Realistic simulation of ophthalmic measurements on normal and diseased eyes is presented. We use clinical data of ametropic and keratoconus patients to construct anatomically accurate three-dimensional eye models and simulate the measurement of a streak retinoscope with all the optical elements. The results show the clinical observations including the anomalous motion in high myopia and the scissors reflex in keratoconus. The demonstrated technique can be applied to other ophthalmic instruments and to other and more extensively abnormal eye conditions. It provides promising features for medical training and for evaluating and developing ocular instruments.
© 2007 Optical Society of America
Since the early 1900’s, analytical eye models have been developed to study the optical performance of human eyes. The early eye models that are used most frequently are those of Gullstrand , von Helmholtz , and Le Grand . These models are anatomically accurate to the first order. To describe higher order aberrations, Lotmar  and Navarro and associates [5–7] introduced asphericities to surfaces of the cornea and lens. Al-Ahdali and El-Messiery  and Liou and Brennan  incorporated the gradient refractive index of the lens. These models utilize measured average optical parameters and characteristics of healthy emmetropic adults. Today, with the advance of high-speed computation, personalized eye models using clinic-measured data from patients become achievable [10, 11 ]. Personalized models provide promising features in assisting ocular surgery [12, 13] and in designing customized spectacle-, contact-, or intraocular lens [14, 15]. With supplementary modifications, these models could predict visual changes under specified environmental or physical conditions. Further, the computer simulation of ophthalmic measurements using personalized models offers a comprehensible tool for medical training. In this paper, we demonstrate simulation of streak retinoscope measurements. Various degrees of ammetropic and keratoconus (KC) models are constructed to simulate the ophthalmic observations.
The retinoscope is a clinical standard device for measuring the refractive state of the eye. Although similar in some respects to the classic spot retinoscope, a contemporary streak retinoscope projects a straight-filament image onto a patient’s eye at a distance of 0.5-1.0 meter. The width of the streak projection is adjustable by moving a condenser lens above the filament in the handle of the device. The retinal reflex is observed by the examiner through a peephole on the scope. When moving the streak projection across the patient’s pupil, the reflex of a myopic or hyperopic eye appears to move with or against the projection motion depending on the position of the condenser lens. Simplified geometrical-optical analysis is normally employed to describe the movement of the reflex in relation to the refractive state of the eye [16–18]. Swaine used a simple model to predict further pupil size influence and the intensity profile of the reflex [19–22]. Higher order aberration that is normally considered a difficulty in retinoscopy interpretation has been investigated more carefully in recent years [23, 24]. Rather than the spot retinoscopy, Smith included the consideration of streak orientation and brightness change in the presence of astigmatism . In this paper, we simulate the streak retinoscopic observation of individual patients under both plane- and concave-mirror operations. Ambiguous observation of the so-called “anomalous with-motion” of the high myopia condition is also produced. Further, the famous scissors reflex of a keratoconus eye is simulated for the first time.
2. Method of simulation
The optics code, ZemaxTM (ZEMAX Development Corporation, Bellevue, WA, USA), was used for the simulation. For both general and personalized eye modeling, parameters of the Navarro wide-angle eye model  were used and then modified in portions for the needs of this work. The personalized keratoconus and ametropic models are described in a previous paper  where patients’ topographies are adopted and the two-step, three-variable iteration procedures were performed. Although the posterior corneal surface is also affected in KC patients, the posterior irregularity was omitted in the modeling. The optical influence of irregular posterior surface was estimated to be no more than 10 to 20% of the anterior influence due to the smaller refractive index difference. Minimization of RMS wavefront error was used as the merit function in the optimization process. The general ametropic eye models were approached in a similar manner for the desired refractions without replacing the corneal topographies. A three-mm aperture stop (3.4 mm entrance pupil) was used in the refraction approach. Since the pupil is typically not large for non-mydriatic visible illumination from the retinoscope, the directional retinal reflection of Stiles-Crawford effect [26–28] was omitted in the modeling in this paper.
