The Mechanism of Human Accommodation As Analyzed by Nonlinear Finite Element Analysis

Ronald A. Schachar, M.D., Ph.D.
Presby Corporation, Dallas, TX

Andrew J. Bax , M.S., P.E.
DRD Technology, Tulsa, OK

Abstract
    The results of non-linear finite element analysis support Schachar’s theory of accommodation and demonstrate that the long-held Helmholtz theory of accommodation is untenable.

Introduction
    Accommodation, the change in focus by the human eye, occurs because its deformable crystalline lens changes optical power.¹  By observing the direction of movement of multiple reflections from the surface of the crystalline lens during human accommodation, it has been demonstrated that the central surface steepens while the peripheral surface of the crystalline lens flattens.^2,3  Moreover, with increasing optical power during accommodation, the spherical aberration of the human crystalline lens decreases.^1,4,5  

    Several theories have been proposed to explain the mechanism of accommodation of the human lens. The more widely accepted Helmholtz theory of accommodation⁶ assumes that the zonules supporting the crystalline lens are under maximal tension when the lens is at minimum optical power. The Helmholtz theory proposes that tension is exerted by the anterior and posterior zonules together or by all three sets of zonules simultaneously. This theory states that the optical power of the crystalline lens is increased by relaxation of the tension on these zonules, while an increase in zonular tension causes a decrease optical power. This theory does not explain the peripheral surface flattening and reduction in spherical aberration that have been reported to occur during accommodation. No working model lens, based upon the Helmholtz’s theory, has been developed that can mimic the properties of the human crystalline lens during accommodation.

    The Schachar theory of accommodation assumes that the equatorial zonules are under minimum tension when the lens is at minimum optical power.^7-11 It states that the equatorial zonules apply increasing tension to the human crystalline lens during accommodation. This increased equatorial zonular tension increases the equatorial diameter of the lens, alters the surface curvatures of the human crystalline lens and, thereby, increases the central optical power of the lens. The Schachar theory proposes that during the accommodative process increasing tension is exerted exclusively by the equatorial zonules.  In Schachar’s theory, the equatorial zonules act similar to skeletal muscle tendons and are the components that transduce the force of the ciliary muscle to change the shape and, thereby, the focal power of the crystalline lens. The anterior and posterior zonules act like the supportive ligaments of skeletal joints and are stabilizing components, which are tense during distance vision and relax during accommodation. A working model of a deformable lens, based on Schachar’s theory of accommodation, has been produced that mimics the properties of the human crystalline lens during accommodation. This working model can alter its optical power by more than 10 diopters and demonstrates a simultaneous reduction in its spherical aberration.^12,13

Methods
    We mathematically modeled the human crystalline lens when in it was under minimal tension for three scenarios according to each of the following theories:

1. The Schachar theory proposes that during accommodation, tension is exerted exclusively by the equatorial zonules and that in the unaccommodated state the tension exerted by these zonules is minimal.
2. The Helmholtz theory proposes that during maximum accommodation the anterior and posterior zonules together are totally relaxed.
3. All three sets of zonules simultaneously are totally relaxed.

    Using nonlinear finite element analysis that incorporated material properties and boundary conditions that closely emulate the human crystalline lens, we calculated the change in the surface curvatures, central thickness, and the amount of force required to produce a given central optical power change in the crystalline lens under the conditions imposed by each of the theories of accommodation. The spherical aberration of the crystalline lens was calculated for each induced change in surface curvature.

Material Properties
    The material properties used in our non-linear finite element model were as follows: Poisson’s ratio for the crystalline lens capsule¹⁴ = 0.47; the elastic modulus for the crystalline lens capsule^15,16 = 1 N/mm²; the crystalline lens capsule was modeled with two layers of elements for a total of 876 elements (the average aspect ratio was approximately 1.5:1). It is a generally accepted that two elements through the thickness of a model are sufficient to capture the out-of-plane bending behavior and that only one element through the thickness is required for in-plane loading. The finite element mesh is shown in Figure 1.

 
Figure 1. The finite element model is comprised of entirely quadrilateral elements with significant mesh refinement near the capsular bag to ensure accuracy (capsular bag outlined in blue).

    The known variation in crystalline lens capsule thickness was included in the analysis.³ Although the crystalline lens is made up of numerous epithelial cells, the epithelial membranes are thin and flexible. From a mechanical point of view, the membranes do not offer any significant mechanical resistance¹⁷, as can easily be verified by breaking the capsule of a young human crystalline lens and rubbing the crystalline lens between the fingers. It has the consistency of a thick gel. Because the crystalline lens consists of approximately 35% protein and 65% water¹⁸, the lens is incompressible and behaves similar to water and other soft biological tissues and has a Poisson’s ratio^19,20 equal to 0.5.

