Modeling internal stress distributions in the human lens: Can opponent theories coexist?

Abstract

The effects of material properties and equatorial stretching forces on the stress distribution and shape profile of human lenses were investigated to see whether support could be found for either or both current theories of accommodation. Finite element analysis was used to create models using shape parameters and material properties from published data. Models were constructed for two lenses of different ages. Material properties were varied to show differences between models with a single elastic modulus and those with different moduli for the cortex and the nucleus. Two levels of stretching forces were applied at the equator. Comparisons between experimental and model profiles were made, and stress distribution patterns were constructed. In all models, stretching produces a flattening in the peripheral curvature of the lens. In the younger lens, model and experimental results show that central curvature at some points is steeper for stretched than for unstretched profiles. In the older lens, gradients are flatter at all central points for stretched model and experimental profiles compared to the unstretched profile. In all models, there is a region of higher stress distribution within the lens that corresponds with the position of an inflection point that appears on the anterior surface and, in the older lens, also on the posterior surface. The results show that equatorial stretching forces can produce shape changes in support of both current theories of accommodation depending on the lens age, shape, and applied force.

Introduction

Explanations for the process of accommodation date back many centuries. The theory that has become the most widely recognized and accepted is that of Helmholtz (1855, cited in Grossman, 1904). Helmholtz believed the lens and capsule to be elastic bodies, implying that the lens must have a “natural shape” (the accommodated form) to which it reverts when the applied force (mediated by the ciliary body) is removed. Acceptance has not been universal; the concept of an active force (contraction of the ciliary muscle) accompanying a relaxation of the lens and zonule has been the subject of continuing scientific consternation. 

Mannhardt (1858), Schoen (1887), and Tscherning (1894, 1895; all cited in Grossman, 1904) opposed Helmholtzian theory on the grounds of Helmholtz' explanation of the forces involved in accommodation. Tscherning (1894) proposed an alternative explanation: During accommodation, the zonule does not relax; the zonular tension then causes the lens shape to change into a conoidal form. Years later, Fincham (1937) raised concerns about the nature of the lens substance and the explanation that the lens adopts its naturally relaxed state when accommodating. Fincham had doubts that the lens substance possessed the requisite elasticity to support the explanation of Helmholtz. 

More recently, opposition to Helmholtz has come from Schachar (1994, 1999), Schachar and Anderson (1995), and Schachar et al. (1996), according to whom the equatorial stretching of the lens should result in an increase in central lens curvature and flattening of the peripheral curvature. The current controversy centers on the shape change in the lens. The material properties of the lens and how these may affect the shape changes are given less consideration. Yet, material properties affect the type and extent of shape alterations, and they become particularly pertinent when age-related changes to the structural proteins of the lens alter tissue properties. A lens that is less pliable (an older lens) will not stretch to the same extent as a younger lens, which has a greater degree of elasticity. 

The material or elastic properties of the lens and its capsule are not easy to measure. Changes in elasticity can be induced by the preparative techniques (e.g., cutting or slicing the tissue), and dehydration of the lens and capsule, once the tissue is removed from the eye, may affect the rheological properties. With respect to these properties, the seminal work of Fisher (1969, 1971) remains the most widely referenced. More recent studies (Beers & van der Heijde, 1994; Czygan & Hartung, 1996; Heys, Cram, & Truscott, 2004; Krag, Olsen, & Andreassen, 1997; Pau & Kranz, 1991; van Alphen & Graebel, 1991; Weeber et al., 2005) have used a variety of approaches and report a range of values. The difficulty with determining the material properties of the entire lens and how these may alter with age is compounded by the fact that the lens has a varied distribution of proteins (Pierscionek & Augusteyn, 1988, 1991), which result in concentration differences and in a refractive index gradient (Pierscionek, 1995a, 1997; Pierscionek & Chan, 1989). 

