Stretchdependent changes in surface profiles of ... - Wiley Online Library

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OPTOMETRY RESEARCH PAPER

Stretch-dependent changes in surface profiles of the human crystalline lens during accommodation: A finite element study Clin Exp Optom 2015; 98: 126–137 Hooman Mohammad Pour*† BSc PhD Sangarapillai Kanapathipillai* BScEng MEngSc PhD Khosrow Zarrabi* PhD Fabrice Manns§|| PhD Arthur Ho†§¶ MOptom PhD FAAO * School of Mechanical and Manufacturing Engineering, The University of New South Wales, Kensington, NSW, Australia † Brien Holden Vision Institute, Kensington, NSW, Australia § Ophthalmic Biophysics Center, Bascom Palmer Eye Institute, University of Miami Miller School of Medicine, Miami, Florida, USA || Biomedical Optics and Laser Laboratory, Department of Biomedical Engineering, University of Miami College of Engineering, Coral Gables, Florida, USA ¶ School of Optometry and Vision Science, The University of New South Wales, Kensington, NSW, Australia E-mail: [email protected]

Submitted: 4 December 2013 Revised: 10 July 2014 Accepted for publication: 8 September 2014

DOI:10.1111/cxo.12263 Background: A non-linear isotropic finite element (FE) model of a 29-year-old human crystalline lens was constructed to study the effects of various geometrical parameters on lens accommodation. Methods: The model simulates dis-accommodation by stretching of the lens and predicts the change in surface profiles of the lens capsule, cortex and nucleus at select states of stretching/accommodation. Multiple regression analysis (MRA) is used to develop a stretchdependent mathematical model relating the lens sagittal height to the radial position of the lens surface as a function of dis-accommodative stretch. A load analysis is performed to compare the finite element results to empirical results from lens stretcher studies. Using the predicted geometrical changes, the optical response of the whole eye during accommodation was analysed by ray-tracing. Results: Aspects of lens shape change relative to stretch were evaluated, including change in diameter, central thickness and accommodation. Maximum accommodation achieved was 10.29 D. From the multiple regression analysis, the stretch-dependent mathematical model of the lens shape related lens curvatures as a function of lens ciliary stretch well (maximum mean-square residual error 2.5 × 10-3 μm, p < 0.001). The results are compared with those from in vitro studies. Conclusions: The finite element and ray-tracing predictions are consistent with Ex Vivo Accommodation Simulator (EVAS) studies in terms of load and power change versus change in thickness. The mathematical stretch-dependent model of accommodation presented may have utility in investigating lens behaviour at states other than the relaxed or fully accommodated states.

Key words: accommodation, crystalline lens, finite element method presbyopia

Accommodation is an opto-mechanical physiological process by which the eye alters its optical power to achieve clear vision from distance to near. The main components of the accommodative system consist of the lens that effects the change in optical focus, suspended at its equator by the zonules to the ciliary body containing the annular ciliary muscle. The empirical study of the characteristics and behaviour of the accommodative system presents technical challenges due to the small dimensions (millimetres to microns), difficulty in measuring mechanical properties of the zonular fibres and the ciliary body in vivo, fine forces (milli-Newtons) involved in the intricately organised ocular tissues and the inaccessible location of the lens Clinical and Experimental Optometry 98.2 March 2015

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inside an intact eye surrounded by the aqueous and vitreous. Despite these challenges, in vivo and ex vivo investigations have been carried out, with various levels of precision, to lend insight into the optical and mechanical aspects of accommodation. While in vivo investigations would be most preferred, these are hindered by additional physical limitations; for example, the view of the peripheral lens (toward its equator) is obscured by the iris. To some extent, these limitations may be circumvented by taking advantage of advanced techniques, including ultrasonic biomicroscopy (UBM)1,2 and magnetic resonance imaging (MRI),3–7 although these imaging devices do require further improvements in resolution.6 Other limitations include the inaccuracy of physi-

cal experimental techniques due to the short period of viability of the post-mortem lens.8 One remaining unmet need is an in vivo method for directly investigating the forces involved in accommodation. An alternative to empirical study is theoretical investigation, of which one approach is the finite element method (FEM). Finite element (FE) models9–12 have been constructed and used to investigate lens behaviour during accommodation. Some have been used to evaluate certain theories of accommodation:9–13 1. to estimate the required dis-accommodative force exerted by the ciliary muscle11 2. to study the age-dependence of the zonular forces as a proposed cause for presbyopia14 © 2015 The Authors

Clinical and Experimental Optometry © 2015 Optometry Australia

Stretch-dependent changes in human crystalline lens during accommodation Pour, Kanapathipillai, Zarrabi, Manns and Ho

Category Discretisation

Details

Reason

Quadratic quadrilateral eight-node element

1. To increase number of calculation nodes

Use of contact elements

To prevent sliding of contacting bodies

Refined elements toward the equatorial region

Accuracy/convergence

Loading steps

Lens is gradually stretched in 10 steps with results at each step outputted

To facilitate investigation of relationship between progression of lens stretch and change in profiles of lens surfaces

