Abstract

A nested shell model of the human lens is developed based on the known anatomical construction of the lens, on the known way in which the lens grows throughout its life, on the measured characteristics of the lens surfaces as a function of the age of the lens, on the measured changes in the shape of the lens during accommodation, and on measured material characteristics of the lens materials, such as density and index of refraction throughout. The observed changes in central surface curvature and thickness force the shell thicknesses to vary in a predicable way and in turn force the shell surface asphericity to take certain values. Thus, in addition to giving the shape of each shell, the model predicts the change expected in the asphericity of the lens surfaces as the lens ages and adds cortical cell layers. Two examples are given, one for a 25-year-old lens and one for a 40-year-old lens, to show how the cortical layers change their shapes throughout the cortex and over time as the lens ages. The performance of the model of this paper is compared to that of two other nested shell models, one where the layers have constant thickness and one where the lens posterior is fixed within the eye over time, to show the superior performance of this model with respect to maintaining a constant refractive error for the eye as the lens ages and grows.

© 2010 Optical Society of America

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References

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  1. A. Gullstrand, “The optical system of the eye,” in Handbuch der Physiogische Optik, 3rd ed., H.von Helmholtz, ed. (Voss Hambery, 1909), Vol. 1, pp. 350–358.
  2. O. Pomerantzeff, H. B. Fish, J. Govidon, and C. L. Schepens, “Wide-angle optical model of the human eye,” Opt. Acta 19, 387–388 (1972).
    [CrossRef]
  3. G. Smith, B. K. Pierscionek, and D. A. Atchison, “The optical modeling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
    [CrossRef] [PubMed]
  4. 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]
  5. D. Siedlecki, H. Kasprzak, and B. K. Pierscionek, “Schematic eye with a gradient-index lens and spherical surfaces,” Opt. Lett. 29, 1197–1199 (2004).
    [CrossRef] [PubMed]
  6. A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. A 24, 2157–2174 (2007).
    [CrossRef]
  7. R. Navarro, F. Palos, and L. Gonzalez, “Adaptive model of the gradient index of the human lens. I. Formulation and model of aging ex vivo lens,” J. Opt. Soc. Am. A 24, 2175–2185 (2007).
    [CrossRef]
  8. K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
    [CrossRef] [PubMed]
  9. A. Glasser, M. A. Croft, and P. L. Kaufman, “Aging of the human crystalline lens and presbyopia,” Int. Ophthalmol. Clin. 41, 1–15 (2001).
    [CrossRef] [PubMed]
  10. I. Fatt, Physiology of the Eye (Butterworths, 1978), p. 85.
  11. 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] [PubMed]
  12. C. E. Jones, D. A. Atchison, R. Meder, and J. L. 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]
  13. 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]
  14. 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 (2005).
    [CrossRef]
  15. S. Norrby, “The Dubbelman eye model analyzed by ray tracing though aspheric surfaces,” Ophthalmic Physiol. Opt. 25, 153–161 (2005).
    [CrossRef] [PubMed]
  16. International Standards Organization (ISO), ISO 19980:2005—Ophthalmic instruments: Corneal topographers (ISO, 2005), Clause 3.15 and Clause 3.4, Note 2.

2007 (2)

2005 (4)

C. E. Jones, D. A. Atchison, R. Meder, and J. L. 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. 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 (2005).
[CrossRef]

S. Norrby, “The Dubbelman eye model analyzed by ray tracing though aspheric surfaces,” Ophthalmic Physiol. Opt. 25, 153–161 (2005).
[CrossRef] [PubMed]

International Standards Organization (ISO), ISO 19980:2005—Ophthalmic instruments: Corneal topographers (ISO, 2005), Clause 3.15 and Clause 3.4, Note 2.

2004 (1)

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]

2001 (3)

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] [PubMed]

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

A. Glasser, M. A. Croft, and P. L. Kaufman, “Aging of the human crystalline lens and presbyopia,” Int. Ophthalmol. Clin. 41, 1–15 (2001).
[CrossRef] [PubMed]

1997 (1)

1991 (1)

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

1978 (1)

I. Fatt, Physiology of the Eye (Butterworths, 1978), p. 85.

1972 (1)

O. Pomerantzeff, H. B. Fish, J. Govidon, and C. L. Schepens, “Wide-angle optical model of the human eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

1909 (1)

A. Gullstrand, “The optical system of the eye,” in Handbuch der Physiogische Optik, 3rd ed., H.von Helmholtz, ed. (Voss Hambery, 1909), Vol. 1, pp. 350–358.

