Abstract

Human crystalline lens has a layered, shell-like structure with the refractive index increasing from cortex to nucleus (gradient index or GRIN structure). Moreover, every iso-indicial layer has a certain curvature which also varies from cortex to nucleus, with a gradient of curvature (G). In the present manuscript, the role of G on the lens power is investigated along with its implications regarding the lens paradox (change of lens power with age) and intra-capsular accommodation mechanism (larger than expected changes of lens power during accommodation compared to a homogenous lens). To this end, a simplified formulation of paraxial lens power based on thin lens approximation is developed and applied to the anterior and posterior parts of the lens. The main theoretical result is that the power of both anterior and posterior lens is given by the sum of the power of a lens with a homogeneous refractive index equal to that of the nucleus and power associated with the contribution of the internal GRIN structure, which depends on G. This general result suggests that the sign of G is fundamental in increasing or decreasing the lens power. We found that the curvature gradient has a strong impact on lens power, helping to explain both the lens paradox and intra-capsular accommodation mechanism.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]

2016 (1)

2015 (2)

S. Jongenelen, J. J. Rozema, and M.-J. Tassignon, EVICR.net, and Project Gullstrand Study Group, “Distribution of the crystalline lens power in vivo as a function of age,” Invest. Ophthalmol. Vis. Sci. 56, 7029–7035 (2015).
[Crossref]

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
[Crossref]

2014 (2)

R. Navarro, “Adaptive model of the aging emmetropic eye and its changes with accommodation,” J. Vis. 14(13), 21 1–17 (2014).
[Crossref]

W. N. Charman and D. A. Atchison, “Age-dependence of the average and equivalent refractive indices of the crystalline lens,” Biomed. Opt. Express 5, 31–39 (2014).
[Crossref]

2012 (1)

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

2011 (1)

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

2010 (1)

2008 (3)

J. A. Díaz, C. Pizarro, and J. Arasa, “Single dispersive gradient-index profile for the aging human lens,” J. Opt. Soc. Am. A 25, 250–261 (2008).
[Crossref]

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “Equivalent refractive index of the human lens upon accommodative response,” Optom. Vis. Sci. 85, 1179–1184 (2008).
[Crossref]

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Visual Sci. 49, 2531–2540 (2008).
[Crossref]

2007 (3)

2005 (1)

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),” Vision Res. 45, 2352–2366 (2005).
[Crossref]

2004 (1)

A. Roorda and A. Glasser, “Wave aberrations of the isolated crystalline lens,” J. Vis. 4, 250–261 (2004).
[Crossref]

2002 (1)

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Explanation of the lens paradox,” Optom. Vis. Sci. 79, 148–150 (2002).
[Crossref]

2001 (1)

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vision Res. 41, 1867–1877 (2001).
[Crossref]

1999 (1)

N. P. Brown, J. F. Koretz, and A. J. Bron, “The development and maintenance of emmetropia,” Eye 13, 83–92 (1999).
[Crossref]

1998 (1)

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

1997 (1)

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64, 887–893 (1997).
[Crossref]

1992 (1)

1990 (1)

B. K. Pierscionek, “Presbyopia-effect of refractive index,” Clin. Exp. Optom. 73, 23–30. (1990).
[Crossref]

1988 (1)

J. F. Koretz and G. H. Handelman, “How the human eye focuses,” Sci. Am. 259, 92–99. (1988).
[Crossref]

1985 (1)

1974 (1)

N. P. Brown, “The change in lens curvature with age,” Exp. Eye Res. 19, 175–183 (1974).
[Crossref]

Arasa, J.

Arrieta, E.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Atchison, D.

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),” Vision Res. 45, 2352–2366 (2005).
[Crossref]

Atchison, D. A.

W. N. Charman and D. A. Atchison, “Age-dependence of the average and equivalent refractive indices of the crystalline lens,” Biomed. Opt. Express 5, 31–39 (2014).
[Crossref]

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Visual Sci. 49, 2531–2540 (2008).
[Crossref]

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Explanation of the lens paradox,” Optom. Vis. Sci. 79, 148–150 (2002).
[Crossref]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A 9, 2111–2117 (1992).
[Crossref]

Augusteyn, R. C.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Bahrami, M.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
[Crossref]

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

Bescós, J.

Borja, D.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Bron, A. J.

N. P. Brown, J. F. Koretz, and A. J. Bron, “The development and maintenance of emmetropia,” Eye 13, 83–92 (1999).
[Crossref]

Brown, N. P.

N. P. Brown, J. F. Koretz, and A. J. Bron, “The development and maintenance of emmetropia,” Eye 13, 83–92 (1999).
[Crossref]

N. P. Brown, “The change in lens curvature with age,” Exp. Eye Res. 19, 175–183 (1974).
[Crossref]

Campbell, M. C.

