Modeling age-related accomodative loss in the human eye

'qat#ematital .",4bdetlin,4. Vol. 7. pp. 1003-1014. 1986 Printed in the U . S . A . All rights reser'.ed.

02"0-0255~86 $3,00 - .00 Copyright ¢ 1986 Pergamon Journals Ltd.

MODELING AGE-RELATED ACCOMMODATIVE LOSS IN THE HUMAN EYE JANE F. KORETZ* Department of Biology Rensselaer Polytechnic Institute Troy, New York 12180-3590, USA

GEORGE H. HANDELMAN Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy. New York 12180-3590, USA

(Received 28 Januao' 1985)

Abstract--A simplified mathematical representation of accommodation--the process by which the eye tbcuses on near objects--is described. This model is based upon assumptions which limit its applicability to the young human eye. but it nevertheless leads to new insights about the physiological mechanism. A hypothesis about the underlying cause(s) of presbyopia--the loss of accommodative amplitude with age--is also presented here, which combines aspects of the expanded accommodative mechanism with the aging characteristics of the human lens. Using a similar mathematical strategy which incorporates new experimental data about lens shape and aging, a more general model of the accommodative mechanism can be developed and used to test the presbyopia hypothesis. INTRODUCTION The h u m a n eye functions as a binocular visual system, consisting of a large fixed refraction contributed by the cornea and a small variable refraction contributed by the crystalline lens. This variable lens contribution arises from controlled changes in lens thickness and radius of curvature along the lens's polar axis (Fig. 1). The posterior surface of the lens rests on the vitreous humor, and both anterior and posterior lens surfaces are indirectly c o n n e c t e d to the collar-shaped ciliary muscle through a filamentous structure called the zonular apparatus[I]. A c c o m m o d a t i o n - - t h e process by which lens shape is altered to allow focusing on near o b j e c t s - - o c c u r s in an unintuitive fashion. When the eye is focused on infinity (for the human eye, about 20 ft), the ciliary muscle is completely relaxed, but the lens is under m a x i m u m stress; it is at its flattest along the polar axis and at its m a x i m u m radial size perpendicular to this axis. During a c c o m m o d a t i o n , the ciliary muscle, which surrounds the lens radially like a collar, contracts in diameter, and its mass distribution m o v e s forward (anteriorly). C o n c o m i t a n t l y , the lens a p p e a r s to go through a controlled elastic r e c o v e r y , becoming steadily thicker along the polar axis and smaller in diameter perpendicular to this axis; its center of mass also m o v e s anteriorly, such that its anterior surface m o v e s closer to the cornea, while its posterior surface remains at about the same distance from the cornea (e.g. [21). * Author to whom correspondence should be addressed. 1003

1004

JANE F. KORETZ AND GEORGE H. HANDELM~N

Zonuiar (Suspensory) Apparatus

Optic N e r v e { Z

Retina Choroid Sclera

J

Ciliary Muscle

Fig. 1. A s c h e m a t i c diagram of the h u m a n eye. shossing the crystalline lens in relation to o t h e r important visual c o m p o n e n t s . W h e n an a c c o m m o d a t i o n occurs, the lens b e c o m e s more sharply curved, and the distance from the cornea to the anterior (front) lens surface decreases. The distance from the cornea to the posterior (back) surface, h o w e v e r , does not change. T h e zonular apparatus, threads of u n k n o w n elastic properties, couples ciliary muscle contraction to lens elastic recovery, but the geometry of the c o n n e c t i o n s has not been unequivocally determined. Note that the polar axis. the axis of s y m m e t r y through the cornea and tens and the visual axis. which c e n t e r s the image on the fovea, are not coincident.