Optical modelling

Optical performance evaluated using ray-tracing method

Paraxial calculations do not give an accurate lens optical power over a wider aperture

2. Required for axi-symmetric analysis of nearly incompressible materials

Table 1. Key modifications in the present model from the Burd model

3. to verify the effectiveness of proposed treatments for presbyopia12 4. to investigate the effects of changing the lens material properties15 5. to examine the importance of including the influence of the vitreous in accommodation16 and 6. to perform sensitivity analyses on the crystalline lens and accommodation.17 While other parameters such as the internal refractive index gradient18 of the lens may also be involved, the principal parameter that determines lens optical power change during accommodation is the change in its surface curvatures. These surfaces include the anterior and posterior interfaces between aqueous, lens capsule, lens cortex, lens nucleus and vitreous. Typically, previous finite element models report the resultant optical power or load at particular instances or stretching steps during dis-accommodation (that is, ciliary muscle relaxation leading to stretching of the zonules and lens). A few finite elementmodelling studies have investigated the relationship between the progression of lens stretch and the resultant change in various aspects of lens surface profiles in terms of curvature or slope. Although some studies have reported the force-dependent variation in certain geometrical parameters (for example, the apical/central curvature radii of lens surfaces),19–21 none of these studies has described the lens shape in terms of point-by-point profile across a substantial extent of the lens as a continuous function of the amount of ciliary stretch. A stretchdependent model of lens surface shape and subsequently, lens optical power, would be of value, as it could provide greater under-

standing of the mechanics and optics of the lens during the process of accommodation. Finite element models have been constructed using lens shapes, forces and optical properties based on ‘typical’ values in the literature. Thus, individual or individualised values are not represented. The objective of the present study was to explore the stretchdependent changes in terms of various geometrical parameters using numerical modelling and to develop a mathematical model of the surface profiles of the lens, the predictions of which can be validated against corresponding results obtained from direct ex vivo lens stretching experiments.22 As lensstretcher systems are capable of providing results for individual eyes, we hoped that this approach would provide first, a method for direct validation of the model and second, a basis for developing individualised models for a specific eye.

METHODS

The Burd finite element model For finite element modelling, the procedure developed by Burd, Judge and Cross,9 henceforth referred to as the ‘Burd model’, was replicated with some modifications/ assumptions that are summarised in Table 1. The Burd Model is well established and much cited, providing predictions for 11, 29 and 45 years of age and is reasonably straightforward to implement. This model can portray most of the lens behaviours during accommodation and has been widely used10–13,16 to investigate aspects of accommodation and presbyopia.

© 2015 The Authors Clinical and Experimental Optometry © 2015 Optometry Australia

In this study, we assumed the 29-year-old model of Burd, Judge and Cross9 as a starting point. Modifications were made to the model for achieving the objective of our study. For brevity, the remainder of the methods section will describe only key features and differences between the two models. The reader is encouraged to refer to Burd, Judge and Cross9 for full details of the original model. Lens deformations during accommodation are dependent on the distribution of material properties.12 In the present model, the lens geometry and material properties from Burd, Judge and Cross9 were adopted as the starting point. While different elastic moduli were assigned to the cortex and nucleus, the materials within each of the cortical and nuclear ‘compartments’ were considered to be isotropic and linear elastic. Time-dependent mechanical phenomena such as creep are beyond the scope of this study and were not taken into account. The model in this study was implemented using the commercial finite element software ANSYS (ANSYS® Academic Research, Release 13.0, ANSYS Inc, Canonberg, Pennsylvania, USA) and the model was solved using ANSYS® Mechanical APDL (ANSYS Parametric Design Language).

Modelling the reference geometry: Discretisation of the model To model the stretch-dependency of lens surface profiles, a higher data density for each position of stretch than the original Burd model was instituted to improve precision. In an attempt to assure convergence and improve precision of solution, to discretise the model, quadratic quadrilateral eight-node elements were used instead of triangular elements to increase the number of calculation nodes. This element type is appropriate for the axi-symmetric analysis of nearly incompressible materials and should produce a more realistic portrayal of the lens. Surface-to-surface three-node contact elements with ‘bonded’ option set were placed in the nucleus-cortex and cortex-capsule contact boundary to prevent the contacting bodies from sliding or being detached. From the clinical macroobservation stand-point, it is familiar to most surgeons that significant manipulations are required to remove clumps of cortex or nucleus (even after phaco-emulsification) and to strip lens epithelial cells from the internal surface of the capsule. These obserClinical and Experimental Optometry 98.2 March 2015

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diameter increase of 0.98 mm) was chosen to maximise overlap of induced lens changes (as opposed to ciliary stretching) in the finite element model with EVAS results for comparison purposes, while also ensuring convergence of the finite element model.

Finite element analysis A static structural analysis is carried out and the numerical model is solved using the ANSYS® FE package. Literature on finite element modelling reports that changes in the lens to either the equatorial force or displacement does not follow a linear response.10,12,15,24 Thus, a large deflection non-linear solution was performed using the ANSYS APDL®.

