Age-dependent Fourier model of the shape of the isolated ex vivo human crystalline lens

Estimation of the full shape of the crystalline lens from OCT: validation using stretched donor lenses

Quantifying human crystalline lens geometry as a function of age and accommodation is important for improved cataract and presbyopia treatments. In previous works we presented eigenlenses as a basis of 3-D functions to represent the full shape of the crystalline lens ex vivo. Also, we presented the application of eigenlenses to estimate the full shape of the lens in vivo from 3-D optical coherence tomography (OCT) images, where only the central part of the lens -visible through the pupil- is available. The current work presents a validation of the use of eigenlenses to estimate in vivo the full shape of dis-accommodated lenses. We used 14 ex vivo crystalline lenses from donor eyes (11-54 y/o) mounted in a lens stretcher, and measured the geometry and the power of the lenses using a combined OCT and ray tracing aberrometry system. Ex vivo, the full extent of the lens is accessible from OCT because the incident light is not blocked by the iris. We measured in non-stretched (fully accommodated) and stretched (mimicking in vivo dis-accommodated lenses) conditions. Then, we simulated computationally in vivo conditions on the obtained ex vivo lenses geometry (assuming that just the portion of the lens within a given pupil is available), and estimated the full shape using eigenlenses. The mean absolute error (MAE) between estimated and measured lens' diameters and volumes were MAE = 0.26 ± 0.18 mm and MAE = 7.0 ± 4.5 mm3, respectively. Furthermore, we concluded that the estimation error between measured and estimated lenses did not depend on the accommodative state (change in power due to stretching), and thus eigenlenses are also useful for the full shape estimation of in vivo dis-accommodated lenses.

Relationship of the cornea and globe dimensions to the changes in adult human crystalline lens diameter, thickness and power with age

It is well known that human crystalline lens shape, dimensions and optical properties change throughout life and influence whole eye refraction. However, it is not clear if lens properties are associated with other ocular parameters. The purpose of the present study was to investigate the relationship of corneal and external globe dimensions with adult lens diameter (LD), lens thickness (LT) and lens power (LP) in order to determine if external factors influence lens properties. Postmortem human eyes (n = 66, age = 20-78 years) were obtained from the Ramayamma International Eye Bank, Hyderabad, India. Globe antero-posterior length (GAPL) and mean (average of horizontal and vertical) diameters of cornea (MCD) and globe (MGD) were measured using digital calipers. Eyes were dissected to produce ocular structures that contain the lens maintained in its accommodating framework, including intact zonules, ciliary body and sections of sclera. Specimens were mounted in a mechanical lens stretching system. LD, LT and LP were measured using high magnification retro-illumination photography, slit illumination photography and Scheiner principle-based optical system respectively in the unstretched (accommodated) state. Relationships between external globe and corneal dimensions and LD, LT or LP were assessed by multiple regression analysis. Age (0.012 ± 0.003 mm/year; p<0.001) and GAPL (0.185 ± 0.045 mm/mm; p<0.001) were significant (p<0.0001) predictors of LD. After adjusting for age-related increases, LD appears to be positively correlated with GAPL. Age (0.010 ± 0.004 mm/year; p = 0.009) and GAPL (-0.143 ± 0.060 mm/mm; p = 0.02) were significant (p = 0.001) predictors of LT. After adjusting for the age-related increase, LT appears to be negatively correlated with GAPL. Only age was a significant predictor of LP (-0.26 ± 0.04 D/year; p<0.001). The results suggest that, apart from aging, lens diameter and thickness are dependent on the anteroposterior length of the eye globe. Lens power is not influenced by globe dimensions.

Morphometric analysis of in vitro human crystalline lenses using digital shadow photogrammetry

There is a great need for accurate biometric data on human lenses. To meet this, a compact tabletop optical comparator, the minishadowgraph, was built for measuring isolated eye lens shape and dimensions while the lens was fully immersed in supporting medium. The instrument was based around a specially designed cell and an illumination system which permitted image recording in both sagittal and equatorial (coronal) directions. Data were acquired with a digital camera and analyzed using a specially written MATLAB program as well as by manual measurements in image analysis software. The possible effect of lens orientation and gravity on the dimensions was examined by measuring dimensions with anterior or posterior surfaces up and by measuring lenses with calipers after removal from the minishadowgraph cell. Dimensions, curvatures and shape factors were obtained for 134 fully accommodated lenses ranging in age from birth to 88 years postnatal. Of these, 41 were from donors aged under 20 years, ages which are generally of limited availability. Thickness and diameter showed the same age-related trends described in previous studies but, for the lenses measured in air, age-dependent differences were observed in thickness (-5 to 0%) and diameter (+5 to 0%), consistent with gravitational sag. Anterior and posterior radii of curvature of the central 3 or 6 mm, depending on lens diameter, increase with age, with the anterior increase greater than the posterior. The anterior surface shape of the neonatal lens is that of a prolate ellipse and the posterior, an oblate ellipse. Both surfaces become hyperbolic after age 20. The data presented here on dimensions, shape and sagging will be of great value in assessing age-related changes in the optical and mechanical performance of the lens. In particular, the comprehensive data set from donors aged under 20 years provides a unique and valuable insight to the changes in size and shape during the early dynamic growth period of the lens.

