1.

Introduction

Accurate measurement of the physical and optical properties of the crystalline lens is essential in understanding the condition of presbyopia which is characterized by the gradual age-related loss of accommodation. These lens properties are closely related to each other, i.e., changes in the physical properties can result in changes in the optical properties.¹ According to the Helmholtz theory of accommodation, contraction of the ciliary muscle releases tension on the zonules and the lens capsule causing the elastic lens capsule to mold the crystalline lens into a more spherical shape which increases its optical power. In the presbyopic eye, the gradual hardening of the lens contents reduces the ability of the lens to change shape and optical power. As a result, focusing on near objects is compromised.

Using various imaging techniques, many previous models have been developed to understand and describe the shape and optical properties of the crystalline lens, both  in vivo and in vitro. These techniques include the use of Purkinje images,^{2, 3} Scheimpflug photography,^{2, 4, 5} shadow photogrammetry,⁶ ultrasonography,⁷ and magnetic resonance imaging (MRI).^{4, 5, 8} A more recent imaging technique, optical coherence tomography (OCT), which was first demonstrated by Fujimoto ⁹ in 1991, has found wide spread use for high-resolution ocular imaging, from the corneal shape through to the retinal structure. This nondestructive, depth-penetrating technique is also useful to obtain cross-sectional and 3D images of the crystalline lens in vivo,^{10, 11} as well as ex vivo. ¹²

Most images of the lens require some form of adjustment to correct for the refractive distortions introduced by the various curved refractive surfaces and different refractive indices within the eye. In a further step, the outline of the lens profile needs to be extracted for further processing and mathematical analysis.¹³ Several mathematical curve fitting techniques to define the lens contour have been described previously. Most of them split the lens contour into anterior and posterior profiles to fit parabolic, conic, polynomial curves, or cosine function to the measured data sets. ^{6, 8, 14, 15, 16, 17, 18, 19} Kasprzak,²⁰ and Urs et al,²¹ however, described the lens contour in a single profile providing continuity of radius curvature using hyperbolic and cosine functions. The mathematical curve fitting method presented in this paper also describes the lens contour in a single profile for different stages of accommodation.

Previous attempts to develop algorithms for the automated edge detection and curve fitting on OCT images of ex-vivo crystalline lenses²² have shown deficiencies in practical use, mainly due to the often poor and unpredictable quality of the acquired OCT images. The purpose of this study is to overcome these deficiencies by implementing more sophisticated analysis tools with direct user intervention to confirm the validity of the detected lens outline. Lens surface fitting functions are also refined to reliably describe the lens contour as a single function for both anterior and posterior surface and compared to other mathematical curve fitting functions. The changes in the lens biometry, such as the diameter and thickness of the lens are analyzed as the lenses are being stretched to simulate accommodation. This helps to understand the changes of the behavior of the lens and ensures that the algorithm is suitable for different shapes of lenses. Radius of curvature and the eccentricity value are compared between the different curve fitting functions across both anterior and posterior surfaces. The results are compared to previous studies and interpreted with respect to their potential impact on age related changes of vision and accommodation.

2.

Methods

All human eyes were obtained and used in compliance with the guidelines of the Declaration of Helsinki for research involving the use of human tissue. All animal eyes were obtained post-mortem following approved institutional animal care guidelines and the study was performed in accordance with a University of Miami approved ACUC protocol.

In determining the properties, crystalline lenses were mounted on the Ex-vivo Accommodation Simulator (EVAS II) device and cross-sectional OCT images were taken after stretching the lens at a fixed distance each time. The forces exerted on the vitro lenses simulate the affect of unaccommodation. The description of the instrument, lens preparation, and lens loading onto the EVAS instruments have been described previously by Ehrmann ^{23, 24} The standard measurement protocol²⁵ consists of four stretching steps of 0.5 mm radially, starting from 0 mm, which corresponds to the fully accommodated state. An OCT image is captured at the initial position and after each stretching step.

In this study, 3 human and 3 baboon lenses were used and analyzed; details are listed in Table 1. The eight sclera segments around the crystalline lens are bonded to eight PMMA shoes and the entire assembly immersed in Dulbecco's modified eagle medium (DMEM) to control and maintain the natural hydration state of the lens. The setup of the OCT system has been previously described by Uhlhorn ¹² It is a time-domain OCT with system resolution of 12 μm and sensitivity of 85 dB. The generated images are recorded with 5000 points per A-line and 500 A-lines per image and loaded into the customized LabWindow/CVI program for analysis. The reflectivity intensity data were converted into linear gray scale values ranging from 0 (black) to 255 (white). The lateral scan width varied between the human and baboon lenses. In DMEM medium with refractive index of 1.345, the axial scan length is 7.45 mm.