The material properties of the lens will affect the distribution of stresses within the tissue during the process of accommodation. As it is very difficult to directly measure these internal stresses in a functioning lens, this study uses models, based on experimental data (Pierscionek, 1993), with varying material properties and stretching forces, to determine how internal stresses may be distributed and how these can affect the surface shape. 

Methods

Three-dimensional lens models were generated using ABAQUS standard version 6.5 based on lens profiles obtained from experiments on two representative human lenses aged 27 and 46 years (Pierscionek, 1993). Three-dimensional models were created so that the angular effect of stretching forces that operate from adjacent ligaments could be taken into account. Lens photographs of experimental data (Pierscionek, 1993), showing different levels of stretch (simulating varying accommodative states), were magnified, and profiles were reproduced in a Cartesian coordinate system. Dimensions were obtained using a calibrating object in the photographs. 

Models were assembled from three separately created sections: lens, capsule, and zonular ligaments. Two cases were considered for each lens. In the first case, the lens was constructed as a single-material body, and in the second, it was constructed as an assembly that is composed of two materials representing the nucleus and the cortex. The proportion of lens representing the nucleus was taken as the volume calculated from a radius of two thirds the size of the total equatorial radius for each lens. As the lens is rotationally symmetric about the optic axis, a quarter of each lens was modeled ( Figure 1). 

Figure 1

 

A meshed model of a quarter of the lens.

The lens was modeled as a solid body with a total of 32,563 three-dimensional continuum elements of the type C3D8R, yielding a total number of 34,800 nodes, whereas the capsule was modeled as a shell using 690 three-dimensional shell elements of the type S4R, yielding 729 nodes. The zonules were modeled as beam elements with one end attached to the capsule and the other bundled to the analytical points from which forces were applied ( Figure 2). 

Figure 2

 

Meshed quarter of a capsule in accommodated state with zonules bundled.

The degrees of freedom were such as to permit translation in three directions for the continuum solid elements and included an additional rotation at each node for the shell elements. The maximum wave-front degree of freedom for an entire model ranged from 3,672 to 8,778. 

The center of the lens was constrained in all directions. 

For a given elastic modulus, the material was assumed to be isotropic and homogenous; damping or viscous effects were not included. The internal profile of the capsule was considered to have the same spatial coordinates as the external profile of the lens. Contact between the lens and capsule was assumed to be frictionless. 

The total force, F _t, was divided equally into smaller forces, F _n, and each was applied from 124 equally spaced loading points located around the lens located at 2.0 mm from the edge of the equator. Each loading point was linked to fives nodes on the capsule, one node on the equator, and two nodes on either side ( Figure 2). Hence, each F _n was, in turn, subdivided into five smaller forces pulling on the capsule. 

Initial models were constructed using the curvatures and dimensions from unstretched lens results (Pierscionek, 1993, Table 1). These models represent lenses in the maximal accommodative state in vitro (Pierscionek, 1993). The values for material properties were taken from Fisher (1969; 1971; shown in Table 2). 

Table 1

 

Dimensions of lenses used for modeling (from Pierscionek, 1993).

	Equatorial radius (mm)	Anterior sagittal thickness (mm)	Posterior sagittal thickness (mm)	Total sagittal thickness (mm)
27-year-old lens	4.26	1.67	2.40	4.07
46-year-old lens	4.41	1.44	2.61	4.05

Table 2

 

Material properties for lens and capsule taken from Fisher ( 1969, 1971).

	Capsule		Cortex		Nucleus
	27 years	46 years	27 years	46 years	27 years	46 years
Single-modulus lens
Young's modulus (N/m 2)	5 × 10 6	3.9 × 10 6	3 × 10 3	4 × 10 3	3 × 10 3	4 × 10 3
Poisson ratio	0.47	0.47	0.47	0.47
Two-modulus lens
Young's modulus (N/m 2)	5 × 10 6	3.9 × 10 6	3 × 10 3	4 × 10 3	6 × 10 2	10 3
Poisson ratio	0.47	0.47	0.47	0.47	0.47	0.47

The extent of lens deformation was tested by applying theoretical forces of different magnitudes to the model. After several runs, three force levels of 0.065, 0.100, and 0.120 N were found to give comparable deformations to those measured in experimental work (Pierscionek, 1993). The respective internal stress distributions for these forces were modeled. The iterative solution technique was linear and based on ABAQUS default FETI (Finite Element Tearing and Interconnecting) with default convergence control. For all cases, convergence of the solution was achieved after five increments with a typical CPU time of 3,144 s. 

