Abstract

A simple, parametric adaptive model of the refractive index distribution of the ex vivo crystalline lens is presented. It assumes conicoid (or nonrevolution quadric in 3D) iso-indical surfaces, concentric with the external surfaces of the lens. The model uses a minimum number of internal structural parameters, while the shape of the iso-indical surfaces adapts automatically to the external geometry. In this way, it is able to adapt and fit individual distributions as well as adapt to the changes of the lens shape and structure with age and accommodation. The model is fit to experimental data for individual eyes spanning ages 7 to 82 years, where for each eye the crystalline lens dimensions and iso-indical index data are known. The analysis demonstrates that only one age-dependent structural parameter is needed to replicate the internal iso-indical index structure, given age-dependent models for the external surfaces. An age-dependent-parameter global model is derived and is shown to predict age-dependent changes in the ex vivo lens power and longitudinal spherical aberration with age.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Gullstrand, "Appendix II" in Helmholtz's Handbuch der Physiologischen Optik, 3rd ed. English translation edited by J.P.Southall (Optical Society of America, 1962) Vol. 1, pp. 351-352.
  2. M. C. W. Campbell, "Measurement of refractive index in an intact crystalline lens," Vision Res. 24, 409-415 (1984).
    [CrossRef] [PubMed]
  3. B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
    [CrossRef]
  4. C. E. Jones, D. A. Atchison, R. Meder, and J. M. 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] [PubMed]
  5. D. Vazquez, E. Acosta, G. Smith, and L. Garner, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. II. The rotationally symmetrical lens," J. Opt. Soc. Am. A 23, 2551-2565 (2006).
    [CrossRef]
  6. S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).
  7. G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
    [CrossRef] [PubMed]
  8. D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," J. Opt. Soc. Am. A 14, 1684-1695 (1995).
  9. 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]
  10. D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, "Schematic eye with a gradient-index lens and aspheric surfaces," Opt. Lett. 29, 1197-1199 (2004)
    [CrossRef] [PubMed]
  11. J. W Blaker, "Toward an adaptive model of the human eye," J. Opt. Soc. Am. 70, 220-223 (1980).
    [CrossRef] [PubMed]
  12. M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
    [CrossRef] [PubMed]
  13. 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] [PubMed]
  14. R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Change with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).
  15. A. Glasser and M. Campbell, "Biometric, optical and physical changes in the isolated human crystalline lens with age in relation to presbyopia," Vision Res. 39, 1991-2015 (1999).
    [CrossRef] [PubMed]
  16. A. Popiolek-Masajada, "Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye," Ophthalmic Physiol. Opt. 19, 41-49 (1999).
    [CrossRef]
  17. N. Brown, "The change in lens curvature with age," Exp. Eye Res. 19, 175-183 (1974).
    [CrossRef] [PubMed]
  18. G. Smith and D. A. Atchison, "The gradient index and spherical aberration of the lens of the human eye," Ophthalmic Physiol. Opt. 21, 317-326 (2001).
    [CrossRef] [PubMed]
  19. J. G. Sivak and R. O. Kreuzer, "Spherical aberration of the crystalline lens," Vision Res. 23, 59-70 (1983).
    [CrossRef] [PubMed]
  20. A. Roorda and A. Glasser, "Wave aberrations of the isolated crystalline lens," J. Vision 4, 250-261 (2004).
    [CrossRef]
  21. J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Accommodation and presbyopia in the human eye. Changes in the anterior segment and crystalline lens with focus," Invest. Ophthalmol. Visual Sci. 38, 569-578 (1997).
  22. J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape at zero-diopter accommodation," J. Opt. Soc. Am. A 18, 265-272 (2001).
    [CrossRef]
  23. M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, "Change in shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2004).
    [CrossRef] [PubMed]
  24. R. Navarro, F. Palos, and L. González, "Adaptive model of the gradient index of the human lens. II. Optics of the accommodating aging lens," (submitted to J. Opt. Soc. Am. A).
  25. H. T. Kasprzak, "New approximation for the whole profile of the human crystalline lens," Ophthalmic Physiol. Opt. 20, 31-43 (2000).
    [CrossRef] [PubMed]
  26. A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
    [CrossRef]
  27. R. Navarro, L. González, and J. L. Hernández, "Optics of the average normal cornea from general and canonical representations of its surface topography," J. Opt. Soc. Am. A 23, 219-232 (2006).
    [CrossRef]
  28. Y. Le Grand and S. G. El Hage, Physiological Optics (Springer-Verlag, 1980).
  29. M. J. Howcroft and J. A. Parker, "Aspheric curvatures for the human lens," Vision Res. 17, 1217-1223 (1977).
    [CrossRef] [PubMed]
  30. R. Navarro, J. Santamaría, and J. Bescós, "Accommodation-dependent model of the human eye with aspherics," J. Opt. Soc. Am. A 2, 1273-1281 (1985).
    [CrossRef] [PubMed]
  31. 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]

