Download Change with age of the refractive index gradient of the

Change With Age of the Refractive Index Gradient of the
Human Ocular Lens
Richard P. Hemenger, Leon F. Garner, and Chuan S. Ooi
Purpose. To study age-related changes in the refractive index distribution of the human ocular
lens.
Methods. Biometric data collected on 48 eyes in subjects ranging in age from 19 to 31 years
and 48 eyes in subjects ranging in age from 49 to 61 years allowed estimation of a single
parameter related to the refractive index distribution of the crystalline lens. The authors
selected a gradient index model of the lens characterized by a fixed index at the lens center,
a somewhat lower fixed index at the surface, and a continuum of index values between center
and surface depending on a single parameter, /?. This parameter was evaluated for each of
the two age groups.
Results. The distributions of the gradient index parameter /3 for the two age groups were
found to be statistically well separated. On average, the older group was found to have an
index gradient that was flatter near the lens center and steeper near the surface, implying a
lower refractive power of about 2 D for representative lens surface curvatures.
Conclusions. It has been observed that surface curvatures and thicknesses of the ocular lens
increase with age, whereas other ocular dimensions apparently do not change, implying a
trend toward myopia. This trend has not been observed. The authors' results are consistent
with and strongly in support of the hypothesis that subtle index changes in the aging lens
compensate to a large extent for changes in surface curvatures. Invest Ophthalmol Vis Sci.
1995; 36:703-707.
.Measurements on extracted crystalline lenses of humans and animals demonstrate conclusively that the
refractive index decreases significantly from the center of the lens to the surface. '~5 Thus, this gradient in
refractive index must be taken into account if one is
to understand thoroughly the optics of the crystalline
lens. Although there is agreement on general characteristics of this index gradient, many questions remain
concerning its details as well as its visual consequences.
We have investigated an interesting related question. What is the role of the index gradient in the
changing optics of the aging crystalline lens? The lens
grows throughout life. In adults, it becomes progressively thicker and acquires steeper surfaces.6 7 Because
other dimensions of the eye change litde with age,
From the Department of Optometry, University of Auckland, Auckland, New
Zealand.
Supported in part by the New Zealand Optomelric Vision Research Foundation.
Submitted for publication July 14, 1994; revised October 11, 1994; accepted October
17, 1994.
Proprietary interest category: N.
Reprint requests: Richard P. Hemenger, Southern California College of Optometry,
2575 Yorba Linda Boulevard, FulkrUm, CA 92631.
one naturally predicts an increase in lens power and
a tendency toward myopia in the older eye. That this
does not happen has sometimes been called the "lens
paradox." 8 How might this paradox be resolved? An
hypothesis that the index gradient of the lens develops
in such a way that lens power changes less widi age
dian might be predicted based only on changes in
surface curvatures has been put forward by Pierscionek.8 We have tested this hypothesis by taking measurements on 96 eyes divided into younger and older
subgroups. Our data allow us to estimate the refractive
index variation for each eye within a particular lens
model. We find a significant difference in the variation
of the lens index between the two age groups. This
result is completely consistent with and provides
strong support for the hypothesis.
METHODS
Two balanced age groups were selected to avoid overlap. The younger group ranged in age from 19 to 31
Investigative Ophthalmology & Visual Science, March 1995, Vol. 36, No. 3
Copyright © Association for Research in Vision and Ophthalmology
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704
Investigative Ophthalmology & Visual Science, March 1995, Vol. 36, No. 3
years (mean, 22.1 years), and the older group ranged
in age from 49 to 61 years (mean, 53.9 years). Within
each group, a quota sampling technique9"11 was used
to obtain a balanced proportion of subjects in the
three refractive categories—those with myopia (equivalent sphere < —0.62 D), those with emmetropia
(equivalent sphere between +0.87 D and —0.50 D),
and those with hyperopia (equivalent sphere > +1.00
D). An equal number of male and female subjects
was assigned to each refractive category to balance
possible gender differences. Volunteers were recruited from the staff and student population of the
University of Auckland and from the general public.
