Aspheric curvatures for the human lens

Prediction of Subjective Refraction From Anterior Corneal Surface, Eye Lengths, and Age Using Machine Learning Algorithms

Purpose

To develop a machine learning regression model of subjective refractive prescription from minimum ocular biometry and corneal topography features.

Methods

Anterior corneal surface parameters (Zernike coefficients and keratometry), axial length, anterior chamber depth, and age were posed as features to predict subjective refractions. Measurements from 355 eyes were split into training (75%) and test (25%) sets. Different machine learning regression algorithms were trained by 10-fold cross-validation, optimized, and tested. A neighborhood component analysis provided features’ normalized weights in predictions.

Results

Gaussian process regression algorithms provided the best models with mean absolute errors of around 1.00 diopters (D) in the spherical component and 0.15 D in the astigmatic components.

Conclusions

The normalized weights showed that subjective refraction can be predicted by only keratometry, age, and axial length. Increasing the topographic description detail of the anterior corneal surface implied by a high-order Zernike decomposition versus adjustment to a spherocylindrical surface is not reflected as improved subjective refraction prediction, which is poor, mainly in the spherical component. However, the highest achievable accuracy differs by only 0.75 D from that of other works with a more exhaustive eye refractive elements description. Although the chosen parameters may have not been the most efficient, applying machine learning and big data to predict subjective refraction can be risky and impractical when evaluating a particular subject at statistical extremes.

Translational Relevance

This work evaluates subjective refraction prediction by machine learning from the anterior corneal surface and ocular biometry. It shows the minimum biometric information required and the highest achievable accuracy.

RESUMEN

Objetivo

El desarrollo de un modelo de regresión de aprendizaje automático prescripción refractiva subjetiva a partir de las características mínimas de la biometría ocular y la superficie corneal.

Métodos

Los parámetros de la superficie corneal anterior (coeficientes de Zernike y queratometría), además de longitudes axiales y de cámara anterior, edades y las refracciones subjetivas no ciclopléjicas de 355 ojos se dividieron en un conjunto de entrenamiento (75%) y otro de test (25%) y se entrenaron diferentes algoritmos de regresión de aprendizaje automático mediante validación cruzada 10 veces, se optimizaron y se probaron sobre el conjunto test.

Resultados

Los algoritmos de regresión del proceso gaussiano proporcionaron los mejores modelos con un error absoluto medio fue de alrededor de 1.00 D en el componente esférico y de 0.25 D en los componentes astigmáticos.

Conclusiones

Los pesos normalizados mostraron que la refracción subjetiva puede predecirse utilizando únicamente la queratometría, la edad y la longitud axial como características. El aumento del detalle de la descripción topográfica de la superficie corneal anterior que supone una descomposición de Zernike de alto orden frente al ajuste a una superficie esferocilíndrica realizado por queratometría no se refleja en una mejora de la predicción de la refracción subjetiva, que es pobre, en cualquier caso, principalmente en el componente esférico. Sin embargo, la máxima precisión alcanzada difiere en sólo 0,75 D de la de otros trabajos con una descripción más exhaustiva de los elementos refractivos del ojo. De todos modos, el aprendizaje automático y los datos masivos aplicados a la predicción de la refracción subjetiva pueden ser arriesgados y poco prácticos cuando se evalúa a un sujeto concreto en los extremos estadísticos, aunque los parámetros elegidos puedan no haber sido los más ineficaces.

Relevancia Traslativa

El trabajo evalúa la predicción de la refracción subjetiva mediante aprendizaje automático a partir de la superficie corneal anterior y la biometría ocular, mostrando la mínima información biométrica requerida y la máxima precisión alcanzable.

