Ray tracing through a schematic eye containing second-order (quadric) surfaces using 4 x 4 matrix notation

Back-calculation of keratometer index based on OCT data and raytracing - a Monte Carlo simulation

PURPOSE

This study aims to develop a raytracing-based strategy for calculating corneal power from anterior segment optical coherence tomography data and extracting the individual keratometer index, which converts the corneal front surface radius to corneal power.

METHODS

A large OCT dataset (10,218 eyes of 8,430 patients) from the Casia 2 (Tomey, Japan) was post-processed in MATLAB (MathWorks, USA). Radius of curvature, asphericity of the corneal front and back surface, central corneal thickness and pupil size (aperture) were used to trace a bundle of rays through the cornea and derive the best focus plane. Corneal power was calculated with respect to the corneal front vertex plane, and the keratometer index was back-calculated using corneal power and front surface radius. Keratometer index was analysed in a multivariate linear model.

RESULTS

The averaged resulting keratometer index was 1.3317 ± 0.0017 with a median of 1.3317 and range from 1.3233 to 1.3390. In a univariate model, only the front surface asphericity affected the keratometer index. The multivariate model for modelling the keratometer index using all 6 input parameters performed very well (RMS error: 5.54e-4, R² : 0.90, significance vs. constant model: <0.0001).

CONCLUSIONS

In the classical calculation, the keratometer index used for converting corneal radius to dioptric power uses several model assumptions. As these assumptions are not generally satisfied, corneal power cannot be calculated from corneal front surface radius alone. Considering all 6 input variables, the linear prediction model performs well and can be used if all input parameters are measured with a tomographer.

Prediction model for best focus, power, and spherical aberration of the cornea: Raytracing on a large dataset of OCT data

Purpose

To analyse corneal power based on a large optical coherence tomography dataset using raytracing, and to evaluate corneal power with respect to the corneal front apex plane for different definitions of best focus.

Methods

A large OCT dataset (10,218 eyes of 8,430 patients) from the Casia 2 (Tomey, Japan) was post-processed in MATLAB (MathWorks, USA). Using radius of curvature, corneal front and back surface asphericity, central corneal thickness, and pupil size (aperture) a bundle of rays was traced through the cornea. Various best focus definitions were tested: a) minimum wavefront error, b) root mean squared ray scatter, c) mean absolute ray scatter, and d) total spot diameter. All 4 target optimisation criteria were tested with each best focus plane. With the best-fit keratometer index the difference of corneal power and keratometric power was evaluated using a multivariate linear model.

Results

The mean corneal powers for a/b/c/d were 43.02±1.61/42.92±1.58/42.91±1.58/42.94±1.59 dpt respectively. The root mean squared deviations of corneal power from keratometric power (n_K = 1.3317/1.3309/1.3308/1.3311 for a/b/c/d) were 0.308/0.185/0.171/0.209 dpt. With the multivariate linear model the respective RMS error was reduced to 0.110/0.052/0.043/0.065 dpt (R² = 0.872/0.921/0.935/0.904).

Conclusions

Raytracing improves on linear Gaussian optics by considering the asphericity of both refracting surfaces and using Snell’s law of refraction in preference to paraxial simplifications. However, there is no unique definition of best focus, and therefore the calculated corneal power varies depending on the definition of best focus. The multivariate linear model enabled more precise estimation of corneal power compared to the simple keratometer equation.

[Combination of lens decentration and tilt in phakic and pseudophakic eyes-Optical simulation of defocus, astigmatism and coma]

BACKGROUND AND PURPOSE

The effect of lens decentration and tilt on retinal image quality has been extensively studied in the past in simulations and clinical studies. The purpose of this study was to analyze the effect of combined lens decentration and tilt on the induction of defocus, astigmatism and coma in phakic and pseudophakic eyes.

METHODS

Simulations were performed with Zemax on the Liou-Brennan schematic model eye. Based on the position of the gradient lens the image plane was determined (best focus). The lens was decentered horizontally from -1.0 mm to 1.0 mm in steps of 0.2 mm and tilted with respect to the vertical axis from -10° to 10° in steps of 2° (in total 121 combinations of decentration and tilt). For each combination of decentration and tilt defocus, astigmatism (in 0/180°) and horizontal coma was extracted from wave front error and recorded for a pupil size of 4 mm. After replacement of the gradient lens with an aberration correcting artificial lens implant model with the equatorial plane of the artificial lens aligned to the equatorial plane of the gradient lens, the simulations were repeated for the pseudophakic eye model.