Research followed the tenets of the Declaration of
Helsinki, informed consent was obtained from each
subject, and the protocol of this study was approved
by the University of Auckland Human Subjects Ethics
Committee.
For each subject, two initial measurements of refractive error were made using a precalibrated Humphrey Autorefractometer Model 530 (Humphrey Instruments, San Leandro, CA) in the auto-plus mode.
The corneal radius of curvature in the vertical meridian was determined three times by means of the precalibrated Humphrey Autokeratometer Model 410.
The ocular accommodation was paralyzed using
two drops of 1% cyclopentolate hydrochloride, which
also induced a mydriasis for the ophthalmophakometry procedures. Three cycloplegic autorefraction measurements were made using the Humphrey Autorefractometer Model 530 in the auto-plus mode. The
average spherocylindrical refractive error was then determined. The spherical equivalent refractive error
was used for all calculations carried out in the following section.
The first, third, and fourth Purkinje images were
recorded, and heights were measured using a video
recording technique described elsewhere.1213 The distance from the targets to the cornea was about 135
mm for measurements of the first and fourth Purkinje
images and about 126 mm for the third. The angular
separation of the two targets was about 5.9° and 6.3°,
respectively. Also, the visual axis was approximately
halfway between the targets and the camera, which
were separated by about 20°. Although the light
sources were not collimated, the calculations of equivalent mirror radii for the two surfaces of the crystalline
lens took this into account.
The cornea was anesthesized with one drop of
1.0% proparacaine hydrochloride. Axial dimensions
were measured with a Teknar (St. Louis, MO) Ophthasonic AB-scan III ultrasonometer.
We have used a biellipsoidal model of the crystalline lens, consisting of two semiellipsoids widi the circular equatorial section of the lens providing a common bounding surface.14 The major semiaxes of the
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0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Normalized Lens Radius
1
FIGURE 1. Distribution of refractive index in a model lens
for three values of the parameter p. The index is a function
of distance from the center of the lens to the surface. Distance units are normalized so that the total distance from
the center to the surface is unity.
two ellipsoids, given the symbol b, are equal to each
other and to the radius of the equatorial section. The
minor semiaxes, given the symbols at and a^, are not
equal in general. The sum of fli and a^ is equal to the
lens thickness, and the two vertex radii of curvature
are given by b z /ai and b 2 /a 2 . The three semiaxes of
the biellipsoidal model are uniquely determined by
specifying the anterior and posterior radii of curvature
of the lens as well as its thickness. Therefore, the equatorial diameter of the lens, 2b, is not an adjustable
parameter and does not correspond to an anatomic
dimension. We do not view this as a problem because
the optical calculations to follow use the paraxial approximation and, therefore, involve lens geometry
only in the immediate vicinity of the optical axis. It is
further assumed that the iso-indicial surfaces of the
lens are of the same form as the outer surface, i.e.,
ellipsoidal. It remains only to specify the mathematical
form of the decline of index from the lens center
to the surface. For this purpose, it is convenient to
introduce a coordinate, r, measured along a line from
the lens center to the surface and normalized so that
the value of r at the surface is unity. In our model,
the dependence of the refractive index on this coordinate is given by,
1.406
- .02)r2 - /?r4.
The index at the center of the lens (r = 0) is given
by 1.406, and the index at the surface (r = 1) is given
by 1.406 - .02 = 1.386. These center and surface
values are those of the Gullstrand no. 1 eye. Varying
the parameter (5 changes the form of the index variation from the lens center to the surface without changing the end points. Some examples of this function
are given in Figure 1, in which it is apparent that as
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705
Age Dependence of Ocular Lens Index
ft
P increases, the index gradient becomes flatter in the
central part of the lens and steeper in the outer part.