The lens was originally believed by Helmholtz to be a deformable bag of fluid (see also [3]), but modern structural studies show that it is instead highly organized internally[4]. It consists of a set of enucleated, crescent-shaped cells, each of which extends in length from the anterior to the posterior of the lens. Each of these cells is connected all along its length to its six nearest neighbors by ball-and-socket or tongue-in-groove joints, as well as by gap junctions: such organization implies that the cells cannot slide past each other during overall lens shape changes. During an individual*s lifetime, these fiber cells are continually being laid down. so that the lens steadily increases in overall size; a plot of average lens thickness versus age is roughly linear, implying that the number and/or size of the cells laid down also increases with age[5]. The ordered aggregate of lens fibers is completely enclosed by the lens capsule, a basement membrane which contains collagen oriented in the tangent plane of the membrane[6, 7]. The zonular apparatus connects the capsule with the ciliary muscle, while the capsule is itself unconnected directly to the lens fibers. In the original bag-of-fluid view of the lens, the capsule was believed to mold the shape of the material it contained, and modern versions of this model (e.g. [8]) also impute ultimate control of lens shape to the capsule. In general, the details of lens ultrastructure and anatomical connections, as well as the events occurring during accommodation, are agreed upon by vision scientists interested in the process. The question of the mechanism by which accommodation is attained, however, is currently not answered unequivocally. The geometry of the zonular connection between the ciliary muscle and the lens capsule is unresolved[9, 10], and little is known about the elastic properties of the zonular fibers themselves; thus it is well-nigh impossible to estimate the pattern of forces acting upon the lens capsule or the lens itself. The relative contribution of the ciliary muscle to lens deformation is itself open to question, with some believing it wholly or almost wholly responsible (e.g. [I 1, 12]), while others suggest that the vitreous itself molds lens shape through the compressive action of musclecoupled structures on it (e.g. [13]). And, as mentioned earlier, there is disagreement as to the importance of the capsule in determination of lens shape at various accommodative states. The mechanism by which accommodation occurs must be known to address in turn an

Age-related accommodative loss

1005

aging problem which affects us all sooner or later--presbyopia. Presbyopia involves the loss of accommodative amplitude with increasing age. A young child exhibits an accommodative range which can extend from 0 to well over 15 diopters (where 1 diopter is the reciprocal of 1-m focal length), but as it gets older, this range decreases. By the time an individual is in his or her midforties, the accommodative range has been reduced to about 0-4 diopters, and at 60 to about 0 to 1-1.5 diopters. Thus an individual with 20/20 vision will eventually need reading glasses and. ultimatel.,,, bifocals or trifocals. Presbyopia is not a life-threatening condition, but it is an annoying, expensive and universal one. A number of theories concerning presbyopia have been presented. The condition has been attributed variously to dehydration of the lens lot some other factor which would change its elastic properties), degeneration of the ciliary muscle or liquification of the vitreous, which also occurs in the midforties (cf. [5. 141). More recently, it has been suggested[15, 16] that presbyopia may arise from changes in the geometry of the system: that is, the increase in lens thickness (without necessarily a change in lens elasticity) combined with changes in the relative location and orientation of the zonular apparatus due to lens growth[17] could result in loss of effective Ishape-changing) forces acting on the lens. In order to address the related problems of th'e accommodative mechanism in the human eye and age-related accommodative loss, we decided to begin our stud,, from the "'vie~point" of the lens itself. Since the geometric relationship of the zonules and ciliary muscle to the lens remains unresolved, and since the elastic properties of the zonules are also unknown, we could not represent the pattern of forces acting upon the lens and go from there to changes in lens shape. Rather, we considered the lens as a body of certain elastic properties[18] whose shape was known before and after a small deformation. If this deformation could be described, then it was possible to define the elements of a strain tensor produced by these increments of displacement, calculate the increments in the stress tensor over the existing stresses and the overall increments in net forces acting on the lens. In this paper we describe the development of a simplified model of accommodation which is accurate for the young human lens[19-21] and its mechanistic implications. Closer study of lens shape as a function of both accommodative state and age[15] has given us the means to generalize this approach to all human lenses, regardless of age. and to approach the problem of presbyopia directly. It has also provided the first direct evidence of lens involvement in this aging phenomenon.

METHODS Definin~ the system We define the lens as a biconvex elastic body with radii of curvature 9, and pp. essentially two spheric sections of differing radii of curvature sandwiched together and rotationally symmetric around the polar axis (Fig. 2). We further assume that. after a small accommodative change (additional deformation), the lens can again be represented as two spheric sections, but with altered radii of curvature. These assumptions appear to be good approximations to the accommodative behavior of the young (e.g. age 11 ) human eye, and to be more accurate for the anterior lens surface than the posterior. With increasing lens age, however, lens curvature seems to diverge increasingly from this simple shape; it has been approximated variously as an ellipsoid[22, 23] and as a paraboloid[15]. The representation being used, then, is for the young human eye. and values for the radius of curvature as a function of accommodative state were obtained from the studies of Brown[2].

