The shape of the human lens nucleus with accommodation

Abstract

Knowledge about geometric properties such as shape and volume and Poisson's ratio of the nucleus can be used in the mechanical and optical modeling of the accommodation process. Therefore, Scheimpflug imaging was used to determine the shape of the human lens nucleus during accommodation in five subjects. To describe the shape of the nucleus, we fitted a parametric model of the cross-sectional geometry to the gradient of the Scheimpflug images using the Hough transform. The geometric model made it possible to estimate the anterior and the posterior central radius, central thickness, equatorial diameter, and cross-sectional area of the nucleus. Assuming that the nucleus is rotationally symmetric, the volume of the nucleus can be estimated by integrating around the circumference. For all five subjects, the results show that during accommodation the nucleus became more convex and that the central thickness increased whereas the equatorial diameter decreased. This decrease in equatorial diameter of the nucleus with accommodation is in accordance with the Helmholtz accommodation theory. Finally, the volume of the nucleus (on average 35 mm ³) showed no significant change during accommodation in any of the subjects, presumably due to the fact that the human nucleus consists of incompressible material with a Poisson's ratio that is near .5.

Introduction

To understand the accommodation mechanism, accurate measurement and description of the geometric changes in the human lens during accommodation are important. It has been shown that the increase in lens thickness during accommodation mainly corresponds to a change in the central thickness of the nucleus, while the thickness of the cortex of the lens remains the same (Brown, 1973; Dubbelman, Van der Heijde, Weeber, & Vrensen, 2003; Koretz, Cook, & Kaufman, 1997; Patnaik, 1967). As a result, it is generally assumed that the changes in the nucleus play a substantial role in the accommodation process. 

Using Scheimpflug imaging it is possible to give an accurate description of the central geometry of the lens with accommodation (Dubbelman, Van der Heijde, & Weeber, 2005). Imaging of the equator of the lens is not possible because it is located behind the iris. However, it is possible to capture the equator of the nucleus. A simple but complete parametric description of the shape of the nucleus during accommodation would be useful for optical (Norrby, 2005) and mechanical modeling (Burd, Judge, & Cross, 2002) of the accommodation system, and it could be used to gain insight into the process of accommodation. In particular, the exact mechanism of accommodation is still a subject of discussion (Atchison, 1995). For instance, Schachar et al. (1996) opposed the conventional Helmholtz theory of accommodation (Von Helmholtz, 1855) by assuming that during accommodation the equatorial diameter of the lens increases, instead of decreasing according to the Helmholtz theory. 

Brown (1973), who made quantitative measurements of the shape of the human nucleus using Scheimpflug photography, made a distinction between anterior and posterior nuclear thickness with respect to the equatorial plane and determined the equatorial diameter of the nucleus. Cook and Koretz (1991, 1998) proposed a method to parametrically estimate the central shape of the lens. This method was also applied to determine the central shape of the nucleus of the lens (Koretz, Cook, & Kaufman, 2001; 2002). However, earlier studies in which Scheimpflug photography was used have not taken into account the fact that the nucleus is imaged through the cornea and the anterior lens surface. As a result, the nucleus appears distorted on the Scheimpflug image, and correction is needed to be able to determine the shape accurately (Dubbelman & Van der Heijde, 2001; Fink, 2005). 

Because the central thickness of the nucleus changes with accommodation, a possible change in lens volume would primarily be expected in the nucleus. Based on magnetic resonance imaging (MRI) measurements Strenk, Strenk, Semmlow, and DeMarco (2004) determined the cross-sectional area (CSA) of the human lens as a function of age and accommodation. Strenk et al. suggested that the lens might be compressed in the unaccommodated state. However, change in the CSA does not necessarily imply change in lens volume (Judge & Burd, 2004). The change in volume of the human lens nucleus has not yet been measured in vivo as a function of accommodation. 

In the present study, Scheimpflug imaging was used to determine the shape of the nucleus as a function of accommodation in five subjects. The Scheimpflug images were corrected for optical distortion, and a pattern recognition technique was applied to determine the shape of the nucleus. The shape was described by a parametric representation from which the CSA, the volume, and the change in volume during accommodation could be estimated. 

