrossRef] [PubMed]

] reports a value of −4.5 ± 2.8 for the anterior conic of the unaccommodated 30 year old lens, and a value of −2.9 ± 3.8 for the posterior conic. Ortiz et al. [25

25. S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3, 2471–2488 (2012). [CrossRef] [PubMed]

] show a range in averaged asphericities of 3 subjects; from −2.57 to −0.43 for the anterior conic, and −1.64 to −0.01 for the posterior conic. For the in vitro lens (assumed to be fully accommodated), Howcroft and Parker [30

30. M. J. Howcroft and J. A. Parker, “Aspheric curvatures for the human lens,” Vis. Res. 17, 1213–1217 (1977). [CrossRef] [PubMed]

] show that both the anterior and posterior surfaces varied widely with hyperbolic, parabolic and elliptical profiles for 60 cadaver eyes. Judging by the figures presented in the work of Borja et al [31

31. D. Borja, D. Siedlecki, A. de Castro, S. Uhlhorn, S. Ortiz, E. Arrieta, J.-M. Parel, S. Marcos, and F. Manns, “Distortions of the posterior surface in optical coherence tomography images of the isolated crystalline lens: effect of the lens index gradient,” Biomed. Opt. Express 1, 1331–1340 (2010). [CrossRef]

], the asphericity of the anterior surface ranges between −14 and +2; that of the posterior surface ranges between −2 and +2. The work of Manns et al. [32

32. F. Manns, V. Fernandez, S. Zipper, S. Sandadi, M. Hamaoui, A. Ho, and J.-M. Parel, “Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses,” Exp. Eye Res. 78, 39–51 (2004). [CrossRef]

] produces an average of 3.27 ± 2.01 for the anterior conic, and −1.64 ± 1.85 for the posterior conic of cadaver lenses. The average in vitro data are difficult to interpret, since the accommodative state is not well-defined. In terms of theoretical modelling, the lens of Smith et al. [10

10. G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optometry Vis. Sci. 85, 330–340 (2008). [CrossRef]

] (based on Liou and Brennan’s [11

11. H.-L. Liou and N. A. Brennan, “Anatomically accurate, finite model eye for optical modeling,” J. Opt. Soc. Am. A. 14, 1684–1695 (1997). [CrossRef]

]) shows a decrease in anterior and posterior conic constants from −0.94 to −0.955 and 0.96 to 0.471, respectively, with 3 D accommodation. The nested shell optical model of Campbell [27

27. C. E. Campbell, “Nested shell optical model of the lens of the human eye,” J. Opt. Soc. Am. A 27, 2432–2441 (2010). [CrossRef]

] indicates that the anterior and posterior conic constants have a value of −5.00 for the 25 year old lens. The finite element model of Lanchares et al. [28

28. E. Lanchares, R. Navarro, and B. Calvo, “Hyperelastic modelling of the crystalline lens: Accommodation and presbyopia,” J. Optometry 5, 110–120 (2012). [CrossRef]

] shows that the anterior and posterior conic constants have values of −4 and −3, respectively, in the unaccommodated eye. For their finite element model, Hermans et al. [33

33. E. Hermans, M. Dubbelman, G. van der Heijde, and R. Heethaar, “Estimating the external force acting on the human eye lens during accommodation by finite element modelling,” Vis. Res. 46, 3642–3650 (2006). [CrossRef] [PubMed]

] use values of −4 and −8 for the conic constants of the anterior surface at 0 D and 8 D respectively, while using a value of −3 for the posterior conic at both accommodative amplitudes; these values are based on the work of Dubbelman and co-workers [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

, 29

29. M. Dubbelman and G. van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vis. Res. 41, 1867–1877 (2001). [CrossRef] [PubMed]

]. The majority of these studies have shown the conic constants to be negative, with Dubbelman et al. [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

] reporting that the conics become more negative with accommodation; our lens example shows a similar trend.

