Schematic Eye Research Articles

Erratum

"Erratum." Clinical and Experimental Optometry, 78(6), p. 222This article refers to:Invited Review Schematic eyes: history, description and applications

Invited Review Schematic eyes: history, description and applications

A schematic eye is a mathematical or physical model that represents the basic optical features of the real eye. By its nature, it is a simplified version and there is a range of schematic eyes that have been developed which offer different levels of simplicity or complexity, from the simplest reduced version that consists of a single refracting surface to the complex model with four aspheric surfaces and a gradient index lens.Schematic eyes have many applications, particular as teaching aids in optics, optometry, ophthalmology, psychology (vision and visual perception) and visual ergonomics. Besides showing the basic optical structure of the eye, the dimensions and other ocular parameters allow one to trace rays through the eye and determine the position, size and orientation of images and assess the aberrations and hence quality of the retinal image. However, there are many other applications which will be discussed in this article.This article will begin with the historical development of our knowledge of the structure and function of the eye from the first recorded writings of the ancient Greeks. It will then examine the range of schematic eyes currently available, describe their properties and applications and finish with a brief discussion of the probable direction of further developments. (Clin Exp Optom 1995; 78: 5: 176–189)

A four-surface schematic eye of macaque monkey obtained by an optical method

Schematic eyes for four Macaca fascicularis monkeys were constructed from measurements of the positions and curvatures of the anterior and posterior surfaces of the cornea and lens. All of these measurements were obtained from Scheimpflug photography through the use of a ray-tracing analysis. Some of these measurements were also checked (and confirmed) by keratometry and ultrasound. Gaussian lens equations were applied to the measured dimensions of each individual eye in order to construct schematic eyes. The mean total power predicted by the schematic eyes agreed closely with independent measurements based on retinoscopy and ultrasound results, 74.2 ± 1.3 (SEM) vs 74.7 ± 0.3 (SEM) diopters. The predicted magnification of 202 μm/deg in one eye was confirmed by direct measurement of 205 μm/deg for a foveal laser lesion. The mean foveal retinal magnification calculated for our eight schematic eyes was 211 ± (SEM) μm/deg, slightly less than the value obtained by application of the method of Rolls and Cowey [ Experimental Brain Research, 10, 298–310 (1970)] to our eight eyes but just 4% more than the value obtained by application of the method of Perry and Cowey [ Vision Research, 12, 1795–1810 (1985)].

Visual Acuity Modeling Using Optical Raytracing of Schematic Eyes

We developed a methodology to predict changes in visual performance that result from changes in the optical properties of the eye. Exact raytracing of schematic eyes was used to calculate the point spread function and the modulation transfer function of the visual system. The Stiles-Crawford effect, photopic response, diffraction, and the retinal contrast sensitivity are included in the model. Visual acuity was predicted by examining the modulation of the resultant retinal image of a bar target and by determining when the modulation falls below a threshold value. Visual acuity was predicted for refractive errors ranging from 0 to 5 diopters and for pupil diameters ranging from 0.5 to 8 mm. Visual acuity predictions were compared to clinically found Snellen visual acuities and were found to be highly correlated (r2 = .909). This modeling technique shows promise as a means of evaluating clinical and surgical procedures before undertaking clinical trails.

Posterior corneal curvature and its influence on corneal dioptric power.

An algorithm for estimation of the posterior corneal curvature is presented and applied on data from normal and keratoconic eyes. Radius of central posterior corneal curvature are demonstrated to be (mean +/- SD) 6.71 +/- 0.23 mm and 5.58 +/- 0.78 mm in normals and keratoconic eyes, respectively. This corresponds to a ratio between posterior and anterior corneal curvature at 0.85 and 0.83 in the groups mentioned. Both these ratios are significantly smaller than the corresponding ratio at 0.88 in Gullstrand's schematic eye which on corneal dioptric power results in offset errors at 0.20 D and 0.46 D in normal and keratoconic eyes. It is further demonstrated that the ratio is not constant over the corneal surface, resulting in central peripheral dioptric offset errors between 0.2 D and -0.31 D in normals and between 0.46 D and -0.38 D in keratoconic eyes. On corneal dioptric power it is finally shown that a variation in the refractive index of aqueous humor has a 30 times larger influence than a similar variation in corneal refractive index.