Age-Related Changes to the Three-Dimensional Full Shape of the Isolated Human Crystalline Lens

Purpose

Studying the full shape crystalline lens geometry is important to understand the changes undergone by the crystalline lens leading to presbyopia, cataract, or failure of emmetropization, and to aid in the design and selection of intraocular lenses and new strategies for correction. We used custom-developed three-dimensional (3-D) quantitative optical coherence tomography (OCT) to study age-related changes in the full shape of the isolated human crystalline lens.

Methods

A total of 103 ex vivo human isolated lenses from 87 subjects (age range, 0–56 years) were imaged using a 3-D spectral-domain OCT system. Lens models, constructed after segmentation of the surfaces and distortion correction, were used to automatically quantify central geometric parameters (lens thickness, radii of curvatures, and asphericities of anterior and posterior surfaces) and full shape parameters (lens volume, surface area, diameter, and equatorial plane position). Age-dependencies of these parameters were studied.

Results

Most of the measured parameters showed a biphasic behavior, statistically significantly increasing (radii of curvature, lens volume, surface area, diameter) or decreasing (asphericities, lens thickness) very fast in the first two decades of life, followed by a slow but significant increase after age 20 years (for all the parameters except for the posterior surface asphericity and the equatorial plane position, that remained constant).

Conclusions

Three-dimensional quantitative OCT allowed us to study the age-dependency of geometric parameters of the full isolated human crystalline lens. We found that most of the lens geometric parameters showed a biphasic behavior, changing rapidly before age 20 years and with a slower linear growth thereafter.

The lens growth process

The factors that regulate the size of organs to ensure that they fit within an organism are not well understood. A simple organ, the ocular lens serves as a useful model with which to tackle this problem. In many systems, considerable variance in the organ growth process is tolerable. This is almost certainly not the case in the lens, which in addition to fitting comfortably within the eyeball, must also be of the correct size and shape to focus light sharply onto the retina. Furthermore, the lens does not perform its optical function in isolation. Its growth, which continues throughout life, must therefore be coordinated with that of other tissues in the optical train. Here, we review the lens growth process in detail, from pioneering clinical investigations in the late nineteenth century to insights gleaned more recently in the course of cell and molecular studies. During embryonic development, the lens forms from an invagination of surface ectoderm. Consequently, the progenitor cell population is located at its surface and differentiated cells are confined to the interior. The interactions that regulate cell fate thus occur within the obligate ellipsoidal geometry of the lens. In this context, mathematical models are particularly appropriate tools with which to examine the growth process. In addition to identifying key growth determinants, such models constitute a framework for integrating cell biological and optical data, helping clarify the relationship between gene expression in the lens and image quality at the retinal plane.

Equatorial wrinkles in the human lens capsule

Equatorial wrinkles, or crenations, have been previously observed around the equator in coronal images of the human ocular lens. However, wrinkles are typically not apparent when the lens is viewed from saggital directions. In the current paper, the existence and geometry of these wrinkles is shown to be consistent with a mechanical model of the isolated lens, in which the capsule is held in a state of residual tension by a spatially uniform internal pressure. The occurrence of equatorial wrinkles is therefore seen to be a mechanical consequence of the spheroidal shape of the lens capsule and an excess intralenticular pressure. New observations are made, on post mortem lenses, on the geometric arrangement of these equatorial wrinkles. These observations indicate a well-defined pattern in which wrinkles exists along meridional lines in the equatorial regions of the lens. A preliminary 'puncture test' is used to demonstrate that the residual stresses within the capsule in the equatorial region of the lens are broadly consistent with the proposed mechanical model of the lens capsule. It is suggested that the presence of equatorial wrinkles may have an influence on the mechanical performance of the capsule during the accommodation process.