The volume of the lens can be compared with the experimental in vivo data presented in the paper by Hermans et al. [18

18. E. A. Hermans, P. J. W. Pouwels, M. Dubbelman, J. P. A. Kuijer, R. G. L. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: An MRI and scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50, 281–289 (2009). [CrossRef]

]. This paper indicates that the lenticular volume of five subjects between 18 and 35 years of age is in the range 150–165 mm3. A study of 27 isolated eyes between 6–82 years old in a paper by Urs et al. [34

34. R. Urs, F. Manns, A. Ho, D. Borja, A. Amelinckx, J. Smith, R. Jain, R. Augusteyn, and J.-M. Parel, “Shape of the isolated ex-vivo human crystalline lens,” Vis. Res. 49, 74–83 (2009). [CrossRef]

] provides a regression of volume versus age, giving a volume of 172.06 ± 11.31 mm3 for the 30 year old eye. The smaller volume given for our example could be attributed to the relatively small value for unaccommodated lens thickness, T0 = 3.18 mm. A study by Koretz and co-workers suggests that the 30 year old lens’ thickness is in the range 3.52–3.67 mm, based on MRI and Scheimpflug imaging, respectively [35

35. J. F. Koretz, S. A. Strenk, L. M. Strenk, and J. L. Semmlow, “Scheimpflug and high-resolution magnetic resonance imaging of the anterior segment: a comparative study,” J. Opt. Soc. Am. A. 21, 346–354 (2004). [CrossRef]

]. For this thickness range, the volume of our lens example would be 154.2–167.9 mm3, which is in line with experimental data.

The theoretical values for surface area show the expected decreasing trend with accommodation. The surface area decrease was calculated as 3.28% over 6 D, which is comparable to the experimental MRI estimate of 4.78% ± 2.29% over the same accommodative range (0–6 D) in the study by Hermans et al. [18

18. E. A. Hermans, P. J. W. Pouwels, M. Dubbelman, J. P. A. Kuijer, R. G. L. van der Heijde, and R. M. Heethaar, “Constant volume of the human lens and decrease in surface area of the capsular bag during accommodation: An MRI and scheimpflug study,” Invest. Ophthalmol. Visual Sci. 50, 281–289 (2009). [CrossRef]

]. This study gives values of 175.9 ± 2.8 mm2 and 167.5±2.9 mm2 for the unaccommdated and 6 D accommodated lens, respectively. The regression formula for 27 isolated lenses provided in Urs et al. [34

34. R. Urs, F. Manns, A. Ho, D. Borja, A. Amelinckx, J. Smith, R. Jain, R. Augusteyn, and J.-M. Parel, “Shape of the isolated ex-vivo human crystalline lens,” Vis. Res. 49, 74–83 (2009). [CrossRef]

] gives a value of 170.5±9.2 mm2; usually, the isolated lenses assume the fully accommodated state. At 6 D, our model predicts a value of 147.6 mm2; however, our initial lens is relatively thin, with a thickness of only 3.18 mm. Bringing the initial lens thickness to average values provided in Koretz et al. [35

35. J. F. Koretz, S. A. Strenk, L. M. Strenk, and J. L. Semmlow, “Scheimpflug and high-resolution magnetic resonance imaging of the anterior segment: a comparative study,” J. Opt. Soc. Am. A. 21, 346–354 (2004). [CrossRef]

] of 3.52–3.67 mm would produce a surface area of 167.3–176.1 mm2 at 6 D. Hence, the initial surface area of our model, and area decrease with accommodation, are comparable to the experimental findings.

For 30 isolated lenses aged 20–69, Urs et al. [36

36. R. Urs, A. Ho, F. Manns, and J.-M. Parel, “Age-dependent fourier model of the shape of the isolated ex vivo human crystalline lens,” Vis. Res. 50, 1041–1047 (2010). [CrossRef] [PubMed]

] provide a regression formula for lenticular diameter, giving a value of 9.2 ± 0.6 mm for the 30 year old lens. For the in vivo lens, the study of Jones et al. [19

19. C. E. Jones, D. A. Atchison, and J. M. Pope, “Changes in lens dimensions and refractive index with age and accommodation,” Opt. Vis. Sci. 84, 990–995 (2007). [CrossRef]

] gives a per-dioptre change in diameter of −0.067 ± 0.030 mm/D, corresponding to −0.402 ± 0.180 mm over 6 D. Their average unaccommodated diameter is given as 9.33 ± 0.33 mm; this represents a percentage decrease in diameter of 4.29% ± 1.93% over 6 D. The diameter of our lens example decreases by 2.52% over the same 6 D. This decrease in diameter with accommodation, together with the initial value of 9.0 mm, is within the range of experimental data.

