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Show ORIGINAL CONTRIBUTION Retinal Image Stability in Head Tremor and Nystagmus: Counterintuitive Observations Brian R. Younge, MD, Jay W. McLaren, PhD, and William L. Brown, OD, PhD Background: Ophthalmoscopy of the fundus in a patient with head tremor is facilitated by image stability, whereas such observation of a patient with nystagmus is made difficult by image movement. Why does this occur? Methods: We offer an explanation of these observations and derive mathematical proofs for both direct and indirect ophthalmoscopes in various states of ametropia. Results: By means of image displacement calculations, we have created a graphical display of refractive error versus image displacement, showing zero displacement in emmetropia in head tremor and exaggerated displacement during nystagmus, regardless of the refractive status of the eye. Conclusions: Simple optics and a mathematical proof explain the clinical observations. The image stability in the patient with head tremor and the image instability in an eye with nystagmus are the consequences of the optical system of the eye. (/ Neuro- Ophthalmol 2007; 27: 107- 114) hen using an ophthalmoscope to examine a patient with a head tremor, the observer perceives the image without difficulty because the image is relatively stable compared with the movement of the pupil. One explanation of this observation is that the eye undergoes counterrotation by an equal amount. During this motion, however, the presence of a normal vestibulo- ocular reflex ( VOR) keeps the optical axis of the eye aligned in one direction. During observation with the ophthalmoscope, the side- to- side motion of the pupil is quite prominent and the lateral displacement of the structures at the back of the eye Department of Ophthalmology, Mayo Clinic, Rochester, Minnesota. This work was presented as a poster at the American Ophthalmological Society Meeting, Sea Island, Georgia, May 21, 2001. This work was supported in part by Research to prevent Blindness, Inc., New York, NY. Address correspondence to Brian R. Younge, MD, Department of Ophthalmology, Mayo Clinic, 200 First Street SW, Rochester, MN 55905; E- mail: BYounge@ mayo. edu translates through the same motion. Nonetheless, if the patient has no refractive error, the image remains stable. In contrast, during a similar examination of a patient with nystagmus, the fundus is difficult to see because the image is unstable and moves in the opposite direction of the eye movement. Observing such rapid eye movements, the examiner is often in awe of the patient's visual acuity, which can be quite good approaching 20/ 25 to 20/ 40 or even better. Why does the optic disc appear stable during head tremor and move violently during nystagmus when in both cases the pupil moves dramatically relative to the ophthalmoscope? Both the stability and the instability are counterintuitive observations explained in this article. METHODS Image Formation by Direct and Indirect Ophthalmoscopes We addressed the optics questions by determining the location of the retinal image in indirect and direct ophthalmoscopy with the use of simple optical properties of lenses. We then determined the lateral displacement of the ophthalmoscopic images 1) during lateral motion through 10 mm while maintaining the optical axis of the eye parallel to the optical axis of the ophthalmoscope, as one might experience during head tremor and 2) during eye rotation through 10° as one might experience during nystagmus. Motion of the retinal image as viewed through an ophthalmoscope can be estimated by considering geometric optics ( Figs. 1- 4). The formation of the aerial image of the retina with an indirect ophthalmoscope and an image directly on the observer's retina with a direct ophthalmoscope has two parts. First, the optical structures of a patient's eye form a real or virtual image of the patient's retina. Second the ophthalmoscope relays that image into a focal plane that can be viewed conveniently by the observer either directly ( direct ophthalmoscope) or with an auxiliary lens ( indirect ophthalmoscope). We estimated displacement of the image formed by the indirect and direct ophthalmoscopes during lateral displacement by head tremor and J Neuro- Ophthalmol, Vol. 27, No. 2, 2007 107 J Neuro- Ophthalmol, Vol. 27, No. 2, 2007 Younge et al Emmetropia r,. = oo Myopia Image of retina r,> 0 FIG. 1. The first image of the patient's retina forms at a distance r, from the cornea. X