Figure 1. Discretised finite element model of the accommodative system with boundary conditions applied. A) single-point ciliary body, accommodative stretch is applied to this point. B and D) anchoring points of the anterior and posterior zonules. C) Anchoring point of the equatorial zonules and the lens equator. The brighter areas represent the lens nucleus and cortex respectively; elements are gradually refined as they approach the lens capsule; the lens polar axis is fixed (shown by roller support symbols) in the horizontal direction due to the lens axi-symmetry (that is, no x-translation of the lens polar axis during stretch).

vations imply a degree of ‘cohesiveness’ between layers of the lens that would prevent sliding between layers. Hence contact elements are used in this model. The geometry is meshed using 9,766 elements (corresponding to 30,577 nodes). The meshed model of the entire lens, the zonular fibres and the ciliary anchoring points are shown in Figure 1.

Boundary conditions and loading process Employing the conventional assumption that, when fully accommodated, the lens is free of stresses; no pre-stresses are applied to the lens in the baseline (unstretched) configuration. As the lens is rotationally symmetric, the lens polar axis is permitted only one degree of freedom (that is, translation in axial (Y) direction, Figure 1) and the required pull (ciliary stretch) for disaccommodation is applied as displacement in the horizontal (X) direction through a single-point ciliary body (Point A in Figure 1). It should be noted that in actuality, the model simulates dis-accommodation (that is, radially stretching, as opposed to releasing the lens). The initial, unstretch position represents the maximally accommodated (near focus) state of the lens while the maximally (end-point) stretched lens Clinical and Experimental Optometry 98.2 March 2015

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Surface profiles fitting

represents the distance focus state. To provide ease of comparability and consistency with previous studies, results are reported here in terms of accommodation and accommodative amplitude; that is, the difference in optical power from the maximally stretched, distance focus state, to the other states of stretch. The difference in power between maximally stretched and unstretched represents the maximum amplitude of accommodation. In the Burd model, the 29-year-old lens ciliary body was stretched to a maximum of 0.36 mm. In this study, the radial ciliary stretch was applied to the single-point ciliary starting from zero as the reference geometry to a maximum value of 0.49 mm in 10 steps with increments of approximately equal size ranging from 39 μm to 61 μm. This greater amount of stretch is commensurate with results of experiments on fresh donor tissues using a lens stretcher apparatus, the Ex Vivo Accommodation Simulator (EVAS) that showed human and non-human primate crystalline lenses can be stretched reversibly when the inner ciliary ring diameter is increased by up to approximately 0.8 mm in diameter,22,23 which is an equivalent of 0.4 mm half-diameter change in the axisymmetric finite element model. The value of 0.49 mm (corresponding to a ciliary ring

The finite element analysis generates the deformed profiles of the anterior and posterior capsule, cortex and nucleus at each step of ciliary body stretch. For each of the 11 steps (reference and 10 stretching steps) and each of the six lens surfaces (that is, aqueous-capsule, capsule-cortex, cortexnucleus, nucleus-cortex, cortex-capsule, capsule-vitreous), Cartesian (X, Y) coordinates are extracted for 24 data points that are located approximately equally spaced from the lens axis (that is, X = 0 mm) to 2.53 mm radial distance. To satisfy identified requirements for functions describing lens surface curvatures,24 sixth-order even polynomials (Equation 1) were used to describe each of the six surfaces at each of 11 states of stretch. An even polynomial was used given the assumed symmetry of the profiles. Values for the polynomial coefficients (a through d) of Equation 1 were determined using a least-squares curve-fitting method to each set of points.

Y = aX 6 + bX 4 + cX 2 + d

(1)

In Equation 1, Y represents the sagittal height (distance along Y-axis) of the surface in millimetres at the radial distance (from the axis) X in millimetres. During preliminary studies, fourth-order and eighth-order even polynomial fittings were also attempted; however, errors of estimate for the fourth-order curves (maximum residuals of 50 μm) were consistently higher (especially for points towards the lens equator) than those for the sixth-order polynomial (maximum residuals of 10 μm). Conversely, errors of estimate for the eighth-order fitting (maximum residuals of nine μm) were consistently similar to those of the sixth-order © 2015 The Authors



Surface number

Surface type

Component

Thickness (mm)

Index

Even polynomial coefficients Second-order (mm-1)

0

Standard

Visual object

Infinity

1.000

1

Paraxial

Accommodation

0.000

1.000

2

Paraxial

Cornea (ideal)

3.000

1.3374

3

Even Asphere

Capsule front

0.721

1.381

4

Even Asphere

Nucleus front

2.680

1.406

5

Even Asphere

Nucleus back

0.733

1.381

6

Even Asphere

Capsule back

15.718‡

1.336

-1.371E-01

7

Standard

Retina

Fourth-order (mm-3)

Power (D)

Sixth-order (mm-5)

0.00† 42.00 8.991E-02

-1.697E-03

-5.404E-05

1.342E-01

2.053E-03

1.649E-04

-1.408E-01

-2.421E-03

4.750E-04

-2.017E-03

-2.004E-04

1.336

‡ is the distance from the posterior lens to the retina set by optimisation of the most-stretched state (distance focus) in the first stage of computation. † is the optical power of the surface representing accommodation that is set by optimisation for all other stretched states in the second stage of computation

Table 2. Prescription of the optical eye model used for calculating change in optical power (accommodation) at each step of stretch. The values shown are for the unstretched (maximally accommodated) 29-year-old lens.