A dimensionless ordered pull-through model of the mammalian lens epithelium evidences scaling across species and explains the age-dependent changes in cell density in the human lens

We present a mathematical (ordered pull-through; OPT) model of the cell-density profile for the mammalian lens epithelium together with new experimental data. The model is based upon dimensionless parameters, an important criterion for inter-species comparisons where lens sizes can vary greatly (e.g. bovine (approx. 18 mm); mouse (approx. 2 mm)) and confirms that mammalian lenses scale with size. The validated model includes two parameters: β/α, which is the ratio of the proliferation rate in the peripheral and in the central region of the lens; and γ_GZ, a dimensionless pull-through parameter that accounts for the cell transition and exit from the epithelium into the lens body. Best-fit values were determined for mouse, rat, rabbit, bovine and human lens epithelia. The OPT model accounts for the peak in cell density at the periphery of the lens epithelium, a region where cell proliferation is concentrated and reaches a maximum coincident with the germinative zone. The β/α ratio correlates with the measured FGF-2 gradient, a morphogen critical to lens cell survival, proliferation and differentiation. As proliferation declines with age, the OPT model predicted age-dependent changes in cell-density profiles, which we observed in mouse and human lenses.

Measurement of Crystalline Lens Volume During Accommodation in a Lens Stretcher

PURPOSE

To determine if the lens volume changes during accommodation.

METHODS

The study used data acquired on 36 cynomolgus monkey lenses that were stretched in a stepwise fashion to simulate disaccommodation. At each step, stretching force and dioptric power were measured and a cross-sectional image of the lens was acquired using an optical coherence tomography system. Images were corrected for refractive distortions and lens volume was calculated assuming rotational symmetry. The average change in lens volume was calculated and the relation between volume change and power change, and between volume change and stretching force, were quantified. Linear regressions of volume-power and volume-force plots were calculated.

RESULTS

The mean (± SD) volume in the unstretched (accommodated) state was 97 ± 8 mm3. On average, there was a small but statistically significant (P = 0.002) increase in measured lens volume with stretching. The mean change in lens volume was +0.8 ± 1.3 mm3. The mean volume-power and volume-load slopes were -0.018 ± 0.058 mm3/D and +0.16 ± 0.40 mm3/g.

CONCLUSIONS

Lens volume remains effectively constant during accommodation, with changes that are less than 1% on average. This result supports a hypothesis that the change in lens shape with accommodation is accompanied by a redistribution of tissue within the capsular bag without significant compression of the lens contents or fluid exchange through the capsule.

Geometry-invariant GRIN lens: finite ray tracing

The refractive index distribution of the geometry-invariant gradient refractive index lens (GIGL) model is derived as a function of Cartesian coordinates. The adjustable external geometry of the GIGL model aims to mimic the shape of the human and animal crystalline lens. The refractive index distribution is based on an adjustable power-law profile, which provides additional flexibility of the model. An analytical method for layer-by-layer finite ray tracing through the GIGL model is developed and used to calculate aberrations of the GIGL model. The result of the finite ray tracing aberrations of the GIGL model are compared to those obtained with paraxial ray tracing. The derived analytical expression for the refractive index distribution can be employed in the reconstruction processes of the eye using the conventional ray tracing methods. The layer-by-layer finite ray tracing approach would be an asset in ray tracing through a modified GIGL model, where the refractive index distribution cannot be described analytically. Using the layer-by-layer finite ray-tracing method, the potential of the GIGL model in representing continuous as well as shell-like layered structures is illustrated and the results for both cases are presented and analysed.

Axial ultrasound B-scans of the entire eye with a 20-MHz linear array: correction of crystalline lens phase aberration by applying Fermat's principle

In ophthalmic ultrasonography the crystalline lens is known to be the main source of phase aberration, causing a significant decrease in resolution and distortion effects on axial B-scans. This paper proposes a computationally efficient method to correct the phase aberration arising from the crystalline lens, including refraction effects using a bending ray tracing approach based on Fermat's principle. This method is used as a basis to perform eye-adapted beamforming (BF), with appropriate focusing delays for a 128-element 20-MHz linear array in both emission and reception. Implementation was achieved on an in-house developed experimental ultrasound scanning device, the ECODERM. The proposed BF was tested in vitro by imaging a wire phantom through an eye phantom consisting of a synthetic gelatin lens anatomically set up in an appropriate liquid (turpentine) to approach the in vivo velocity ratio. Both extremes of accommodation shapes of the human crystalline lens were investigated. The performance of the developed BF was evaluated in relation to that in homogeneous medium and compared to a conventional delay-and-sum (DAS) BF and a second adapted BF which was simplified to ignore the lens refraction. Global expectations provided by our method with the transducer array are reviewed by an analysis quantifying both image quality and spatial fidelity, as well as the detrimental effects of a crystalline lens in conventional reconstruction. Compared to conventional array imaging, the results indicated a two-fold improvement in the lateral resolution, greater sensitivity and a considerable reduction of spatial distortions that were sufficient to envisage reliable biometry directly in B-mode, especially phakometry.