Figure 1 illustrates the retinoscope elements with corresponding parameters in the computation. From the light source, the elements include a filament light source (0.03 mm × 3 mm), a rectangular aperture (2 mm × 4 mm), a condenser lens (20 mm focal length), a beam splitter with window aperture (10 mm × 14 mm), and a circular peephole (3 mm diameter). The distances between each element are specified in the figure. A movable sleeve was included that allows the examiner to vertically move the lens and change the convergence of the streak projection. The wavelength was set at 555 nm and 0.5 or 0.67 meter working distance was assumed. Coordinate breaks (including coordinate shifts and rotations) were used to move or rotate the streak beam across the entrance pupil of the model eye. Double-pass image analysis through the model eye was performed under the assumption of perfect diffusive retinal reflection/scattering. Multiple reflections and scattering were omitted. An aberration-free imaging system was used to simulate the examiner’s eye behind the peephole. The focus plane of the examiner’s eye was set on the corneal surface of the model eye.
As in the real condition, four effective apertures were involved in this retinoscopic simulation. These apertures were the small aperture in front of the filament, the window on the beam splitter (along both paths), the pupil of the eye (along both paths), and the peephole of the observation. Ray aiming was applied to ensure that all of the vignetting or cut-off effects were encountered when using coordinate breaks.
3. Results and discussion
3.1 Sleeve position: Plane mirror and concave mirror operations
The retinoscope sleeve position defines the plane- and concave-mirror operations. Figure 2 shows the simulation result when the retinoscope sleeve moves vertically across a fifteen mm distance. The corresponding streak projections on the center of patient’s eye are illustrated on the left column. The illustrated eye in each image has a 3.4 mm pupil and an 11 mm iris. Each image is scaled 10 cm by 10 cm. The false colors represent the relative intensity distribution. As the sleeve moves upward, the convergence of projection increases. Because the condenser lens has a focal-length of 20-mm, the filament image is sharply focused at a sleeve height of h=21 mm. The sleeve-down position (h<21mm) corresponds to the “plane mirror” position, and sleeve-up, the “concave mirror” position. When the sleeve moves all the way up, the projection shape tends to reveal the rectangular filament window.
The measurement simulation was performed for five refractive conditions of hyperopia of +2, and myopia of -1, -2, -4, and -6 D (diopter=meter -1), as indicated at the top of each column. The pupil is 3.4 mm diameter. Since the observing distance is d ob=0.5 meter, the retina surface of the -2D eye is conjugate to the window of retinoscope. Neutralization occurs at any sleeve location for this refractive condition.
Notice that the retinal strip-reflex is often in poor contrast and hard to observe if the images are in gray-level instead of the false-color illustration. It is especially so for a refractive condition that is close to neutralization and when the sleeve location is away from h=21 mm. Streak reflex is more easily observed under concave-mirror operation for high myopia and under plane-mirror operation for hyperopia.
In the streak retinocope, the filament is imaged by a condenser lens. The location of the filament image, l, has an important effect and is indicated beside the figure. From the patient’s viewpoint, the light source (filament image) changes with the sleeve position. When the lens is located at the lowest position, 10-mm above the filament, the image of the filament is about 47 mm behind the peephole. As the sleeve moves upward, the light source image rapidly moves farther away from the patient. The light source moves to infinity as the sleeve glides into the 20-mm height position. When the sleeve moves above the 20-mm position, i.e. into the concave-mirror condition, the filament image appears to be on the patient side of the peephole. At h=21 mm, the filament image is at about 10 cm in front of the patient’s eye. When the sleeve is pushed farther upward, the light source image moves toward back to the retinoscope. This light source location, in relation to the peep-hole position, determines the reflex motion, the direction, and speed of the reflex movement.
3.2 Streak rotation: observation of cylinder
In retinoscopy, astigmatism is often observed by rotating the streak projection. When rotating the streak, two distinctive astigmatic appearances are the variation in reflex brightness and strip thickness. When the streak projection is aligned with one of the two major meridians, the thickness and the brightness appear to be either optimized or minimized. Illustrated in gray scale and, more clearly, in false color, Fig. 3 shows the retinal reflex of an eye with the prescription of (S+1.00, C+2.00, X90). The dotted arrow line in each image indicates the orientation of the projection. The sleeve location was set at 18 mm. The reflex thickness and intensity variations are obtained. A third astigmatic appearance is the skew or break phenomenon, which shows the misaligned motions between the projection and reflex streak. This is also clearly shown in the simulated images where the beams are not aligned with meridians at either 180 or 90 degrees. The streak reflex appears to be misaligned to the streak projection.