Fig. 14

Diameter (a) and thickness (b) values at each step of stretch for human lens H1. (*) showing p<0.05 from the Step 0.

Table 1

Details of lenses used for analysis. PMT: Post Mortem Time.

Typical OCT images of a baboon crystalline lens at different stages of accommodation using EVAS are shown in Figs. 1a and 1b.

Fig. 1

Raw OCT images of baboon crystalline lens; (a) Step 0–0 mm stretched; (b) Step 4–2 mm stretched.

3.

Results

The following results represent data for three sets of baboon and three sets of human lens at five different steps of the stretching with the EVAS instrument. All 30 OCT images were processed by the same operator, using the semiautomated process of determining the lens contour and shape as described above. Full and detailed results are presented for two typical set of data from one human and one baboon eye. Key numerical parameters are summarized for all eyes to illustrate variability and shape changes with lens stretching. Angular tilt of lens images was generally very small, ranging from 0.5 deg to a maximum of 8.0 deg.

3.1.

Edge Detection

The manual detection process facilitates the selection of points, which correspond closest to the edge contour of the lens. Figure 2 shows an ellipse which is determined by the two cursors enclosing the lens. Raw edge points are selected manually along each spoke and the obtained data set stored for further processing.

Fig. 2

Snapshot of the edge detection process, red spokes indicates periphery and blue spokes indicates central of the lens. Each spokes contains edge points. Cyan spoke shows the currently selected spoke and cyan point shows currently manually selected edge points along the spoke. All the points before the currently selected cyan spoke are manually selected by the user and points after the currently selected cyan spoke have not been manually selected yet showing points from automated edge detection process (rotating in anticlockwise direction).

3.2.

Lens Contour

After detecting all the lens edge points along the spoke, they were converted into polar coordinates as shown in Fig. 3.

Fig. 3

Graph plot of pixel points converted to polar coordinates; Empty circle points: edge point in polar coordinates 90 deg shifted; solid square points: Fourier cosine curve from edge points 90 deg shifted; cross points: Fourier cosine curve 0 deg shifted.

3.3.

Fourier Cosine Series

The polar coordinate is fitted with the Fourier cosine curve and is converted back to the x-y pixel point coordinate form. Now the contour of the crystalline lens is described as a mathematical description of the Fourier cosine series. An example of a Fourier cosine curve overlaying on the rotated image of the lens is shown in Figs. 4, 5. Note the different lateral scan width between the human lens (20 mm) and the baboon lens (10 mm).

Fig. 4

Baboon lens B3 showing Fourier cosine curve along Step 0 (a) and Step 4 (b).

Fig. 5

Human lens H1 showing Fourier cosine curve on Step 0 (a) and Step 4 (b).

3.4.

Physical Lens Shape

From the raw data there was a trend of increase in the lens diameter and decrease in the lens thickness while the lenses were stretched. The trend was more obvious in baboon lenses compared to the human lenses, Fig. 6. After fitting Fourier cosine, polynomial, and elliptical functions to the raw data points, thickness, and diameter values were obtained again from each of the curve methods. Little variation was found between raw, cosine, and polynomial thickness, but the elliptical curve fit gave consistenly lower thickness values as shown in Fig. 7. This difference was statistically significant for all the baboon lenses at all steps, but less than half for the human lenses.

Fig. 6

Lens thickness and diameter for all three sets of human (a) and baboon (b) across the 2 mm stretch.

Fig. 7

Thickness values of four different curve functions at different stages of the stretches for human (a) and baboon (b) lenses. 1st column: raw points; 2nd column: Fourier cosine points; 3rd column: polynomial points; 4th column: ellipse curve points.

3.5.

Lens Surface Curvature

Different curvatures were obtained from the manual detected points of the OCT images along the anterior and posterior surfaces by fitting Fourier cosine curve, polynomial curve, and the ellipse curve across the entire diameter of the lens. A typical example of the OCT image with different curvatures plotted is shown in Figs. 8, 9.

Fig. 8

Different curve fitting functions on the image. Yellow points are polynomial; green points are ellipse points along the purple ellipse curve fit; red points are cosine points; black points are manual detected points.[H1 lens Step 0 (a) and Step 4 (b)].

Fig. 9

Different curve fitting functions on the image. Green points are ellipse points along the purple ellipse curve fit; Red points are cosine points; Yellow points are polynomial; Black points are manual points. [B3 lens Step 0 (a) and Step4 (b)].

Figures 10, 11 outline each curve fitting functions on separate images for the lens shown in Fig. 8 of Step 0 and Step 4, respectively.