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

Charman, W. N.

de Castro, A.

Díaz, J. A.

Dubbelman, M.

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “Equivalent refractive index of the human lens upon accommodative response,” Optom. Vis. Sci. 85, 1179–1184 (2008).
[Crossref]

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “The shape of the human lens nucleus with accommodation,” J. Vis. 7(10), 1–10 16 (2007).
[Crossref]

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vision Res. 41, 1867–1877 (2001).
[Crossref]

El Hage, S. G.

Y. Le Grand and S. G. El Hage, Physiological Optics (Springer-Verlag, 1980).

Gambra, E.

Glasser, A.

A. Roorda and A. Glasser, “Wave aberrations of the isolated crystalline lens,” J. Vis. 4, 250–261 (2004).
[Crossref]

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

Goncharov, A.

Goncharov, A. V.

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

González, L.

Gullstand, A.

A. Gullstand, “How I found the mechanism of intracapsular accommodation,” Nobel Lecture (Dec. 11, 1911).

A. Gullstand, “Mechanism of accommodation,” in Handbuch der Physiologischen Optic, H. Helmholtz von, ed. (1909), Appendix IV, pp. 383–415.

Handelman, G. H.

J. F. Koretz and G. H. Handelman, “How the human eye focuses,” Sci. Am. 259, 92–99. (1988).
[Crossref]

Heethaar, R. M.

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “Equivalent refractive index of the human lens upon accommodative response,” Optom. Vis. Sci. 85, 1179–1184 (2008).
[Crossref]

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “The shape of the human lens nucleus with accommodation,” J. Vis. 7(10), 1–10 16 (2007).
[Crossref]

Hermans, E. A.

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “Equivalent refractive index of the human lens upon accommodative response,” Optom. Vis. Sci. 85, 1179–1184 (2008).
[Crossref]

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “The shape of the human lens nucleus with accommodation,” J. Vis. 7(10), 1–10 16 (2007).
[Crossref]

Ho, A.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Hoshino, M.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
[Crossref]

Jones, C.

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),” Vision Res. 45, 2352–2366 (2005).
[Crossref]

Jongenelen, S.

S. Jongenelen, J. J. Rozema, and M.-J. Tassignon, EVICR.net, and Project Gullstrand Study Group, “Distribution of the crystalline lens power in vivo as a function of age,” Invest. Ophthalmol. Vis. Sci. 56, 7029–7035 (2015).
[Crossref]

Kasthurirangan, S.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Visual Sci. 49, 2531–2540 (2008).
[Crossref]

Koretz, J. F.

N. P. Brown, J. F. Koretz, and A. J. Bron, “The development and maintenance of emmetropia,” Eye 13, 83–92 (1999).
[Crossref]

J. F. Koretz and G. H. Handelman, “How the human eye focuses,” Sci. Am. 259, 92–99. (1988).
[Crossref]

Le Grand, Y.

Y. Le Grand and S. G. El Hage, Physiological Optics (Springer-Verlag, 1980).

Maceo, B. M.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Manns, F.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Marcos, S.

Markwell, E. L.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Visual Sci. 49, 2531–2540 (2008).
[Crossref]

Meder, R.

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),” Vision Res. 45, 2352–2366 (2005).
[Crossref]

Moffat, B. A.

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Explanation of the lens paradox,” Optom. Vis. Sci. 79, 148–150 (2002).
[Crossref]

Nankivil, D.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Navarro, R.

Ortiz, S.

Palos, F.

Parel, J. M.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Pierscionek, B.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
[Crossref]

Pierscionek, B. K.

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64, 887–893 (1997).
[Crossref]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A 9, 2111–2117 (1992).
[Crossref]

B. K. Pierscionek, “Presbyopia-effect of refractive index,” Clin. Exp. Optom. 73, 23–30. (1990).
[Crossref]

Pizarro, C.

Pope, J.

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),” Vision Res. 45, 2352–2366 (2005).
[Crossref]

Pope, J. M.

S. Kasthurirangan, E. L. Markwell, D. A. Atchison, and J. M. Pope, “In vivo study of changes in refractive index distribution in the human crystalline lens with age and accommodation,” Invest. Ophthalmol. Visual Sci. 49, 2531–2540 (2008).
[Crossref]

B. A. Moffat, D. A. Atchison, and J. M. Pope, “Explanation of the lens paradox,” Optom. Vis. Sci. 79, 148–150 (2002).
[Crossref]

Regini, J.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
[Crossref]

Roorda, A.

A. Roorda and A. Glasser, “Wave aberrations of the isolated crystalline lens,” J. Vis. 4, 250–261 (2004).
[Crossref]

Rozema, J. J.

S. Jongenelen, J. J. Rozema, and M.-J. Tassignon, EVICR.net, and Project Gullstrand Study Group, “Distribution of the crystalline lens power in vivo as a function of age,” Invest. Ophthalmol. Vis. Sci. 56, 7029–7035 (2015).
[Crossref]

Santamaría, J.