In principle, our data on each eye supply enough
information to determine the index parameter /?. To
do this, however, presents an algebraic problem. From
the size of the fourth Purkinje image, one knows the
radius, R^, of the equivalent mirror associated with
the posterior surface of the lens. However, the actual
radius of curvature of the posterior lens surface, R3,
depends on the index gradient of the lens, i.e., it
depends on the parameter 0 in our model. The problem then is to find values of P and R3 that yield the
known radius of the equivalent mirror. Similarly, the
distance, z, from the anterior cornea to the image of
a distant object is known from ultrasound measurements and from the refractive error. Again, one must
find values of P and R3 that yield the correct image
position. Thus, one has two conditions and two unknowns related by highly nonlinear equations. The
approach used here is to assume that for the relatively
small variations in parameters from eye to eye, the
changes in these quantities are linearly related. Expanding a function f{x,y) of two variables in a Taylor
series about the values xu,yu, keeping only linear
terms, gives
f(x,y)
- i(xo,yo)
= di/dX(x
-xo)+
df/dy(y
- yo).
The partial derivatives are to be evaluated at the
point Xa^o. Let /?o, R30, Reo» and Zo be a typical set of
values of parameters that are related through the exact equations for a particular eye. We then assume that
for small changes in one or mere of these quantities,
- 0O) + a 12 (R 3 - R30) = Re - P
~ R30) = z - zo.
I
The coefficients, Oy, in these equations are therefore partial derivatives of R^ or z with respect to /3 or
R3. They were determined by several exact calculations for a particular eye using the ray-tracing method
of Sharma et al.15 The coefficients an and 021 were
found by recording changes in R,, and z due to changes
in P while holding R3 constant. Similarly, a^ and a^2
were found by varying R3. The resultant coefficients
are assumed to have the same values for all eyes in
the study. Thus, for example, we have assumed that
the change in R,, due to a small change in P is independent of the small variation in parameters from eye
to eye. The problem reduces therefore to specifying
convenient starting values for /?0 and R30, then finding
die corresponding quantities R<,o and Zo for each eye
by ray tracing. These will, of course, not be in
agreement with the experimentally determined values, R,. and z. Substitution of these six quantities into
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
Gradient Index Parameter (ft)
FIGURE 2. Distributions of values of the parameter 0 for 48
subjects in the younger group {filled bars) and for 48 subjects
in the older group (unfilled bars). The mean value of P for
the older group (mean = +.0078) is larger than that for
the younger group (mean = —.0033), implying a steeper
index gradient, on average, for the older group.
the linear equations gives "correct" values for /? and
R3. Our assumptions were justified by substituting calculated values for P and R3 back into the exact relationships, verifying that the experimentally determined values for R,. and z resulted. If necessary, the
above equations can be iterated. We found satisfactory
accuracy witfi a single application of these equations.
The coefficient values used by us were au = 0.798, a-n
= 40.925, ffl,2 = 0.714, and 022 = 0.427.
RESULTS
The gradient index parameter /? has been calculated
as described above from measurements made on a
group of 48 younger eyes and on a group of 48 older
eyes. The resultant two distributions of this parameter
are shown in Figure 2. Visually, die two distributions
are well separated. This impression is strengthened by
applying a Mest that rejects the hypothesis diat the
means are equal at die .0001 level. Notice that the
mean value of P for the older subjects is more positive
than that for die younger subjects so diat the corresponding index gradient for the older subjects is
steeper in die outer lens than that for the younger
subjects (Fig. 1). A steeper index gradient under the
conditions of this model (fixed indices at the lens
center and at the lens surface) implies a refractive
power less than that for the same lens widi a less steep
gradient. The magnitude of diis effect has been estimated by calculating lens power for typical lens dimensions and for the values of P corresponding to die
mean values for the older subjects (P = +.0078 +
.0077) and for the younger (P = -.0033 + .0095).
We find a difference in lens power of between 1.5 to
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TABLE l.