Customized eye modeling for optical quality assessment in myopic femto-LASIK surgery

Refractive surgery is recognized as an effective method for myopia treatment, but it can induce night vision disturbances such as glare. We present an eye modeling method for the optical quality assessment in response to the structural changes in the eyes by femto-LASIK surgery. Customized eye models were built from the measurements of 134 right eyes pre- and post-operatively. Optical performance was evaluated using spot diagrams, point spread functions (PSFs), modulation transfer functions (MTFs), and chromatic aberrations at various fields (0°-30°), different pupil diameters (2-6 mm), and initial myopias (- 1.25 to - 10.5 D). Pupil size and initial myopia are the two major factors that affect visual performance of post-operative eyes. The results of spot diagrams, PSFs, and MTFs indicated that post-operative visual performance deteriorated as the visual field and pupil size increased, and it was significantly influenced by initial myopia. Post-operative chromatic aberrations were also affected by initial myopia. As pupil size increased, the post-operative longitudinal chromatic aberrations tended to decrease slightly, while the transverse chromatic aberrations remained similar. The use of eye modeling for refractive surgery assessment could possibly provide a more personalized surgical approach, could improve the prediction accuracy of refractive surgery outcomes, and promote the invention and development of better surgical methods.

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.

Ray tracing 3D spectral scenes through human optics models

Scientists and engineers have created computations and made measurements that characterize the first steps of seeing. ISETBio software integrates such computations and data into an open-source software package. The initial ISETBio implementations modeled image formation (physiological optics) for planar or distant scenes. The ISET3d software described here extends that implementation, simulating image formation for three-dimensional scenes. The software system relies on a quantitative computer graphics program that ray traces the scene radiance through the physiological optics to the retinal irradiance. We describe and validate the implementation for several model eyes. Then, we use the software to quantify the impact of several physiological optics parameters on three-dimensional image formation. ISET3d is integrated with ISETBio, making it straightforward to convert the retinal irradiance into cone excitations. These methods help the user compute the predictions of optics models for a wide range of spatially rich three-dimensional scenes. They can also be used to evaluate the impact of nearby visual occlusion, the information available to binocular vision, or the retinal images expected from near-field and augmented reality displays.

Schematic eye models to mimic the behavior of the accommodating human eye

A simplified version of the human eye is known as a schematic eye model. Since the first attempts in the middle of the 19th century, numerous approaches describing new schematic eye models have been introduced. Some are able to describe the accommodation ability of the human eye. Accommodated schematic eyes could be of great interest because they explain the functionality of the human eye and they are easy to use for research purposes. Purposes include the design and testing of multifocal ophthalmic intraocular lenses, the evaluation of the effect of optical aberrations on retinal image quality, and the study of the optical performance of the eye at different distances after some refractive surgical procedures. This paper reviews and summarizes the most important features and details of accommodated schematic eye models that have been proposed in the past years.

The eye lens: age-related trends and individual variations in refractive index and shape parameters

The eye lens grows throughout life by cell accrual on its surface and can change shape to adjust the focussing power of the eye. Varying concentrations of proteins in successive cell layers create a refractive index gradient. The continued growth of the lens and age-related changes in proteins render it less able to alter shape with loss of capacity by the end of the sixth decade of life. Growth and protein ageing alter the refractive index but as accurate measurement of this parameter is difficult, the nature of such alterations remains uncertain. The most accurate method to date for measuring refractive index in intact lenses has been developed at the SPring-8 synchrotron. The technique, based on Talbot interferometry, has an X-ray source and was used to measure refractive index in sixty-six human lenses, aged from 16 to 91 years. Height and width were measured for forty-five lenses. Refractive index contours show decentration in some older lenses but individual variations mask age-related trends. Refractive index profiles along the optic axis have relatively flat central sections with distinct micro-fluctuations and a steep gradient in the cortex but do not exhibit an age-related trend. The refractive index profiles in the equatorial aspect show statistical significance with age, particularly for lenses below the age of sixty that had capacity to alter shape in vivo. The maximum refractive index in the lens centre decreases slightly with age with considerable scatter in the data and there are age-related variations in sagittal thickness and equatorial height.

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.

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

PURPOSE

To develop an age-dependent mathematical model of the zero-order shape of the isolated ex vivo human crystalline lens, using one mathematical function, that can be subsequently used to facilitate the development of other models for specific purposes such as optical modeling and analytical and numerical modeling of the lens.