    Finally, the incompressibility of the interior of the crystalline lens was modeled by using two separate methods. A numerical scheme to adjust the pressure within the capsule to insure that the enclosed volume was conserved.^19,21,22 The incompressibility of the crystalline lens was also modeled by using hyper-elastic elements and assigning Mooney-Rivlin constants, A = B = 10^-7 N/mm², with a Poisson’s ratio = 0.4999999.

Boundary Conditions
    The boundary conditions used in the model were as follows: The thickness and radii of curvatures for the crystalline lenses of a 19 and 29 year old, as measured by Brown²³, for the un-accommodated state, were used for the analysis. Axisymmetry perpendicular to the optic axis of the crystalline lens was assumed. The anterior and posterior surfaces were fit with four separate polynomials. Proceeding on the surfaces from the optic axis for 3mm, 14th order polynomials were fit to the anterior and posterior surfaces, respectively. The remainders of the anterior and posterior surfaces were fit with different 14th order polynomials. The fits had R² = 0.9999998.

    To simulate when tension is applied to the crystalline lens equator by the equatorial zonules, a gradually increasing displacement was applied to one point on the equator at the equatorial plane. To simulate tension applied to the crystalline lens equator by the anterior and posterior zonules, or simultaneously by all three sets of zonules, a displacement was applied along the equatorial plane where these zonules meet.

    The zonules were attached to the crystalline lens at a point on the anterior capsule 0.37 mm from the equator to emulate the anterior zonules.²⁴; at a point on the equator at the equatorial plane, in order to emulate the equatorial zonules; and at a point on the posterior capsule 0.25 mm from the equator to emulate the posterior zonules.²⁵ The points of attachments of the anterior and posterior zonules from the equator were varied from 0.37 mm to 1.42 mm, and from 0.25 mm to 1.0 mm, respectively.

    The angle made between the anterior and posterior zonules at the equatorial plane was 35°. The angle made by the anterior and posterior zonules for the same points of attachment to the crystalline lens was varied from 5° to 90°.

    To emulate Schachar’s theory of accommodation, the initial central optical power of the crystalline lens was 18 diopters and 15 diopters (the un-accommodated state) for the 19-year-old and 29-year-old crystalline lenses, respectively. The central thickness, equatorial radius, and the anterior and posterior central radii of curvatures, were 3.67 mm, 4.3 mm, 12.8 mm, 7.1 mm for the 19-year-old crystalline lens, and 3.71 mm, 4.3 mm, 13.4 mm, 9.4 mm for the 29-year-old crystalline lens, respectively.  Tension was applied by just the equatorial zonules, since according to Schachar’s theory the equatorial zonules are the active components and the anterior and posterior zonules have been observed clinically to actually curl.²⁶

    To emulate Helmholtz’s theory of accommodation, in which the lens power is maximum when the zonules are relaxed, the initial central optical power was set at 30.5 diopters for the 19-year-old crystalline lens. We used 33 diopters of central optical power for the 29-year-old crystalline lens in order to make direct comparisons with the study of Burd, and coworkers.¹⁹ The central thickness, equatorial radius, and the anterior and posterior central radii of curvatures, were 4.13 mm, 4.2 mm, 7.75 mm , 4.14 mm for the 19-year-old crystalline lens, and 4.13 mm, 4.3 mm, 6.5 mm, 4.0 mm for the 29-year-old crystalline lens, respectively.

Procedures
    The analysis was performed using ANSYS 5.6,²⁷ a general-purpose non-linear finite element computer program that is widely used in mechanical and civil engineering. The amount of force necessary to produce a given amount of equatorial displacement was calculated. The optical power of the crystalline lens was determined by using the thick lens formula.28 An optical computer program, Zemax EE,29 was used to evaluate the longitudinal spherical aberration of various levels of crystalline lens accommodation.

Results
    When a displacement is imposed exclusively by the equatorial zonules on the crystalline lens, the shape changes (Fig 2) as the central anterior and posterior curvatures steepen, the center initially thickens and then, with equatorial displacements greater than 100 microns, it thins; and the peripheral surfaces flatten.

Figure 2. The change in the shape of the surfaces of the 29-year-old un-accommodated (----) human crystalline lens as a result of a 35 micron outward displacement (^___) of the equatorial zonules.  Similar shape changes are obtained with the 19-year-old crystalline lens.