RESULTS

For the lens positioned according to the Liou-Brennan schematic model eye the simulation yielded a defocus of 0.026 dpt/-0.001 dpt, astigmatism of -0.045 dpt/-0.018 dpt, and a coma of -0.015 µm/0.047 µm for phakic/pseudophakic eyes. Maximum values were observed for a horizontal decentration of 1.0 mm and a tilt with respect to the vertical axis of 10° with 1.547 dpt/2.982 dpt for defocus, 0.971 dpt/1.871 dpt for astigmatism, and 0.441 µm/1.209 µm for coma. Maximum negative values occurred in phakic/pseudophakic eyes with -0.293 dpt/-1.224 dpt for defocus, for astigmatism -0.625 dpt/-0.663 dpt and for coma -0.491 µm /-0.559 µm, respectively.

CONCLUSION

In this simulation study the effect of a combination of lens decentration in horizontal direction and tilt with respect to the vertical axis on defocus, astigmatism and horizontal coma was analyzed. The results may help to describe in clinical routine if with a decentered or tilted artificial lens implant the postoperative refraction does not match the target refraction or the resulting astigmatism after cataract surgery is not fully explained by measurement of corneal astigmatism.

A model of the entrance pupil of the human eye

The aperture stop of the iris is subject to refraction by the cornea, and thus an outside observer sees a virtual image: the “entrance pupil” of the eye. When viewed off-axis, the entrance pupil has an elliptical form. The precise appearance of the entrance pupil is a consequence of the anatomical and optical properties of the eye, and the relative positions of the eye and the observer. This paper presents a ray traced model eye that provides the parameters of the entrance pupil ellipse for an observer at an arbitrary location. The model is able to reproduce empirical measurements of the shape of the entrance pupil with good accuracy. I demonstrate that accurate specification of the entrance pupil of a stationary eye requires modeling of corneal refraction, the misalignment of the visual and optical axes, and the non-circularity of the aperture stop. The model, including a three-dimensional ray tracing function through quadric surfaces, is implemented in open-source MATLAB code.

Simplifying numerical ray tracing for two-dimensional non circularly symmetric models of the human eye

Ray tracing is a powerful technique to understand the light behavior through an intricate optical system such as that of a human eye. The prediction of visual acuity can be achieved through characteristics of an optical system such as the geometrical point spread function. In general, its precision depends on the number of discrete rays and the accurate surface representation of each eye's components. Recently, a method that simplifies calculation of the geometrical point spread function has been proposed for circularly symmetric systems [Appl. Opt.53, 4784 (2014)]. An extension of this method to 2D noncircularly symmetric systems is proposed. In this method, a two-dimensional ray tracing procedure for an arbitrary number of surfaces and arbitrary surface shapes has been developed where surfaces, rays, and refractive indices are all represented in functional forms being approximated by Chebyshev polynomials. The Liou and Brennan anatomically accurate eye model has been adapted and used for evaluating the method. Further, real measurements of the anterior corneal surface of normal, astigmatic, and keratoconic eyes were substituted for the first surface in the model. The results have shown that performing ray tracing, utilizing the two-dimensional Chebyshev function approximation, is possible for noncircularly symmetric models, and that such calculation can be performed with a newly created Chebfun toolbox.

Fast ray-tracing of human eye optics on Graphics Processing Units

We present a new technique for simulating retinal image formation by tracing a large number of rays from objects in three dimensions as they pass through the optic apparatus of the eye to objects. Simulating human optics is useful for understanding basic questions of vision science and for studying vision defects and their corrections. Because of the complexity of computing such simulations accurately, most previous efforts used simplified analytical models of the normal eye. This makes them less effective in modeling vision disorders associated with abnormal shapes of the ocular structures which are hard to be precisely represented by analytical surfaces. We have developed a computer simulator that can simulate ocular structures of arbitrary shapes, for instance represented by polygon meshes. Topographic and geometric measurements of the cornea, lens, and retina from keratometer or medical imaging data can be integrated for individualized examination. We utilize parallel processing using modern Graphics Processing Units (GPUs) to efficiently compute retinal images by tracing millions of rays. A stable retinal image can be generated within minutes. We simulated depth-of-field, accommodation, chromatic aberrations, as well as astigmatism and correction. We also show application of the technique in patient specific vision correction by incorporating geometric models of the orbit reconstructed from clinical medical images.

Automatic intraocular lens segmentation and detection in optical coherence tomography images

We present a new algorithm for automatic segmentation and detection of an accommodative intraocular lens implanted in a biomechanical eye model. We extracted lens curvature and position. The algorithm contains denoising and fan correction by a multi-level calibration routine. The segmentation is realized by an adapted canny edge detection algorithm followed by a detection of lens surface with an automatic region of interest search to suppress non-optical surfaces like the lens haptic. The optical distortion of lens back surface is corrected by inverse raytracing. Lens geometry was extracted by a spherical fit. We implemented and demonstrated a powerful algorithm for automatic segmentation, detection and surface analysis of intraocular lenses in vitro. The achieved accuracy is within the expected range determined by previous studies. Future improvements will include the transfer to clinical anterior segment OCT devices.