2006 (3)

2005 (1)

C. E. Jones, D. A. Atchison, R. Meder, and J. M. 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] [PubMed]

2004 (3)

D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, "Schematic eye with a gradient-index lens and aspheric surfaces," Opt. Lett. 29, 1197-1199 (2004)
[CrossRef] [PubMed]

M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, "Change in shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2004).
[CrossRef] [PubMed]

A. Roorda and A. Glasser, "Wave aberrations of the isolated crystalline lens," J. Vision 4, 250-261 (2004).
[CrossRef]

2003 (1)

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape at zero-diopter accommodation," J. Opt. Soc. Am. A 18, 265-272 (2001).
[CrossRef]

G. Smith and D. A. Atchison, "The gradient index and spherical aberration of the lens of the human eye," Ophthalmic Physiol. Opt. 21, 317-326 (2001).
[CrossRef] [PubMed]

2000 (1)

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

1999 (2)

A. Glasser and M. Campbell, "Biometric, optical and physical changes in the isolated human crystalline lens with age in relation to presbyopia," Vision Res. 39, 1991-2015 (1999).
[CrossRef] [PubMed]

A. Popiolek-Masajada, "Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye," Ophthalmic Physiol. Opt. 19, 41-49 (1999).
[CrossRef]

1997 (2)

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]

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Accommodation and presbyopia in the human eye. Changes in the anterior segment and crystalline lens with focus," Invest. Ophthalmol. Visual Sci. 38, 569-578 (1997).

1995 (2)

R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Change with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).

D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," J. Opt. Soc. Am. A 14, 1684-1695 (1995).

1992 (1)

1991 (1)

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

1989 (1)

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

1985 (1)

1984 (1)

M. C. W. Campbell, "Measurement of refractive index in an intact crystalline lens," Vision Res. 24, 409-415 (1984).
[CrossRef] [PubMed]

1983 (1)

J. G. Sivak and R. O. Kreuzer, "Spherical aberration of the crystalline lens," Vision Res. 23, 59-70 (1983).
[CrossRef] [PubMed]

1980 (1)

1977 (1)

M. J. Howcroft and J. A. Parker, "Aspheric curvatures for the human lens," Vision Res. 17, 1217-1223 (1977).
[CrossRef] [PubMed]

1974 (1)

N. Brown, "The change in lens curvature with age," Exp. Eye Res. 19, 175-183 (1974).
[CrossRef] [PubMed]

1969 (1)

S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).

Acosta, E.

Artal, P.

Atchison, D. A.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. 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] [PubMed]

G. Smith and D. A. Atchison, "The gradient index and spherical aberration of the lens of the human eye," Ophthalmic Physiol. Opt. 21, 317-326 (2001).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," J. Opt. Soc. Am. A 14, 1684-1695 (1995).

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] [PubMed]

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

Augusteyn, R. C.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Berrio, E.

Bescós, J.

Blaker, J. W

Borja, D.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Brennan, N. A.

Brown, N.

N. Brown, "The change in lens curvature with age," Exp. Eye Res. 19, 175-183 (1974).
[CrossRef] [PubMed]

Campbell, M.

A. Glasser and M. Campbell, "Biometric, optical and physical changes in the isolated human crystalline lens with age in relation to presbyopia," Vision Res. 39, 1991-2015 (1999).
[CrossRef] [PubMed]

Campbell, M. C. W.

M. C. W. Campbell, "Measurement of refractive index in an intact crystalline lens," Vision Res. 24, 409-415 (1984).
[CrossRef] [PubMed]

Chan, D. Y. C.

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

Cook, C. A.

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape at zero-diopter accommodation," J. Opt. Soc. Am. A 18, 265-272 (2001).
[CrossRef]

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Accommodation and presbyopia in the human eye. Changes in the anterior segment and crystalline lens with focus," Invest. Ophthalmol. Visual Sci. 38, 569-578 (1997).

Denham, D B.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Dubbelman, M.

M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, "Change in shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2004).
[CrossRef] [PubMed]

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

El Hage, S. G.

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

Fernandez, V.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Garner, L.

Garner, L. F.

R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Change with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).

Glasser, A.