represents the image of a part of the retina on the central optical axis of the eye. Image of retina Hyperopia r,< 0 during rotation of the globe by nystagmus. Displacement of this image was used to assess steadiness of the view of the patient's retina. An image of the patient's retina forms at a distance from the eye that depends on the patient's refractive status ( Fig. 1). For an emmetropic eye, the image of the retina appears at infinity; rays that originate from an axial point on the retina leave the cornea parallel to the optical axis of the eye and have zero vergence. In a myopic eye, however, rays converge after they leave the cornea, and a real image of the retina forms at a distance rt from the cornea. In a hyperopic eye, rays diverge after they exit the cornea and a virtual image of the retina appears at a distance rt behind the cornea. In myopia and hyperopia, the distance to the image can be determined from the vergence of rays at the cornea as 1 ( 1) where D is the refractive correction of the eye in diopters. In both states of ametropia, the refractive correction is the negative of the vergence of rays that form the conjugate image. If the eye is myopic, D is < 0 and rt is positive ( image lies to the right in Fig. 1); if the eye is hyperopic, D is > 0 and rt is negative ( virtual image appears to the left in Fig. 1). Condensing lens Aerial image of retina Image of retina v:- * n _ Focal point of condensing lens o = ( r;- s) FIG. 2. Vergences and distances to images from the condensing lens of an indirect ophthalmoscope. The image of the retina is formed by the optical components of the patient's eye in the absence of the condensing lens and is the object to the condensing lens. X represents the image of a part of the retina on the central optical axis of the eye. / is distance to image; o is distance to object; r, is distance from cornea to image, s is distance from cornea to condensing lens; V, is vergence for image; V0 is vergence for object. 108 © 2007 Lippincott Williams & Wilkins Retinal Image Stability JNeuro- Ophthalmol, Vol. 27, No. 2, 2007 Condensing lens Aerial image of retina ' % d„ '--£/„ First image of retina FIG. 3. Lateral translation of a myopic eye during observation with an indirect ophthalmoscope. When the patient's eye moves laterally through distance d0 but does not rotate, the first image also moves through distance d0. The aerial image moves laterally through distance dh The ratio of d, to d0 depends on the refractive state of the patient. X represents the image of a part of the retina on the central optical axis of the eye. fd is focal length of condensing lens. In both cases, this image is the object to the ophthalmoscope. If the patient's eye is displaced laterally from the axis of the ophthalmoscope ( during head tremor) or rotated ( during nystagmus), the image of the patient's retina is also displaced from the optical axis. The corresponding motion of the ophthalmoscopic image can be determined by considering geometric optics. Indirect Ophthalmoscope Head Tremor The vergence of rays entering the condensing lens of an indirect ophthalmoscope, V0, is the inverse of the distance to the object, o, for that lens and for an ametropic eye: 1 V0=- ( 2) From Figure 2, the object distance o is r, minus the distance between the patient's cornea and the condensing lens, s. From Equation 1, the object distance can be expressed as ° = - ( j> + s) ( 3) The vergence of rays exiting the condensing lens is V0 plus the power of the lens, P; the distance from the condensing lens to the aerial image is the inverse of this vergence, expressed as i = --- ( 4) V0+ P V ' Equation 4 shows that the aerial image of the retina of an emmetropic eye ( V0 = 0) appears at the focal point of the condensing lens. For a myopic eye, the image appears between the lens and the focal point, and for a hyperopic eye, the image appears beyond the focal point. When the patient's eye moves laterally with its axis remaining parallel to the axis of the condensing lens, as FIG. 4. Displacement of the aerial image viewed through an indirect ophthalmoscope during rotation of the patient's eye with nystagmus. When the eye rotates about the center of rotation ( CR), displacement of the first image d0 is proportional to the cosine of the angle 6. X represents the image of a part of the retina on the central optical axis of the eye. d, is distance image moves; /' is distance to image; o is distance to object; r, is distance from cornea to image; s is distance from cornea to condensing lens; Scr is distance from CR to cornea; V, is vergence for image; V0 is vergence for object. 