Al-Ghoul, K. J.

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

Atchison, D. A.

C. E. Jones, D. A. Atchison, R. Meder, and J. L. 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, B. K. Pierscionek, and D. A. Atchison, “The optical modeling of the human lens,” Ophthalmic Physiol. Opt. 11, 359–369 (1991).
[CrossRef] [PubMed]

Brennan, N. A.

Costello, M. J.

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

Croft, M. A.

A. Glasser, M. A. Croft, and P. L. Kaufman, “Aging of the human crystalline lens and presbyopia,” Int. Ophthalmol. Clin. 41, 1–15 (2001).
[CrossRef] [PubMed]

Dainty, C.

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 (2005).
[CrossRef]

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]

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] [PubMed]

Fatt, I.

I. Fatt, Physiology of the Eye (Butterworths, 1978), p. 85.

Fish, H. B.

O. Pomerantzeff, H. B. Fish, J. Govidon, and C. L. Schepens, “Wide-angle optical model of the human eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Freel, C. D.

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

Glasser, A.

A. Glasser, M. A. Croft, and P. L. Kaufman, “Aging of the human crystalline lens and presbyopia,” Int. Ophthalmol. Clin. 41, 1–15 (2001).
[CrossRef] [PubMed]

Goncharov, A. V.

Gonzalez, L.

Govidon, J.

O. Pomerantzeff, H. B. Fish, J. Govidon, and C. L. Schepens, “Wide-angle optical model of the human eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Gullstrand, A.

A. Gullstrand, “The optical system of the eye,” in Handbuch der Physiogische Optik, 3rd ed., H.von Helmholtz, ed. (Voss Hambery, 1909), Vol. 1, pp. 350–358.

Jones, C. E.

C. E. Jones, D. A. Atchison, R. Meder, and J. L. 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.

Kaufman, P. L.

A. Glasser, M. A. Croft, and P. L. Kaufman, “Aging of the human crystalline lens and presbyopia,” Int. Ophthalmol. Clin. 41, 1–15 (2001).
[CrossRef] [PubMed]

Kusak, A. J.

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

Kusak, J. R.

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

Liou, H. L.

Meder, R.

C. E. Jones, D. A. Atchison, R. Meder, and J. L. 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]

Navarro, R.

Nordgren, R. K.

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

Norrby, S.

S. Norrby, “The Dubbelman eye model analyzed by ray tracing though aspheric surfaces,” Ophthalmic Physiol. Opt. 25, 153–161 (2005).
[CrossRef] [PubMed]

Palos, F.

Pierscionek, B. K.

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

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

Pomerantzeff, O.

O. Pomerantzeff, H. B. Fish, J. Govidon, and C. L. Schepens, “Wide-angle optical model of the human eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Pope, J. L.

C. E. Jones, D. A. Atchison, R. Meder, and J. L. 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]

Schepens, C. L.

O. Pomerantzeff, H. B. Fish, J. Govidon, and C. L. Schepens, “Wide-angle optical model of the human eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Siedlecki, D.

Smith, G.

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

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 (2005).
[CrossRef]

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]

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] [PubMed]

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 (2005).
[CrossRef]

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)

K. J. Al-Ghoul, R. K. Nordgren, A. J. Kusak, C. D. Freel, M. J. Costello, and J. R. Kusak, “Structural evidence of human nuclear fiber compaction as a function of ageing and cataractogenisis,” Exp. Eye Res. 72, 199–214 (2001).
[CrossRef] [PubMed]

Int. Ophthalmol. Clin. (1)

A. Glasser, M. A. Croft, and P. L. Kaufman, “Aging of the human crystalline lens and presbyopia,” Int. Ophthalmol. Clin. 41, 1–15 (2001).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Ophthalmic Physiol. Opt. (2)

S. Norrby, “The Dubbelman eye model analyzed by ray tracing though aspheric surfaces,” Ophthalmic Physiol. Opt. 25, 153–161 (2005).
[CrossRef] [PubMed]

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

Opt. Acta (1)

O. Pomerantzeff, H. B. Fish, J. Govidon, and C. L. Schepens, “Wide-angle optical model of the human eye,” Opt. Acta 19, 387–388 (1972).
[CrossRef]

Opt. Lett. (1)

Vision Res. (4)

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] [PubMed]

C. E. Jones, D. A. Atchison, R. Meder, and J. L. 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. 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]

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 (2005).
[CrossRef]

Other (3)

International Standards Organization (ISO), ISO 19980:2005—Ophthalmic instruments: Corneal topographers (ISO, 2005), Clause 3.15 and Clause 3.4, Note 2.