Method

The sample population consisted of five subjects, between the ages of 16 and 32 years, who had no ocular abnormalities, diabetes mellitus, cataract, or previous ocular surgery. Experiments were performed with the understanding and written consent of each subject, according to the tenets of the Declaration of Helsinki. Images of the anterior segment of the eye were obtained with the Topcon SL-45 Scheimpflug camera, the film of which was replaced by a CCD-camera (St-9XE, SBIG astronomical instruments) with a dynamic range of 16 bits of grey values (512 × 512 pixels, pixel size 20 × 20 μm, magnification: 1×). Ocular axial length was measured with the IOL Master (®Zeiss), which is based on partial coherence interferometry (Drexler et al., 1998), and the ocular refractive error was measured with the IRX3 aberrometer (Imagine Eyes Optics, Orsay, France). 

To capture the image of the complete nucleus, we dilated the pupil of the right eye of each subject with two drops of 5% phenylephrine HCl. The subjects were instructed to fixate on a fixation light in the Scheimpflug camera, while the slit of the camera was aligned along the optical axis of the right eye. Next, the subjects fixated with the left eye on a Maltese star, the position of which can be adjusted horizontally and vertically by remote control until the subject reports that the fixation light of the Scheimpflug camera is superimposed on the center of the Maltese star, after which the internal fixation light of the camera was turned off. The subject was then asked to focus on the Maltese star and three images were obtained. Scheimpflug images of the right eye as a function of accommodation were also obtained. For these images, we adhered to the same procedure, except that in order to induce accommodation the power of a lens positioned 19 mm in front of the left eye was increased in steps of −1 D. The measurements were performed until the subject indicated that it was no longer possible to obtain a sharp image of the star. Using the position and power of the different lenses placed in front of the eye, the accommodation stimulus was converted to the accommodation at the corneal plane (Rabbetts, 1998) (Figure 1). 

Figure 1

 

Scheimpflug image of the cornea and lens of a 28-year-old subject.

Postprocessing of the Scheimpflug images

Based on the gradient of the central densitogram, Dubbelman et al. (2003) defined the central thickness of the nucleus from the edges of the C3 zone, according to the Oxford classification system (Sparrow, Bron, Brown, Ayliffe, & Hill, 1986). In the present study, the complete nucleus was also defined by the boundaries of the C3 zone, the identification of which was based on the gradient of a Scheimpflug image. Therefore, a Canny edge filter (Canny, 1986) was used to compute the gradient G, with G_x and G_y the components in x- and y-directions (radial and axial directions, respectively). The Canny edge filter was composed of the directional derivatives of a Gaussian filter (standard deviation σ = 1 pixel) and was convoluted with the original image. Cook and Koretz (1998) showed experimentally that for the detection of internal lens boundaries, the inverse magnitude of the gradient I_grad provides good contrast: 

$I_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}}=\frac{1}{\sqrt{G_x^2+G_y^2}}.$

(1)

To improve contrast, we used thresholding and gamma correction (factor 0.5) to map the values of I_grad to values between zero (black) and one (white). Figure 2 gives an example of the result of this procedure for the Scheimpflug image as shown in Figure 1. 

Figure 2

 

Visualization of the Canny edge filtered inverse magnitude gradient I_grad, σ = 1, gamma correction factor = .5 and thresholding. Black corresponds to a high gradient, and thus a large change in grey value between adjacent pixels in the original image.