109 J Neuro- Ophthalmol, Vol. 27, No. 2, 2007 Younge et al during head tremor in a patient with a working VOR, the motion of the image can be determined by considering lateral magnification by the condensing lens. If the eye moves laterally through distance dm the first image also moves through distance d0, and the aerial image formed by the condensing lens moves through distance dt ( Fig. 3). The ratio of dt to da is the lateral magnification of the condensing lens and is equal to the ratio of the image distance i to the object distance o. From Equations 2 and 4, magnification can be expressed as dj = 1 d0 1+ Po ( 5) From Equation 3, this ratio can be expressed in terms of the refractive error, the power of the condensing lens, and the distance between the condensing lens and the eye as follows: dj = 1 d0 l- P(± + s) ( 6) From Equation 6, the image displacement for a given lateral displacement of the patient's eye can be determined. If the patient is emmetropic, D becomes 0 and djd0 converges to 0. Nystagmus In a patient with nystagmus, the globe and the first image of the retina rotate about the center of rotation of the eye ( Fig. 4). The distance from the entrance of the condensing lens to the first image is determined along the axis of the condensing lens and is related to the angular position of the eye, 9, as o = ( rt + s„) cos6 - ( s + s„) ( 7) where Scr is the distance from the center of rotation of the eye to the cornea. With substitution of Equation 7 into Equation 5, the lateral magnification is ± = 1 d0 1+ P[( ri + scr) cos 6 - ( s + scr)} ( 8) Equation 8 is analogous to Equation 6, expressed in terms of the angular displacement of the first image of the patient's retina. The displacement of the object, d0, can be expressed in terms of the angle of rotation, 9: d0 = ( rt + scr) sin6 ( 9) With substitution of Equation 9 for d0 in Equation 8 and recognition that r, is equal to the negative inverse of A the ratio of image displacement to the sine of angular position of the patient's eye is dj Scr sin# 1 + P[( scr- jj) cos# - ( sc. If the patient is emmetropic and D is zero, the image displacement, dt, is equal to ( tan 6) 1 P, or the focal length of the condensing lens multiplied by tan 9. Direct Ophthalmoscope Head Tremor Image formation with a direct ophthalmoscope is similar to that with the indirect ophthalmoscope; the patient's retina is viewed with the help of the lens in the ophthalmoscope. The power of the lens is selected to compensate for the refractive status of the patient and to place the image of the patient's retina at a convenient distance from the observer for viewing. To simplify this analysis, assume that a power has been chosen that places the image at distance i behind the ophthalmoscope. Often this distance is chosen as the least distance of distinct vision for the observer, 25 cm behind the ophthalmoscope ( assuming that the observer can accommodate to 25 cm), although it can be any distance. This situation is analogous to that of the indirect ophthalmoscope, except that the virtual image of the patient's retina produced by the lens of the ophthalmoscope always appears at distance i behind the ophthalmoscope from the observer ( Fig. 5). Displacement of this image during a lateral translation of the patient's eye and eye rotation can be determined in the same way as with the indirect ophthalmoscope. Consider a lateral displacement d0 of the patient's eye ( Fig. 6). The object distance can be determined from Equation 3, and the lateral magnification is di_ d0 When the patient is emmetropic ( ie, D - 0), this ratio is zero. With the direct ophthalmoscope, the angular displacement of the image of the retina is easier to understand than the displacement dt because dt is only useful in the context of the image distance i. The angular displacement 7 is the inverse tangent of the image displacement divided by the image distance, dji. From Equation 11, 1 D ( 11) y = tan ( 12) ( 10) Equation 12 gives the angular displacement of the image through the direct ophthalmoscope regardless of the distance to the image viewed by the observer, which is the distance determined by the ophthalmoscopic lens selected by the observer. Nystagmus Image motion during nystagmus as viewed through the direct ophthalmoscope is analogous to motion of the aerial image in an indirect ophthalmoscope. Displacement of the image