    For both the 19- and 29-year-old crystalline lenses, the increase in central optical power and central thickness as a function of increase in outward equatorial displacement induced exclusively by the equatorial zonules is given in Figures 3 and 4, respectively. The force required for a given amount of equatorial displacement and increase in central optical power as a result of tension applied exclusively by the equatorial zonules is within the physiological force range of the ciliary muscle (Fig 5). Equatorial displacement produced by the equatorial zonules decreases longitudinal spherical aberration (Figs 6, 7).

	Figure 5. The force required by the ciliary muscle to produce an equatorial displacement when the force is exclusively transduced by the equatorial zonules.

    When tension is initially applied to the equator of the 19- and 29-year-old crystalline lenses by the anterior and posterior zonules or simultaneously by all three sets of zonules, there is a small increase in central optical power, with a decrease in central thickness (Figs 8-11). This increase in central optical power is independent of the radial distance of attachment of the anterior and posterior zonules from the crystalline equator within the range examined, of the angle that the anterior and posterior zonules make with the equatorial plane between 5 and 90°, and of the method for modeling the incompressibility of the crystalline lens.
 

Figure 8. The change in the shape of the surfaces of the accommodated 29-year-old crystalline lens (----) as a result of a 35 micron outward displacement (___) of the anterior and posterior zonules.  Although the central thickness decreases during the first 35 microns of outward displacement, the central curvatures of the anterior and posterior surfaces of the crystalline lens steepen (see Fig 9).   Similar changes in shape are obtained with the 19-year-old crystalline lens.

Figure 9. The change in the shape of the surfaces of the accommodated human crystalline lens (----) as a result of a 35 micron simultaneous outward displacement (___) of all three sets of zonules.  Although the central thickness decreases, during the first 35 microns of outward displacement, the central curvatures of the anterior and posterior surfaces of the crystalline lens steepen. Similar changes in shape are obtained with the 19-year-old crystalline lens.

	Figure 10. The change in central optical power as a result of an outward displacement of the anterior and posterior zonules. Note, that for the first 65 microns of outward displacement of the anterior and posterior zonules, for both the 29 year old and 19 year old, there is an increase in central optical power of the crystalline lens. A similar increase in central optical power occurs with an initial simultaneous outward displacement of all three sets of zonules.
Figure 11. The central thickness of the crystalline lens decreases as a result of outward displacement of the anterior and posterior zonules. Conversely, relaxation these zonules results in an increase in central thickness. Similar changes in central thickness occur with a simultaneous outward displacement or relaxation of all three sets of zonules.

    As more tension is exerted by the anterior and posterior zonules or simultaneously by all three sets of zonules the central optical power does decrease; however, to achieve changes in central optical power consistent with a normal amplitude of accommodation,³⁰ the force required exceeds the range of force that the ciliary muscle can apply^31,32 (Fig 12). The spherical aberration increases with total relaxation of the anterior and posterior zonules or all three sets of zonules (Figs 13,14).

	Figure 12. The force required by the ciliary muscle to produce an outward displacement of the anterior and posterior zonules. Note that the force to produce a normal amplitude of accommodation is greater than the physiological force range of the ciliary muscle. A simultaneous outward displacement of all three sets of zonules requires a similar amount of force.
Figure 13. Plots of the longitudinal spherical aberration of the 29 year old as the anterior and posterior zonules relax. Note the minimal change in spherical aberration until at maximum central optical power, when the zonules are totally relaxed, the spherical aberration increases. A similar increase in spherical aberration occurs with simultaneous total relaxation of all three sets of zonules.
	Figure 14. Plots of the longitudinal spherical aberration of the 19 year old as the anterior and posterior zonules relax. Note the minimal change in spherical aberration until at maximum central optical power, when the zonules are totally relaxed, the spherical aberration increases. A similar increase in spherical aberration occurs with simultaneous total relaxation of all three sets of zonules.