METHODS

Profiles of whole isolated human lenses (n=30) aged 20-69, were measured from shadow-photogrammetric images. The profiles were fit to a 10th-order Fourier series consisting of cosine functions in polar-co-ordinate system that included terms for tilt and decentration. The profiles were corrected using these terms and processed in two ways. In the first, each lens was fit to a 10th-order Fourier series to obtain thickness and diameter, while in the second, all lenses were simultaneously fit to a Fourier series equation that explicitly include linear terms for age to develop an age-dependent mathematical model for the whole lens shape.

RESULTS

Thickness and diameter obtained from Fourier series fits exhibited high correlation with manual measurements made from shadow-photogrammetric images. The root-mean-squared-error of the age-dependent fit was 205 microm. The age-dependent equations provide a reliable lens model for ages 20-60 years.

CONCLUSION

The contour of the whole human crystalline lens can be modeled with a Fourier series. Shape obtained from the age-dependent model described in this paper can be used to facilitate the development of other models for specific purposes such as optical modeling and analytical and numerical modeling of the lens.

Shape of the isolated ex-vivo human crystalline lens

PURPOSE

To develop an age-dependent mathematical model of the isolated ex-vivo human crystalline lens shape to serve as basis for use in computational modeling.

METHODS

Profiles of whole isolated human lenses (n=27) aged 6 to 82, were measured from shadow-photogrammetric images. Two methods were used to analyze the lenses. In the two curves method (TCM) the anterior and posterior surfaces of the lens were fit to 10th-order even polynomials and in the one curve method (OCM) the contour of one half-meridional section of the lens was fit to 10th-order polynomials. The age-dependence of the polynomial coefficients was assessed. The analysis was used to produce an age-dependent polynomial model of the whole lens shape.

RESULTS

The root mean squared errors for the fits ranged from 11 to 70 microm for the OCM, 9 to 27 microm for the posterior surface of the TCM and 8 to 134 microm for the anterior surface of the TCM. The coefficients of the OCM did not display a significant trend with age. The 2nd-, 6th- and 10th-order coefficients of the anterior surface of the TCM decreased with age while the 8th-order coefficient increased. For the posterior surface of the TCM, the 8th-order coefficient significantly decreased with age and the 10th-order coefficient increased. The age-dependent equations of both the models provide a reliable model from age 20 to 60. The OCM model can be used for lenses older than 60 as well.

CONCLUSION

The shape of the whole human crystalline lens can be accurately modeled with 10th-order polynomial functions. These models can serve to improve computational modeling, such as finite element (FE) modeling of crystalline lenses.

The Optical Design of the Human Eye: a Critical Review

Cornea, lens and eye models are analyzed and compared to experimental findings to assess properties and eventually unveil optical design principles involved in the structure and function of the optical system of the eye. Models and data often show good match but also some paradoxes. The optical design seems to correspond to a wide angle lens. Compared to conventional optical systems, the eye presents a poor optical quality on axis, but a relatively good quality off-axis, thus yielding higher homogeneity for a wide visual field. This seems the result of an intriguing combination of the symmetry design principle with a total lack of rotational symmetry, decentrations and misalignments of the optical surfaces.

Optical power of the isolated human crystalline lens

PURPOSE

To characterize the age dependence of isolated human crystalline lens power and quantify the contributions of the lens surfaces and refractive index gradient.

METHODS

Experiments were performed on 100 eyes of 73 donors (average 2.8 +/- 1.6 days postmortem) with an age range of 6 to 94 years. Lens power was measured with a modified commercial lensmeter or with an optical system based on the Scheiner principle. The radius of curvature and asphericity of the isolated lens surfaces were measured by shadow photography. For each lens, the contributions of the surfaces and the refractive index gradient to the measured lens power were calculated by using optical ray-tracing software. The age dependency of these refractive powers was assessed.

RESULTS

The total refractive power and surface refractive power both showed a biphasic age dependency. The total power decreased at a rate of -0.41 D/y between ages 6 and 58.1, and increased at a rate of 0.33D/y between ages 58.1 and 82. The surface contribution decreased at a rate of -0.13 D/y between ages 6 and 55.2 and increased at a rate of 0.04 D/y between ages 55.2 and 94. The relative contribution of the surfaces increased by 0.17% per year. The equivalent refractive index also showed a biphasic age dependency with a decrease at a rate of -3.9 x 10(-4) per year from ages 6 to 60.4 followed by a plateau.