A. Roorda and A. Glasser, "Wave aberrations of the isolated crystalline lens," J. Vision 4, 250-261 (2004).
[CrossRef]

A. Glasser and M. Campbell, "Biometric, optical and physical changes in the isolated human crystalline lens with age in relation to presbyopia," Vision Res. 39, 1991-2015 (1999).
[CrossRef] [PubMed]

González, L.

R. Navarro, L. González, and J. L. Hernández, "Optics of the average normal cornea from general and canonical representations of its surface topography," J. Opt. Soc. Am. A 23, 219-232 (2006).
[CrossRef]

R. Navarro, F. Palos, and L. González, "Adaptive model of the gradient index of the human lens. II. Optics of the accommodating aging lens," (submitted to J. Opt. Soc. Am. A).

Guirao, A.

Gullstrand, A.

A. Gullstrand, "Appendix II" in Helmholtz's Handbuch der Physiologischen Optik, 3rd ed. English translation edited by J.P.Southall (Optical Society of America, 1962) Vol. 1, pp. 351-352.

Hemenger, R. P.

R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Change with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).

Hernández, J. L.

Ho, A.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Howcroft, M. J.

M. J. Howcroft and J. A. Parker, "Aspheric curvatures for the human lens," Vision Res. 17, 1217-1223 (1977).
[CrossRef] [PubMed]

Iwata, K.

S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).

Jones, C. E.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. 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] [PubMed]

Kasprzak, H.

Kasprzak, H. T.

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

Kaufman, P. L.

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape at zero-diopter accommodation," J. Opt. Soc. Am. A 18, 265-272 (2001).
[CrossRef]

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Accommodation and presbyopia in the human eye. Changes in the anterior segment and crystalline lens with focus," Invest. Ophthalmol. Visual Sci. 38, 569-578 (1997).

Koretz, J. F.

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape at zero-diopter accommodation," J. Opt. Soc. Am. A 18, 265-272 (2001).
[CrossRef]

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Accommodation and presbyopia in the human eye. Changes in the anterior segment and crystalline lens with focus," Invest. Ophthalmol. Visual Sci. 38, 569-578 (1997).

Kreuzer, R. O.

J. G. Sivak and R. O. Kreuzer, "Spherical aberration of the crystalline lens," Vision Res. 23, 59-70 (1983).
[CrossRef] [PubMed]

Le Grand, Y.

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

Liou, H.-L.

Manns, F.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Meder, R.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. 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] [PubMed]

Nagata, R.

S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).

Nakao, S.

S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).

Navarro, R.

Ono, T.

S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).

Ooi, C. S.

R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Change with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).

Palos, F.

R. Navarro, F. Palos, and L. González, "Adaptive model of the gradient index of the human lens. II. Optics of the accommodating aging lens," (submitted to J. Opt. Soc. Am. A).

Parel, J.-M.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Parker, J. A.

M. J. Howcroft and J. A. Parker, "Aspheric curvatures for the human lens," Vision Res. 17, 1217-1223 (1977).
[CrossRef] [PubMed]

Piers, P.

Pierscionek, B. K.

D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, "Schematic eye with a gradient-index lens and aspheric surfaces," Opt. Lett. 29, 1197-1199 (2004)
[CrossRef] [PubMed]

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] [PubMed]

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

Pope, J. M.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. 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] [PubMed]

Popiolek-Masajada, A.

A. Popiolek-Masajada, "Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye," Ophthalmic Physiol. Opt. 19, 41-49 (1999).
[CrossRef]

Roorda, A.

A. Roorda and A. Glasser, "Wave aberrations of the isolated crystalline lens," J. Vision 4, 250-261 (2004).
[CrossRef]

Rosen, A. M.

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

Santamaría, J.

Siedlecki, D.

Sivak, J. G.

J. G. Sivak and R. O. Kreuzer, "Spherical aberration of the crystalline lens," Vision Res. 23, 59-70 (1983).
[CrossRef] [PubMed]

Smith, G.

Van der Heijde, G. L.

M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, "Change in shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2004).
[CrossRef] [PubMed]

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

Vazquez, D.

Vrensen, G. F. J. M.

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

Weeber, H. A.

M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, "Change in shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2004).
[CrossRef] [PubMed]

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

Exp. Eye Res. (1)

N. Brown, "The change in lens curvature with age," Exp. Eye Res. 19, 175-183 (1974).
[CrossRef] [PubMed]

Invest. Ophthalmol. Visual Sci. (2)

R. P. Hemenger, L. F. Garner, and C. S. Ooi, "Change with age of the refractive index gradient of the human ocular lens," Invest. Ophthalmol. Visual Sci. 36, 703-707 (1995).