Irregularity of the posterior corneal surface during applanation using a curved femtosecond laser interface and microkeratome cutting head

PURPOSE

To evaluate the irregularity of the posterior corneal surface and intrastromal dissection during the preparation of donor tissue for Descemet stripping automated endothelial keratoplasty (DSAEK) using a curved interface femtosecond laser and microkeratome.

METHODS

Sixteen human donor corneas unsuitable for transplantation were divided into two groups: a femtosecond (FS) laser group (n=7) using the VisuMax femtosecond laser (Carl Zeiss Meditec) and a microkeratome group (n=9) using the Amadeus II microkeratome (Ziemer Ophthalmic Group). The corneas were fixed on artificial anterior chambers. Horizontal cross-sections were obtained using spectral-domain optical coherence tomography prior to applanation, during applanation, as well as during and after intrastromal dissection at 450-μm corneal depth. The posterior surface and the dissection line were evaluated for irregularity by fitting a second-order polynomial curve using regression analysis and obtaining the root-mean-square error (RMSE). Groups were compared using analysis of variance.

RESULTS

The RMSE of the posterior surface prior to applanation was 9.7 ± 3.1 μm in the FS laser group and 10.2 ± 2.3 μm in the microkeratome group. The RMSE increased to 50.7 ± 9.4 μm and 20.9 ± 6.1 μm during applanation and decreased again to 10.6 ± 1.4 μm and 8.1 ± 1.8 μm after applanation in the FS laser and microkeratome groups, respectively. The RMSE of the intrastromal cut was 19.5 ± 5.7 μm in the FS laser group and 7.7 ± 3.0 μm in the microkeratome group (P<.001).

CONCLUSIONS

Our results show significantly greater irregularity with the curved interface femtosecond laser-assisted cleavage compared to microkeratome-assisted corneal dissection, possibly due to applanation-derived deformation of the posterior cornea.

Customized aspheric IOL design by raytracing through the eye containing quadric surfaces

PURPOSE

The purpose of the present study was to demonstrate a method of how to calculate intraocular lenses with a customized asphericity and how to apply this strategy to clinical examples in cases where biometric data of the cornea (front and back surface topography) as well as distances in the eye are known.

METHODS

(1) we demonstrated an algebraic method for tracing a bundle of rays through a schematic eye containing surfaces which can be represented by 2nd order surfaces (quadric surfaces), and (2) we introduced a strategy for customization of the lens' back surface for compensating the optical path length differences of the rays from object to image in terms of a wave front correction while predefining the lens front surface.

RESULTS

The presented method was applied to three working examples: example 1 referred to a centered optical system with a spherical cornea (front and back surfaces) and a predefined spherical lens front surface, example 2 referred to a centered optical system with aspherical surfaces for the corneal front and back surfaces and a predefined spherical lens front surface, and example 3 referrred to a non-centered system with a decentered aspherical cornea (front and back surface), and a predefined spherical lens front surface. The parameterized ray intersection points with the lens back surface were optimized in terms of equalizing the ray path lengths and a quadric surface was fitted to these ray intersection points to characterize the customized lens. The fitting error, ray spot diagram, and the optical path length of the rays are provided.

CONCLUSION

This simple calculation strategy may be the first step in developing individual aspherical lenses, which have the potential to fully compensate spherical aberrations based on individual measures of the eye.

Magnifications of single and dual element accommodative intraocular lenses: paraxial optics analysis

PURPOSE

Using an analytical approach of paraxial optics, we evaluated the magnification of a model eye implanted with single-element (1E) and dual-element (2E) translating-optics accommodative intraocular lenses (AIOL) with an objective of understanding key control parameters relevant to their design. Potential clinical implications of the results arising from pseudophakic accommodation were also considered.

METHODS

Lateral and angular magnifications in a pseudophakic model eye were analyzed using the matrix method of paraxial optics. The effects of key control parameters such as direction (forward or backward) and distance (0 to 2 mm) of translation, power combinations of the 2E-AIOL elements (front element power range +20.0 D to +40.0 D), and amplitudes of accommodation (0 to 4 D) were tested. Relative magnification, defined as the ratio of the retinal image size of the accommodated eye to that of unaccommodated phakic (rLM(1)) or pseudophakic (rLM(2)) model eyes, was computed to determine how retinal image size changes with pseudophakic accommodation.

