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

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2013 (4)

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]

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]

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]

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. Express4, 466–480 (2013).
[CrossRef] [PubMed]

2012 (3)

E. Lanchares, R. Navarro, and B. Calvo, “Hyperelastic modelling of the crystalline lens: Accommodation and presbyopia,” J. Optometry5, 110–120 (2012).
[CrossRef]

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

S. Ortiz, P. Pérez-Merino, E. Gambra, A. de Castro, and S. Marcos, “In vivo human crystalline lens topography,” Biomed. Opt. Express3, 2471–2488 (2012).
[CrossRef] [PubMed]

2011 (1)

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).

2010 (4)

2009 (4)

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]

E. Gambra, L. Sawides, C. Dorronsoro, and S. Marcos, “Accommodative lag and fluctuations when optical aberrations are manipulated,” J. Vision9, 1–15 (2009).
[CrossRef]

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]

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]

2008 (3)

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]

N. López-Gil, V. Fernández-Sánchez, R. Legras, R. Montés-Micó, F. Lara, and J. L. Nguyen-Khoa, “Accommodation-related changes in monochromatic aberrations of the human eye as a function of age,” Invest. Ophthalmol. Visual Sci.49, 1736–1743 (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 (6)

Y. Wang, Z.-Q. Wang, H.-Q. Guo, Y. Wang, and T. Zuo, “Wavefront aberrations in the accommodated human eye based on individual eye model,” Optik118, 271–277 (2007).
[CrossRef]

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]

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]

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]

A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. A24, 2157–2174 (2007).
[CrossRef]

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. A24, 2175–2185 (2007).
[CrossRef]

2006 (1)

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]

2005 (4)

J. L. Alió, P. Schimchak, H. P. Negri, and R. Montées-Micó, “Crystalline lens optical dysfunction through aging,” Ophthalmology112, 2022–2029 (2005).
[CrossRef] [PubMed]

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]

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]

S. Plainis, H. S. Ginis, and A. Pallikaris, “The effect of ocular aberrations on steady-state errors of accommodative response,” J. Vision5, 466–477 (2005).
[CrossRef]

2004 (8)

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vision4, 272–280 (2004).
[CrossRef]

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]

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).

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

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]

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]

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]

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]

2003 (3)

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. A20, 1155–1163 (2003).
[CrossRef]

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

C. Hazel, M. Cox, and N. Strang, “Wavefront aberration and its relationship to the accommodative stimulus-response function in myopic subjects,” Opt. Vis. Sci.80, 151–158 (2003).
[CrossRef]

2002 (4)

S. Ninomiya, T. Fujikado, T. Kuroda, N. Maeda, Y. Tano, T. Oshika, Y. Hirohara, and T. Mihashi, “Changes of ocular aberration with accommodation,” Am. J. Ophthalmol.134, 924–926 (2002).
[CrossRef] [PubMed]

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]

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

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]

2001 (3)

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]

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]

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]

2000 (3)

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

J. He, S. Burns, and S. Marcos, “Monochromatic aberrations in the accommodated human eye,” Vis. Res.40, 41–48 (2000).
[CrossRef] [PubMed]

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).

1998 (3)

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]

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).

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]

1997 (1)

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]

1995 (2)

D. A. Atchison, M. J. Collins, C. F. Wildsoet, J. Christensen, and M. D. Waterworth, “Measurement of monochromatic ocular aberrations of human eyes as a function of accommodation by the howland aberroscope technique,” Vis. Res.35, 313–323 (1995).
[CrossRef] [PubMed]

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vis. Res.35, 1157–1163 (1995).
[CrossRef] [PubMed]

1993 (1)

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).

1992 (1)

1985 (1)

1983 (1)

J. Sivak and R. Kreuzer, “Spherical aberration of the crystalline lens,” Vision Research23, 59–70 (1983).
[CrossRef] [PubMed]

1980 (1)

1979 (1)

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]

1977 (1)

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

1973 (1)

1963 (1)

T. Jenkins, “Aberrations of the eye and their effects on vision: 1. spherical aberration,” Br. J. Physiol. Opt.20, 59 (1963).
[PubMed]

1949 (1)

1947 (1)

1801 (1)

T. Young, “On the mechanism of the eye,” Philos. Trans. R. Soc. London91, 23–88 (1801).
[CrossRef]

Alió, J. L.

J. L. Alió, P. Schimchak, H. P. Negri, and R. Montées-Micó, “Crystalline lens optical dysfunction through aging,” Ophthalmology112, 2022–2029 (2005).
[CrossRef] [PubMed]

Amano, S.

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).
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Amano, Y.

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]

Amelinckx, A.

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]

Applegate, R. A.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vision4, 272–280 (2004).
[CrossRef]

Arrieta, E.

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).
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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. Express1, 1331–1340 (2010).
[CrossRef]

Artal, P.

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]

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]

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]

Ashman, R.

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]

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),” Vis. Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

Atchison, D. A.