Investigative Ophthalmology & Visual Science, March 1995, Vol. 36, No. 3
Post Hoc Comparison of Refractive Error and Ocular Component Values
Sphere refractive error (D)
Equivalent sphere (D)
Refractive astigmatism (D)
Equivalent corneal radius (D)
Anterior chamber depth (mm)
Lens thickness (mm)
Vitreous chamber depdi (mm)
Total axial length (mm)
Anterior lens radius (mm)
Anterior lens power (D)
Posterior lens radius (mm)
Posterior lens power (D)
Equivalent lens power (D)
Calculated ocular refractive power (D)
f
Younger Adults
Mean ± SD
Older Adults
Mean ± SD
+0.09
-0.13
-0.44
7.78
3.81
3.58
16.14
23.52
11.21
7.74
6.45
13.45
20.93
60.35
+0.16
-0.08
-0.46
7.77
3.39
4.33
15.70
23.42
9.30
9.39
6.15
14.05
23.04
62.14
±
±
±
±
±
±
±
±
±
±
±
±
±
±
1.09
1.09
0.23
0.24
0.27
0.19
0.67
0.67
1.08
0.75
0.56
1.24
1.67
1.82
±
±
±
±
±
±
±
±
±
±
±
±
±
±
0.94
0.94
0.22
0.26
0.30
0.28
0.83
0.89
1.13
1.18
0.52
1.23
1.89
2.21
Mean Diff
Young-Old
F-Test
-0.07
-0.05
+0.02
+0.01
+0.42
-0.75
+0.44
+0.10
+ 1.91
-1.65
+0.29
-0.60
-2.11
-1.79
0.104
0.069
0.242
0.060
50.169*
242.208*
7.594*
0.366
71.237*
66.549*
6.534*
5.585*
33.628*
18.608*
Significant at the 95% level (a = 0.05).
2 D, depending on the lens dimensions, with the more
positive value for 0 (steeper gradient) having the
smaller power.
DISCUSSION
From our measurements and for a particular lens
model, we have found a significant difference in the
variation of the lens index gradient between two age
groups. Our results are entirely consistent with and
strongly in support of the hypothesis of Pierscionek8
and its elaboration by Smith et al.16 She proposed that
the apparent contradiction between increasing lens
surface curvatures and the failure of a tendency of the
aging eye toward myopia could be resolved with subtle
changes in the lens gradient index. The notion that
surface curvatures and index gradient of the lens
change with age so as to compensate for each other,
thus maintaining a nearly constant refractive error, is
clearly of interest.
Our data confirm the existence of a lens paradox
that is then resolved by allowing the distribution of
refractive index in the lens to differ between the two
age groups. Statistical analysis of the data has been
performed elsewhere." The results appear in Table
1. Thus, it is shown that there are no significant differences in axial length (mean axial lengths 23.52 mm
and 23.42 mm) or corneal curvature (mean corneal
radii of curvature 7.78 mm and 7.77 mm) between
the younger and older groups.1718 Specifically, the hypothesis that axial length decreases widi age, thus
compensating for increased lens power,10 is not supported by these data. However, the older group does
show a significandy larger mean lens power of approximately 2 D if one assumes die same refractive index
distribution for the two groups. An age-dependent re-
fractive index distribution is the only apparent plausible resolution.
How model dependent are our results? The existence of a gradient in the refractive index of the crystalline lens is firmly established. Although there may
be agreement on broad features of this gradient, the
details remain uncertain. We could easily have contrived any number of gradient index models for use
in this study. However, we think that whatever model
we might have used would have led to much the same
result, i.e., significantly different refractive index distributions in the two age groups.
Each of the data used in this study is, of course,
subject to error. Zadnik et al,19 for example, have
made a thorough study of repeatability errors for the
biometric measurements used by us. We argue that
because of the balance in selection of subjects in the
younger and older groups, the data for each group
are subject to the same errors. If so, our result diat
the distributions of the parameter /3 for younger and
older subjects are separate is unaffected.
It is likely that a better understanding of the visual
consequences of the gradient index will follow only
from in vivo studies of the aberrations of the crystalline
lens. This might be accomplished, for example, by
measuring the aberrations of the eye, then optically
removing the effects of the cornea. Recent developments in devices for measuring aberrations of the
eye,20 as well as corneal topography, seem to make
such an approach feasible.
Key Words
crystalline lens, gradient index, aging changes, refractive
power
Acknowledgment
The authors thank Dr. George Smith for helpful suggestions
during preparation of the manuscript.
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Age Dependence of Ocular Lens Index
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