To investigate the changes in central curvature of the anterior surface (pertinent to the theories of Helmholtz and Schachar), were fitted second-order polynomial curves to all central anterior profiles (model and experimental) and calculated the gradients. The accuracy of the simulated models was assessed by comparing the volumes calculated for the lens models to the volumes calculated from the respective experimental profiles. Volumes were determined by rotating the profiles (experimental and model) about the y-axis to generate a solid of revolution (representing the anterior or posterior sections of the lens separately) and using the method of cylindrical shells to calculate the volume generated (Thomas, 1973). 

Results

Figure 3 shows the unaccommodated profiles for the 27-year-old lens with a single-modulus value (given in Table 2) under two different loading forces: (a) 0.065 N and (b) 0.120 N. These are compared with the experimental profiles that most closely match the models: 0.6 mm ( Figure 3a) and 1 mm ( Figure 3b). (The radial stretch distances for the experimental profiles describe the distance moved by radially positioned stepper-motor-driven jaws that clasped the ciliary muscle and imparted a stretching force on the lens [Pierscionek, 1993].) 

Figure 3

 

Deformed profiles of a 27-year-old lens for a single elastic modulus for stretching forces of (a) 0.065 N and (b) 0.120 N.

The posterior surface shape is unaltered; as with the experimental results, all deformations are seen at the anterior surface. There is a slight deviation at the extreme end between the model and the experimental profiles for the case with the lower applied force ( Figure 3a). A slight inflection point is seen on the anterior lens surface, about two thirds of the distance from the lens center, in the modeled profiles for both levels of applied force, and the curvature in the periphery flattens with force application ( Figures 3a and 3b). It is hard to discern how stretching changes the curvature at the anterior pole (there is no change at the posterior pole). There appears to be a slight increase in curvature for the smaller stretching force in both experimental and modeled data ( Figure 3a) but a flattening for the greater degree of stretch ( Figure 3b). 

Figure 4 shows the respective stress distribution patterns that correspond to profiles in Figures 3a and 3b. There is an annulus of high stress concentration, around the equatorial plane, located about two thirds of the way from the center of the lens. The color key next to each model shows the range of stress values (as von Mises stress in MPa = N/m ² × 10 ⁶). The annulus of maximum stress is surrounded by progressively lower stress levels, marked in orange and then yellow, which extend into the center of the lens. With the higher level of applied force ( F _t = 0.120 N, Figure 4b), the stress concentration spreads toward the center of the lens and the maximum stress region decreases in size. 

Figure 4

 

Stress distributions (as von Mises stress in MPa = N/m ² × 10 ⁶) for a 27-year-old lens for a single modulus for stretching forces of (a) 0.065 N and (b) 0.120 N.

Similar forces were applied to the model of the 46-year-old lens. The same amount of force produces less deformation in the older lens than in its younger counterpart. In this case, there is a very close match between a modeled force of 0.065 N and an experimental stretch of 1 mm movement of the jaws ( Figure 5a) and a modeled force of 0.120 N and an experimental stretch of 2 mm movement of the jaws ( Figure 5b). As with the 27-year-old lens, there is no change in the posterior lens surface shape for an applied force of 0.065 N ( Figure 5a), which compares to experimental data. There is a slight shift in the modeled posterior profile for a force of 0.100 N ( Figure 5b). A very slight inflection in the surface shapes can be seen on the anterior and posterior surfaces of this model and on the anterior surface of the experimental profile for the greater stretching force ( Figure 5b). There is also a small degree of peripheral flattening as the lens is stretched. Changes in central curvature with stretching are very slight but appear to show a slight flattening in curvature for both applied forces ( Figures 5a and 5b). The other notable feature about the profile in Figure 5b is the small posterior shift in the position of the equator in the stretched model. 