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Accommodation and presbyopia in the human eye. Changes in the anterior segment and crystalline lens with focus," Invest. Ophthalmol. Visual Sci. 38, 569-578 (1997).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

J. Vision (1)

A. Roorda and A. Glasser, "Wave aberrations of the isolated crystalline lens," J. Vision 4, 250-261 (2004).
[CrossRef]

Jpn. J. Clin. Ophthalmol. (1)

S. Nakao, T. Ono, R. Nagata, and K. Iwata, "Model of refractive indices in the human crystalline lens," Jpn. J. Clin. Ophthalmol. 23, 903-906 (1969).

Ophthalmic Physiol. Opt. (4)

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

A. Popiolek-Masajada, "Numerical study of the influence of the shell structure of the crystalline lens on the refractive properties of the human eye," Ophthalmic Physiol. Opt. 19, 41-49 (1999).
[CrossRef]

G. Smith and D. A. Atchison, "The gradient index and spherical aberration of the lens of the human eye," Ophthalmic Physiol. Opt. 21, 317-326 (2001).
[CrossRef] [PubMed]

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

Opt. Lett. (1)

Optom. Vision Sci. (1)

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

Vision Res. (8)

C. E. Jones, D. A. Atchison, R. Meder, and J. M. 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] [PubMed]

M. C. W. Campbell, "Measurement of refractive index in an intact crystalline lens," Vision Res. 24, 409-415 (1984).
[CrossRef] [PubMed]

J. G. Sivak and R. O. Kreuzer, "Spherical aberration of the crystalline lens," Vision Res. 23, 59-70 (1983).
[CrossRef] [PubMed]

M. Dubbelman, G. L. Van der Heijde, H. A. Weeber, and G. F. J. M. Vrensen, "Changes in the internal structure of the human crystalline lens with age and accommodation," Vision Res. 43, 2363-2375 (2003).
[CrossRef] [PubMed]

A. Glasser and M. Campbell, "Biometric, optical and physical changes in the isolated human crystalline lens with age in relation to presbyopia," Vision Res. 39, 1991-2015 (1999).
[CrossRef] [PubMed]

A. M. Rosen, D B. Denham, V. Fernandez, D. Borja, A. Ho, F. Manns, J.-M. Parel, and R. C. Augusteyn, "In vitro dimensions and curvatures of human lenses," Vision Res. 46, 1002-1009 (2006).
[CrossRef]

M. Dubbelman, G. L. Van der Heijde, and H. A. Weeber, "Change in shape of the aging human crystalline lens with accommodation," Vision Res. 45, 117-132 (2004).
[CrossRef] [PubMed]

M. J. Howcroft and J. A. Parker, "Aspheric curvatures for the human lens," Vision Res. 17, 1217-1223 (1977).
[CrossRef] [PubMed]

Other (3)

R. Navarro, F. Palos, and L. González, "Adaptive model of the gradient index of the human lens. II. Optics of the accommodating aging lens," (submitted to J. Opt. Soc. Am. A).

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

A. Gullstrand, "Appendix II" in Helmholtz's Handbuch der Physiologischen Optik, 3rd ed. English translation edited by J.P.Southall (Optical Society of America, 1962) Vol. 1, pp. 351-352.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Cemented doublet model of the crystalline lens: r a n t and r p o s are anterior and posterior sides of the iso-indical surface. The conic interface is formed by the intersections of these two sides of the iso-indical surfaces.

Fig. 2
Fig. 2

Axial distribution of refractive index for four out of the eight lenses analyzed, corresponding to 7, 27, 50, and 82 years. Each plot compares the experimental data (diamonds) with the results of individual (continuous curve) and global (dashed curve) fits by the adaptive GRIN model. Point n 0 (open circle) represents the position of the interface obtained in the individual fit.

Fig. 3
Fig. 3

Iso-indical surfaces of the GRIN model for the eight lenses obtained by fitting each individual lens (upper row) versus that of the resulting aging lens model obtained by a global fit (lower row). The global model shows a monotonic smooth evolution with age, while individual variability is patent in the upper row.

Fig. 4
Fig. 4

Exponent p versus age for the different lens samples. Diamonds represent the data points, and the solid curve the result obtained through global fit. We can see a clear outlier (circle corresponding to the 40-year-old lens). The dashed curve is the result of a least-squares fit when considering this point, whereas the continuous curve is the final result after removing this outlier.