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]

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]

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]

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vis. Res.35, 1157–1163 (1995).
[CrossRef] [PubMed]

D. A. Atchison, M. J. Collins, C. F. Wildsoet, J. Christensen, and M. D. Waterworth, “Measurement of monochromatic ocular aberrations of human eyes as a function of accommodation by the howland aberroscope technique,” Vis. Res.35, 313–323 (1995).
[CrossRef] [PubMed]

G. Smith, D. A. Atchison, and B. K. Pierscionek, “Modeling the power of the aging human eye,” J. Opt. Soc. Am. A9, 2111–2117 (1992).
[CrossRef] [PubMed]

Augusteyn, R.

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]

Bahrami, M.

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

Barnett, J. K.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vision4, 272–280 (2004).
[CrossRef]

Bedggood, P.

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]

Berny, F.

Berrio, E.

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]

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]

Bescós, J.

Birkenfeld, J.

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]

Blaker, J. W.

Borja, D.

Brennan, N. A.

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]

Bullimore, M. A.

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]

Burd, H.

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

Burd, H. J.

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).

Burns, S.

J. He, S. Burns, and S. Marcos, “Monochromatic aberrations in the accommodated human eye,” Vis. Res.40, 41–48 (2000).
[CrossRef] [PubMed]

Calver, R.

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]

Calvo, B.

E. Lanchares, R. Navarro, and B. Calvo, “Hyperelastic modelling of the crystalline lens: Accommodation and presbyopia,” J. Optometry5, 110–120 (2012).
[CrossRef]

Campbell, C. E.

Campbell, M. C.

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]

Candia, O. A.

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]

Cheng, H.

H. Cheng, J. K. Barnett, A. S. Vilupuru, J. D. Marsack, S. Kasthurirangan, R. A. Applegate, and A. Roorda, “A population study on changes in wave aberrations with accomodation,” J. Vision4, 272–280 (2004).
[CrossRef]

Choi, J. A.

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).

Christensen, J.

D. A. Atchison, M. J. Collins, C. F. Wildsoet, J. Christensen, and M. D. Waterworth, “Measurement of monochromatic ocular aberrations of human eyes as a function of accommodation by the howland aberroscope technique,” Vis. Res.35, 313–323 (1995).
[CrossRef] [PubMed]

Collins, M. J.

D. A. Atchison, M. J. Collins, C. F. Wildsoet, J. Christensen, and M. D. Waterworth, “Measurement of monochromatic ocular aberrations of human eyes as a function of accommodation by the howland aberroscope technique,” Vis. Res.35, 313–323 (1995).
[CrossRef] [PubMed]

M. J. Collins, C. F. Wildsoet, and D. A. Atchison, “Monochromatic aberrations and myopia,” Vis. Res.35, 1157–1163 (1995).
[CrossRef] [PubMed]

Cox, M.

C. Hazel, M. Cox, and N. Strang, “Wavefront aberration and its relationship to the accommodative stimulus-response function in myopic subjects,” Opt. Vis. Sci.80, 151–158 (2003).
[CrossRef]

Cox, M. J.

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]

Cross, J.

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

Daaboul, M.

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]

Dainty, C.

de Castro, A.

DeMarco, J. K.

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).
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der Heijde, G. V.

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).
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Dorronsoro, C.

E. Gambra, L. Sawides, C. Dorronsoro, and S. Marcos, “Accommodative lag and fluctuations when optical aberrations are manipulated,” J. Vision9, 1–15 (2009).
[CrossRef]

Dubbelman, M.

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]

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]

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]

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]

El Hage, S. G.

Escobar, D. P.

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]

Fernandez, V.

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]

Fernández-Sánchez, V.

N. López-Gil, V. Fernández-Sánchez, R. Legras, R. Montés-Micó, F. Lara, and J. L. Nguyen-Khoa, “Accommodation-related changes in monochromatic aberrations of the human eye as a function of age,” Invest. Ophthalmol. Visual Sci.49, 1736–1743 (2008).
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Fujikado, T.

S. Ninomiya, T. Fujikado, T. Kuroda, N. Maeda, Y. Tano, T. Oshika, Y. Hirohara, and T. Mihashi, “Changes of ocular aberration with accommodation,” Am. J. Ophthalmol.134, 924–926 (2002).
[CrossRef] [PubMed]

Gambra, E.

Garner, L. F.

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]

Garriott, R.

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).

Gerometta, R.

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]

Ginis, H. S.

S. Plainis, H. S. Ginis, and A. Pallikaris, “The effect of ocular aberrations on steady-state errors of accommodative response,” J. Vision5, 466–477 (2005).
[CrossRef]

Giovanzana, S.

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]

Glasser, A.

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]

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]

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] [PubMed]

A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. A24, 2157–2174 (2007).
[CrossRef]

González, L.

González, L. M.

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]

Guirao, A.

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]

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]

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]

Gullstrand, A.

A. Gullstrand, Appendix IV of Treatise on Physiological Optics, vol. 1 (Dover Phoenix Editions, 2005).

Guo, H.-Q.

