With substitution of −1/D for ri and Equation 9 for do, the ratio of di to sin θ is expressed as Equation 14:
Angular displacement γ can be determined in a similar way, as was used for head tremor:
When the patient is emmetropic and D is 0, one can show that di is i tan θ or that the angular displacement γ is equal to θ.
Estimate of Image Motion
Motion of the aerial image viewed through an indirect ophthalmoscope during lateral translation through 10 mm (head tremor) and during rotation through 10° (nystagmus) was calculated at refractive errors between −12 D and 12 D by using Equations 6 and 10. We assumed that 1) the condensing lens was placed 20 mm from the patient's eye, 2) the power of the lens was 20 D, and 3) the center of rotation was 15 mm posterior to the surface of the cornea.
The angular displacement of the retinal image viewed through a direct ophthalmoscope for the same lateral translation and nystagmus was also calculated for refractive errors between −12 D and 12 D by using Equations 12 and 15. We assumed that the ophthalmoscope lens was held 10 mm from the patient's cornea.
Figure 8 illustrates displacement of the aerial image of the patient's retina during head tremor, which displaces the eye by 10 mm, and nystagmus with eye rotation through 10°, as determined by Equations 6 and 10. During head tremor, the image from a myopic eye was displaced in the same direction as the direction of motion of the eye and was diminished as the refractive correction decreased. In hyperopic eyes, the image was displaced in the opposite direction from the eye movement, and displacement decreased as the refractive correction decreased. In emmetropic eyes, lateral displacement was zero; the image of the patient's retina was stable even while the eye was translated from side to side.
In contrast, image displacement during nystagmus was least in eyes with the greatest myopic error and increased as refractive correction decreased through emmetropia and continued to increase as the correction became more hyperopic. The image in an emmetropic eye moved through approximately 9 mm for a 10° rotation of the globe. The image of the patient's retina moved during nystagmus regardless of the patient's refractive status.
Movement of the retinal image viewed through a direct ophthalmoscope during a 10-mm lateral displacement of the patient's eye varied with the refractive error of the patient (Fig. 9). For myopic eyes, the image rotated through a positive angle, which displaced the image in a direction opposite to the actual motion. As the myopic correction decreased, image motion decreased. Lateral motion of the image disappeared in emmetropic eyes but reappeared in the same direction as the motion (negative angle) in hyperopic eyes.
The apparent angular displacement of the retinal image during nystagmus ranged from approximately 13 to 8° for a 10° angular rotation of the eye, depending on the refractive status (Fig. 9). The image was displaced through a positive angle, the same direction as the original rotation. During nystagmus, the image always moved regardless of the patient's refractive status.
The observer easily perceives a head tremor along with the movement of the head and eye. Yet on looking through the moving pupil, one can easily examine the inside of the eye with a direct ophthalmoscope because the image of the optic disc and vessels is quite stable relative to the movement of the pupil itself. One might think that the explanation for this observation is that the head and eye move in opposite directions by an equal amount, so that their motions cancel. Counterrotation does stabilize the eye, but this is not the explanation for the stability of the image of the fundus. In fact, the patient's cornea and retina are displaced laterally during head rotation. Therefore, one could ask why the optic disc does not appear to move as the front of the eye does and whether the stability is counterintuitive.
The compensatory VOR holds the eyes on the fixation target at optical infinity, rotating the eyes in an exactly opposite direction from the head rotation and maintaining a stable direction of gaze. However, the eye translates through a small arc, because the center of head rotation and center of the eye are not colocated. During observation with the direct ophthalmoscope, this movement can be seen easily in a patient with a head tremor as a side-to-side shift of the pupil. The shift of the pupil is accompanied by exactly and precisely the same shift as the other structures in the eye, including the optic disc and fundus. The visual axis also shifts but remains parallel to its original axis, held by the VOR. However, the image of the optic disc and fundus does not shift if the patient is emmetropic. In various degrees of ametropia, there is a damped image shift, as illustrated by the slope of the line in Figure 9.
In emmetropia, the image shift is zero with either direct or indirect ophthalmoscopy in a patient with a head tremor. The VOR maintains the patient's gaze directed ahead, although the optical axis moves left and right of the examiner's axis of gaze. The optics diagrammed in Figures 1 through 7 illustrate this point, and the proof is in Figure 9, derived from Equation 12. The combined optics of the eye and ophthalmoscope and the axial stability of the eye by the VOR explain the stable image. If the VOR is delayed or not functioning properly, the image shifts, and this can be seen as very small movements through the ophthalmoscope (1,2). The stability of the retinoscopic image can be explained in terms of the optics used every day in ophthalmology. This observation has not been described, and we have shown the mathematical proof.
During examination of a patient with congenital or acquired nystagmus, the image of the optic disc and vessels moves back and forth so rapidly that detailed examination is difficult if not impossible. In this situation, the eye oscillates about its own center of rotation. Patients with congenital nystagmus may have a small compensatory head movement but not enough to compensate for the eye rotation. They can see fairly well because of brief foveation periods in which the eye is transiently still. The amplitude of nystagmus diminishes in such patients when fixating at a near target, thus permitting reading vision that is often better than the distance equivalent. The observer looking into the back of an eye of a patient with congenital nystagmus has difficulty seeing retinal and optic nerve details because the amplitude of the image movement with the direct ophthalmoscope is magnified. Both direct and indirect ophthalmoscopic images exhibit similar shifts of the images, but the magnification factors differ-the magnification factor with the direct ophthalmoscope is approximately 15x and with the indirect ophthalmoscope it is 3x (with a 20-D viewing lens). Thus, retinal detail in a patient with nystagmus or in an uncooperative patient can be visualized better with the indirect ophthalmoscope, albeit with less magnification. In nystagmus the eye rotates, but the center of rotation does not translate from side to side. Because of the rotation of the eye's optical axis, the view of the retina through an ophthalmoscope moves left to right and back with each eye cycle. Examination of the moving image of the fundus is difficult because the amplitude of the image displacement is exaggerated.
Thus, the ophthalmoscopic appearance of a stable fundic image in a patient with a head tremor is a direct result of the optical properties of the geometric optics of the eye and ophthalmoscope. In fact, the fundus moves from side to side precisely with the patient's head. However, the eye's optical system moves with the fundus and, because of the VOR, the direction of the optical axis of the eye does not change. Conversely, the amplitude of the image movement in a patient with nystagmus is exaggerated by the changing optical axis of the patient's eye, and the image movement is in the same direction as the retinal movement.
1. Zee DS. Ophthalmoscopy in examination of patients with vestibular disorders. Ann Neurol
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2. Meyer-Arendt JR. Introduction to Classical and Modern Optics, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall; 1984.