Figure 5

 

Deformed profiles of a 46-year-old lens for a single elastic modulus for stretching forces of (a) 0.065 N and (b) 0.120 N.

In the three-dimensional stress distribution diagrams for the 46-year-old lens (seen in Figures 6a and 6b), the annulus of high stress concentration within the equatorial plane is located about two thirds of the way from the center of the lens, as for the younger lens model ( Figures 4a and 4b). For a force of 0.065 N, this annulus of maximum stress is smaller and displaced more anteriorly than in the younger model. The anterior displacement of the maximum stress region is more marked for the higher applied force of 0.100 N ( Figure 4b). Here, the maximum force is very close to the surface of the lens. The stresses are less evenly distributed and decrease more rapidly with progression into the center of this lens than in its younger counterpart. 

Figure 6

 

Stress distributions (as von Mises stress in MPa = N/m ² × 10 ⁶) for a 46-year-old lens for a single modulus for stretching forces of (a) 0.065 N and (b) 0.120 N.

Figure 7 shows the anterior and posterior profiles of the 27-year-old lens modeled with a lower elastic modulus for the nucleus than for the cortex using the values of Fisher (1971; Table 2). With a lower value for the nuclear modulus, modeled forces of 0.065 and 0.100 N correspond most closely to experimental deformations of 1- and 2-mm radial movements, respectively (Figures 7a and 7b). As for the single-modulus case (Figures 3a and 3b), there is no change in the posterior surface shape with applied force, which corresponds to the experimental results. An inflection on the anterior surface of the model profiles is again evident for both levels of force (Figures 7a and 7b), and a hint of this is seen in the experimental profile for the higher force (Figure 7b). This inflection is less well defined than in the single-modulus case (Figures 3a and 3b). Peripheral flattening is seen in model profiles for both levels of stretch. Changes in central curvature are barely discernable, but there appear to be very slight increases in central curvature for both levels of force (Figures 7a and 7b). 

Figure 7

 

Deformed profiles of a 27-year-old lens for a model with different elastic moduli in the cortex and the nucleus for stretching forces of (a) 0.065 N and (b) 0.100 N.

Figures 8a and 8b show the stress distribution patterns corresponding to the profiles shown in Figures 7a and 7b, respectively. With a two-modulus lens, there is a band of higher stresses distributed across the cortical region from the anterior to posterior surface between the equatorial edge of the nucleus and the equator ( Figures 8a and 8b). The highest stresses that are reached are substantially lower than those seen in the single-modulus case. A similar pattern of stress distribution is seen for forces of 0.065 and 0.100 N. 

Figure 8

 

Stress distributions (as von Mises stress in MPa = N/m ² × 10 ⁶) for a 27-year-old lens for different elastic moduli in the cortex and the nucleus for stretching forces of (a) 0.065 N and (b) 0.100 N.

The 46-year-old lens was also modeled with separate moduli for the cortex and the nucleus using the values shown in Table 2. The profiles are shown in Figure 9. Again, this model was more deformable than the model for a single-material lens ( Figure 5) but with little difference between degrees of stretch. Both 0.065- and 0.100-N forces produced a degree of stretch most comparable to the experimental result for the 2-mm radial movement with the better match between experimental and modeled data seen for the applied force of 0.100 N ( Figure 9). The application of force shifts the modeled posterior surface slightly in an anterior direction compared to the experimental profile, and there is a slight posterior shift in the position of the equator. As in the single-modulus case, stretching causes a flattening of the lens in the periphery but to a lesser degree than for the single-modulus models. A change in central curvature is difficult to distinguish, but there appears to be a flattening in both sets of data (model and experimental). There is a hint of an inflection on the posterior surface ( Figure 9). 