CONCLUSIONS

The lens power decreases with age, due mainly to a decrease in the contribution of the gradient. The use of a constant equivalent refractive index value to calculate lens power with the lens maker formula will underestimate the power of young lenses and overestimate the power of older lenses.

Wide-field schematic eye models with gradient-index lens

We propose a wide-field schematic eye model, which provides a more realistic description of the optical system of the eye in relation to its anatomical structure. The wide-field model incorporates a gradient-index (GRIN) lens, which enables it to fulfill properties of two well-known schematic eye models, namely, Navarro's model for off-axis aberrations and Thibos's chromatic on-axis model (the Indiana eye). These two models are based on extensive experimental data, which makes the derived wide-field eye model also consistent with that data. A mathematical method to construct a GRIN lens with its iso-indicial contours following the optical surfaces of given asphericity is presented. The efficiency of the method is demonstrated with three variants related to different age groups. The role of the GRIN structure in relation to the lens paradox is analyzed. The wide-field model with a GRIN lens can be used as a starting design for the eye inverse problem, i.e., reconstructing the optical structure of the eye from off-axis wavefront measurements. Anatomically more accurate age-dependent optical models of the eye could ultimately help an optical designer to improve wide-field retinal imaging.

Adaptive model of the gradient index of the human lens. I. Formulation and model of aging ex vivo lenses

A simple, parametric adaptive model of the refractive index distribution of the ex vivo crystalline lens is presented. It assumes conicoid (or nonrevolution quadric in 3D) iso-indical surfaces, concentric with the external surfaces of the lens. The model uses a minimum number of internal structural parameters, while the shape of the iso-indical surfaces adapts automatically to the external geometry. In this way, it is able to adapt and fit individual distributions as well as adapt to the changes of the lens shape and structure with age and accommodation. The model is fit to experimental data for individual eyes spanning ages 7 to 82 years, where for each eye the crystalline lens dimensions and iso-indical index data are known. The analysis demonstrates that only one age-dependent structural parameter is needed to replicate the internal iso-indical index structure, given age-dependent models for the external surfaces. An age-dependent-parameter global model is derived and is shown to predict age-dependent changes in the ex vivo lens power and longitudinal spherical aberration with age.

In vitro dimensions and curvatures of human lenses

The purpose of this study was to determine dimensions and curvatures of excised human lenses using the technique of shadowphotogrammetry. A modified optical comparator and digital camera were used to photograph magnified sagittal and coronal lens profiles. Equatorial diameter, anterior and posterior sagittal thickness, anterior and posterior curvatures, and shape factors were obtained from these images. The data were used to calculate lens volumes, which were compared with the lens weights. Measurements were made on 37 human lenses ranging in age from 20 to 99 years. These showed that lens dimensions and the anterior radius of curvature increase linearly throughout adult life while posterior curvature remains constant. The relative shape (or aspect ratio) of the posterior lens is unchanged through adult life since both equatorial diameter and posterior thickness increase at the same rate. The ratio of anterior thickness to posterior thickness is constant at 0.70. It is suggested that in vivo forces alter the apparent location of the lens equator, that the in vitro lens shape corresponds to the maximally accommodated shape in vivo and that the shapes of the accommodated and unaccommodated lens progressively converge toward each other due to lens growth with age, with a convergence point located near the age of total loss of accommodation (55-60 years). Together, these observations provide additional support for the Helmholtz theory of accommodation.

On the prediction of optical aberrations by personalized eye models

PURPOSE

The purpose of this study is to develop and analyze a method to obtain optical schematic models of individual eyes. Each model should be able to reproduce the measured monochromatic wave aberration with high fidelity.