Fig. 10

Different curve fitting functions outlined on separate images. Green points (a) are ellipse points along the purple ellipse curve fit; red points (b) are cosine points; yellow points (c) are polynomial; black points (d) are manual points. (H1 lens Step 0).

Fig. 11

Different curve fitting functions outlined on separate images. Green points (a) are ellipse points along the purple ellipse curve fit; red points (b) are cosine points; yellow points (c) are polynomial; black points (d) are manual points. (H1 lens Step 4).

The error curve values were calculated separately for the anterior and posterior surfaces and are shown in Fig. 12a for human and Fig. 12b for baboon. All the points across the full diameter range of the lens were used in the calculation. The order for the Fourier cosine and polynomial curves was limited to 14 to avoid high frequency surface ripple effects. For both species, the elliptical fit improves as the lens is stretched. For the baboon lenses, the residual curve fit error for the ellipse was around twice as large compared to the Fourier cosine and polynomial curves.

Using data points of each curve function obtained, best ellipse and best circle was fitted across 5 mm diameter from the center of the lens, to determine the radius of curvature and the eccentricity value for the anterior and posterior lens contour. Radius of curvature values are shown in Fig. 13a for one human and Fig. 13b for one baboon sample lens. Average results from three human lenses and three baboon lenses of the best circle and ellipse fits are summarized in Tables 2, 3, respectively.

Fig. 13

Radius of curvatures for both anterior and posterior surfaces for one human H1 (a) and one baboon B1 (b) lenses using best ellipse fitting across four different curve methods.

Table 3

Averaged calculation of the radius (r) obtained from best circle, central radius thickness (r0) and eccentricity value (p) obtained from best ellipse fit on the four different curve functions for the three baboon lenses, unstretched and fully stretched.

	Anterior			Posterior
	Circle	Ellipse		Circle	Ellipse
Baboon	r (mm)	r0 (mm)	p	r (mm)	r0 (mm)	p
Step 0 Raw	6.18	8.84	4.49	4.95	5.49	1.70
Cosine	6.23	8.45	3.92	4.86	5.20	1.40
Poly	6.77	8.74	4.44	4.88	5.46	1.82
Ellipse	6.07	8.82	4.50	4.92	5.48	1.71
Step 4 Raw	7.64	12.23	7.94	5.50	5.94	1.63
Cosine	7.54	11.75	7.22	5.43	5.72	1.37
Poly	8.82	12.18	7.95	5.45	5.83	1.64
Ellipse	7.97	12.22	7.95	5.48	5.92	1.62

Table 2

Averaged calculation of the radius (r) obtained from best circle, central radius thickness (r0), and eccentricity value (p) obtained from best ellipse fit on the four different curve functions for the three human lenses, unstretched and fully stretched.

	Anterior			Posterior
	Circle	Ellipse		Circle	Ellipse
Human	r (mm)	r0 (mm)	p	r (mm)	r0 (mm)	p
Step 0 Raw	6.02	10.66	5.87	5.23	5.35	1.12
Cosine	6.16	10.03	5.42	5.13	5.33	1.22
Poly	6.82	10.44	5.98	5.23	5.12	0.85
Ellipse	7.08	10.63	6.23	5.22	5.33	1.12
Step 4 Raw	6.46	11.88	6.93	5.35	5.47	1.12
Cosine	6.77	11.67	6.71	5.34	5.32	1.01
Poly	7.70	12.15	7.50	5.37	5.36	0.99
Ellipse	7.81	11.86	6.93	5.35	5.45	1.12

4.

Discussion

The variable and low signal to noise ratio of OCT images had been a challenge for automated edge detection algorithms. Outliers and missing data points can lead to incorrect lens parameter extraction in subsequent processing steps. The semimanual edge detection method described in this study eliminates this problem by providing instant graphical feedback to the operator on selected raw data points as they are displayed and overlaid on the OCT image. Although this process is initially more time consuming, the subsequent processing steps become more reliable and require less error checking. The consistency of this process was verified and is sufficient to detect small changes in lens thickness and diameter associated with lens stretching.

In this current study, we generated radial spokes across the entire lens emitting each line from the center of the lens for detecting edges. Chien et al,¹⁹ also used radial spokes, generating radial lines emitting from the origin starting at pole of anterior surface to the pole of posterior surface equally spaced at 2 deg.

Most of the studies have described the lens profile separately for anterior and posterior surfaces using different functions. However Kasprak,²⁰ using a hyperbolic cosine function modulated by hyperbolic tangent functions, mathematically described the lens as a single profile. Chien et al¹⁹ used a cosine function to fit the lens profile, however applied the function separately giving two sets of coefficients for anterior and posterior surfaces. As shown in this study, higher order Fourier cosine curve is suitable to describe the outline of crystalline lenses of different species and accommodation states as a single continuous profile.