5. Analysis of lenticular optical power and spherical aberration

It is worth noting that Eqs. 7 were generated for accommodative amplitude of the whole eye. Taking the cornea and lens as two refracting surfaces in a simplified schematic eye, their relative separation implies that a total ocular power change of 8 D requires a lenticular power change of approximately 10 D. An approximate thin-lens equation—differing by only 1.4% from the exact equation—was used to verify the change in lenticular power F versus accommodation [13

13. M. Bahrami and A. V. Goncharov, “Geometry-invariant gradient refractive index lens: analytical ray tracing,” J. Biomed. Opt. 17, 055001 (2012). [CrossRef] [PubMed]

]:
F=nsnaqRa+2P2P1(ncns)(1Ra+1Rp)+nsnvitRp,
where naq and nvit are the refractive indices of the aqueous and vitreous, respectively (naq = nvit = 1.336), and the radii Ra and Rp are given in metres; P is the exponent characterising the GRIN structure of the lens. Table 1 indicates the approximate 10 D lenticular power change between the minimally and maximally accommodated states, in accordance with the 8 D total ocular power change. Note that a 30 year old subject might not be able to accommodate by 8 D, due to the onset of presbyopia. Taking into account accommodative lag and expected accommodative ability at 30 years of age, the model probably overestimates the change in lens power of the 30 year old lens.

Further validation of the method was achieved through calculation of the total lenticular spherical aberration (SA). It is clear that an accurate model of the human crystalline lens will need to account for the trend in lenticular SA as well as the change in refractive power. Total ocular SA is known to become less positive with accommodation [10

10. G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optometry Vis. Sci. 85, 330–340 (2008). [CrossRef]

, 37

37. A. Ivanoff, “On the influence of accommodation on spherical aberration in the human eye, an attempt to interpret night myopia,” J. Opt. Soc. Am. 37, 730–731 (1947). [CrossRef] [PubMed]

49

49. Y.-J. Li, J. A. Choi, H. Kim, S.-Y. Yu, and C.-K. Joo, “Changes in ocular wavefront aberrations and retinal image quality with objective accommodation,” J. CataractRefractive Surg. 37, 835–841 (2011).

], a trend first observed by Young in 1801 [50

50. T. Young, “On the mechanism of the eye,” Philos. Trans. R. Soc. London 91, 23–88 (1801). [CrossRef]

]; see Table 3 for a summary of recent experimental data for changes in ocular SA (Zernike coefficient Z40). However, the change in lenticular SA with accommodation is not widely documented in the literature—most studies report changes in total ocular aberrations only. In general, it is difficult to disentangle the SA contribution of the human lens from that of the cornea when measuring in vivo. In the majority of cases, no distinction is made between the posterior corneal surface and the lens, so that the optical effects of these internal optics are considered together. In this case, the lenticular SA is found as the difference between ocular and anterior corneal SA, as in the study by Li et al. [49

49. Y.-J. Li, J. A. Choi, H. Kim, S.-Y. Yu, and C.-K. Joo, “Changes in ocular wavefront aberrations and retinal image quality with objective accommodation,” J. CataractRefractive Surg. 37, 835–841 (2011).

]. Despite this limitation, we shall try to compare the lens model prediction with experimental data.

Table 3:. Experimental changes in lenticular and ocular SA ( Z40) per dioptre. All data are scaled down to a 3 mm pupil diameter from their measured pupil diameters (given in mm).

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Before calculating the SA output of the current tool, it was first necessary to assess the contribution of conic constant to SA. According to the third-order theory applied in this paper, the SA of a surface or GRIN medium depends only on radius and conic constant. This definition is fine for surfaces which are pure conicoids of revolution, or when the pupil size is limited to the paraxial region. The solid blue curve in Fig. 4 shows the change of the SA wavefront aberration coefficient (W4,0) calculated using the first Seidel sum (SI) [13

13. M. Bahrami and A. V. Goncharov, “Geometry-invariant gradient refractive index lens: analytical ray tracing,” J. Biomed. Opt. 17, 055001 (2012). [CrossRef] [PubMed]

]:
W4,0=18SI.