RESULTS

Both lateral and angular magnifications increased with increased power of the front element in 2E-AIOL and amplitude of accommodation. For a 2E-AIOL with front element power of +35 D, rLM(1) and rLM(2) increased by 17.0% and 16.3%, respectively, per millimetre of forward translation of the element, compared to the magnification at distance focus (unaccommodated). These changes correspond to a change of 9.4% and 6.5% per dioptre of accommodation, respectively. Angular magnification also increased with pseudophakic accommodation. 1E-AIOLs produced consistently less magnification than 2E-AIOLs. Relative retinal image size decreased at a rate of 0.25% with each dioptre of accommodation in the phakic model eye. The position of the image space nodal point shifted away from the retina (towards the cornea) with both phakic and pseudophakic accommodation.

CONCLUSION

Power of the mobile element, and amount and direction of the translation (or the achieved accommodative amplitude) are important parameters in determining the magnifications of the AIOLs. The results highlight the need for caution in the prescribing of AIOL. Aniso-accommodation or inter-ocular differences in AIOL designs (or relative to the natural lens of the contralateral eye) may introduce dynamic aniseikonia and consequent impaired binocular vision. Nevertheless, some designs, offering greater increases in magnification on accommodation, may provide enhanced near vision depending on patient needs.

Assessment and statistics of surgically induced astigmatism

The aim of the thesis was to develop methods for assessment of surgically induced astigmatism (SIA) in individual eyes, and in groups of eyes. The thesis is based on 12 peer-reviewed publications, published over a period of 16 years. In these publications older and contemporary literature was reviewed(1). A new method (the polar system) for analysis of SIA was developed. Multivariate statistical analysis of refractive data was described(2-4). Clinical validation studies were performed. The description of a cylinder surface with polar values and differential geometry was compared. The main results were: refractive data in the form of sphere, cylinder and axis may define an individual patient or data set, but are unsuited for mathematical and statistical analyses(1). The polar value system converts net astigmatisms to orthonormal components in dioptric space. A polar value is the difference in meridional power between two orthogonal meridians(5,6). Any pair of polar values, separated by an arch of 45 degrees, characterizes a net astigmatism completely(7). The two polar values represent the net curvital and net torsional power over the chosen meridian(8). The spherical component is described by the spherical equivalent power. Several clinical studies demonstrated the efficiency of multivariate statistical analysis of refractive data(4,9-11). Polar values and formal differential geometry describe astigmatic surfaces with similar concepts and mathematical functions(8). Other contemporary methods, such as Long's power matrix, Holladay's and Alpins' methods, Zernike(12) and Fourier analyses(8), are correlated to the polar value system. In conclusion, analysis of SIA should be performed with polar values or other contemporary component systems. The study was supported by Statens Sundhedsvidenskabeligt Forskningsråd, Cykelhandler P. Th. Rasmussen og Hustrus Mindelegat, Hotelejer Carl Larsen og Hustru Nicoline Larsens Mindelegat, Landsforeningen til Vaern om Synet, Forskningsinitiativet for Arhus Amt, Alcon Denmark, and Desirée and Niels Ydes Fond.

simEye: Computer-based simulation of visual perception under various eye defects using Zernike polynomials

We describe a computer eye model that allows for aspheric surfaces and a three-dimensional computer-based ray-tracing technique to simulate optical properties of the human eye and visual perception under various eye defects. Eye surfaces, such as the cornea, eye lens, and retina, are modeled or approximated by a set of Zernike polynomials that are fitted to input data for the respective surfaces. A ray-tracing procedure propagates light rays using Snell's law of refraction from an input object (e.g., digital image) through the eye under investigation (i.e., eye with defects to be modeled) to form a retinal image that is upside down and left-right inverted. To obtain a first-order realistic visual perception without having to model or simulate the retina and the visual cortex, this retinal image is then back-propagated through an emmetropic eye (e.g., Gullstrand exact schematic eye model with no additional eye defects) to an output screen of the same dimensions and at the same distance from the eye as the input object. Visual perception under instances of emmetropia, regular astigmatism, irregular astigmatism, and (central symmetric) keratoconus is simulated and depicted. In addition to still images, the computer ray-tracing tool presented here (simEye) permits the production of animated movies. These developments may have scientific and educational value. This tool may facilitate the education and training of both the public, for example, patients before undergoing eye surgery, and those in the medical field, such as students and professionals. Moreover, simEye may be used as a scientific research tool to investigate optical lens systems in general and the visual perception under a variety of eye conditions and surgical procedures such as cataract surgery and laser assisted in situ keratomileusis (LASIK) in particular.