1006

JANE F. KORETZ AND GEORGE H. HANDELM~N

Z

~

-l-i(0,0,0)

T (0,0,0) (O,O,-o)

~

r

Polar Axis

Fig. 2. The lens before and alter an accommodation, represented schematically in a cylindrical coordinate system. Changes in shape have been exaggerated, la) The lens before accommodation.The anterior lens surface is located a distance a from the origin (the intersection of the polar axis with the cornea, which remains fixed in position). and the lens is of thickness h. Anterior and posterior lens curvatures are represented by spheric sections of radius p,, and pp. respectively. (b) The lens after an accommodation. Note that the lens is more sharply curved on both surfaces, that the equatorial radius has decreased in extent and that the lens is thicker along the polar axis. While the distance from the origin to the anterior lens surface has decreased to ~. the distance to the posterior surface has remained unchanged. Our second assumption is c o n c e r n e d with internal lens movements during deformation. As mentioned earlier, the number and distribution of connections between adjacent lens fibers argues against a significant degree of sliding. It is therefore assumed that lens shape changes arise from a redistribution of fiber cytoplasm within each fiber. The net result of this redistribution during an increase in a c c o m m o d a t i o n is a movement of lens mass anteriorly and, ultrastructurally, a change in the spacing of adjacent lens fiber boundaries. Since it has been shown that the lens fiber membranes are sheathed with strongly bound actin filaments (e.g. [24]), which are flexible but not extensible, we further assume that meridional lengths are unchanged with the relative change in spacing of the fiber boundaries. (Concomitant surface area changes appear to be small: we estimate a total surface area change of about 0.3% for a 2-diopter accommodation.) Recent estimation of changes in arc length with a c c o m m o d a t i o n , based upon experimental data[15], appears to validate this assumption. Our final assumptions are concerned with the elastic properties of the lens material itself. Fisher[18] has measured the Y o u n g ' s Modulus of Elasticity for lenses of varying ages in the direction parallel and perpendicular to the polar axis, and his results indicate that the lens is an anisotropic body. Initially, our formulation treated the lens as isotropic, because there was only a 20% difference in the magnitudes of the two moduli, but we later incorporated this difference into the model. The Poisson ratios and bulk moduli have never been measured, though the high water content of the lens suggests a Poisson ratio near 0.5; we therefore tested different values within the range of 0.3-0.5, and were able to narrow down the possibilities using a priori information about a c c o m m o d a t i o n not incorporated into the model.

Derivation o f a simple accommodation model Consider the elastic body described previously located in a spherical or cylindrical coordinate space (Fig. 2). The origin has been placed at the cornea, since it is fixed in place; the location of the anterior lens surface where it crosses the symmetry axis is then (0.0, - a ) and after a small a c c o m m o d a t i o n (0, 0, -?~), where a is larger than ~7in absolute magnitude. The radius of curvature of the anterior lens surface also changes during an

Age-related accommodative loss

1007

a c c o m m o d a t i o n , from # to ~. Since the body is rotationally s y m m e t r i c around the axis, its deformation can be examined in a plane passing through the axis of s y m m e t r y and a meridian; thus a generic point on the surface or internally on a nested curve can be defined by coordinate pairs. Ir. z) and (ri, z~). respectively, before a c c o m m o d a t i o n , and (?, 2) and (?~, 2~), respectively, after a small a c c o m m o d a t i o n (Fig. 3a). The ratio of the radii of curvature before and after a c c o m m o d a t i o n will be represented by k = pO,

where 0 -< k - l ~ 1. We have made the a s s u m p t i o n s that the arc representing the anterior surface is circular both before and after a small a c c o m m o d a t i o n and that the arc length from the axis to a material point on the arc is unchanged during deformation. Thus the arc length l and l are equal, and p0 = ~0

or

0 = kO,

where 0 is the angle subtended at the center of the undeformed arc and 0 the angle subtended at the center of the deformed arc. For a point located s radial units below the surface before an a c c o m m o d a t i o n and ~ after a small a c c o m m o d a t i o n , similar arguments hold. That is to say. if p~ = p - s

and

~

= ~ - 7,

then preservation of arc length requires that

Since 0 = kO, then F~ = (l/k)p~. T h e r e f o r e , the difference of the location of a generic point before and after a small a c c o m m o d a t i o n in spherical coordinates is

-Pl

-

Pt

=

(l/k

l)pl.