Figure 9

 

Deformed profiles of a 46-year-old lens for a model with different elastic moduli in the cortex and the nucleus for a stretching force of 0.100 N.

As in its younger counterpart, in the 46-year-old two-modulus lens model, the stresses are distributed over a larger area than for the single-modulus case, and the range of stresses is smaller ( Figure 10). There is no region of very high stress but a more even distribution of higher stresses in the cortical region stretching from anterior to posterior surface between the equator and the nucleus. 

Figure 10

 

Stress distributions (as von Mises stress in MPa = N/m ² × 10 ⁶) for a 46 year old lens for different elastic moduli in cortex and nucleus for a stretching force of 0.100N.

The changes in central curvature were determined by fitting second-order polynomial curves to all anterior profiles (from Figures 3, 5, 7, and 9), from the center to the point of inflection to approximately 3 mm from the center, and by calculating gradients at incremental points. In all cases, curves fitted the data with an R ² ≥ .995. The coefficients of x ² and x are shown in Table 3 together with the gradients at 0.2, 0.4, 0.6, and 1.0 mm radial distance from the center. 

Table 3

 

Coefficients of second-order polynomials fitted to central anterior profiles (up to 3 mm from the center) and gradients at 0.2, 0.4, 0.6, and 1.0 mm radial distance from the center.

	Coefficient of x 2	Coefficient of x	Gradient at x = 0.2 mm	Gradient at x = 0.4 mm	Gradient at x = 0.6 mm	Gradient at x = 1.0 mm
27-year-old single-modulus lens
Unstretched	−0.0748	−0.0134	−0.0433	−0.0732	−0.1032	−0.1630
Model 0.065 N	−0.0627	−0.0230	−0.0481	−0.0732	−0.0982	−0.1484
Experimental 0.6 mm	−0.0873	−0.0017	−0.0366	−0.0715	−0.1065	−0.1763
Model 0.120 N	−0.0533	−0.0198	−0.0411	−0.0624	−0.0838	−0.1264
Experimental 1.0 mm	−0.0562	0.0043	−0.0182	−0.0407	−0.0631	−0.1081
27-year-old two-modulus lens
Unstretched	−0.0748	−0.0134	−0.0433	−0.0732	−0.1032	−0.1630
Model 0.065 N	−0.0589	−0.0247	−0.0483	−0.0718	−0.0954	−0.1425
Experimental 1.0 mm	−0.0562	0.0043	−0.0182	−0.0407	−0.0631	−0.1081
Model 0.100 N	−0.0426	−0.039	−0.0560	−0.0731	−0.0901	−0.1242
Experimental 2.0 mm	−0.0349	0.0011	−0.0129	−0.0268	−0.0408	−0.0687
46-year-old single-modulus lens
Unstretched	−0.0317	−0.0592	−0.0719	−0.0846	−0.0909	−0.1226
Model 0.065 N	−0.0362	−0.0343	−0.0488	−0.0633	−0.0777	−0.1067
Experimental 1.0 mm	−0.0414	0.0017	−0.0149	−0.0314	−0.0480	−0.0811
Model 0.120 N	−0.0336	−0.0266	−0.0400	−0.0535	−0.0669	−0.0938
Experimental 2.0 mm	−0.0307	−0.0457	−0.0580	−0.0703	−0.0825	−0.1071
46-year-old two-modulus lens
Unstretched	−0.0317	−0.0592	−0.0719	−0.0846	−0.0909	−0.1226
Model 0.100 N	−0.0248	−0.0353	−0.0452	−0.0551	−0.0651	−0.0849
Experimental 2.0 mm	−0.0307	−0.0457	−0.0580	−0.0703	−0.0825	−0.1071

For the 27-year-old lens, the experimental stretch of 0.6 mm and the matched model for the 0.065-N force (single modulus) indicate that both experimental and model data have slightly steeper gradients at certain points than does the unstretched profile. For the two-modulus case, the 27-year-old models for both 0.065- and 0.100-N forces have slightly steeper gradients at 0.2 mm radial distance from the center than does the unstretched case. For the older lens, the unstretched profiles have steeper gradients than the stretched experimental profile and the single- and two-modulus model profiles, at all points tested. 