Parametric description of geometry

Kasprzak (2000) proposed an analytical function that describes the whole axisymmetric lens profile in an accommodated and a disaccommodated state. This function could also be used to describe the geometry inside the lens. However, a disadvantage of this method is the high number of parameters that have to be fitted simultaneously on rotational symmetric data. Therefore, in the present study parametric geometry was used, which has the advantage of describing a nonsymmetrical cross-section of the whole nucleus. Furthermore, this type of geometry makes it possible to estimate the anterior and the posterior sides of the nucleus separately, and therefore there are less parameters that have to be fitted simultaneously. Figure 3 illustrates the geometric model that consists of central anterior and posterior parabolics: 

$y=y_0+\frac{{(x-x_0)}^2}{2R},$

(2)

with (x₀, y₀) as the apex position and R as the central radius of curvature at an aperture of 3.5 mm. The positions of left (x_EQL, y_EQL) and right (x_EQR, y_EQR) equators were used to form a nucleus shape, using four curves that closed the periphery of the nucleus. It was assumed that the curve describing the outline of the nucleus was continuous, that the derivative of this curve was continuous, and that the y derivative was zero at the equators. We chose to describe the parametric curve with as few parameters as possible. Therefore, nucleus geometry was closed with four polynomials of the third order. The four curves in the periphery were described with the following equation: 

$x_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{y}}=x_{\mathrm{E}\mathrm{Q}}-a{(y_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{y}}-y_{\mathrm{E}\mathrm{Q}})}^2-b{(y_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{y}}-y_{\mathrm{E}\mathrm{Q}})}^3,$

(3)

with (x_EQ,y_EQ) as the position of left or right equator and (x_perifery, y_perifery) as the coordinates of curves in the periphery. 

Figure 3

 

Geometric representation of the nucleus of the human lens; X- and Y-axes are representing the radial and the axial directions, respectively.

Omitting the term of the first order in Equation 3 guaranteed that the y derivative was zero at the equator ( y _perifery = y _EQ). For every curve in the periphery, we constrained the parameters a and b by the necessary continuity of function and derivative at the interception point with the central parabolic:  

$\frac{a=(3x_{\mathrm{E}\mathrm{Q}}-3x_i)(x_0-x_i)+R(y_{\mathrm{E}\mathrm{Q}}-y_i)}{(x_0-x_i){(y_i-y_{\mathrm{E}\mathrm{Q}})}^2},$

(4)

 

$b=\frac{(2x_0-2x_i)(x_i-x_{\mathrm{E}\mathrm{Q}})+R(y_i-y_{\mathrm{E}\mathrm{Q}})}{(x_0-x_i){(y_i-y_{\mathrm{E}\mathrm{Q}})}^3},$

(5)

with ( x _i, y _i) as the position of the interception point between the central parabolic (3.5 mm aperture) and the specific curve in the periphery. 

Parametric estimate of geometry using the Hough transform

To fit the nucleus geometry to the gradient image I _grad, we applied a Hough transform, similar to the Hough transform proposed by Cook and Koretz (1991, 1998), to the Scheimpflug image. Movie 1 shows a typical example of this fitting procedure for the 28-year-old subject. 

 

Movie 1

 

Hough transform procedure to fit the parametric geometry of the nucleus to the gradient image I_grad.

First, it was necessary to make an initial estimate of the anterior and the posterior parabolic parameters and the left and right equator positions. On the boundary of the nucleus, points in the gradient image I _grad were selected manually by mouse to obtain this initial estimate. Subsequently, two parabolics were fitted through these selected points with the Levenberg–Marquardt method. Furthermore, the initial parabolics were plotted, and mouse dragging of the equator positions was possible with a real-time update of the curves closing the periphery. The left and right equator positions were determined manually, based on the darkest transition in equatorial direction of the gradient image. It was possible to make an extra visual check of equator position because the curves in the periphery should lie on the darkest pad between equator and interception point. 

With the parametric geometry description, it is possible to execute the anterior and the posterior Hough transform separately. For the anterior and the posterior side, we formed a three-dimensional matrix, containing values around the initial estimate of central parabolic parameters x _{0 start}, y _{0 start}, and R _start:  

$arraylll{x}_{0\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}-4:x_{0\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}+4\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{1}\mathrm{-}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{s}{y}_{0\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}-4:y_{0\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}+4\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{1}\mathrm{-}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{s}{R}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}-30:R_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}+30\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{3}\mathrm{-}\mathrm{p}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{s}array.$

 