-

- 0 = (k -

I)0.

with qJ unchanged. The displacement v e c t o r U in spherical coordinates is (tg,, . + , .~)[25]: each c o m p o n e n t can be denoted in this case as Up

=

(l/k

-

1)p,,

U , = 0. U, = pl(O - 0). The c o m p o n e n t s of the strain tensor in spherical coordinates are then found to be %p = Oup/Opl = I / k -

1,

e0o = (I/pl)0u./00 + tg,/pl = k + l/k - 2, %e, = (Pl sin 0 ) - I 0 u + / a ~ = I/k-

1 + (k-

+

(l/pl)llp + ( l / p l ) u o COt 0

I)0 cot 0,

t

cl

0,O)

?i

7’

L

r

Age-related accommodative loss

1009

and in cylindrical coordinates e~, =

-1

-

l/k s i n 0 sin k0 * cos 0 c o s k 0 ,

~+,~ = - 1

+ l / k sin kO/sin 0.

%: = - I

+ I/kcosOcoskO,

er: = 0.5(I/k -

1)sin[0(k + l)],

F o r the isotropic, h o m o g e n e o u s , elastic case, the relationship between the incremental stress and incremental strain tensor is straightforward[19, 20]. The average Y o u n g ' s Modulus E is a multiplier in all stress tensor elements and can be factored out for results in dimensionless form. These stress tensor elements are in turn used to determine the incremental traction, or incremental force per unit area. on each of the four surfaces of a wedge (Fig. 3b), an a s y m m e t r i c three-dimensional segment of the anterior lens of rotational width ,X~b. Ultimately. the tractions can be used to determine the reduced incremental forces acting on each of the four surfaces by' integrating o v e r each surface, and overall change in applied force by vectorial addition of each surface's contributions. The only unknown remaining is the value for the Poisson ratio: evaluating the model with different values for this p a r a m e t e r using a priori experimental observations of a c c o m m o d a t i o n , the best fit a p p e a r e d to be cr = 0.46. The same d e v e l o p m e n t was followed for the anisotropic case[21], and all equations remained unchanged under these conditions except for the representation of the stress tensor elements. Following the d e v e l o p m e n t of Lekhnitskii[26] for a rotationally symmetric body which is elastically anisotropic parallel and perpendicular to the axis of symmetry, there are five p a r a m e t e r s to be defined: the Y o u n g ' s Moduli and Poisson ratios in the two directions and a shear modulus between the two. Inverting his expressions for the strain tensor elements and gathering terms, the stress tensor elements in cylindrical coordinates are ,,, T+,~ =

C[(1

+ B){,,

C[Be,,

+

+ Be++ + o':(1 + 2 B ) e : - ] ,

(1 + B)e++ + o-:(1 + 2B)e::].

"r:.- = E:{(cr:/AE:)({,.,. + e++) + [1 + (2cr:2/AEz)]e::}, ,,-: - [E:/(I + cr:)]e,. . . . .

-+ -

,,!,: = 0.

where A B =

1

O"r

20"~

E,.

E,.

E:

~,.

,4.

and C = E,/(I

*- ~ ) .

The values of E , and E: are obtained from Fisher[18] and the Poisson ratios were each varied in 0.02 steps between 0.30 and 0,50 for consideration of 121 possible solutions.

1010

JANE F. KORETZ AND GEORGE H. HANDEL~,IAN

(The shear modulus C, which depends on the value of the Poisson ratio in the transverse direction, was also implicitly varied.) Criteria for selecting among the possible cases were supplied by experimental description of the accommodative process: e.g. at maximum accommodation, there can be no further effective (shape-changing) forces applied. It should be noted that, at 0 diopters accommodation, the lens is under maximum stress. so that increased accommodation is associated with elastic relaxation of the lens. We cannot assume, however, that the maximum accommodative state is associated with complete relaxation of applied forces. At present, we have no observational data on the shape of the lens under zero internal stress. As a result, we are presently unable to solve the classic problem starting from no stress and arriving at a given state of accommodation. Thus we are only able to find the incremental stresses above an unknown initial state as the accommodation is changed incrementally. It is hoped that future experimental information will allow us to reverse the problem.