Improvements on Littmann's method of determining the size of retinal features by fundus photography

Littmann's formula relating the size of a retinal feature to its measured image size on a telecentric fundus camera film is widely used. It requires only the corneal radius, ametropia, and Littmann's factor q obtained from nomograms or tables. These procedures are here computerized for practitioners' convenience. Basic optical principles are discussed, showing q to be a constant fraction of the theoretical ocular dimension k', the distance from the eye's second principal point to the retina. If the eye's axial length is known, three new methods of determining q become available: (a) simply reducing the axial length by a constant 1.82 mm; (b) constructing a personalized schematic eye, given additional data; (c) ray tracing through this eye to extend calculations to peripheral retinal areas. Results of all these evaluations for 12 subjects of known ocular dimensions are presented for comparison. Method (a), the simplest, is arguably the most reliable. It shows good agreement with Littmann's supplementary procedure when the eye's axial length is known.

The visual system of the Florida garfish, Lepisosteus platyrhincus (Ginglymodi). II. Cornea and lens.

The cornea of the Florida gar, Lepisosteus platyrhincus (Ginglymodi) was examined at the scanning and transmission electron microscopic levels. In addition, the schematic eye of the garfish was revealed by frozen sectioning of the whole orbit in the horizontal and transverse planes. The lens is spherical, obeys Matthiessen's ratio, and is supported by a dorsal suspensory ligament and a ventral retractor lentis muscle. The cornea, devoid of a spectacle, is comprised anteriorly of an epithelium (eight to ten cells thick) and covered by a layer of flattened cells up to 26 microns in diameter. On the scanning electron microscope, these cells appear to be covered in microplicae and microvilli. Beneath the epithelium lies a granular basement membrane abutting a true Bowman's layer, composed of a random arrangement of collagen fibrils with no keratocytes. The corneal stroma constitutes 54% of the total thickness and contains 55-65 collagen fibril lamellae, oriented perpendicular to neighbouring lamellae. Scattered keratocytes, containing large amounts of mitochondria, lipid droplets and glycogen granules lie in between the perpendicularly oriented lamellae. Posterior to the stroma is a thin and partially broken basement membrane (no true Descemet's membrane exists), adjacent to a monolayered endothelium covered in microvilli. In the periphery, an autochthonous layer is found between the stroma and the endothelium. Stromal pigment granules, enveloped in large nucleated cells, act as a non-occlusible yellow filter in the dorsal cornea. Functional correlations are made and the presence and/or thickness of corneal structures discussed in relation to the evolution of the vertebrate cornea.

Myopia following penetrating keratoplasty for keratoconus.

The frequent occurrence of spherical myopia after penetrating keratoplasty for keratoconus is partly the result of the excessive dioptric power of the grafted cornea which occurs when the diameter selected for the donor button is greater than the diameter of the host incision. This excessive power could be reduced by eliminating disparity between the diameters of the graft and host. To determine what proportion of the myopia in these eyes would persist as a result of axial myopia the axial lengths of 60 patients grafted for keratoconus and 25 emmetropic controls were compared. A keratometry, objective refraction, and contact probe ultrasonic biometry were performed on all eyes. A comparison of the results with a representational schematic eye indicated that the mean spherical refractive error of the grafted keratoconic eyes (-4.83 dioptres) was the combined effect of steepness of the corneal graft (mean radius of curvature 7.46 mm) and an abnormally great axial length (mean 24.84 mm). The increased axial length was mainly the result of elongation of the posterior segment of the globe with a small contribution from an increased anterior chamber depth. Though axial myopia is common in keratoconus, a further study of 70 keratoconic eyes that had not been grafted showed no statistically significant correlation between the posterior segment length and the severity of corneal ectasia. These data suggest that even if excessive corneal power is eliminated after penetrating keratoplasty for keratoconus the associated axial myopia would still produce a mean spherical refractive error of at least -2.8 dioptres.

Relationship between axial length and chromatic refraction of the eye

A new mathematical expression is derived to provide an estimate, alpha x, of the axial length of the human schematic eye. Approximations in the expression are made after considering effects in the Gullstrand-Emsley schematic eye. Parameters required to solve the expression are, for two wavelengths, the refractive indices of the humours, the focal power of the cornea and refractive errors in two states of accommodation. Unlike recognized optical methods, additional intraocular parameters are not required. For emmetropic variants of the Gullstrand-Emsley schematic eye with axial lengths between 19 and 27 mm, alpha x is within 0.06 to 0.26 mm of the actual value. Furthermore, alpha x is essentially independent of variations in the refractive indices of the crystalline lens. However, the estimate of alpha x is critically dependent on the accuracy of values assigned to the parameters. For that reason, the model is considered impractical for clinical application.