2

1

4

3

7

6

5

Figure 2. Layout of the accommodative system optical model. Numbered surfaces correspond to those of Table 2 together with their relative position (thickness column of the table). Surfaces 1 and 2 are co-located surfaces representing the optical power provided by accommodation and the idealised cornea. Surface 3 is the anterior capsule; Surfaces 4 and 5 the anterior and posterior cortex-nucleus interfaces and Surface 6 is the posterior capsule. During optical analysis, four rings of light rays are traced from distant object (left) to the retina (7 at far right). suggesting possible redundancy or at least numerical inefficiency.

Optical power calculation For each of the 11 stretching steps, the optical power of the lens was calculated using an eye model constructed in an optical ray-tracing software (Zemax version 12, Radiant Zemax, Redmond, Washington, USA). Table 2 provides the prescription of this optical model eye for the initial (unstretched reference) 29-year-old lens. We assumed that the capsule does not contribute to the optical power of the lens. Thus, the refractive interfaces between

aqueous-lens and lens-vitreous were considered to be the anterior aqueous-capsule (Surface 3) and posterior capsule-vitreous (Surface 6) surfaces, while the capsulecortex surfaces were ignored. In this way, the sixth-order even polynomials fitted to the four lens surfaces (anteriorly: aqueouscapsule and cortex-nucleus and posteriorly: nucleus-cortex and capsule-vitreous) at each stretch step, together with the position of the vertices of the surfaces, were used to define the refracting surfaces (Surfaces 3 to 6) of the lens. The sixth-order even polynomial surfaces were implemented by using the ‘even aspheric’ surface type in Zemax.


The natural lens receives light rays that have already been refracted by the cornea, that is, it does not operate at infinite conjugate ratio. We included this effect in the optical model by including an ideal thin lens (Surface 2) to represent corneal refraction, modelled using the ‘paraxial’ surface type in Zemax. This thin lens ‘cornea’ has a focal length in air of 23.8 mm, which is approximately equivalent to the typical corneal power of 42 D and is placed 3.0 mm in front of the anterior capsular surface of the lens to represent the distance from the cornea to the lens (that is, the anterior chamber). To avoid confusion, it is worth noting that the ‘paraxial’ lens type in Zemax is a fullaperture lens (that is, not limited to paraxial rays) that has ideal optical characteristics. In particular, its equivalent focal length at any ray-height is the same as its paraxial focal length (hence its name). Thus, ray-tracing is conducted at finite ray-heights, not just paraxially. This type of surface was used to direct positive vergence incident rays onto the lens surface without introducing any aberrations. The refractive indices assumed for the lens components were based on recent studies,25 while refractive indices for the aqueous and vitreous were values adopted from an established eye model.26 The lens equivalent index has been shown by recent studies to remain constant during accommodation.27–30 Figure 2 shows the layout of the eye model in the unstretched state. Clinical and Experimental Optometry 98.2 March 2015

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Figure 3. Surface profile change of the lens at four accommodative steps: A) baseline geometry; B) S = 200 μm, C) S = 360 μm, D) S = 490 μm.

Change in thickness (mm)

0

optical power of the paraxial thin lens (Surface 1) representing amplitude of accommodation to zero (that is, infinite focal length). Next, we optimised the optical model using the merit function system to find the best position (vitreous to retina distance, thickness for Surface 6) for the retina. The retinal position is then fixed for the remainder of the optical computation. In the second stage, for each stretching step including the initial unstretched state in turn, we optimised the optical model using the merit function system by varying the power of the paraxial thin lens (Surface 1) representing accommodative amplitude. The negative of the value for the optical power found by optimisation represented the accommodative power for that stretch state.

Stretch-dependent function for surface profiles

-0.1 -0.2

Examination of the coefficients from the sixth-order polynomial fit of the lens surfaces plotted against ciliary body stretch suggested that a reasonable fit of surface profile against amount of stretch may be achieved using a fourth-order function of stretch. Thus, a stretch-dependent function relating sagittal height, Y and radial distance X to amount of ciliary stretch S (in millimetres) was constructed according to Equation 2.

-0.3 -0.4 -0.5 -0.6 Total T Cortex T Nucleus T

-0.7 -0.8 -0.9 0

0.1

0.2

0.3

0.4

0.5

Stretch (mm) 3

Figure 4. Change in nucleus, cortex and total thickness versus accommodative stretch. Approximately 70 per cent of the change in total lens thickness is due to the change in thickness of the nucleus.