An analytical method for predicting the geometrical and optical properties of the human lens under accommodation

We present an analytical method to describe the accommodative changes in the human crystalline lens. The method is based on the geometry-invariant lens model, in which the gradient-index (GRIN) iso-indicial contours are coupled to the external shape. This feature ensures that any given number of iso-indicial contours does not change with accommodation, which preserves the optical integrity of the GRIN structure. The coupling also enables us to define the GRIN structure if the radii and asphericities of the external lens surfaces are known. As an example, the accommodative changes in lenticular radii and central thickness were taken from the literature, while the asphericities of the external surfaces were derived analytically by adhering to the basic physical conditions of constant lens volume and its axial position. The resulting changes in lens geometry are consistent with experimental data, and the optical properties are in line with expected values for optical power and spherical aberration. The aim of the paper is to provide an anatomically and optically accurate lens model that is valid for 3 mm pupils and can be used as a new tool for better understanding of accommodation.

Adjustable internal structure for reconstructing gradient index profile of crystalline lens

Employing advanced technologies in studying the crystalline lens of the eye has improved our understanding of the refractive index gradient of the lens. Reconstructing and studying such a complex structure requires models with adaptable internal geometry that can be altered to simulate geometrical and optical changes of the lens with aging. In this Letter, we introduce an optically well-defined, geometrical structure for modeling the gradient refractive index profile of the crystalline lens with the advantage of an adjustable internal structure that is not available with existing models. The refractive index profile assigned to this rotationally symmetric geometry is calculated numerically, yet it is shown that this does not limit the model. The study provides a basis for developing lens models with sophisticated external and internal structures without the need for analytical solutions to calculate refractive index profiles.

Geometry-invariant gradient refractive index lens: analytical ray tracing

A new class of gradient refractive index (GRIN) lens is introduced and analyzed. The interior iso-indicial contours mimic the external shape of the lens, which leads to an invariant geometry of the GRIN structure. The lens model employs a conventional surface representation using a coincoid of revolution with a higher-order aspheric term. This model has a unique feature, namely, it allows analytical paraxial ray tracing. The height and the angle of an arbitrary incident ray can be found inside the lens in a closed-form expression, which is used to calculate the main optical characteristics of the lens, including the optical power and third-order monochromatic aberration coefficients. Moreover, due to strong coupling of the external surface shape to the GRIN structure, the proposed GRIN lens is well suited for studying accommodation mechanism in the eye. To show the power of the model, several examples are given emphasizing the usefulness of the analytical solution. The presented geometry-invariant GRIN lens can be used for modeling and reconstructing the crystalline lens of the human eye and other types of eyes featuring a GRIN lens.

Contribution of the crystalline lens gradient refractive index to the accommodation amplitude in non-human primates: in vitro studies

The purpose of this study was to determine the contribution of the gradient refractive index to the change in lens power in hamadryas baboon and cynomolgus monkey lenses during simulated accommodation in a lens stretcher. Thirty-six monkey lenses (1.4-14.1 years) and twenty-five baboon lenses (1.8-28.0 years) were stretched in discrete steps. At each stretching step, the lens back vertex power was measured and the lens cross-section was imaged with optical coherence tomography. The radii of curvature for the lens anterior and posterior surfaces were calculated for each step. The power of each lens surface was determined using refractive indices of 1.365 for the outer cortex and 1.336 for the aqueous. The gradient contribution was calculated by subtracting the power of the surfaces from the measured lens power. In all lenses, the contribution of the surfaces and gradient increased linearly with the amplitude of accommodation. The gradient contributes on average 65 ± 3% for monkeys and 66 ± 3% for baboons to the total power change during accommodation. When expressed in percent of the total power change, the relative contribution of the gradient remains constant with accommodation and age in both species. These findings are consistent with Gullstrand's intracapsular theory of accommodation.