The volumes and deviations between volumes of model simulations from experimental data are shown in Table 4 for the cases where any deviations between the model and experimental profiles were found ( Figures 3, 5, 7, and 9). This occurred for all anterior profiles ( Figures 3, 5, 7, and 9) and one posterior profile ( Figure 9). Volumes were calculated for the radial distance available from the range of experimental data (up to 3.5 mm for the 27-year-old lens' anterior profile, up to 4.0 mm for the 46-year-old lens' anterior profile, and up to 2.75 mm for the 46-year-old lens' posterior profile). Table 4 shows that models closely matched experimental data; the maximum volumetric difference of model simulations from respective experimental data was 6.6% of the experimental volume. 

Table 4

 

Volumes of anterior and posterior sections of simulated model lenses and respective experimental data showing percentage deviation of model from experiment.

	Experiment (mm 3)	Model (mm 3)	Deviation (%)
Anterior section
27-year-old single-modulus lens
Experimental 0.6 mm, model 0.065 N	37.4	39.4	5.3
Experimental 1.0 mm, model 0.120 N	34.63	34.625	0.01
27-year-old two-modulus lens
Experimental 1.0 mm, model 0.065 N	34.63	36.93	6.6
Experimental 2.0 mm, model 0.100 N	34.66	32.82	5.3
46-year-old single-modulus lens
Experimental 1.0 mm, model 0.065 N	46.79	44.8	4.25
Experimental 2.0 mm, model 0.120 N	39.71	40.9	3.06
46-year-old two-modulus lens
Experimental 2.0 mm, model 0.100 N	39.71	38.81	2.3
Posterior section
46-year-old two-modulus lens
Experimental 2.0 mm, model 0.100 N	50.07	49.63	0.88

Discussion

In any experimental investigation of lens change with accommodation, it is difficult to observe the entire accommodative system. Biomicroscopic studies on living eyes provide the best images of curvature change, but the presence of the iris blocks the view of the equator where the force is applied (Brown, 1973, 1974; Koretz, Cook, & Kaufman, 1997; Koretz, Handelman, & Brown, 1984). In vitro studies that have tried to simulate the process of accommodation provide similarly limited information about the ciliary muscle action (Glasser & Campbell, 1998; Pierscionek, 1993, 1995b). Magnetic resonance imaging (MRI) permits a view of the whole lens and the ciliary muscle without optical distortions (Koretz, Strenk, Strenk, & Semmlow, 2004; Strenk et al., 1999). It is more difficult, however, to control for vergence movements when the eye accommodates and MRI has a lower resolution and magnification compared with biomicroscopic imaging. In addition, the properties and processes of a living system are subject to change, both short term, with the functional dynamics of the system, and long term, as the system ages. This, coupled with the fact that there will always be individual variations in dimensions and material properties of tissues and systems of the body, suggests that there may be another, less well recognized reason why certain explanations for accommodation endure despite evidence that supports other, seemingly opponent, theories. It is possible that both the Helmholtzian explanation and that of Schachar are correct depending on the biometry of the system, the material properties, and the direction and strength of forces (Pierscionek, 2005). All of these, in turn, may vary with individuals and with age. The complexities and inherent inaccuracies in experimental studies that have sought to measure rheological properties of ocular components suggest that no findings can, as yet, be treated as definitive. 