A merit function system was constructed for optimisation of the model eye to determine the refractive power of the eye model at each lens stretch step. This merit function system is based on minimising the root mean square distance of four rings of light rays assuming axial symmetry. An entrance pupil diameter of 5.0 mm was used throughout. The above criteria for best focal position (root mean square spot size) and pupil diameter were selected to provide comparability with the lens stretching system22 against which the model’s predictions may be validated. Thus, it should be noted that the model portrays the opto-mechanical response of the lens itself and not ocular accommodation, as it does not include the influences of Clinical and Experimental Optometry 98.2 March 2015

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4

Y = ∑ ∑ k2i , j X 2i S j

(2)

i =0 j =0

The coefficients for this function were computed using a least-squares multiple linear regression model.

such known phenomena as Stiles-Crawford effect, polychromatic response of the eye (as are incorporated in, for example, the visual Strehl ratio20) and pupil miosis. The amount of optical accommodation corresponding to each step of lens stretch was computed by placing an additional ideal thin lens (Surface 1) in the same plane as the idealised cornea. Thus with this arrangement, the calculated optical power and subsequent accommodation amplitude, are referenced to the corneal surface. The computation involved two stages. At the first stage, the maximally stretched condition (that is, 0.49 mm of stretch) is assumed to correspond to distance focus (that is, unaccommodated). Thus, we set the

RESULTS Figure 3 shows the vertical deformation (that is, surface profile change) in the lens at four steps of stretch. Consistent with the Helmholtzian theory,31 when ciliary stretch is transmitted to the lens equator via the zonules (that is, during dis-accommodation) the central lens curvatures become flatter, resulting in the central lens thickness to reduce and the lens diameter to increase (Figure 3). Maximum accommodation achieved was 10.29 D (referred at the corneal plane). By least-squares (bivariate) curve-fitting, changes in diameter (d), central thickness (T) and accommodation (A) can be © 2015 The Authors



D

Anterior capsule 2.4

Sagittal height (mm)


A 2.2 2.0 1.8 1.6 1.4 0

0.5

1.0 1.5 2.0 Radial position (mm)

B

-1.6 -1.8 -2.0 0

2.5

2.4 2.2 2.0 1.8 1.6 1.4 0

0.5


C

0.5

E

Anterior cortex



-1.4

-2.2

1.2

1.2 1.0 0.8 0.6 0.4 0.2

2.5

Posterior cortex -1.2 -1.4 -1.6 -1.8 -2.0 0

0.5

F

Anterior nucleus 1.4


-1.0

2.5



Posterior capsule -1.0 -1.2


2.5

Posterior nucleus -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4

0

0.5


2.5

0

0.5


2.5

Figure 5. Profile changes of each surface with ciliary stretch. In all diagrams, the steepest profile (greatest sagittal height) corresponds to the unstretched (accommodated) lens.

described as an approximately linear function of lens stretch (S in mm) with standard error of estimate less than 0.6 μm, respectively (Equations 3, 4 and 5).

d (mm ) = 1.5998 S + 8.6256

(3)

T (mm ) = -1.6927 S + 4.0732

(4)

A ( D ) = -21.4418 S + 9.5985

(5)

The anterior and posterior zonules do not pull in an exactly vectorally radial direction. In addition, their zonular points of insertion at the capsule, together with the configuration of the lens and the elasticity of the lens cortex at the region of zonular insertion, render the response slightly curvilinear. A second-order fit to the response provides a substantially improved fit with standard error of estimates of less than 0.2 μm (Equations 6, 7 and 8).

d (mm) = -0.1526 S2 + 1.6685 S + 8.6271 (6) T (mm) = 0.8737 S2 - 2.1363 S + 4.0819

(7)

A (D) = 18.724 S2 - 30.002 S + 9.6712

(8)

Figure 4 shows the variation in the lens total, nuclear and cortical thickness distributions versus accommodative stretch. Results show that the change in the nuclear thickness accounts for approximately 70 per cent of the change in the total lens thickness at any amount of stretch. The changes in profile of individual surfaces with ciliary stretch are plotted in Figure 5. The polynomial coefficients based on Equation 1 are found at each state of the ciliary stretch. Equation 1 fitted to each of the six surfaces returned a worst-case meansquared residual error of 2.5 × 10-3 μm (for posterior cortex, p < 0.001). These coefficients were used as input to the optical model. Surface fitting coefficients for the threedimensional stretch-dependent function (Equation 2) are listed in Table 3. The bolded values with asterisks are coefficients that contributed statistically significantly (p < 0.01) to the surface profile with stretching.


Regarding interpretation of Figure 5 and many of the subsequent figures, the curves plotted are for the absolute position of the surface points at their state of stretch. Since during stretching, the radial position of any given point changes (increases), points with the same radial distance but at different states of stretch are not corresponding points, that is, they are not from the same original unstretched point. Figure 6 shows the plots for the anterior and posterior lens surfaces using the resultant three-dimensional stretch-dependent mathematical model (Equation 2). Figure 7 shows the distribution of surface slope in the lens versus radial position. In all lens profiles, increasing the ciliary stretch causes the absolute slope at any point to decrease. From the surface slope and radial distance, the tangential and sagittal radii of curvature of the lens are calculated as a function of radial distances. The tangential radius (rt) is the instantaneous radius of curvature along the profile of the surface at a given point and is calculated using Equation 9.