To extend the validity of the third-order theory, we have to account for the figuring conicoid functions including the B-coefficient contribution to SA. Thus it is necessary to reconsider our description of the surface shape at intermediate pupil heights. When using only pure conicoids to represent the lens’ surfaces, the conic constant is responsible for the overall shape of the lens at large heights. It has been shown that the lens shape is more complex than this simple conicoid [53

53. G. Smith, D. A. Atchison, D. R. Iskander, C. E. Jones, and J. M. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vis. Res. 49, 2442–2452 (2009). [CrossRef] [PubMed]

55

55. S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013). [CrossRef]

]. To provide more realistic lens shapes, we use a figuring conicoid featuring a higher-order aspheric term and B-coefficient; see Eq. (1).

The method employed in the work of Dubbelman et al. (2005) [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

] was to fit a pure conicoid function to the experimental images of the lens’ surface over a 6 mm diameter; this gave an approximation to the true, highly-aspheric shape of the lens’ surface, with associated approximate conic constant for that surface. However, if a figuring conicoid was used, the B-coefficient would assume a certain value, and would alter the value of the approximate conic constant determined initially. The conic constants obtained using the B-coefficient are figuring conics Ka and Kp, which are suitable for use in third-order aberration analysis with small pupils.

However, to approximate the central and intermediate region of the lens, we need to modify the conic constants so that the pure conicoid of revolution more closely follows the figuring conicoid’s aspheric surface. The modified values for the conic constants ( Ka* and Kp*) can be derived by choosing ρ1 and solving equation Eq. (1) for a corresponding sag, z1, that defines the point (z1, ρ1) at which our new, pure conicoid will intersect the figuring conicoid:
ρ12=2Rz1(1+K)z12+Bz13.
The approximate conic constant can be found by solving for K* in the equation:
ρ12=2Rz1(1+K*)z12,thus:K*=KBz1.

Table 2 contains the figuring and approximate conic constants predicted by the current tool. Note that the approximate conics were obtained by fitting (a pure conicoid) to the lens’ surface at a height ρ1 = 2.5 mm, which corresponds to the point Ma in Fig. 3. This height was chosen so that the internal iso-indicial contours of the GRIN structure which have not yet plateaued to the central refractive index are still suitably approximated by the pure conicoid for ray-tracing at 3 mm pupil diameter. For example, we can see in Fig. 3(a) that the ray at entrance pupil height of 1.5 mm intersects the internal contour at point Min at a height that is scaled down from the initial 2.5 mm height of the point Ma on the external surface. Fitting at larger heights would deteriorate the quality of the fit; thus, analysis at very large heights requires numerical ray-tracing.

Fig. 3: Fitting the figuring conicoid (green) with a pure conicoid (blue) of conic constant Ka* at intersection point Ma, ρ = 2.5 mm (a). The red curve is a pure conicoid with conic constant Ka; (b): the difference in sag between green, red and blue curves for different lens heights.

Fig. 4: Change in ocular (disks) and lenticular (squares) SA; marker size indicates measured pupil diameter. All data are scaled to 3 mm pupil, and are compiled from Tables 2 & 3.

With the approximate conics, the predicted lenticular SA agrees with the well-known trend of decreasing SA with accommodation (becoming more negative). Validation of the tool was extended by analysing the decrease of SA per dioptre over a range of accommodative amplitudes. That is, the decreases in SA over the ranges 0–2 D, 0–4 D, 0–6 D and 0–8 D were each fitted with a line. The slopes of these linear fits are provided in Table 2. Given the size of the error bars and noticeable spread in experimental data [10

10. G. Smith, P. Bedggood, R. Ashman, M. Daaboul, and A. Metha, “Exploring ocular aberrations with a schematic human eye model,” Optometry Vis. Sci. 85, 330–340 (2008). [CrossRef]

], the decrease in Zernike Z40 predicted by the current tool is in agreement with the data listed in Table 3. The predicted rate of change in SA is in line with experimental data for the available accommodative range of 0–6 D. For this range, the output of the model shows a non-linear trend. Fitting with a second-order polynomial gives: W4,0(A) = 0.0262 − 0.0128A − 0.0145A2, where A is accommodation in dioptres.