RESULTS AND DISCUSSION

Elimination of nonphysiologically consistent solutions For the more physiologically accurate anisotropic lens model, 121 cases had to be evaluated[21]. As discussed earlier, these different solutions to the equations arise from lack of information about the Poisson ratio for the lens in the two directions of anisotropy. A priori information about the accommodative process which has not been explicitly incorporated into the model allowed us to eliminate all but one of the cases successfully. The 121 solutions to the equations fall into four general groups (Fig. 2, [21]). Region 1: There is less than a 1% net change in force per 2-D accommodation. Region 2: There are negative (compressive) overall net force changes in the radial (r) and axial (z) directions. Region 3: The overall radial force change is negative, while the axial force change is positive. Region 4: The overall radial force change is positive, while the axial force change is negative. Two of these regions can immediately be discarded, based on lack of agreement with qualitative observations of the accommodative process. As described in the Introduction, an increase in accommodation is accompanied by a decrease in radial width, implying a concomitant decrease in the radial forces acting upon the lens. Since Region 4 is characterized by an increase in this parameter with increasing accommodation, it is eliminated. Another observation, made by Brown[27] on a human subject with a "'key-hole" defect of the iris using slit-lamp photography, indicates that at maximum accommodative amplitude the zonules appear to hang limply. Since. in Region 1, there is almost no change in overall force per 2-diopter accommodative increase, it too can be eliminated. In all, these conditions eliminate about one-third of the possible solutions. In Regions 2 and 3, the net overall force (the vector sum of the total radial force and total axial force changes) for each 2-D accommodative change is directed toward the polar axis. If it is assumed that, for the wedge shaped portion of the anterior lens that we are considering, this force is added vectorially to the force applied by the zonules to this solid, then its magnitude and direction will lead either to an increase or a decrease of zonular force with increasing accommodation (cf. [20]). An increase is unacceptable[27]; a decrease is, of course, acceptable, but can be further constrained such that anterior zonular tension goes to zero at the maximum accommodative range to be expected of the age-I 1 eye whose parameters we are using (in this case about 14 diopters). With these further constraints, all of Region 2 is eliminated, as well as all but one case

Age-related accommodati,.e Ios~

101[

in Region 3. In this case. where err = 0.42 and ~: = 0.46. the zonular force goes to zero by about 15 diopters accommodation. The comparatively high values for the Poisson ratios are consistent with the value of 0.46 found for the isotropic case, and tend to confirm the generally held view that, for biological substances of high water content, the volume changes under physiological stresses would be small. The difference between these values in the radial and axial directions may also reflect the different ratio of cytoplasm to membrane in the two directions.

Implications of the force distribution It must be emphasized that this computer-based representation of accommodation in the young human eye is implicitly restricted in ways other than an age-related sense. The way it was set up. and the elasticity data employed, make it a representation of the lens fiber aggregate without the capsule. Furthermore, because we evaluated the forces acting on a solid segment from the anterior portion of the lens only. we are indirectly getting information on the forces acting on the posterior portion of the lens from that surface of the segment which would abut the posterior. Since the only anatomical structures impinging on the anterior lens are the capsule and the anterior zonular apparatus, we are justified in attributing any overall forces or changes in overall forces to these structures: this allowed us to select among the various solutions for the one presented in the previous section. With these caveats in mind, it is nevertheless possible to formulate a more detailed qualitative mechanism of accommodation than that presented in the Introduction, tentatively assigning roles to each of the structures anatomically and geometrically related to the lens fiber aggregate. These assignments are based in part on the nature of the structures themselves and in part on the distribution of the applied forces over the anterior lens segment.

The capstde Calculations of the magnitude of the tractions in the axial and radial directions as a function of either theta or radial distance from the polar axis for the isotropic and anisotropic cases show what would be expected. That is to say, the radial traction is zero at the polar axis, increasing in magnitude with increasing distance from the axis. The axial traction, in contrast, is at its maximum at r = 0 and decreases steadily in magnitude with increasing r. For the isotropic case, both tractions were negative in direction (directed into the lens fibers), as was the normal; further, the normal exhibited approximately the same magnitude independent of radial location (Fig. 2, [19]). This suggested to us that the capsule acted as a force distributor, converting the discrete stress transmitted from the ciliary muscle through the zonules to their capsular attachment into an even force over the lens surface. The traction in the tangential direction would be largely dissipated without effect, partially because the capsule is not directly attached to the lens fibers and partly because its modulus is about three orders of magnitude greater than that of the lens aggregate itself[6, 28]. The anisotropic case exhibits similar behavior, although the magnitudes differ.