Testing variations in material properties, shape parameters, and forces can be conducted using models. To test the feasibility of a model, it should be compared to experimental findings and it should produce a result that is physiologically viable. This study has created models based on experimental data from two lenses of different ages and looked at the changes in internal stress distributions for different levels of stretching force and for variations in material properties. Finite element modeling of the lens has been conducted in previous studies (Burd, Judge, & Cross, 2002; Hermans, Dubbelman, van der Heijde, & Heethaar, 2006; Martin, Guthoff, Terwee, & Schmitz, 2005; Schachar & Bax, 2001). The models produced varying results in central curvature change. However, in two of the studies (Burd et al., 2002; Martin et al., 2005), there were discontinuities in curvatures at the points of force application. Such discontinuities do not occur in vivo or in experimentation with in vitro lenses. Previous studies, using two-dimensional finite element models, applied forces at no more than three points from a given point of application (Burd et al., 2002; Martin et al., 2005; Schachar & Bax, 2001). Hermans et al. (2006) constructed an elegant three-dimensional lens model, based on a corrected Scheimpflug image of a 29-year-old lens, with the objective of ascertaining whether there is any difference in shape change depending on the position of the zonule. They modeled three separate force configurations and found that, whether zonular force is centralized at the equator or applied additionally from the anterior and posterior lens surfaces, the requisite change in accommodation could be produced (Hermans et al., 2006). 

The models used in this study are also three-dimensional, and the applied forces are distributed over five sections. There are no discontinuities in surface shape, and models are closely matched to experimental data in which the force is applied in the same direction as in the model, that is, radially and within the equatorial plane (Pierscionek, 1993). 

The values for Young's modulus of elasticity were taken from the work of Fisher (1971) who reported that the elastic modulus of the cortex was higher than that of the nucleus. The implication of this finding is that the nucleus is the more pliable of the two lens sections. This would appear to be at odds with the measurements of refractive index (Pierscionek, 1995a, 1997; Pierscionek & Chan, 1989), protein (Fagerholm, Philipson, & Lindström, 1981), and water gradients (Siebinga, Vrensen, de Mul, & Greve, 1991) that show a higher protein density in the nuclear region. Burd, Wilde, and Judge (2006) applied a modeling approach to test the results of Fisher and questioned the assumptions made by Fisher with regard to the corticonuclear boundary. The results of Burd et al. are not conclusive, suggesting that there may not be a significant difference in stiffness between the cortex and the nucleus but that inhomogeneity, in elastic properties, cannot be excluded. 

A lower elastic modulus in the nucleus needs to have a structural explanation. The nucleus has a greater concentration of proteins than does the cortex, but this does not necessarily give any indication of the relative elasticity of the two sections. The pertinent difference is that the ratio of free to protein-bound water is higher in the nucleus than in the cortex (Lahm, Lee, & Bettelheim, 1985). Studies on skin elasticity have shown that age-related sagging and a decrease in Young's modulus are correlated with a decrease in protein-bound water (Gniadecka et al., 1998). It is not implausible to suggest that the age-related structural changes in the lenticular protein–water relationship may have a similar functional effect to that of the structural changes in skin. This would support the findings of Fisher (1971) and may provide a structural explanation for a lower elastic modulus in the nucleus compared to that of the cortex. 

This study firstly considered models with a single modulus of elasticity to see what effect varying equatorial stretching force would have on stress distribution within the lens and how this could affect the surface profile. The most interesting finding is that with a single elasticity modulus, the model has an annulus of maximum stress that runs approximately around the corticonuclear border. This concentration of maximum stress is redistributed, when the elasticity modulus of the nucleus is less than that of the cortex, as a band of higher stress running from the anterior to the posterior surface, with its center located, as for the annulus, near the corticonuclear border. This shows that for a structure shaped like the eye lens and subjected to equatorial stretching force, the impact of the stretching force is greatest in a region approximately two thirds of the radial distance from the center. The radial position of these high stresses corresponds to the location of the surface inflection seen in most of the models. In the older lens, the inflection is less pronounced and the region of maximum stress is more anteriorly positioned than in the younger lens. 