The unaccommodated lens with approximate conics shows a value of SA (W4,0) = 0.0262 μm, which corresponds to the Zernike coefficient of primary SA: Z40=W4,0/(65)=0.0020μm. This value is slightly positive, while the value obtained using the figuring conics is negative (W4,0 = −0.0880 μm); this negative value is valid for much smaller pupil diameters. Typically, the human lens shows a negative value at 0 D. Studies have found that, in general, positive corneal SA is partly cancelled by negative SA of the lens or internal optics [56

56. S. G. El Hage and F. Berny, “Contribution of the crystalline lens to the spherical aberration of the eye,” J. Opt. Soc. Am. 63, 205–211 (1973). [CrossRef] [PubMed]

65

65. J. L. Alió, P. Schimchak, H. P. Negri, and R. Montées-Micó, “Crystalline lens optical dysfunction through aging,” Ophthalmology 112, 2022–2029 (2005). [CrossRef] [PubMed]

]; we stress here the generality of those results. However, a paper by Millodot and Sivak shows that the lenticular SA generally adds to that of the cornea [66

66. M. Millodot and J. Sivak, “Contribution of the cornea and lens to the spherical aberration of the eye,” Vis. Res. 19, 685–687 (1979). [CrossRef] [PubMed]

]. Papers by Artal et al. and Glasser and Campbell show compensation in young eyes, with augmentation in older eyes [67

67. P. Artal, E. Berrio, A. Guirao, and P. Piers, “Contribution of the cornea and internal surfaces to the change of ocular aberrations with age,” J. Opt. Soc. Am. A. 19, 137–143 (2002). [CrossRef]

, 68

68. A. Glasser and M. C. Campbell, “Presbyopia and the optical changes in the human crystalline lens with age,” Vis. Res. 38, 209–229 (1998). [CrossRef] [PubMed]

]. Furthermore, more recent papers by He et al. [69

69. J. C. He, J. Gwiazda, F. Thorn, and R. Held, “Wave-front aberrations in the anterior corneal surface and the whole eye,” J. Opt. Soc. Am. A 20, 1155–1163 (2003). [CrossRef]

], and Salmon and Thibos [70

70. T. O. Salmon and L. N. Thibos, “Videokeratoscope-line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A. 19, 657–669 (2002). [CrossRef]

] show both compensatory and additive roles of the internal optics in ocular SA production. A paper by He et al. shows that the lens reduces the total RMS wavefront aberration of the cornea, but individual Zernike corneal aberrations were not always compensated by the lens [71

71. J. He, E. Ong, J. Gwiazda, R. Held, and F. Thorn, “Wave-front aberrations in the cornea and the whole eye,” Invest. Ophthalmol. Visual Sci. 41, S105 (2000).

]. Kelly and co-workers also found that corneal SA was not always reduced by the lens [64

64. J. E. Kelly, T. Mihashi, and H. C. Howland, “Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye,” J. Vision 4, 262–271 (2004). [CrossRef]

].

We can see that there is large inter-subject variability in the experimental values for the SA of internal optics at 0 D; thus, we should not expect that lenticular SA is always negative. For the particular choice of initial lens geometry for this model [25

25. S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express 3, 2471–2488 (2012). [CrossRef] [PubMed]

], the value of lenticular SA at 0 D is slightly positive when analysing the lens at a 3 mm pupil. More important is the trend in SA with accommodation, which is in agreement with experimental data. The approximating conics are more valid for this intermediate pupil diameter.

These approximate conics can be compared to those reported in the work of Dubbelman et al. (2005) [24

24. M. Dubbelman, G. V. der Heijde, and H. Weeber, “Change in shape of the aging human crystalline lens with accommodation,” Vis. Res. 45, 117–132 (2005). [CrossRef]

]. Therein provided is a linear regression for the change per dioptre in conic of the lens’ anterior surface, as a function of the anterior conic at 0 D, K0a:
ΔKa=(0.630.07×K0a)A,
where A is accommodation in dioptres. For this, we use K0a*=0.83, corresponding to an experimental ΔKa*=4.58 over 8 D; this can be compared with the change ΔKa*=5.79 predicted by the current tool, and given in Table 2.