The order number for fitting Fourier cosine series is determined by the mse value. This iteration process was implemented to optimize the curve fitting function as different orders may be required to fit different type of species as well as for different stages of the stretch. Higher order Fourier cosine series generally provide a closer fit to the lens contour, but also introduce small ripples which compromise their application for further optical analysis, including ray tracing and analysis of higher order aberrations.

To measure how well each curvature fitted to the manually detected points, the radial mean square errors between the fitted curve and the manually detected points was calculated. In general, as exemplified in Fig. 12, for accommodated and un-accommodated lenses, the ellipse fit showed the highest error for baboon lenses. This poor curve fitting is also reflected in the extracted lens thickness values whereby the thickness data obtained from the fitted elliptical curves was significantly less compared to other curve fittings. For human lenses, the ellipse fit showed similar error values compared to other curve fits for all different steps, which implies human lens’ are more elliptical in shape than the baboon lens.

Fig. 12

Error values calculated for each curve function across the full lens diameter for one human H1 (a) and one baboon B1(b) lenses. Anterior and posterior surfaces were calculated separately.

The posterior lens surface in all the images in this study are not corrected for the difference in refractive index between crystalline lens and DMEM, hence the results may vary from other studies which have been corrected^{22, 26} or do not suffer from optical distortions such as MRI⁴ or shadow photogrammetry.⁶ Our results for the human lens diameter and the thickness measurements obtained from the four different curve functions (Fourier cosine, polynomial, ellipse, and raw) is in broad agreement with previous studies by Pierscionek and Augusteyn,²⁷ Jones et al,⁸ Rosen et al,⁶ and Urs ²¹ The thickness value for baboon (B1) lens was 4.32 mm which was slightly higher compared to other studies which was around 3.75 mm using phakometry Purkinje imaging technique of two 9 year old rhesus monkeys²⁸ and slit-lamp Scheimpflug photography of rhesus monkeys.²⁹

As the lenses were stretched, there was a trend of increase in lens diameter and decrease in lens thickness. This supports the Helmholtz theory of accommodation. Also, there was an increase in the radius of curvature values for anterior and posterior surfaces in both in human and baboon lenses, although least noticeable for the 65 years old human lens (H2). The anterior surface showed a steeper increase than the posterior surface in both the human and baboon species. This contradicts Chien et al³⁰ who showed a decrease in the radius of curvature values as the lenses were stretched, but agrees well with most other authors.^{5, 28, 31}

The radius of curvature value at the unaccommodated state for human lenses had been reported by various studies.^{2, 4, 5, 16} These results for the anterior and posterior radius of curvatures are very similar to our measurements. Using the ellipse fit, the average value of radii of curvatures across the four different curve fit method at the unaccommodated human lenses was 10.17±1.55 and 5.19±0.01 mm for anterior and posterior, respectively. Rosen et al⁶ reported slightly lower anterior radius of curvature 7.54±2.34 mm but similar posterior radius of curvature 5.5±0.9 mm. Schachar,³² however, reported higher posterior radius of curvature of 7.1±1.0 mm. The differences may be explained by the varying central diameter range selected which Schachar used to fit the elliptical curves.

The raw data average radius of curvatures for three baboon lenses was 8.84±0.68 mm for anterior surface was and 5.49±1.27 mm for the posterior surface at the un-accommodated state. This value was lower compared to the study Rosales ²⁸ performed, which was 11.11±1.58 and 6.64±0.62 mm for anterior and posterior, respectively. The study performed by Rosales used rhesus monkeys which had undergone removal of the iris and implantation of stimulating electrodes into the EW nucleus. These may contribute in the difference in the results performed in our current study.

5.

Conclusion

In this study, modified methods and algorithms were presented to obtain more accurate contours from crystalline lens OCT images compared to previous attempts.²² The process of semimanual detection of edge points along the contour of the lens provides reliable raw data points for further mathematical description and analysis. Describing the lens contour with a single continuous profile by fitting a Fourier cosine series is a valid method for various shapes and species of lenses and is useful to investigate the changes in lens shape with age and accommodation to better understand the condition of presbyopia. The measured flattening of the central anterior and posterior human and baboon lens profiles as the lens is stretched, supports the Helmholtz theory of accommodation. Analyzing different species (human and baboon) at different stages of the stretch also emphasize the ability for the outlined algorithms to work for different types and shapes of the lenses. This analysis and description of the lens shape can be used to further characterize the optical properties of the lens, in particular optical power, aberrations, and gradient refractive index.