Discussion
    A valid mechanism for accommodation must satisfy all of the known properties of the crystalline lens during accommodation. The central surfaces steepen,^6,23 the central thickness increases,^6,23,38 the peripheral surfaces flatten,^2,3 the spherical aberration decreases,^1,4,5 and the anterior and posterior zonules are totally relaxed during maximum accommodation.²⁶ The force required to produce the central optical change must not exceed the range of force that the ciliary muscle can apply.^31,32

    The nonlinear finite element analysis demonstrates that only tension exclusively exerted by the equatorial zonules, with the consequential increase in equatorial diameter, satisfies all of the above known properties of the accommodative process (Table 1). The predicted increase in equatorial diameter is consistent with ultrasound biomicroscopy measurements that demonstrated that the crystalline lens equator moves towards the sclera during pharmacologically controlled accommodation.³³

    The counterintuitive phenomenon of an increase in central optical power with equatorial stretching is consistent with the changes in radius of curvature induced by stretching the equator of an air-filled¹¹ or gel-filled Mylar balloon.³⁴ It is also consistent with the steepening of the central vertical radius of curvature of the cornea which occurs as a result of tight sutures placed at the 12:00 meridian of a cataract wound.³⁵

    The analysis demonstrates that at the point when the anterior and posterior zonules or all three sets of zonules totally relax, the central optical power of the crystalline lens would decrease, not increase. A simple analogous model readily demonstrates the paradoxical phenomenon of central steepening, an increase in central optical power, with equatorial stretching induced by the anterior and posterior zonules. Use an 8-inch air- or gel-filled Mylar balloon. Attach strings or ¾ inch by 3-inch strips of Scotch tape to the equator of the balloon in the approximate relative position that the anterior and posterior zonules are attached to the crystalline lens, and pull on these analogous zonules, with an outward displacement the reflection from the center of the balloon minifies, demonstrating that the center of balloon is steepening (Fig 15).

    These findings are in contradiction to Burd’s non-linear finite element analysis,¹⁸ which did not include the equatorial zonules that are present throughout life.³⁶ We hypothesize that Burd and associates¹⁹ did not truly maintain a constant volume, as required for the incompressibility of the crystalline lens. We show in Figure 16 that only if the volume is allowed to decrease by approximately 4% can an initial tension applied by the anterior and posterior zonules or simultaneously by all three sets of zonules produce a decrease in central optical power. Conversely, only if the volume of the crystalline lens could increase by more than approximately 4% during accommodation, can total relaxation of the anterior and posterior zonules or all three sets of zonules account for the increase in central optical power that occurs during human accommodation. As no evidence, or even a theoretical basis, for a change in the volume of the crystalline lens of this magnitude during accommodation, relaxation of the anterior and posterior zonules or all three sets of zonules cannot account for the mechanism of accommodation.

Figure 16. The change in central optical power of the 29-year-old crystalline lens that would occur, if the volume decreased when the anterior and posterior zonules or all three sets of zonules are outwardly displaced 35 microns. Only if the crystalline lens volume decreased approximately 4% is it possible for an initial tension, applied by the anterior and posterior zonules or all three sets of zonules, to produce a decrease in central optical power.

Conclusions
    Mathematical analysis using either our present nonlinear or linear finite element models¹⁹ or Rayleigh’s method³⁷ or an analytical solution³⁹ demonstrates, that when tension is applied to the equator of the human crystalline lens by the equatorial zonules, its equatorial diameter increases and there is a large increase in central optical power with peripheral flattening. The force necessary for the equatorial zonules to develop sufficient tension to achieve a given central optical power change in the crystalline lens is consistent with the physiological force range of the ciliary muscle. We have demonstrated, by using a standard optical computer program, that tension applied by the equatorial zonules results in a decrease in spherical aberration.

    Conversely, our nonlinear finite analysis, which incorporates the variation in crystalline lens capsule thickness, demonstrates that tension initially applied by either the anterior and posterior zonules or simultaneously by all three sets of zonules results in an increase in central optical power. These findings are in contradiction to the predictions of Helmholtz’s theory, which states when initial tension is applied to the equator of the crystalline lens, its optical power should decrease. Because only tension applied exclusively by the equatorial zonules can satisfy all the constraint requirements of accommodation, the equatorial zonules must be the active components in human accommodation.

    In summary, nonlinear finite element analysis using proper boundary conditions predicts reality. Using this technique, we have shown that to account for the observed large increase in central optical power, for the decrease in spherical aberration that occurs during accommodation, and the force limitations of the ciliary muscle, both tension applied by the equatorial zonules and the equatorial diameter of the crystalline lens must both increase during accommodation.

    These findings are consistent with Schachar’s theory of accommodation that holds that an increase in equatorial zonular tension, simultaneous relaxation of the anterior and posterior zonules, an increase in the crystalline lens equatorial diameter, and a decrease in spherical aberration during human accommodation. The analysis demonstrates that the Helmholtz theory of accommodation is untenable (Table 2).

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