3.3 With and against motion in ametropia
Figures 4 and5 show the retinal reflex motion under the plane mirror (sleeve located at h=19 mm) and the concave mirror (h=21 mm) operations, respectively. The upper row in each simulation illustrates the projection that moves across the pupil along one major meridian of the eye. Three personalized eye models  are used in the simulation. In upper Fig. 4 is presented the reflex of a mild myopic eye, MY1 with prescription of (S-1.50D, C+0.25D, X180) and a best-correction RMS wavefront aberration (WA) of 0.111 μm in the 3.4 mm pupil. The streak projection moves along the -1.5D meridian at a working distance of 667 mm. The neutralization or reversal appearance is clearly seen. The intensity variation shows the larger high-order aberration of this eye.
In middle Fig. 4 and upper Fig. 5 are illustrated measurements of an hyperopic eye, HY2, of (S+2.55D, C+0.5D, X10) and best-correction RMS WF of 0.079 μm. The streak projection moves along the 10-degree meridian. The characteristics of with-motion in the plane-mirror setting and against -motion in concave-mirror operation are clearly shown. Similarly, in lower Figs. 4 and 5, the observations of a myopic eye model, MY3, with prescription of (S-6.0D, C+0.75D, X70) and best-correction RMS WF of 0.117 μm are predicted and the against-motion and with-motion behaviors, respectively, in plane- and concave-mirror operations are clearly demonstrated. One observation to be noticed is the so-called “cut-off” phenomenon that occurs when the edge of projection falls inside the pupil. This effect is present in the against-motion cases in lower Fig. 4 and upper Fig. 5. In these two sets of images, the appearance of the edge doesn’t affect the judgment on reflex movement. However, at certain conditions, anomalous reflex motions occur.
The anomalous motion is often observed at high myopic or accommodative conditions in infants or patients with large pupils. This phenomena was first reported by Borish in 1970  and named the cut-off phenomena. Later, Howland in 1978  and Mutti in 2004  investigated the geometric causes of this anomalous motion. Figure 6 shows the simulation of such observation in the myopic eye, MY3, with 5.65 mm pupil. Under plane-mirror operation, the myopic eye should be against-motion, but because of the edge-effect that occurs at sideways, 5, 7, and 9 mm, the reflex motion appears as with-motion. If one looks carefully at the center images without edge influence, the movement of reflex, although not clear, is against-motion as it should be. This is more evident from the false-color images.
3.4 Scissors reflex in the keratoconus patient
The simulation results of the keratoconus (KC) eye are shown in Fig. 7.This KC eye has a protruding cone of about 60 μm in the lower left quadrant in its topography . The manifest refraction is (S-6.00D, C+6.00D, X135), and the best-correction RMS WA is 1.994 μm. The upper set of images in Fig. 7 shows the result of rotating the retinoscope projection at a distance of 0.5 meter. Although the refraction of -6.00D is significant, the strip-shaped reflex is not observed. Instead, a typical keratoconus “shadow” appears in the retinoscopic reflex. The irregular intensity distribution shows the significant high-order aberration and especially the coma of this eye. The lower set of images shows the so-called scissors reflex of KC eye as the projection moves along the meridian of 135 degree. The opening and closing movements of a pair of scissors is clearly shown
The realistic simulation and illustration of retinoscopic measurement are presented. We demonstrated the theoretical prediction of the ophthalmic measurement using three-dimensional ray tracing and anatomically accurate eye models. Both the personalized eye model and general symmetric eye models are employed. The technique can be applied to predict the ophthalmic measurements of eyes of various conditions. The application can be directly used for medical training and for evaluating performance of ophthalmic instruments.
This study was partially supported by ARMY TATRC Grant W81XWH-05-1-0409, the University of Tennessee Space Institute’s Center for Laser Applications, and a NASA Space Grant Fellowship.
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