Fig. 5
Fig. 5

Change of refractive power of the lens model with age. There is a linear ( R 2 = 0.9999 ) decline of power with age. The horizontal thin line represents the power of the lens from the classic Gullstrad–Le Grand’s eye model.

Fig. 6
Fig. 6

Change of the equivalent homogeneous refractive index with age. The horizontal line represents the 1.42 value of Le Grand’s model. There is a progressive decline, probably related to the increase of exponent p in the GRIN distribution.

Fig. 7
Fig. 7

Plots of the longitudinal LSA, in diopters, versus pupil radius, for different possible conic constants Q: spheres ( Q a n t = Q p o s = 0 ) ; paraboloids ( Q a n t = Q p o s = 1 ) ; and experimental values of Howcroft and Parker [29] ( Q a n t = 3.1316 , Q p o s = 1 ) . The dashed curve represents the homogeneous lens model of Navarro et al. [30].

Fig. 8
Fig. 8

Change of the longitudinal LSA with age for Howcroft and Parker [29] conic constants ( Q a n t = 3.1316 , Q p o s = 1 ) . The dashed curve represents the homogeneous lens model of Navarro et al. [30]. There is a clear trend of a monotonic decrease of SA with age.

Tables (1)

Tables Icon

Table 1 Parameters of the GRIN Models for Eight Lenses of Different Ages a

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

n ( r ) = n 0 + δ n r 2 p ,
n a n t ( r a n t ) = n o + δ n ( 1 r a n t 2 ( z , ω ) ) p
n p o s ( r p o s ) = n o + δ n ( r p o s 2 ( z , ω ) ) p .
r a n t 2 = 1 f a n t ( ( z Δ a n t ) 2 a a n t 2 + ϵ a n t ω 2 b a n t 2 O a n t ) ,
r p o s 2 = 1 f p o s ( ( z Δ p o s ) 2 a p o s 2 + ϵ p o s ω 2 b p o s 2 O p o s ) ,
Δ a n t = ϵ a n t a a n t ,
Δ p o s = t ϵ p o s a p o s ,
O a n t = Δ a n t 2 a a n t 2 , f a n t = t a n t 2 2 t a n t Δ a n t a a n t 2 .
O p o s = ( Δ p o s t a n t ) 2 a p o s 2 , f p o s = t p o s 2 2 t p o s ( Δ p o s t a n t ) a p o s 2 .
n a n t ( z i , ω i ) = n p o s ( z i , ω i ) ,
[ 1 f p o s a p o s 2 + 1 f a n t a a n t 2 ] z i 2 2 [ Δ p o s f p o s a p o s 2 + Δ a n t f a n t a a n t 2 ] z i + [ 1 f p o s b p o s 2 + 1 f a n t b a n t 2 ] ω i 2 + [ 2 Δ p o s t a n t t a n t 2 f p o s a p o s 2 ] 1 = 0 .
Z i = β ± β 2 4 α ( γ ω i 2 + δ ) 2 α .
n a n t ( z , ω ) = n o + δ n ( 1 1 f a n t ( z 2 2 Δ a n t z a a n t 2 + ϵ a n t ω 2 b a n t 2 ) ) p for ( z a n t , ω a n t ) ( z , ω ) < ( z i , ω i ) ,
n p o s ( z , ω ) = n o + δ n ( 1 f p o s ( z 2 t a n t 2 2 Δ p o s ( z t a n t ) a p o s 2 + ϵ p o s ω 2 b p o s 2 ) ) p for ( z i , ω i ) ( z , ω ) < ( z p o s , ω p o s ) ,
p = 1.1 × 10 7 age 4 + 2.85 , t a n t = 0.6 t .
n ( z , ω ) = n 00 + n 01 z + n 02 z 2 + n 20 ω 2 .
n 00 = n 0 + δ n ; n 01 = 2 Δ a n t δ n f a n t a a n t 2 ; n 02 = δ n f a n t a a n t 2 ; n 20 = ϵ a n t δ n f a n t b a n t 2
n 00 = n 0 ; n 01 = 2 δ n ( t a n t Δ p o s ) f p o s a p o s 2 ; n 02 = δ n f p o s a p o s 2 ; n 20 = ϵ p o s δ n f p o s b p o s 2
r a n t 2 = 1 f a ( ( z Δ a n t ) 2 a a n t 2 + ϵ a n t x 2 b a n t 2 + ϵ a n t y 2 c a n t 2 O a n t ) ,
r p o s 2 = 1 f p o s ( ( z Δ p o s ) 2 a p o s 2 + ϵ p o s x 2 b p o s 2 + ϵ p o s y 2 c p o s 2 O p o s ) .

Metrics