The concentration of stresses around the corticonuclear boundary, in the two-modulus model, suggests that there may be some resistance to the stretching force in this region. This would result in a greater effect of the force on the outer part of the lens, flattening the peripheral curvature, and a concomitant push of material into the inner lens, causing an increase in central curvature. Such a shape change (a peripheral flattening and an increase in central curvature with an inflection separating the two “regions”) has been seen in microtubules (with shapes akin to that of the eye lens) when stretching forces were applied around the equator (Fygenson, Marko, & Libchaber, 1997), and it concurs with the theory of Schachar. 

Curves fitted to the central anterior surface showed that, for the 27-year-old lens with a single modulus, the 0.6-mm experimental stretch and the matched model for the 0.065-N force have gradients at some points that are slightly steeper than those for the unstretched lens ( Table 3). This would support Schachar's theory. For the higher level of force, in the model and experimental data, gradients are flatter at all points than in the unstretched case, concurring with the theory of Helmholtz. For the two-modulus 27-year-old model, both levels of force showed that, at 0.2 mm radial distance from the center, there was a slightly steeper gradient than that for the unstretched profile ( Table 3). For the 46-year-old lens, the unstretched profile had steeper gradients than the model and experimental profiles at all points tested ( Table 3), in accordance with Helmholtz' theory. In all cases, there is a decrease in peripheral curvature with applied stretching force. 

In essence, for a younger lens model (less curved surface and lower elastic moduli), the results support the theory of Schachar if the model has separate moduli values for the cortex and the nucleus. This applies for both levels of stretching force tested. For a single-modulus model of the younger lens, Schachar's theory is supported for a lower level of stretching force; for the greater force, the findings concur with the theory of Helmholtz. For the older lens model, irrespective of force applied or whether there is a single modulus or dual moduli, the results support Helmholtz. 

The findings suggest that the manner in which the lens shape changes when it is stretched by a force applied in the equatorial region depends on the initial surface curvatures (prior to force application), the material properties, and the applied force. When the lens is at the limit of its accommodative capacity and approaches the age of onset of presbyopia, as for the 46-year-old lens model, the small degree of shape change that can be induced with stretching follows the predictions of Helmholtz. In a younger, flatter, and more pliable lens, depending on force and material properties, the theory of Schachar may apply. With regard to how initial shape may affect shape change with stretching, Schachar and Fygenson (2006) found that biconvex structures that are spherical, oval, or spherocylindrical, when subjected to equatorial stretching, will produce shapes predicted by Helmholtz. However, “long oval” objects, when stretched equatorially, show support for Schachar. 

It should be noted that models created in this study, similarly to those created in previous work (Burd et al., 2002; Hermans et al., 2006; Martin et al., 2005; Schachar & Bax, 2001), are based on lenses stretched in the equatorial direction or only slightly angled away from the equator. In the living eye, the stretching forces operate in more than one direction as the zonule has a number of directionally varying components that differ widely in direction of stretch (Bron, Tripathi, & Tripathi, 1997). If the controversy over the theories of Helmholtz and Schachar is to be resolved, future work needs to determine which of the zonular forces is strongest in vivo, whether this varies with age, and how this may impact upon the change of lens shape in lenses of different ages. 

Conclusions

The material properties of the lens affect the stress distributions within the tissue. Stretching a biconvex object such as the lens along its equatorial plane concentrates internal stresses in a region located about two thirds of the distance from the center. This is likely to cause an inflection on the lens surface and flattening of the peripheral curvature. The effect on central curvature may depend on the age of the lens and on the degree of applied force. 

Acknowledgments

The authors gratefully acknowledge Essilor International for supporting this study. 

Commercial relationships: none. 

Corresponding author: B. K. Pierscionek. 

Email: [e-mail]. 

Address: Department of Biomedical Sciences, University of Ulster, Cromore Road, Coleraine BT52 1SA, UK.