Y. Wang, Z.-Q. Wang, H.-Q. Guo, Y. Wang, and T. Zuo, “Wavefront aberrations in the accommodated human eye based on individual eye model,” Optik118, 271–277 (2007).
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Gwiazda, J.

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. A20, 1155–1163 (2003).
[CrossRef]

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).

Hamaoui, M.

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]

Hazel, C.

C. Hazel, M. Cox, and N. Strang, “Wavefront aberration and its relationship to the accommodative stimulus-response function in myopic subjects,” Opt. Vis. Sci.80, 151–158 (2003).
[CrossRef]

He, J.

J. He, S. Burns, and S. Marcos, “Monochromatic aberrations in the accommodated human eye,” Vis. Res.40, 41–48 (2000).
[CrossRef] [PubMed]

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).

He, J. C.

Heethaar, R.

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]

Heethaar, R. M.

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]

Held, R.

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. A20, 1155–1163 (2003).
[CrossRef]

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).

Hemenger, R. P.

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).

Hermans, E.

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]

Hermans, E. A.

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]

Hirohara, Y.

S. Ninomiya, T. Fujikado, T. Kuroda, N. Maeda, Y. Tano, T. Oshika, Y. Hirohara, and T. Mihashi, “Changes of ocular aberration with accommodation,” Am. J. Ophthalmol.134, 924–926 (2002).
[CrossRef] [PubMed]

Ho, A.

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]

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]

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]

Howcroft, M. J.

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

Howland, H. C.

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]

Iskander, D. R.

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]

Ivanoff, A.

Jain, R.

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]

Jenkins, T.

T. Jenkins, “Aberrations of the eye and their effects on vision: 1. spherical aberration,” Br. J. Physiol. Opt.20, 59 (1963).
[PubMed]

Jiang, H.

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),” Vis. Res.45, 2352–2366 (2005).
[CrossRef] [PubMed]

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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).
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[CrossRef]

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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]

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[CrossRef]

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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).
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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).
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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).
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[CrossRef]

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Figures (4)

Fig. 1:
Fig. 1:

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

Fig. 2:
Fig. 2:

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

Fig. 3:
Fig. 3:

Fitting the figuring conicoid (green) with a pure conicoid (blue) of conic constant K a * 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:
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.

Tables (3)

Tables Icon

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.

Tables Icon

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.

Tables Icon

Table 3: Experimental changes in lenticular and ocular SA ( Z 4 0) per dioptre. All data are scaled down to a 3 mm pupil diameter from their measured pupil diameters (given in mm).

Equations (19)

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n ( r ) = c 0 + c p r p ,
n ( ζ ) = n c + ( n s n c ) ( ζ 2 ) P ,
ρ 2 = 2 R z ( 1 + K ) z 2 ,
ρ 2 = 2 R z ( 1 + K ) z 2 + B z 3 .
2 R a ( T a + Z c ) Q a ( T a + Z c ) 2 + B a ( T a + Z c ) 3 = 2 R p ( T p Z c ) Q p ( T p Z c ) 2 + B p ( T p Z c ) 3 .
B a = 2 3 Q a ( T a + Z c ) R a ( T a + Z c ) 2 and B p = 2 3 Q p ( T p Z c ) R p ( T p Z c ) 2 .
Z c = 1 Q p Q a [ T a Q a + T p Q p 2 ( R a + R p ) + T 2 Q a Q p 4 T ( R a Q p + R p Q a ) + 4 ( R a + R p ) 2 ] ,
V = π T a Z c ρ a 2 ( z ) d z + π Z c T p ρ p 2 ( z ) d z ,
V = 1 6 π [ 5 R p Z p 2 + 5 R a Z a 2 Q p Z p 3 Q a Z a 3 ] .
R a ( A ) = R 0 a + Δ R a = R 0 a + ( 0.35 0.084 R 0 a ) A , R p ( A ) = R 0 p + Δ R p = R 0 p + ( 0.37 0.082 R 0 p ) A , T ( A ) = T 0 + Δ T = T 0 + ( 0.0436 ) A ,
Z a ( A ) = Z 0 a Δ ACD ( A ) .
K a = π ( T Z a ) ( R p ( T Z a ) R a Z a ) 6 V 0 π T Z a 2 + 5 R a Z a 1 , and
K p = π Z a ( R a Z a R p ( T Z a ) ) 6 V 0 π T ( T Z a ) 2 + 5 R p T Z a 1 .
F = n s n aq R a + 2 P 2 P 1 ( n c n s ) ( 1 R a + 1 R p ) + n s n vit R p ,
W 4 , 0 = 1 8 S I .
ρ 1 2 = 2 R z 1 ( 1 + K ) z 1 2 + B z 1 3 .
ρ 1 2 = 2 R z 1 ( 1 + K * ) z 1 2 , thus : K * = K B z 1 .
W 4 , 0 = 6 5 Z 4 0 .
Δ K a = ( 0.63 0.07 × K 0 a ) A ,

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