(

df (x ) 1+ dx rt = d 2 f (x ) dx 2

3 2 2

)

(9)

The sagittal radius (rs) is the radius of curvature in a plane perpendicular to the profile. Geometrically, it is the distance from the given point to the intersection of the normal at that point and the lens axis. It is calculated using Equation 10.

⎞ ⎛ 1 ⎟ ⎜ 2 rs (x ) = x + ⎜ 2 f (x ) + df (x ) ⎟ ⎟ ⎜⎝ dx ⎠

2

(10)

Figures 8 and 9 show respectively the local distribution of the instantaneous (tangential) and axial (sagittal) radii of curvature for each surface. These two definitions of radius are relevant to the optical power contribution of the lens surfaces. For example, the sagittal radius is inversely related to the on-axis power of the lens at a given radial distance from the axis (which optically equates to the ray-height). The tangential radii of curvature in the capsule and cortex surfaces increase with Clinical and Experimental Optometry 98.2 March 2015

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Anterior lens i Capsule

0 0

2.0397*

Posterior lens

Cortex

Nucleus

Capsule

Cortex

Nucleus

2.0302*

1.3193* -2.0942* -2.0784* -1.3610*

0 1 -0.0900* -0.0903* -0.1343*

0.1408*

0.1409*

0.1371*

0 2

0.0017*

0.0016* -0.0021*

0.0024*

0.0027*

0.0020*

0 3

0.0001

0.0001

-0.0002* -0.0005* -0.0005*

0.0002*

1 0 -1.9014* -1.9005* -1.5980* 1 1

0.1887*

0.1919*

0.3091*

0.2828* -0.0432

0.3007*

0.1813*

-0.0348

-0.0903*

1 2 -0.0035

-0.0039

-0.0093

-0.0124

-0.0159

-0.006

1 3 -0.0006

-0.0006

0.0006

0.0008

0.0012

-0.0008

4.8869*

3.5111*

3.5668*

2 0

4.7737*

4.7830*

2 1 -0.2309

-0.2522

2 2 -0.0132

-0.0087

0.0272

0.0031

0.0003

2 3

0.0032

-0.4925* -0.0456

3.5231*

-0.1204

-0.0433

-0.0066

0.0227

-0.0110

0.002

-0.0015

0.0047

3 0 -7.5634* -7.5837* -7.9770* -7.0496* -7.2100* -7.184* 3 1

0.0001

0.0583

0.4356

-0.1050

0.1191

-0.0206

3 2

0.0562

0.0426

-0.0166

0.0637

-0.0277

0.0637

3 3 -0.0071

-0.0068

-0.005

-0.0083

0.003

-0.0113

4 0

5.1571*

5.1749*

4 1

0.1733

0.119

-0.1138

0.1616

-0.0577

-0.0403

-0.0101

-0.0629

0.029

0.0051

0.0063

0.0078

-0.0036

4 2 -0.0534 4 3

0.0055

5.4763*

5.1669*

5.3205*

5.2335* 0.0950 -0.064 0.0094

Table 3. Surface fitting coefficients for Equation 2. Coefficients in bold with asterisks are statistically significant at p < 0.01. The values for j represent orders for stretch (S), while 2i represents the orders for radial distance from axis (X).

increase in the radial position. This is similar to the geometry of the initial lens shape (that is, flatter near the poles and steeper near the equatorial region). In the anterior nucleus, the radius of curvature undergoes a dramatic decrease with increase in radial position. In contrast, the radius of curvature in the posterior nucleus first increases slightly and then decreases with increase in the radial position (except for the S = 0.00 mm profile, which has constant radius of curvature since the nucleus was initially modelled as a circular curve).

DISCUSSION A finite element model of the crystalline lens based on previous numerical studies has been constructed to predict quantitatively the changes in the surface profiles of the lens Clinical and Experimental Optometry 98.2 March 2015

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2 1.5 Sagittal height (mm)

k2i,j j

1 0.5 0 -0.5 -1 -1.5 -2 0 0.1 -2

0.2 -1

0.3

0

1 Radial distance (mm)

2

0.4

Stretch (mm)

Figure 6. Plot of the three-dimensional stretch-dependent mathematical model (Equation 2) relating sagittal height of the six lens surfaces as a function of radial distance from geometrical axis and amount of stretch. The lens equatorial plane is at zero sagittal height. A two-dimensional (plane) slice at a selected stretch value provides the predicted profile for the six surfaces at that state of stretch. Note that due to its thinness, the capsule thickness (upper-most and bottom-most surfaces) cannot be resolved in this plot.