Relationship between axial length and chromatic refraction of the eye

A new mathematical expression is derived to provide an estimate, ax, of the axial length of the human schematic eye. Approximations in the expression are made after considering effects in the Gullstrand-Emsley schematic eye. Parameters required to solve the expression are, for two wavelengths, the refractive indices of the humours, the focal power of the cornea and refractive errors in two states of accommodation. Unlike recognized optical methods, additional intraocular parameters are not required. For cmmetropic variants of the Gullstrand-Emsley schematic eye with axial lengths between 19 and 27 mm, ax is within 0.06 to 0.26 mm of the actual value. Furthermore, ax is essentially independent of variations in the refractive indices of the crystalline lens. However, the estimate of ax is critically dependent on the accuracy of values assigned to the parameters. For that reason, the model is considered impractical for clinical application.

Construction of a model eye and its applications

Construction details are given of a model eye, based on the Bennett and Rabbetts schematic eye. It incorporates a cornea, lens, and spherical fund us. Distilled water filled the anterior and vitreous chambers. By means of a micrometer screw, the vitreous chamber depth can be precisely varied to produce axial ametropia from +11 to −17D. Readings taken over the greater part of this range with a Topcon Auto refractor RM-A6500 were found to be repeatable and reproducible within ±0.25 D. The model eye was used to investigate the relationship between the actual size of a fundus feature and its photographic image in two different fundus cameras. With a Zeiss Oberkochen camera of lelecentric design the magnification was found lo remain constant whatever the degree of axial ametropia, whereas with a Carl Zeiss Jena camera the magnification varied linearly with amelropia. A technique developed by Littmann for determining the actual size of a retinal feature when using a fundus camera of telecentric design is discussed briefly.

Construction of a model eye and its applications

Construction details are given of a model eye, based on the Bennett and Rabbetts schematic eye. It incorporates a cornea, lens, and spherical fundus. Distilled water filled the anterior and vitreous chambers. By means of a micrometer screw, the vitreous chamber depth can be precisely varied to produce axial ametropia from +11 to -17 D. Readings taken over the greater part of this range with a Topcon Autorefractor RM-A6500 were found to be repeatable and reproducible within +/- 0.25 D. The model eye was used to investigate the relationship between the actual size of a fundus feature and its photographic image in two different fundus cameras. With a Zeiss Oberkochen camera of telecentric design the magnification was found to remain constant whatever the degree of axial ametropia, whereas with a Carl Zeiss Jena camera the magnification varied linearly with ametropia. A technique developed by Littmann for determining the actual size of a retinal feature when using a fundus camera of telecentric design is discussed briefly.

The effect of accommodation on retinal image size.

The apparent size of an object is diminished when accommodation of the eye moves inward to a position closer to the observer than to a viewed object. This phenomenon is called accommodation micropsia. Using schematic eyes, we investigated change in retinal image size caused by a change in accommodation. The use of schematic eyes is also discussed and is justified. The calculated magnitude of this diminution for four schematic eyes ranged from unity at infinity to a maximum of 0.98 (-2%) at about 12.0 diopters (D). For distances at which accommodation micropsia is typically observed (about 2.0 D), retinal minification is less than 0.997 (-0.3%). Thus changes in the size of the retinal image attributable to accommodation are virtually negligible when compared with the observed reduction of 3% to 33%. This suggests that accommodation micropsia is mediated almost entirely by processes other than those involving the optics of the eye.

Optical performance of the super-reversed intraocular lens

We compared the aniseikonia, Seidel aberrations, spot diagrams, and peripheral refractive power errors of a schematic eye with a super-reversed intraocular lens (IOL) and with conventional IOLs. The results did not indicate that the super-reversed IOL performs better optically than the other IOLs, primarily because of its poorer on-axis image performance. However, this disadvantage can be overcome by aspherizing the super-reversed IOL.