The zonular apparatus Analysis of the radial and axial forces acting on each of the four surfaces of the asymmetric solid segment{21] indicates that most of the force calculated to be applied to the lens can be supplied by the zonular apparatus. On the anterior lens surface, these overall incremental forces are both negative, consistent with the geometry of the connections

1012

JANE F. KORETZ AND GEORGE H. HANDEL,MAN

and the properties of the lens capsule. On the posterior, in contrast, the forces, which are both positive, cannot be supplied by the zonular apparatus alone. The force in the axial direction is much larger than the radial force: while there is disagreement about the geometry of the posterior zonules, it is impossible for any zonular arrangement to supply these types of force. It is therefore assumed that, on the posterior surface, the vitreous as well as the zonular apparatus contributes to the accommodative change. The vitreous, in our interpretation, acts largely as a reactive support, but the possibility of a small active contribution cannot presently be eliminated. With increasing accommodation, the ciliary muscle moves slightly anterior, as well as "tightening the collar." At the same time, the lens becomes thicker along the polar axis and more sharply curved (smaller overall radius of curvature). As a result, the geometry of the lens-zonule connection changes such that further applied force acts in an altered vector direction. These alterations can be calculated from the model for the anterior zonules using the overall force vector if the initial geometry at 0 diopters is known. This cannot be done for the posterior at the present time, both because the geometry is the subject of disagreement and because we cannot determine how much of the positive axial force comes from the vitreous contribution.

The development of presbyopia Recent analysis of slit-lamp photographs of four human subjects--aged 1I. 19, 29 and 45--taken through each subject's accommodative range at two diopter intervals has provided some clues about the relationship of lens growth and development to presbyopia[15]. It was found that all discernible lens curves (anterior and posterior lens surfaces, boundaries between adjacent zones of discontinuity, anterior and posterior boundaries between the lens cortex and nucleus) could be fitted to parabolas with high accuracy irrespective of lens age or accommodative state. For each pre-presbyopic lens, there was a linear relationship between curve location within the lens and curvature whose slope was independent of accommodative state but dependent on lens age. In the presbyopic (age 45) lens, in contrast, this empirical relationship appears to break down. While the posterior portion of the lens exhibits the same type of empirical relationship as the younger lenses, the anterior half seems to be " f r o z e n " into a configuration which is nonlinear and nonchanging over the small accommodative range left to the subject. In terms of the simple model of accommodation we have presented, we would suggest that presbyopia is a primarily geometric disease. With increasing subject age, the lens becomes larger and thicker. At the same time, the anterior zonules appear to change their relative point of attachment to the anterior lens capsule: in fact, the surface diameter of the zonular ring remains unchanged, indicating that the capsule adjusts to increased lens size by growth around the radius of the lens[17]. As a result, both the point of application and the angle of application of the zonular forces have been altered greatly from the young human eye. Remembering that, at 0 diopters, the lens is under maximum stress, it is likely that there is no effective relaxation of the anterior zonular force: i.e. the altered configuration would direct the ciliary muscle force primarily tangential to the capsule, where it would be dissipated without effect on anterior lens shape. This explanation can stand on its own without invoking changes in lens elastic properties, sudden degeneration of the ciliary muscle, or other factors, and is consistent both with what is currently known about the lens and with our recent results from the analysis of slit-lamp data. CONCLUSIONS The computer-based model of accommodation described here is limited to the anterior portion of young human lenses, in part because of simplifying assumptions and in part

Age-related a c c o m m o d a t i v e loss

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because of a dearth of physical data about the lens. Despite these limitations, we have been able to expand our understanding of the mechanism of accommodation in the young human eye through its use, especially in tentatively assigning functions to lens-related structures. On the basis of these assignments and recent data on the presb.~opic lens, an alternative explanation of presbyopia has been formulated. Both the accommodation model and the model for presbyopia are experimentally testable. Analysis of the slit-lamp photographs has provided us with a means to formulate a truly general accommodation model, since all the c u r v e s - - b o t h anterior and posterior--that we were able to discern could be fitted to a parabolic shape. When we are finished deriving a set of expressions based on these studies. ,xe will be able to consider either the anterior or posterior of the lens or both together for all lens ages, and address the question of the development of presbyopia directly. Acknowled',,ments--The authors are grateful for the support of Grant EY02195 from the National E,,e institute for this ~ o r k .

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26. S. G. Lekhnitskii. Theory of Elasticity of an Anisotropic Elastic Body, pp. 1-68. Holden-Day. Sa~: Francisco (1963). 27. N. P. Brown. The shape of the lens equator. Exp. Eye Res. 20, 571-576 (1974). 28. R. F. Fisher, Elastic constants of the human lens capsule. J. Physiol. 201, 1-19 (1969).