METHODS

First, we choose a generic eye model as the input guess and then apply a two-stage customization procedure. Stage 1 consists of replacing, in the initial generic model, those anatomic and optical parameters with experimental data measured on the eye under analysis. The set of experimental data was that provided by a standard clinical preoperative examination, namely lens topography, ultrasound biometry, and total wave aberration. Then, the second stage is to find the unknown lens structure that would reproduce the measured wave aberration through optical optimization. Two totally different initial eye models have been compared; one considers a simpler constant refractive index for the lens, whereas the second model has a gradient-index (GRIN) lens.

RESULTS

This automatic customization method has been applied to 19 eyes with different degrees of spherical ametropia (from +0.4 D to -8 D). Two models have been obtained for each eye (constant and gradient index lens). The results were highly satisfactory, with 100% convergence, and with average RMS prediction errors approximately lambda/100. This is one order of magnitude lower than typical measurement errors. The models with a constant refractive index lens tended to overestimate surface curvatures, whereas for the GRIN model, lens surfaces were too flat.

CONCLUSIONS

The proposed method is highly efficient and robust giving a high-fidelity reproduction of the wavefront in all cases attempted so far. Limitations found in reproducing the geometry of the lens seem to be associated with the use of inaccurate models of its refractive index.

Change in shape of the aging human crystalline lens with accommodation

The objective was to measure the change in shape of the aging human crystalline eye lens in vivo during accommodation. Scheimpflug images were made of 65 subjects between 16 and 51 years of age, who were able to accommodate at least 1D. The Scheimpflug images were corrected for distortion due to the geometry of the camera and the refraction of the cornea and anterior lens surface, which is necessary to determine the real shape of the lens. To ensure accurate correction for the refraction of the anterior lens surface, the refractive index of the crystalline lens must be determined. Therefore, axial length was also measured, which made it possible to calculate the equivalent refractive index of the lens and possible changes in this index during accommodation. The results show that during accommodation there is a decrease in both the anterior and the posterior radius of the lens, although the change in mm per diopter of the latter is much smaller. The increase in lens thickness with accommodation is higher than the decrease in the anterior chamber depth, indicating that the posterior lens surface moves backwards with accommodation. During accommodation the anterior lens surface becomes more hyperbolic. Furthermore, an increase in the equivalent refractive index during accommodation was determined.

New optical scheme for a polarimetric-based glucose sensor

A new optical scheme to detect glucose concentration in the aqueous humor of the eye is presented. The ultimate aim is to apply this technique in designing a new instrument for, routinely and frequently, noninvasively monitoring blood glucose levels in diabetic patients without contact (no index matching) between the eye and the instrument. The optical scheme exploits the Brewster reflection of circularly polarized light off of the lens of the eye. Theoretically, this reflected linearly polarized light on its way to the detector is expected to rotate its state of polarization, owing to the presence of glucose molecules in the aqueous humor of a patient's eye. An experimental laboratory setup based on this scheme was designed and tested by measuring a range of known concentrations of glucose solutions dissolved in water. (c) 2004 Society of Photo-Optical Instrumentation Engineers.

Radius of curvature and asphericity of the anterior and posterior surface of human cadaver crystalline lenses

The purpose of this study was to measure the shape of the anterior and posterior surface of human cadaver lenses in situ using a corneal topography system. Measurements were performed on 13 pairs of eyes using the PAR Corneal Topography System (PAR-CTS). The age of the donors ranged from 46 to 93 years, with an average age of 76.4 years. Anterior lens topography was measured after excision of the cornea and iris. Posterior lens topography was measured after sectioning the posterior segment and adherent vitreous. The PAR-CTS files providing raw surface height were exported for analysis. In each surface, 18 meridians separated by 10 degrees were fitted using conic sections to obtain values of the apical radius of curvature (R) and shape factor (p). The average apical radius of curvature and asphericity were R=10.15+/-1.39mm and p=4.27+/-1.39 for the anterior surface and R=-6.25+/-0.79mm and p=-0.64+/-1.85 for the posterior surface. A significant variation of the radius of curvature and shape factor as a function of the meridian angle (lens astigmatism) was found in some lenses. Contrary to previous findings, the anterior lens surface was found to steepen toward the periphery.

Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography

PURPOSE

To obtain the shape of the posterior corneal surface in a healthy population, using Scheimpflug photography corrected for distortion due to the geometry of the Scheimpflug imaging system and the refraction of the anterior corneal surface.

METHODS

The posterior corneas of 83 subjects, ranging in age from 16 to 62 years, were measured in the vertical meridian using corrected Scheimpflug photography. The aspherical shape of the anterior Corneal surface was also determined in conjunction with the correction of Scheimpflug images.

RESULTS

The average radius of the anterior corneal surface was 7.87 +/- 0.27 mm (SD), while the average radius of the posterior corneal surface was 6.40 +/- 0.28 mm (SD). The ratio between the posterior and the anterior radius of curvature was 0.81 +/- 0.02. The asphericity of the anterior and the posterior corneal surfaces was 0.82 +/- 0.18 and 0.62 +/- 0.27, respectively. The asphericity of the posterior corneal surface decreased significantly with age. The posterior/anterior asphericity ratio is also dependent on age and was 0.98 +/- 0.17 at 16 years of age and 0.53 +/- 0.30 at 62 years of age.

CONCLUSION

Corrected Scheimpflug photography is an appropriate technique for measuring the radius and asphericity of the posterior corneal surface. The asphericity of the posterior corneal surface changes with age.

Ostrich ocular optics

The optical structure of the eyes of ostriches (Struthio camelus; Struthionidae; Struthioniformes) was determined by the construction of a schematic eye model for paraxial optics. The eye is large (axial length = 38 mm) and of globose shape with an anterior focal length (posterior nodal distance) of 21.8 mm. The optical design of the eye is such that the lens and cornea contribute equally to its total optical power. Interspecific comparison shows that optically the ostrich eye is a larger scaled version of the eyes of common starlings (Sturnus vulgaris) and an owl (Strix aluco).

Model of the optical system of the human eye during accommodation

A model of the human eye with a gradient index crystalline lens is presented. The crystalline lens shape at different accommodation levels is described by a single function which is a combination of hyperbolic cosine functions and hyperbolic tangent functions. Using the experimental data published in the literature a model of the variations of the external lens shape was created. Formulae for the lens shape parameters and gradient index distribution for different accommodation levels are given.

The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox

Scheimpflug slit images of the crystalline lens are distorted due to the refracting properties of the cornea and because they are obliquely viewed. We measured the aspheric curvature of the lens of 102 subjects ranging in age between 16 and 65 years and applied correction for these distortions. The procedure was validated by measuring an artificial eye and pseudophakic patients with intraocular lenses of known dimensions. Compared to previous studies using Scheimpflug photography, the decrease of the radius of the anterior lens surface with age was smaller, and the absolute value for the radius of the anterior and posterior lens surface was significantly smaller. A slight decrease of the posterior lens radius with age could be demonstrated. Generally, front and back surfaces were hyperbolic. Axial length was measured of 42 subjects enabling calculation of the equivalent refractive index of the lens, which showed a small, but highly significant decrease with age. These new findings explain the lens paradox and may serve as a basis for modelling the refractive properties of the lens.

The thickness of the aging human lens obtained from corrected Scheimpflug images

Commonly, measurements of lens thickness are performed using A-scan ultrasonography or slitlamp Scheimpflug photography. Both techniques have their drawbacks in the study of presbyopia: ultrasonography requires the velocity of sound in the lens which may change with age, whereas Scheimpflug photography requires knowing the refractive index of the lens to enable correction of the photographs for the distortion due to the refraction of the cornea and lens. By combining Scheimpflug photography and axial optical eye-length measurements, we were able to individually correct the Scheimpflug images for distortion and calculate the refractive index and thickness of the human lens. Lens thickness of 90 subjects ranging in age between 16 and 65 years was measured, and an average increase of 24 microm/year was found. This value is consistent with ultrasonographic measurements assuming an age-independent velocity of sound in the lens of 1641 m/s. The posterior lens surface recedes from the cornea with age, and this backward movement does not differ significantly from the forward movement of the anterior lens surface.