Normal development of refractive state and ocular component dimensions in the tree shrew ( Tupaia belangeri)

The normal development of refractive state, ocular components and simple visually-guided behaviors was examined in maternally-reared tree shrews. Six groups consisting of 5 animals each were anesthetized and examined after 0, 15, 30, 45, 60 and 75 days of normal binocular visual exposure. Measures in the 75-day group provided values for an improved schematic eye of the tree shrew. Cycloplegic refraction showed a marked hyperopia (+25 D) at eye opening which decreased rapidly during the first 15 days of visual exposure and stabilized near the value (+5 D) expected in an eye of this axial length (approx. 7.8 mm). Corneal radius increased slightly during development. Anterior segment depth, measured by A-scan ultrasonography, seemed to complete most of its development at an earlier age (15-30 days of visual exposure) than did other ocular parameters. Lens thickness increased steadily throughout development. Vitreous chamber depth increased rapidly until 15 days of visual exposure, and then decreased because the lens thickness increased more rapidly than axial length. Crude orienting to, and following of, large objects developed shortly after eye opening (median age at onset, 5 and 6 days, respectively). Triggered visual placing responses developed at about the same time that the refractive state completed the rapid drop from highly hyperopic values. The slowed rate of ocular development after 15 days of visual exposure may be related to increased retinal activity that is permitted by neural maturation and by the presence of a relatively well-focussed retinal image. The increased activity may influence the final dimensions of the eye to coordinate the axial length with the focal length of the eye.

The spherical aberration of aphakic eyes corrected with intra‐ocular lenses

The spherical aberration of eyes corrected with intra-ocular lenses is investigated using a model eye with realistic levels of corneal asphericity. The results indicate that the aberration is intermediate between that of paraxial schematic eyes and real eyes. By using standard optical aberration theory, it is shown that for a plano-convex lens with the curved surface facing the cornea, the aberration is similar to that of normal phakic eyes and therefore probably too low to be of any clinical significance. However, for other lens orientations or designs, the level of aberration is usually higher and may lead to a refractive error varying with pupil size and a loss of acuity with large pupil diameters.

Ocular dimensions and schematic eyes of freshwater and sea turtles

Measurements were made of the ocular dimensions from living and frozen eyes of one species of freshwater turtle, Pseudemys scripta elegans, and of three species of marine turtles, Chelonia mydas, Dermochelys cariacea, and Eretmochelys imbricata. Estimates of refractive error by retinoscopy were also obtained with eyes in air and under water. The results suggest that unaccommodated eyes of all four species are approximately emmetropic in air but strongly hyperopic in water. Schematic eyes were calculated for each species in both air and water.

Effect of Iris Displacement on Oblique Astigmatism in Aphakic Eyes

Aphakic eyes possess considerably less oblique astigmatism than phakic eyes. The possibility that posterior iris displacement after lens extraction accounts for the observed reduction in oblique astigmatism is tested in this study. Oblique astigmatism was calculated for a phakic and an aphakic schematic eye. With no iris displacement little reduction of oblique astigmatism occurred contrary to experimental findings. However, introduction of varying levels of iris displacement reduced this aberration to values found experimentally. Posterior iris displacement (mean = 1.19 mm) was confirmed in vivo by pre- and postoperative anterior chamber pachometry in eight patients.

Optimal Astigmatism to Enhance Depth of Focus after Cataract Surgery

A small amount of myopic astigmatism can enhance the depth of focus of the pseudophakic eye, optimally providing at least 20/30 visual acuity for both near and distance fixation. For given spherocylindrical refractive errors and fixation distances, the cross-sectional area of Sturm's conoid at the retina was calculated for a schematic eye. These data were used to determine the optimal astigmatic error needed to obtain maximum depth of focus and least theoretical blur for any given spherical equivalent refractive error. Optimal depth of focus was obtained when the plus cylindrical component equaled negative sphere - 0.25 diopters. The near and distance visual acuities of ten pseudophakic patients with induced refractive errors were highly correlated with this model. Low myopic astigmatism after cataract surgery may represent an alternative to multifocal intraocular lenses by providing spectacle independence.

Determination of the nodal point position in the pseudophakic eye

The optics of the schematic eye have been described by Gullstrand and Listing (System of Ophthalmology (ed. H. Kimpton), Vol. 5, C. V. Mosby, London (1970], data being averaged from a series of adult eyes with a crystalline lens. A ray of light from the height of any given object passing, undeviated, through the nodal point will determine the height of its image on the retina. The nodal point positions, therefore, are fundamental in determining image size, and any eye-to-eye variation in a patient will control the amount of aniseikonia. The optics of the pseudophakic eye are described together with a scientific method for determining the nodal point positions, and the implications are discussed.