6. Discussion and conclusion

The modelled change in SA with accommodation is an important finding, since it represents the first clear understanding of how lenticular SA can decrease with accommodation according to Helmholtz’s theory. It is widely known that increasing surface curvatures cause more positive SA, and so the accommodative decrease in lenticular radii should produce more positive SA. With this tool, we can analytically predict that the conic constants become more negative, and this in turn gives more negative SA of the lens. Our method employs the simple physical lens constraints of constant volume and fixed equator to analytically solve for the conic constants. With these conic constants, we calculate the accommodative changes in SA by ray-tracing through the lens’ GRIN medium. Since the GRIN medium follows the external surface geometry, this model takes into account both surface asphericity and GRIN structure when calculating lenticular SA.

The change in lenticular SA in vivo with accommodation is not widely documented in the literature, since it is difficult to disentangle the SA contribution of the human lens from that of the cornea; most studies report changes in total ocular aberrations only. Usually, the optical effect of the posterior cornea and lens are considered together. In this case, the lenticular SA is found as the difference between ocular and anterior corneal SA. In spite of this obvious limitation, we have compared the lens model prediction with experimental data, and found that the model prediction is in agreement with the data.

With this method, it is possible to predict analytically the accommodative changes in various lenticular parameters such as surface area, equatorial diameter, refractive power and SA. We invite others to use this tool for future refinement as new data become available.

An important aspect of this new method is the constancy of lenticular volume with accommodation. With the derived formulae for figuring conic constants, it is possible to preserve the volume of the lens when modelling in ray-tracing software, where the specification of volume is currently not directly available. This will therefore allow modelling of a physically realistic lens in ray-tracing software, and give new insight into the accommodative process. If the conic constants of the lens are given as a function of accommodation, one could revisit the dependency on experimental data for accommodative changes in lenticular radii and thickness, and instead use these geometrical parameters to keep the lens’ volume constant. Moreover, if it is found that the lenticular volume is not constant with accommodation, one could relax the constant volume condition.

The future aim of ocular modelling is the development of a model which characterises the geometrical, optical and biomechanical properties of the ageing lens lens under accommodation [72

72. H. Burd, S. Judge, and J. Cross, “Numerical modelling of the accommodating lens,” Vis. Res. 42, 2235–2251 (2002). [CrossRef] [PubMed]

]. We hope that an understanding of how the geometrical and optical properties of the lens change with accommodation will help to consolidate, in future, more information regarding the accommodation process.

Acknowledgments

The authors wish to acknowledge the funding of The Irish Research Council Embark Initiative, application RS/2012/351. This research was also supported by Fight for Sight under grant 1319/1320.

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G. Smith, D. A. Atchison, D. R. Iskander, C. E. Jones, and J. M. Pope, “Mathematical models for describing the shape of the in vitro unstretched human crystalline lens,” Vis. Res. 49, 2442–2452 (2009). [CrossRef] [PubMed]

54.

H. T. Kasprzak, “New approximation for the whole profile of the human crystalline lens,” Ophthalmic Physiol. Opt. 20, 31–43 (2000). [CrossRef] [PubMed]

55.

S. Giovanzana, R. A. Schachar, S. Talu, R. D. Kirby, E. Yan, and B. K. Pierscionek, “Evaluation of equations for describing the human crystalline lens,” J. Mod. Opt. 60, 406–413 (2013). [CrossRef]

56.

S. G. El Hage and F. Berny, “Contribution of the crystalline lens to the spherical aberration of the eye,” J. Opt. Soc. Am. 63, 205–211 (1973). [CrossRef] [PubMed]

57.

J. Sivak and R. Kreuzer, “Spherical aberration of the crystalline lens,” Vision Research 23, 59–70 (1983). [CrossRef] [PubMed]

58.

A. Tomlinson, R. P. Hemenger, and R. Garriott, “Method for estimating the spheric aberration of the human crystalline lens in vivo.” Invest. Ophthalmol. Visual Sci. 34, 621–629 (1993).

59.

P. Artal and A. Guirao, “Contributions of the cornea and the lens to the aberrations of the human eye,” Opt. Lett. 23, 1713–1715 (1998). [CrossRef]

60.

T. Salmon and L. Thibos, “Relative contribution of the cornea and internal optics to the aberrations of the eye,” Optometry Vis. Sci. 75, 235 (1998).

61.