at 25 cm from the direct ophthalmoscope can 110 © 2007 Lippincott Williams & Wilkins Retinal Image Stability J Neuro- Ophthalmol, Vol. 27, No. 2, 2007 FIG. 5. Vergences and distances when viewing a patient's retina through a direct ophthalmoscope. The optical power of direct ophthalmoscopes can be adjusted to compensate for the refractive state of the patient so that the retinal image can be placed behind the patient's eye at a convenient position for viewing. The assumption is that the image is placed at a distance / from the ophthalmoscope lens. X represents the image of a part of the retina on the central optical axis of the eye. o is distance to the object; r, is distance from cornea to image; s is distance from cornea to condensing lens. be determined from the lateral magnification. With consideration of the virtual object distance from the ophthalmoscope lens in Figure 7, Equation 5 can be rewritten as Equation 13: di i d0 ( n + s„) cos 9-{ s + s„) ( 13) With substitution of- l/£> for r, and Equation 9 for d0. the ratio of dt to sin 9 is expressed as Equation 14: " i ' \$ cr Jj) sin 9 ( scr - 5) cos 9 - ( scr + s) ( 14) Angular displacement 7 can be determined in a similar way, as was used for head tremor: y = tan sind( sc, i) { scr - 5) COS 9 - ( Scr + S) ( 15) When the patient is emmetropic and D is 0, one can show that dt is i tan 9 or that the angular displacement 7 is equal to 9. 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. RESULTS Indirect Ophthalmoscope 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 FIG. 6. Lateral displacement during head tremor moves the first image d0 and the second image d, from the optical axis of the ophthalmoscope. When the patient is myopic, as shown, the ophthalmoscopic image " o moves in the direction opposite to the translation of the eye. The displacement of the ophthalmoscopic image can also be expressed as the angular displacement 7, which is constant for any image distance and a given displacement d0. X represents the image of a part of the retina on the central optical axis of the eye. d, is distance image moves; d0 is distance object moves; / is distance to image; o is distance to object; r, is distance from cornea to image; s is distance from cornea to condensing lens. Ill J Neuro- Ophthalmol, Vol. 27, No. 2, 2007 Younge et al FIG. 7. Eye rotation during nystagmus moves the first image d0 and the second image d, from the optical axis of the ophthalmoscope, x represents the image of a part of the retina on the central optical axis of the eye. Scr is distance from center of rotation to cornea. Other abbreviations as in the legend to Figure 6. 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 hy-peropic 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. Direct Ophthalmoscope 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. FIG. 8. Displacement of aerial image viewed through an indirect ophthalmoscope during eye translation through 10 mm from head tremor ( light line) and during eye rotation through 10° from nystagmus ( dark line). The horizontal dashed line represents no displacement of the image, and the vertical dashed line indicates displacements in an emmetropic patient. Although there was no motion of the aerial image during lateral head movements when the patient had no refractive error, the image moved dramatically during nystagmus regardless of the refractive status. - 4 - 2 0 2 4 Refractive error ( diopters) 12 112 © 2007 Lippincott Williams & Wilkins Retinal Image Stability JNeuro- Ophthalmol, Vol. 27, No. 2, 2007 Refractive error ( diopters) FIG. 9. Angular displacement of retinal image viewed through a direct ophthalmoscope during eye translation through 10 mm from head tremor ( light line) and during eye rotation through 10° from nystagmus ( dark line). Although there was no motion of the aerial image during lateral head movements when the patient had no refractive error, the image moved dramatically during nystagmus regardless of the refractive status. 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. DISCUSSION 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 113 J Neuro- Ophthalmol, Vol. 27, No. 2, 2007 Younge et al 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. REFERENCES 1. Zee DS. Ophthalmoscopy in examination of patients with vestibular disorders. Ann Neurol 1978; 3: 373^ k 2. Meyer- Arendt JR. Introduction to Classical and Modern Optics, 2nd ed. Englewood Cliffs, NJ: Prentice- Hall; 1984. 114 © 2007 Lippincott Williams & Wilkins |