A. Gullstrand, “The optical system of the eye,” in Handbuch der Physiogische Optik, 3rd ed., H.von Helmholtz, ed. (Voss Hambery, 1909), Vol. 1, pp. 350–358.

I. Fatt, Physiology of the Eye (Butterworths, 1978), p. 85.

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Figures (1)

Fig. 1
Fig. 1

Illustration of the creation of a new cortical cell layer by adding to a new constant thickness layer (a constant thickness shell) with thickness t const a lenslet with a diameter 2 R and central thickness h that increases the central curvature of the new shell to the desired value while at the same time increasing the central thickness of the cortex by the desired amount d c t .

Tables (3)

Tables Icon

Table 1 Parameter Values Used in the Eye Model for the Examples Given a

Tables Icon

Table 2 Results for the Three Types of Nested Shell Models Examined a

Tables Icon

Table 3 Apical Radii of Curvature and the Conic Constants for the Nested Shell Surfaces of This Model for a 25-Year-Old Lens and a 40-Year-Old Lens a

Equations (35)

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d n t = n t t t .
d n t = n t a c c a c c ,
a c c e q = ( n t t ) ( n t a c c ) t .
d r = r a c c a c c .
d r c o m p ( t ) = r a c c ( n t t ) ( n t a c c ) t .
r c o m p i = r i + d r c o m p ( t ) ,
r c o m p i = r i + r a c c ( n t t ) ( n t a c c ) t .
r c o m p t r a c c ( n t t ) ( n t a c c ) ,
r c o m p i = r i + r c o m p t t .
d p = ( p a c c ) a c c .
d p = ( p a c c ) ( n t t ) ( n t a c c ) t .
r ( t ) = r 0 + r t t + r a c c a c c .
c t ( t ) = c t 0 + c t t t .
r ( t ) = r 0 + r t t .
r const = r c o m p i + d i .
r const = r i + r c o m p t t + d i .
d i = c t const t t i ,
r const = r i + r c o m p t t i + c t const t t i ,
r const = r i + ( r c o m p t + c t const t ) t i .
h = R 2 ( x 2 + y 2 ) 2 K ,
h = R 2 ( x 2 + y 2 ) 2 ( 1 r i + 1 1 r const ) = R 2 ( x 2 + y 2 ) 2 ( r const r i + 1 r i + 1 r const ) .
h = R 2 ( x 2 + y 2 ) 2 ( ( r i + ( r c o m p t + c t const t ) t i ) i ( r i + r t t i ) ( r i + r t t i ) ( r i + ( r c o m p t + c t const t ) t i ) ) ,
h = R 2 ( x 2 + y 2 ) 2 ( r c o m p t + c t const t r t ) t i ( r i + r t t i ) ( r i + ( r c o m p t + c t const t ) t i ) .
h ( 0 , 0 ) = R 2 2 ( r c o m p t + c t const t r t ) t i ( r i + r t t i ) ( r i + ( r c o m p t + c t const t ) t i ) = ( c t t c t const t ) t i ,
R 2 2 ( r c o m p t + c t const t r t ) = ( c t t c t const t ) ( r i 2 + r i t i ( r c o m p t + c t const t + r t ) + ( r c o m p t + c t const t ) r t t i 2 ) .
R 2 2 ( c t const t + r c o m p t r t ) = ( c t t c t const t ) ( r i 2 ) .
c t const t = c t t r i 2 + ( r t r c o m p t ) R 2 2 R 2 2 + r i 2 .
n t ( t ) = n t 0 + n t t t ,
n t 0 = 2.11   mm ,
n t t = 0.003   mm / yr ,
n t a c c = ( 0.058 0.005 t )   mm / D .
n t ( t , a c c ) = n t 0 + n t t t + a c c ( 0.058 0.005 t ) .
a c d ( t ) = 3.87 0.01 years D ( 0.048 0.0004 years ) .
n nucleus ( years ) = 1.4204 0.000 051 years .
axial   length ( years ) = corneal   thickness + a c d ( years ) + a c t ( years ) + n t ( years ) + p c t ( years ) + r f .

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