P. Artal, A. Guirao, E. Berrio, and D. R. Williams, “Compensation of corneal aberrations by the internal optics in the human eye,” J. Vision 1, 1–8 (2001). [CrossRef]

62.

G. Smith, M. J. Cox, R. Calver, and L. F. Garner, “The spherical aberration of the crystalline lens of the human eye,” Vis. Res. 41, 235–243 (2001). [CrossRef] [PubMed]

63.

S. Amano, Y. Amano, S. Yamagami, T. Miyai, K. Miyata, T. Samejima, and T. Oshika, “Age-related changes in corneal and ocular higher-order wavefront aberrations,” Am. J. Ophthalmol. 137, 988–992 (2004). [CrossRef] [PubMed]

64.

J. E. Kelly, T. Mihashi, and H. C. Howland, “Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye,” J. Vision 4, 262–271 (2004). [CrossRef]

65.

J. L. Alió, P. Schimchak, H. P. Negri, and R. Montées-Micó, “Crystalline lens optical dysfunction through aging,” Ophthalmology 112, 2022–2029 (2005). [CrossRef] [PubMed]

66.

M. Millodot and J. Sivak, “Contribution of the cornea and lens to the spherical aberration of the eye,” Vis. Res. 19, 685–687 (1979). [CrossRef] [PubMed]

67.

P. Artal, E. Berrio, A. Guirao, and P. Piers, “Contribution of the cornea and internal surfaces to the change of ocular aberrations with age,” J. Opt. Soc. Am. A. 19, 137–143 (2002). [CrossRef]

68.

A. Glasser and M. C. Campbell, “Presbyopia and the optical changes in the human crystalline lens with age,” Vis. Res. 38, 209–229 (1998). [CrossRef] [PubMed]

69.

J. C. He, J. Gwiazda, F. Thorn, and R. Held, “Wave-front aberrations in the anterior corneal surface and the whole eye,” J. Opt. Soc. Am. A 20, 1155–1163 (2003). [CrossRef]

70.

T. O. Salmon and L. N. Thibos, “Videokeratoscope-line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A. 19, 657–669 (2002). [CrossRef]

71.

J. He, E. Ong, J. Gwiazda, R. Held, and F. Thorn, “Wave-front aberrations in the cornea and the whole eye,” Invest. Ophthalmol. Visual Sci. 41, S105 (2000).

72.

H. Burd, S. Judge, and J. Cross, “Numerical modelling of the accommodating lens,” Vis. Res. 42, 2235–2251 (2002). [CrossRef] [PubMed]

OCIS Codes
(080.1010) Geometric optics : Aberrations (global)
(330.4460) Vision, color, and visual optics : Ophthalmic optics and devices
(330.5370) Vision, color, and visual optics : Physiological optics
(080.2468) Geometric optics : First-order optics
(330.7322) Vision, color, and visual optics : Visual optics, accommodation
(330.7326) Vision, color, and visual optics : Visual optics, modeling

ToC Category:
Vision and Visual Optics

History
Original Manuscript: December 3, 2013
Revised Manuscript: February 19, 2014
Manuscript Accepted: March 18, 2014
Published: April 28, 2014

Citation
Conor J. Sheil, Mehdi Bahrami, and Alexander V. Goncharov, "An analytical method for predicting the geometrical and optical properties of the human lens under accommodation," Biomed. Opt. Express 5, 1649-1663 (2014)
http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-5-5-1649


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  61. P. Artal, A. Guirao, E. Berrio, and D. R. Williams, “Compensation of corneal aberrations by the internal optics in the human eye,” J. Vision1, 1–8 (2001). [CrossRef]
  62. G. Smith, M. J. Cox, R. Calver, and L. F. Garner, “The spherical aberration of the crystalline lens of the human eye,” Vis. Res.41, 235–243 (2001). [CrossRef] [PubMed]
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  64. J. E. Kelly, T. Mihashi, and H. C. Howland, “Compensation of corneal horizontal/vertical astigmatism, lateral coma, and spherical aberration by internal optics of the eye,” J. Vision4, 262–271 (2004). [CrossRef]
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  71. J. He, E. Ong, J. Gwiazda, R. Held, and F. Thorn, “Wave-front aberrations in the cornea and the whole eye,” Invest. Ophthalmol. Visual Sci.41, S105 (2000).
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