capsule, cortex and nucleus. Using these predictions, optical power calculations were performed with ray-tracing to determine the amount of accommodation with lens stretch. An additional advantageous feature of finite element modelling is that it facilitates predicting the accommodative response for designing an accommodating intra-ocular lens, when a mechanical lens stretcher such as EVAS is not available.39,40 A novel three-dimensional mathematical model relating the lens surface profile to lens stretch was developed and fitted to the results of the finite element analysis. Results suggest that even with an idealised, purely horizontal ciliary stretch, a small posterior shift occurs during disaccommodation. This can be seen in the slight posterior shift in the equatorial plane of the lens in Figure 3 and suggests that, in the converse direction, a concomitant ante-

rior shift occurs during accommodation, although the vertical component of this shift in the finite element prediction is small, at around 80 μm, while in the correct direction, the predicted shift is small compared to values in the literature.9,10,13,16 It is worth emphasising that the antero-posterior movement of the model on stretching is in ‘free space’, constrained only by the anchoring point of the zonules at the ciliary body. This is to maintain comparability with lens stretching systems, which also do not substantially constrain the motion of the lens in the axial direction. To further improve the modelling of the antero-posterior movement of the lens on accommodation, additional components and/or parameters in the finite element model may be required. This becomes a necessity if modelling the operation of certain designs of ‘accommodating intraocular lenses’.32 © 2015 The Authors



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Figure 7. Local slope variation across the lens at different levels of ciliary stretch for the six lens surfaces.

The results in Figure 4 show that the predicted change in nuclear thickness accounts for approximately 70 per cent of the change in the total lens thickness at any amount of stretch, is close to the experimental results of Patnaik33 and Dubbelman and colleagues.34 Using Scheimpflug imaging, Dubbelman and colleagues34 found that there were no measurable changes in the cortical thickness during accommodation, while the cortex accounts for 30 per cent of the thickness change in the finite element model. The reason for this difference could be that in their experimental study on real lenses, the

gradient mechanical properties of the cortex and nucleus are included, whereas in our finite element model we assign uniform Young’s moduli to the lens. It could also be that the optical and mechanical cortex and nucleus do not correspond to exactly the same definition of the anatomical regions as seen in vivo by Scheimpflug imaging. It can be seen from Figure 7 (C, F) that the ciliary stretch substantially alters the slope of the nuclear profiles near the equator; however, it changes only slightly the slope of the nucleus near the polar region. Further, Figure 8 shows that while the radii of curva-


ture at the pole steadily increase with stretch, the radii of curvature near the equatorial region first decrease very slightly before increasing. The results show that there are reductions of 53 and 25 per cent in the central anterior and posterior surface curvatures, respectively, when the lens is disaccommodated with 0.49 mm stretch. It has been well established and reproduced by the present model that the lens continuously flattens in curvature over its central region (the most optically relevant region) during dis-accommodation. Figures 8 and 9 also show that both the tangential and sagittal radii increase at all points on dis-accommodation, with only the exception of the anterior lens capsule and cortex. These latter decrease in tangential radius on stretching at radial distances above about 2.0 mm.

Reconciliation with lens stretcher results At corresponding states of stretch, the surface profiles predicted with the current model compare well with those of the Burd model, verifying that our model with the modifications is behaving appropriately. This study extends on the work of Burd, Judge and Cross9 using a three-dimensional surface-fitting mathematical model to relate lens surface profiles to state of stretch, thereby realising a model, from which surface shapes may be interpolated for any desired state of stretch. This extension from a finite element model, predicting lens changes against amount of stretch, enables the comparison of the model against published empirical data on lens shape versus stretch and in particular, those produced from lens stretcher apparatuses.22,23 Figure 10 compares various responses of the EVAS experiments on eyes from donors of age around 30 years23 to our numerical modelling results that assumed a 29-year-old lens. The apparatus results show a thickness change of 0.08 mm per dioptre of accommodation, which is favourably comparable with a value of 0.06 mm per dioptre found in the literature.23 Also, the finite element model is consistent with the literature, which found a linear relationship between the accommodative force and the lens diameter. In particular, Augusteyn and colleagues23 found a change in diameter of approximately four microns per milli-Newton force, which is in close agreement to the value of 6.25 microns of diameter change per milli-Newton force from our model. Clinical and Experimental Optometry 98.2 March 2015

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Figure 8. Distribution of instantaneous (tangential) radius of curvature across the lens at different levels of ciliary stretch for the six lens surfaces.

There is a number of factors, which could explain the difference between the finite element model and EVAS results. First, the zonular arrangement in the model does not portray the natural arrangement observed by scanning electron microscopy and confocal microscopy in both adult human and young non-human primates. In addition, the zonules do not all terminate at one point at the ciliary body, as the model approximates. Instead the anterior zonules pass through the ciliary processes and have Clinical and Experimental Optometry 98.2 March 2015

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branches that are attached to the ciliary processes, while the posterior zonules are part of and surround the hyaloid membrane35 that encases the vitreous body. In addition, recent ultrasonic biomicroscopic studies performed in rhesus monkeys show posterior zonules that seem to connect directly to the pars plana of the ciliary body. In reality, anterior and posterior zonules terminate at the pars plana–pars plicata region.36 The distribution of zonular attachments to the lens capsule and ciliary body are complex.