In this way, all possible combinations of x ₀, y ₀, and R formed (9 × 9 × 21 =) 1701 parabolics around the first estimate. The value for the Hough transform of a specific set ( x ₀, y ₀, R) was composed by evaluating the line integral of the gradient image I _grad along that specific corresponding parabolic. To fit not only the central parabolic, but also the complete anterior and posterior geometry of the nucleus, we included the anterior and the posterior curves in the periphery in the evaluation of the line integral. For every combination of ( x ₀, y ₀, R), we carried out the Hough transform, resulting in a three-dimensional Hough transform matrix. The global minimum in the Hough transform matrix corresponds to the parameters that describe an edge. If the initial estimate was close enough, the global minimum of the Hough transforms corresponded to the best fit of parametric geometry of the nucleus. 

Correction for distortion

The inclined position of the CCD camera, according to the Scheimpflug principle, caused distortion (Type I). Furthermore, the light rays that form the image of the nucleus are refracted by the cornea and the anterior side of the lens (Type II). The correction for both types of distortion was performed with custom-developed software written in C++ (Dubbelman et al., 2005). Figure 4 illustrates the influence of the correction algorithm on the original image. Before correction, the parametric model was fitted with the Hough transform. After correction, anterior and posterior central parabolics were fitted at an aperture of 3.5 mm (green curves) with the Levenberg–Marquardt method to describe the corrected geometry. 

Figure 4

 

Nucleus detection before and after correction for Types I and II distortions (subject of 28 years old).

The index of refraction of the lens is needed to determine the path of the light rays that originate from inside the lens and that are refracted at the anterior lens surface. An equivalent index of refraction of the lens was estimated from the ocular axial length and refractive error at 0 D of accommodation stimulus (Dubbelman et al., 2005). It was assumed that the equivalent index of refraction did not change during accommodation. 

Calculation of the volume

The estimation of cross-sectional parametric curves leads to a non-axisymmetric geometry because the axis defined by the anterior and the posterior apex locations does not need to be perpendicular to the equatorial plane defined by both equator positions. Unlike the estimate, the results are given under the assumption of a nucleus that could be described by axisymmetric geometry. Therefore, the equatorial diameter of the nucleus was estimated by the distance between both equator positions. Subsequently, the thickness of the nucleus was divided into an anterior and posterior thickness (half?) by the line drawn through both equators. To determine the CSA enclosed by the geometry of the nucleus, we subtracted the integrals over the anterior and the posterior nucleus curves ( Equations 2 and 3):  

$\mathrm{C}\mathrm{S}\mathrm{A}=\underset{x_{\mathrm{E}\mathrm{Q}\mathrm{L}}}{\overset{x_{\mathrm{E}\mathrm{Q}\mathrm{R}}}{\int }}{f}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}\left(x\right)-\underset{x_{\mathrm{E}\mathrm{Q}\mathrm{L}}}{\overset{x_{\mathrm{E}\mathrm{Q}\mathrm{R}}}{\int }}{f}_{\mathrm{a}\mathrm{n}\mathrm{t}}\left(x\right),$

(6)

with f _ant( x) and f _post( x) as the anterior and the posterior curves, respectively, as a function of the radial position ( x). The volume of a solid of revolution with respect to an axis of rotation can be determined according to the following equation (Arfken & Weber, 2001): 

$V=\pi \int f{\left(y\right)}^2\mathrm{d}y,$

(7)

with f(y) as the axisymmetric cross-section curve as a function of the axial position (y) with respect to the axis of rotation. To calculate the volume of the nucleus, we choose to rotate along the vertical axis (x = 0 in Figure 3) that was defined by the mean of anterior and posterior apex locations. Because the nucleus is not symmetric around this rotation axis, we calculated the volume for both the left and the right half of the nucleus using Equation 7. Finally, the mean of these two volumes was taken to obtain the volume of the nucleus. 

Results

A typical estimate of the shape of the nucleus during accommodation for a 16-year-old subject is shown in Movie 2. During accommodation, the nucleus of the lens becomes thicker, while the equatorial diameter decreases and both the anterior and the posterior central radii of curvature decrease. 