Accuracy and resolution of in vitro imaging based porcine lens volumetric measurements

There is considerable interest in determining lens volume in the living eye. Lens volume is of interest to understand accommodative changes in the lens and to size accommodative IOLs (A-IOLs) to fit the capsular bag. Some studies have suggested lens volume change during accommodation. Magnetic Resonance Imaging (MRI) is the only method available to determine lens volume in vivo. MRI is, by its nature, relatively low in temporal and spatial resolution. Therefore analysis often requires determining lens volume from single image slices with relatively low resolution on which only simple image analysis methods can be used and without repeated measures. In this study, 7 T MRI scans encompassing the full lens volume were performed on 19 enucleated pig eyes. The eyes were then dissected to isolate and photograph the lens in profile and the lens volumes were measured empirically using a fluid displacement method. Lens volumes were calculated from two- and three-dimensional (2D and 3D) MR and 2D photographic profile images of the isolated lenses using several different analysis methods. Image based and actual measured lens volumes were compared. The average image-based volume of all lenses varied from the average measured volume of all lenses by 0.6%-6.4% depending on the image analysis method. Image analysis methods that use gradient based edge detection showed higher precision with actual volumes (r(2): 0.957-0.990), while threshold based segmentation had poorer correlations (r(2): 0.759-0.828). The root-mean-square (RMS) difference between image analysis based volumes and fluid displacement measured volumes ranged from 8.51 μl to 25.79 μl. This provides an estimate of the error of previously published methods used to calculate lens volume. Immobilized, enucleated porcine eyes permit improved MR image resolution relative to living eyes and therefore improved image analysis methods to calculate lens volume. The results show that some of the accommodative changes in lens volume reported in the literature are likely below the resolution limits of imaging methods used. MRI, even with detailed image analysis methods used here, is unlikely to achieve the resolution required to accurately size an A-IOL to the capsular bag.

Semiautomated analysis of optical coherence tomography crystalline lens images under simulated accommodation

Presbyopia is an age related, gradual loss of accommodation, mainly due to changes in the crystalline lens. As part of research efforts to understand and cure this condition, ex vivo, cross-sectional optical coherence tomography images of crystalline lenses were obtained by using the Ex-Vivo Accommodation Simulator (EVAS II) instrument and analyzed to extract their physical and optical properties. Various filters and edge detection methods were applied to isolate the edge contour. An ellipse is fitted to the lens outline to obtain central reference point for transforming the pixel data into the analysis coordinate system. This allows for the fitting of a high order equation to obtain a mathematical description of the edge contour, which obeys constraints of continuity as well as zero to infinite surface slopes from apex to equator. Geometrical parameters of the lens were determined for the lens images captured at different accommodative states. Various curve fitting functions were developed to mathematically describe the anterior and posterior surfaces of the lens. Their differences were evaluated and their suitability for extracting optical performance of the lens was assessed. The robustness of these algorithms was tested by analyzing the same images repeated times.

Age-dependent variation of the Gradient Index profile in human crystalline lenses

PURPOSE: To reconstruct the gradient index (GRIN) profile of human crystalline lenses ex-vivo using Optical Coherence Tomography (OCT) imaging with an optimization technique and to study the dependence of the GRIN profile with age. METHODS: Cross-sectional images of nine isolated human crystalline lenses with ages ranging from 6 to 72 (post mortem time 1 to 4 days) were obtained using a custom-made OCT system. Lenses were extracted from whole cadaver globes and placed in a measurement chamber filled with preservation medium (DMEM). Lenses were imaged with the anterior surface up and then flipped over and imaged again, to obtain posterior lens surface profiles both undistorted and distorted by the refraction through the anterior crystalline lens and GRIN. The GRIN distribution of the lens was described with three variables by means of power function, with variables being the nucleus and surface index, and a power coefficient that describes the decay of the refractive index from the nucleus to the surface. An optimization method was used to search for the parameters that produced the best match of the distorted posterior surface. RESULTS: The distorted surface was simulated with accuracy around the resolution of the OCT system (under 15 µm). The reconstructed refractive index values ranged from 1.356 to 1.388 for the surface, and from 1.396 to 1.434 for the nucleus. The power coefficient ranged between 3 and 18. The power coefficient increased significantly with age, at a rate of 0.24 per year. CONCLUSION: Optical Coherence Tomography allowed optical, non-invasive measurement of the 2-D gradient index profile of the isolated human crystalline lens ex vivo. The age-dependent variation of the changes is consistent with previous data using magnetic resonance imaging, and the progressive formation of a refractive index plateau.

Distortions of the posterior surface in optical coherence tomography images of the isolated crystalline lens: effect of the lens index gradient

We quantify the posterior surface distortions in optical coherence tomography (OCT) images of isolated crystalline lenses. The posterior radius of curvature and asphericity obtained from OCT images acquired with the beam incident first on the anterior, and then the posterior, surface were compared. The results were compared with predictions of a ray-tracing model which includes the index gradient. The results show that the error in the radius of curvature is within the measurement reproducibility and that it can be corrected by assuming a uniform refractive index. However, accurate asphericity values require a correction algorithm that takes into account the gradient.