Biometric, optical and physical changes in the isolated human crystalline lens with age in relation to presbyopia

The biometric, optical and physical properties of 19 pairs of isolated human eye-bank lenses ranging in age from 5 to 96 years were compared. Lens focal length and spherical aberration were measured using a scanning laser apparatus, lens thickness and the lens surface curvatures were measured by digitizing the lens profiles and equivalent refractive indices were calculated for each lens using this data. The second lens from each donor was used to measure resistance to physical deformation by providing a compressive force to the lens. The lens capsule was then removed from each lens and each measurement was repeated to ascertain what role the capsule plays in determining these optical and physical characteristics. Age dependent changes in lens focal length, lens surface curvatures and lens resistance to physical deformation are described. Isolated lens focal length was found to be significantly linearly correlated with both the anterior and posterior surface curvatures. No age dependent change in equivalent refractive index of the isolated lens was found. Although decapsulating human lenses causes similar changes in focal length to that which we have shown to occur when human lenses are mechanically stretched into an unaccommodated state, the effects are due to nonsystematic changes in lens curvatures. These studies reinforce the conclusion that lens hardening must be considered as an important factor in the development of presbyopia, that age changes in the human lens are not limited to the loss of accommodation that characterizes presbyopia but that the lens optical and physical properties change substantially with age in a complex manner.

Anatomically accurate, finite model eye for optical modeling

There is a need for a schematic eye that models vision accurately under various conditions such as refractive surgical procedures, contact lens and spectacle wear, and near vision. Here we propose a new model eye close to anatomical, biometric, and optical realities. This is a finite model with four aspheric refracting surfaces and a gradient-index lens. It has an equivalent power of 60.35 D and an axial length of 23.95 mm. The new model eye provides spherical aberration values within the limits of empirical results and predicts chromatic aberration for wavelengths between 380 and 750 nm. It provides a model for calculating optical transfer functions and predicting optical performance of the eye.

The optics of the eye-lens and lenticular senescence. A review

Although the lens of the eye is structurally a biological tissue, it functions as an optical element providing one third of the refracting power of the human eye, and a variable focus in younger years. Throughout a life-time the optical properties of the eye-lens alter, resulting in changes in function: there is a gradual depletion of the focussing amplitude from infancy to middle age, and a loss of transmittance in the later decades of life. The optical properties of the lens depend on its power, which in turn is determined by its physical dimensions (curvatures and thickness) and its refractive index as well as transmissivity and the organization of its internal components. The power of the functional lens is, however, modifiable by virtue of the lens being attached via the zonule to the ciliary muscle. The contraction and relaxation of the latter respectively increases and decreases lens power in accordance with innervations determined by the physical distance of external objects to be imaged on the retina. This review will consider many of these features and how alterations in any of them may lead to changes in lenticular function. However, as we have recently devoted a detailed study to presbyopia [1] its mechanism will not be considered here.

A simple view of age-related changes in the shape of the lens of the human eye

Data describing the radius of curvature of the anterior surface of the human lens have been re-examined for (i) far-accommodated lenses and (ii) near-accommodated or excised lenses. It is found that, with increasing age, the curvatures converge to an intermediate value. Taking this together with the body of work on the mechanics of accommodative lens changes, a perspective is suggested which affords a simple view of curvature changes with age. The only datum in the literature is found to fit reasonably well with this perspective.

Age-related response of human lenses to stretching forces

Five human lenses of varying ages were subjected to radial stretching forces which mimic the action of the ciliary muscle in vivo. Although the number of lenses was small, it was found that the human lens becomes more resistant, with age, to radial stretching forces and that lenses over the age of 50 years showed very little response to stretch. In prepresbyopic lenses, stretching has a greater effect on the shape of the anterior surface than it does on the shape of the posterior surface. The five lenses studied showed individual variations in lens size and shape and a trend to increased curvature with age, particularly marked for the anterior surface.