 

Movie 2

 

Change in nucleus geometry of the 16-year-old subject between an accommodation stimulus of 0 and 10.4 D.

Figure 5 shows the anterior and the posterior radius of the nucleus of the 16-year-old subject as a function of accommodation stimulus. Figure 6 shows the anterior and the posterior thickness, as well as the ratio between these entities, and Figure 7 shows the equatorial diameter. Figures 8 and 9 illustrate the CSA and the volume change with accommodation for the 16-year-old subject. Linear regression showed that the CSA increased significantly ( p < .0001, r = .71) with accommodation, but that there was no significant change ( p = .61, r = −.08) in volume of the nucleus. Moreover, none of the other subjects showed any significant change in lens volume ( p > .05), and the mean volume was 35 mm ³. If no volume transport takes place in or out of the nucleus, the constant volume implies that the material inside the lens nucleus can be assumed to be incompressible. 

Figure 5

 

Anterior and posterior central nucleus radius as a function of accommodation stimulus for the 16-year-old subject.

Figure 6

 

Anterior and posterior nuclear thickness and the ratio between anterior and posterior nuclear thickness as a function of accommodation stimulus for the 16-year-old subject.

Figure 7

 

Nuclear equatorial diameter as a function of accommodation stimulus for the 16-year-old subject.

Figure 8

 

CSA as a function of accommodation stimulus for the 16-year-old subject.

Figure 9

 

Volume as a function of accommodation stimulus for the 16-year-old subject.

The results of the regression analysis for all subjects are shown in Table 1. With increasing accommodation stimulus, a continuous reduction of the anterior and the posterior central radii of the nucleus, on average with 0.14 and 0.07 mm/D, respectively, is demonstrated. There was a continuous total thickening of on average 0.04 mm/D with accommodation, while the equatorial diameter continuously decreased on average by 0.05 mm/D. For the 16- and 24-year-old subjects, linear regression analysis made it clear that the ratio between anterior and posterior thickness did not change with accommodation ( p > .05). However, for the other three subjects there was a significant change in the ratio of anterior and posterior thickness during accommodation. On average, the mean value for this ratio at 0 D was .95. 

Table 1

 

Regression parameters as a function of accommodation stimulus (D) for all subjects, with p < .001 unless otherwise stated.

Age (y)	Max. stimulus (D)		Anterior radius (mm)	Posterior radius (mm)	Anterior thickness (mm)	Posterior thickness (mm)	ANT/POST thickness ratio	Equatorial diameter (mm)	CSA (mm 2)	Volume (mm 3)
16	10.4	Intercept	4.62 (± 0.05)	4.02 (± 0.05)	1.28 (± 0.01)	1.20 (± 0.01)	1.07 (± 0.01)	5.87 (± 0.02)	10.52 (± 0.07)	39.2 (± 0.4)
		Slope	−0.147 (± 0.009)	−0.094 (± 0.008)	0.018 (± 0.002)	0.017 (± 0.001)	0.0001 (± 0.002)	−0.027 (± 0.004)	0.07 (± 0.011)	0.03 (± 0.05)
		p					0.98			0.61
		r	−0.94	−0.89	0.79	0.92	0.00	−0.76	0.71	0.08
19	8.2	Intercept	4.14 (± 0.08)	3.09 (± 0.04)	1.10 (± 0.02)	1.26 (± 0.01)	0.88 (± 0.02)	6.12 (± 0.04)	9.55 (± 0.06)	35.0 (± 0.3)
		Slope	−0.149 (± 0.017)	−0.041 (± 0.009)	0.028 (± 0.004)	0.012 (± 0.003)	0.014 (± 0.005)	−0.065 (± 0.009)	0.079 (± 0.013)	−0.06 (± 0.07)
		p					0.005			0.38
		r	−0.87	−0.65	0.82	0.61	0.53	−0.81	0.79	−0.18
24	7.4	Intercept	3.77 (± 0.04)	3.35 (± 0.05)	1.25 (± 0.02)	1.24 (± 0.01)	1.00 (± 0.02)	5.99 (± 0.01)	10.18 (± 0.07)	37.2 (± 0.3)
		Slope	−0.161 (± 0.010)	−0.088 (± 0.010)	0.021 (± 0.005)	0.029 (± 0.003)	−0.005 (± 0.004)	−0.080 (± 0.008)	0.085 (± 0.016)	−0.13 (± 0.08)
		p					0.32			0.11
		r	−0.96	−0.85	0.66	0.87	−0.19	−0.88	0.71	−0.31
28	7.4	Intercept	3.67 (± 0.04)	2.59 (± 0.03)	1.14 (± 0.01)	1.22 (± 0.01)	0.93 (± 0.01)	5.59 (± 0.02)	8.81 (± 0.04)	29.5 (± 0.2)
		Slope	−0.145 (± 0.010)	−0.045 (± 0.006)	0.029 (± 0.001)	0.009 (± 0.002)	0.016 (± 0.002)	−0.040 (± 0.004)	0.059 (± 0.008)	−0.06 (± 0.04)
		p				0.003				0.12
		r	−0.94	−0.79	0.98	0.54	0.77	−0.88	0.80	−0.30
32	7.4	Intercept	3.73 (± 0.04)	3.03 (± 0.06)	1.16 (± 0.01)	1.34 (± 0.01)	0.87 (± 0.01)	5.69 (± 0.02)	9.84 (± 0.06)	34.3 (± 0.3)
		Slope	−0.113 (± 0.009)	−0.070 (± 0.013)	0.026 (± 0.002)	0.015 (± 0.003)	0.010 (± 0.002)	−0.038 (± 0.004)	0.081 (± 0.013)	0.03 (± 0.06)
		p								0.69
		r	−0.93	−0.74	0.95	0.75	0.64	−0.87	0.78	0.08