Moreover, these fibres exhibit varied directionality and interlacing,37 making a faithful model difficult to construct. Further, it is known that the positions of distribution of zonular attachment changes with continuous lens growth as a result of aging.38 A model that reproduces the effect of zonular transmission of forces from the ciliary body to the lens at a microscopic level remains a key challenge. Figures 10A and 10B compare the load versus diameter-change and load versus thickness-change response from the finite element model and EVAS experiments for selected individual eyes between 29 to 31 years old. It may be seen that although the experimental data on thickness change suffer from a relatively large amount of scatter, the response found from our model is qualitatively consistent with the EVAS measurements in terms of magnitude of change and linearity. Both the finite element model and EVAS measurements of Augusteyn and colleagues23 suggest a linear response with the accommodative load. The discrepancy could be due to less accurate input data on the lens material properties but it should be noted that lens diameter is not the most robust parameter by which to assess accommodative response; change in diameter is relatively small and measurements of lens diameter in EVAS have limited precision. Additionally, due to scarcity of data, the vitreous and hyaloid membrane were not modelled in this study, which could affect the outcomes, as they are directly in contact with the lens. Discrepancy in the predicted load may also be due to the possible difference between the modelled zonular architecture and reality. Modelling the exact zonular geometry is not possible for at least two reasons: first, the zonular architecture is not known accurately and a natural randomness exists with different eyes having different zonular geometry. Second, the limitations inherent in finite element modelling reduce the similarity of the modelled zonules and the real architecture to some extent. From another point of view, in the finite element model, due to local deformations and stress concentration at the zonular insertion points (equatorial region) the outputted diameter is not exact and may be affected by the stress concentration at the tip of the zonules. It can be seen from Figure 10C that the thickness-change versus accommodative response (power change) of the FE model compares very well with that of EVAS. It should be noted that the © 2015 The Authors



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CONCLUSIONS

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Figure 9. Distribution of axial (sagittal) radii of curvature across the lens at different ciliary stretches for the six lens surfaces.

accommodation amplitude calculated in the current model refers to the corneal plane, while the lens stretch (EVAS) optical calculations refer to the mid-plane (approximate equatorial plane) of the lens sample. The difference in effectivity due to the different reference plane positions (around 5.0 mm maximum), at the largest accommodation found (around 10 D), amounts to a maximum discrepancy in power of less than 0.5 D, which is negligible.

Future models could combine the stretchdependency of the current model with age-dependency to construct a comprehensive stretch and age-dependent mathematical model capable of covering the effect of ciliary stretch with aging to provide an improved basis for understanding presbyopia. With further refinement of the model, the ability for validation of model predictions against direct empirical results of individual lenses, hopefully will provide a platform, from which individualised models for specific lens and eyes may be developed. From this, new knowledge founded on understanding the finer, individual differences may ensue.

Doubtless, further enhancements to the model are needed. Experimental data demonstrated that core (nucleus and cortex) elasticity is not constant in the lens but varies from the lens centre to the outer lens.18,34 This gradient in mechanical properties may need to be modelled in unison with the known gradient refractive index of the lens. To fully model the opto-mechanical behaviour of the lens, their effects should be taken into account.


A numerical finite element model based on a previous model has been constructed and used to characterise the stretch-dependent changes in the surface profiles of the anterior and posterior lens capsule, cortex and nucleus. A function relating the profiles of the capsule, cortex and nucleus to the amount of ciliary stretch was obtained by multiple linear regression of the model output. The results of the study are in agreement with recent ex vivo studies in terms of load and power change versus change in thickness; however, more work is needed to improve the prediction relating load and diameter. This model can then provide the basis for a quasi-analytical solution for the optics of accommodation. ACKNOWLEDGEMENTS

The authors dedicate this study to the memory of Jerry Chen (1978–2009), who initiated much of our program in finite element modelling that led to this work. The authors gratefully acknowledge Professors Jean-Marie Parel (University of Miami) and Robert Augusteyn (Brien Holden Vision Institute) for their encouragement, guidance and constructive comments on the manuscript, Dr Thomas John Naduvilath for assistance with statistical analyses and Dr Ashik Mohamed (LV Prasad Eye Institute) for providing us with the EVAS data. The study was supported in part by National Eye Institute Grants 2R01EY014225 and 1R01EY021834, and P30EY014801 (Center Grant) and an unrestricted grant from Research to Prevent Blindness. Clinical and Experimental Optometry 98.2 March 2015

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A 160 140 120 Load (mN)

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Figure 10. Comparison of finite element modelling predictions with individual eye results from Ex Vivo Accommodation Simulator (EVAS) studies. In all graphs, the hollow square marks the finite element model data. Dashed lines are linear trends for each of the five sets of EVAS results. A) Load versus change in lens diameter, B) load versus change in lens thickness, C) thickness change versus accommodation. REFERENCES 1. Schachar RA, Kamangar F. Computer image analysis of ultrasound biomicroscopy of primate accommodation. Eye 2006; 20: 226–233.

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