Determination and modeling of the 3-D gradient refractive indices in crystalline lenses

A simple method of constructing 3-D gradient refractive-index profiles in crystalline lenses is proposed. The input data are derived from 2-D refraction measurements of rays in the equatorial plane of the lens. In this paper, the isoindicial contours within the lens are modeled as a family of concentric ellipses; however, other physically more appropriate models may also be constructed. This method is illustrated by using it to model the 3-D refractive-index profile of a bovine lens.

A method for raytracing through schematic eyes with off-axis components

A linear algebraic method for tracing skew rays through aspheric surfaces is described. The advantage of this method is that the refracting surfaces can be translated and rotated with respect to each other. The method is used to investigate the optical factors affecting the location of Purkinje images.

Accommodation-dependent model of the human eye with aspherics

We consider a schematic human eye with four centered aspheric surfaces. We show that by introducing recent experimental average measurements of cornea and lens into the Gullstrand-Le Grand model, the average spherical aberration of the actual eye is predicted without any shape fitting. The chromatic dispersions are adjusted to fit the experimentally observed chromatic aberration of the eye. The polychromatic point-spread function and modulation transfer function are calculated for several pupil diameters and show good agreement with previous experimental results. Finally, from this schematic eye an accommodation-dependent model is proposed that reproduces the increment of refractive power of the eye during accommodation. The variation of asphericity with accommodation is also introduced in the model and the resulting optical performance studied.

A schematic eye for the mouse, and comparisons with the rat

The thicknesses and spheric and aspheric curvatures of the optic components were measured from cross-sections of frozen eyes of C57B1/6J mice. The equivalent refractive index of the crystalline lens was obtained from its back-vertex power in albumin. Refractive indices of the cornea, aqueous and vitreous humors were obtained by refractometry or interferometry at four wavelengths across the visible spectrum. The measurements parallel earlier ones on the hooded rat. The eyes of the mouse and rat differ mainly in size, by a linear scale factor of 1.9-2.0, and only slightly in refractive index. Thus refraction, chromatic aberration, and retinal illumination are easily compared in the two species. An analysis of the contribution of each optical surface to refraction may facilitate extrapolation to other strains of mice. Chromatic aberration is discussed with respect to depth of field and the retinoscopy artefact.

Light distribution on the retina of a wide-angle theoretical eye

In a theoretical eye with spherical and aspheric surfaces, the retinal illumination is calculated if a Ganzfeld luminance field is used. The resulting retinal light distribution is nearly homogeneous over the whole retina. The homogeneity is not much influenced by the size of the optical surfaces. The corresponding retinal area and the luminous flux entering the eye are calculated as functions of the size of the visual field. The values of the length of the light path through the crystalline lens and of the angle of incidence on the retina are described as functions of the angle in the visual field.

Aspheric curvatures, refractive indices and chromatic aberration for the rat eye

Thicknesses and curvatures of the optic components of enucleated eyes were determined in both transverse and sagittal planes. The cornea and lens surfaces are closer approximations to spheres than are the retina and choroid. Refractive indices of the cornea, aqueous and vitreous humours were obtained by two techniques at eight wavelengths across the visible spectrum. Photography of light beams passing through the crystalline lens immersed in albumin gave the "back vertex power", and thus (assuming homogeneity) the "equivalent" refractive index. With this assumption, which is supported by additional modelling, calculations for the eyes of six individual rats show chromatic aberration across the spectrum of 5.8 (5.5-6.0) diopter. Longitudinal chromatic aberration may make a significant contribution to image blur.

Model of the accommodative mechanism in the human eye

The crystalline lens of the age 11 human eye has been modelled mathematically, using simplified assumptions about lens curvature, internal organization and elasticity. From this representation, expressions for description of strain and stress during accommodation have been obtained. Solution of these equations indicates that the lens capsule acts as a force distributor, spreading tension applied by the suspensory apparatus evenly over the surface of the underlying lens material. It also becomes clear that the vitreous body provides an essential support function during the accommodative process. Finally, the relative contribution of lens-associated structures has been determined for five different values of the Poisson ratio. In order for accommodation to occur by relaxation of zonular tension, this value must be greater than 0.38; with an additional constraint of the net axial force equalling zero during a small accommodative change, the Poisson ratio equals 0.46.