Table 2 shows geometry parameter values for the 28-year-old subject before and after correction for the two types of distortion (Types I and II). This example indicates that correction for distortion does have a significant influence on the estimated shape of the nucleus. 

Table 2

 

Uncorrected and corrected (Types I and II) geometry parameters of the nucleus for the 28-year-old nonaccommodating subject.

	Anterior radius (mm)	Posterior radius (mm)	Anterior thickness (mm)	Posterior thickness (mm)	ANT/POST thickness ratio	Equatorial diameter (mm)	CSA (mm 2)	Volume (mm 3)
Uncorrected	5.30	4.49	1.20	1.16	1.03	6.92	10.95	45.5
Corrected	3.67	2.59	1.14	1.22	0.93	5.59	8.81	29.5

To relate the changes in shape of the nucleus to the overall changes in lens shape, we investigated the ratio between the anterior radius of the nucleus and those of the whole lens. Figure 10 shows for all subjects the average ratio between the radius of the anterior nucleus and anterior lens surface and that of the posterior nucleus and posterior lens surface as a function of accommodation stimulus. On average, the ratio for the anterior and the posterior surface is .39 and .51, respectively. This ratio can be used in optical and mechanical modeling to assume a shape of the nucleus when only the shape of the whole lens is available. 

Figure 10

 

Average ratio between the anterior and the posterior radius of the nucleus and those of the whole lens as a function of accommodation stimulus.

Discussion and conclusion

In the present study, nucleus shape detection using Scheimpflug imaging, gradient imaging techniques, and a parametric geometric description fitted with the Hough transform was performed in five subjects of different age. Corrections were applied for the distortions inherent to the Scheimpflug principle and the refraction of cornea and lens. These corrections are necessary because the distortions appeared to have a significant influence on the determination of lens geometry. In accordance with the findings of Koretz et al. (2002) and Dubbelman et al. (2005), with increasing accommodation stimulus a reduction of anterior and posterior radius of curvature was found. With increasing accommodation stimulus, the equatorial diameter decreased and the anterior and posterior thickness of the nucleus increased, which was also in agreement with the Scheimpflug studies carried out by Brown (1973) and Dubbelman et al. (2003). The ratio between anterior and posterior thickness with respect to the equatorial plane of the nucleus was approximately .95, and in some cases it changed slightly during accommodation. Based on the central radii and equator positions, the entire geometry of the nucleus could be described with a parametric shape that made it possible to estimate the CSA and the volume (on average 35 mm³). The CSA of the nucleus increased with accommodation but no significant change of volume of the nucleus was found. Therefore, the nucleus can be assumed to be incompressible with a Poisson's ratio near .5 if no volume transport occurs in or out of the nucleus. This finding seems to be in contrast with Strenk et al. (2004), in which it was concluded that the volume of the lens should increase with accommodation because of the increase of CSA that was measured. Nevertheless, an increase of CSA does not necessarily imply a change of volume (Judge & Burd, 2004) as also has been shown for the nucleus in the present study. 