Sheil, C.

Siedlecki, D.

Smith, G.

Southall, J. P. C.

J. P. C. Southall, Trans.: Helmholtz’s Treatise on Physiological Optics (Dover, 1962).

Tassignon, M.-J.

S. Jongenelen, J. J. Rozema, and M.-J. Tassignon, EVICR.net, and Project Gullstrand Study Group, “Distribution of the crystalline lens power in vivo as a function of age,” Invest. Ophthalmol. Vis. Sci. 56, 7029–7035 (2015).
[Crossref]

Uesugi, K.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
[Crossref]

Uhlhorn, S.

B. M. Maceo, F. Manns, D. Borja, D. Nankivil, S. Uhlhorn, E. Arrieta, A. Ho, R. C. Augusteyn, and J. M. Parel, “Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies,” J. Vis. 11(13), 23 1–13 (2011).
[Crossref]

Van der Heijde, G. L.

M. Dubbelman and G. L. Van der Heijde, “The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox,” Vision Res. 41, 1867–1877 (2001).
[Crossref]

Van der Heijde, R.

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “Equivalent refractive index of the human lens upon accommodative response,” Optom. Vis. Sci. 85, 1179–1184 (2008).
[Crossref]

E. A. Hermans, M. Dubbelman, R. Van der Heijde, and R. M. Heethaar, “The shape of the human lens nucleus with accommodation,” J. Vis. 7(10), 1–10 16 (2007).
[Crossref]

Yagi, N.

B. Pierscionek, M. Bahrami, M. Hoshino, K. Uesugi, J. Regini, and N. Yagi, “The eye lens: age-related trends and individual variations in refractive index and shape parameters,” Oncotarget 6, 1–13 (2015).
[Crossref]

Biomed. Opt. Express (2)

Clin. Exp. Optom. (1)

B. K. Pierscionek, “Presbyopia-effect of refractive index,” Clin. Exp. Optom. 73, 23–30. (1990).
[Crossref]

Exp. Eye Res. (2)

N. P. Brown, “The change in lens curvature with age,” Exp. Eye Res. 19, 175–183 (1974).
[Crossref]

B. K. Pierscionek, “Refractive index contours in the human lens,” Exp. Eye Res. 64, 887–893 (1997).
[Crossref]

Eye (1)

N. P. Brown, J. F. Koretz, and A. J. Bron, “The development and maintenance of emmetropia,” Eye 13, 83–92 (1999).
[Crossref]

Invest. Ophthalmol. Vis. Sci. (1)

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Figures (6)

Fig. 1.
Fig. 1.

Examples of iso-indicial contours for different values of the curvature gradient parameter G.

Fig. 2.
Fig. 2.

Paraxial power of a four-surface lens model against the curvature gradient parameter G: Red line and asterisks represent theoretical (thin lens approximation) computation; blue line and circles correspond to Zemax ray tracing. The yellow dot is the power of the homogeneous lens.

Fig. 3.
Fig. 3.

Paraxial power of the continuous GRIN lens versus the curvature gradient parameter, computed using the thin lens approximation (blue line). Zemax results for the GRIN (brown dot) and homogeneous lens (red asterisk) models are also included.

Fig. 4.
Fig. 4.

Equivalent refractive index of the GRIN lens as a function of the curvature gradient parameter.

Fig. 5.
Fig. 5.

Increase of lens power due to accommodation as a function of the curvature gradient parameter.

Fig. 6.
Fig. 6.

Difference between the power of the four-surface and GRIN models versus the curvature gradient parameter.

Equations (14)

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r(z)=RsGz,C(z)=1RsGz.
Plens=Pant+PposeeffectivePantPpos,
P=(nsn0)Cs+i=1N(ni+1ni)Ci=i=0N(ni+1ni)Ci,
P=(nsn0)Cs+0tantdn(z)dzC(z)dz,
(n1n0)+(n2n1)+(n3n2)++(nN+1nN)=nN+1n0=nnn0.
P=(nnn0)Cs=PHOM,
C=1RsGz=Cs1CsGzCs+GCs2z+G2Cs3z2+=Cs+Gf(z),
P=(nsn0)Cs+0tantdn/dz(Cs+Gf(z))dz=(nnn0)Cs+G0tantdn/dzf(z)dz.
P=PHOM+PGRIN=PHOM+GpGRIN,
pGRIN=0tantdn/dzf(z)dz.
Plens=PHOMlens+PGRINlenstnn(PHOMant*PGRINpos+PHOMpos*PGRINant).
nant(z)=nn(nnns)(1ξant)p,
npos(z)=nn(nnns)(ξpos)p,
Aeye(1ePc)Alens,

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