In the literature, no in vivo quantitative estimation of the geometry of the nucleus using another technique than Scheimpflug photography has yet been reported. It is therefore only possible to make quantitative comparisons between the results of the present study and the earlier Scheimpflug measurements reported by Brown (1973) and Koretz et al. (2001). Brown measured the central thickness and the equatorial diameter of the nucleus of the human lens, and Koretz et al. measured the anterior and the posterior radius. However, in these Scheimpflug studies no explicit correction for the Type II distortions have been carried out. Nevertheless, a comparison between the regression analysis performed by Koretz et al., the measurements made by Brown, and the results of the present study was made at 0 D accommodation stimulus. The results of the anterior and the posterior radius reported by Koretz et al. (on average R_ant = 3.6 mm, R_post = 3.4 mm) and the results of our study (on average R_ant = 4.0 mm, R_post = 3.2 mm) were of the same order. Furthermore, the central anterior and posterior thickness measurements (on average 1.2 and 1.3 mm, respectively) were in agreement with those reported by Brown (on average 1.3 and 1.4 mm, respectively), which result in a ratio of .93 between anterior and posterior thickness. Although no Type II correction has been applied in these former studies only the estimated equatorial diameter (on average 5.8 mm) was approximately 15% lower than the Scheimpflug measurements reported by Brown (on average 7.0 mm). Table 2 shows that the determination of the thickness of the nucleus was not influenced by distortion. This could be explained by the image formation of the central part of the lens. A chief ray that originates from the central lens area crosses the cornea at an almost perpendicular angle and is only slightly refracted. The chief ray continues to the image plane of the camera and forms the image. A chief ray in the peripheral part of the lens crosses the cornea under a more oblique angle, resulting in more image distortion. As a result, the difference found in the equatorial diameters can be explained by the fact that the peripheral equators were more distorted by the refraction of cornea and lens (Type II). 

Koretz et al. (2001) approximated the nuclear volume by the solid of revolution of the central anterior and posterior boundaries. Assuming that the scale of Figure 2 in Koretz et al. should be multiplied by a factor of 1000, Koretz found a mean volume of approximately 40 mm³. This slightly higher result can probably be explained by the image distortion (Type II), as already indicated in the above section, and the simplification of nucleus geometry. 

The equatorial diameter of the complete lens is invisible with Scheimpflug photography because it is located behind the iris. Schachar and Bax (2001) built a nonlinear finite element model of a deformable lens and estimated that equatorial diameter should increase during accommodation. However, with Scheimpflug imaging the equator of the nucleus is visible, and in the present study the equatorial diameter of the nucleus showed a continuous decrease with accommodation. This change of equatorial diameter is in accordance with the MRI study of Strenk et al. (1999) who found also a decrease of equatorial diameter of the whole lens with accommodation. As it is unlikely that the equatorial diameter of the nucleus decreases while that of the whole lens increases with accommodation, it can be concluded that the results are in agreement with the Helmholtz accommodation theory (Von Helmholtz, 1855). 

Acknowledgments

Supported by the SenterNovem grant “Young eyes for elderly people” (IS 043081) and Advanced Medical Optics (AMO Groningen B.V.). 

Commercial relationships: none. 

Corresponding author: E. A. Hermans. 

Email: [e-mail]. 

Address: Department of Clinical Physics and Informatics, VU University Medical Center, PO Box 7057, 1007 MB Amsterdam, The Netherlands.