COMPIJI'ER K>DEL OF BINOCULAR ALIGNMENT

Joel M. Miller

Smith-Kettlewell Institute of Visual Sciences Medical Research Institute 2232 Webster Street San Francisco, CA 94115 SUMMARY ~

A model is described of normal and abnormal coordination of the eyes that is useful in diagnosing eye position disorders (strabism~s ) and predicting the effects of extraocular muscle surgery. The model calculates all forces on the eye, finding muscle force from an empirical function of length and innervation. The length and unit movement vector are determined for each muscle from the orbital geometry, but also depend on muscle force. It is possible to account for globe translation. Innervation, unknown in practice, is calculated by the model. Calculated innervations can drive eyes having varied mechanical properties. Muscle paths are based on recent imaging studies in monkeys and humans. The muscle force model is realistic for extreme muscle lengths. The model is implemented under UNIX, and has been ported to PDP-11, VAX, and MASSCOMP computers. INTRODUCTION For vision to be normal, movements of the two eyes must be precisely coordinated over an adequate range. A diverse group of disorders , called strabismus or squint, are characterized by failure of binocular coordination. The result is that in at least some positions of gaze the two e yes fix different points in space : The patient may see double (diplopia), or may suppress vision in one eye: stereoscopic depth perception is lost.

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terms of innervations, the six muscles are three ~ ~ ant~gonistic or reciprocally innervated pairs : the J horizontal (lateral and medial) recti the vertical (superior and inferior) obliques. Gi~en the inner· vation to one member of a pair the innervation to the other is determined. This fact is very helpful: Innervations are initially unknown quantities and reciprocal innervation reduces their number f;om 6 to 3. The other eyeball to the arrangement of simplicity, w~

elastic structures which couple the orbit are the optic nerve and an elastic sheaths or fascias which, f or lump together.

The various orbital and muscul ar dimensions have been me asured in cadavers by Volkman [4] and Nakagawa [5] . The model incorporates what we believe t o be the most reliable average values.

Fi r s t , I wi ll provi de s ome background for the non-specia li s t; the reader may a lso wi s h to consult a textbook on ocul ar motilit y and strabismus, e.g., [1, 2 , 3]. Then, I wi l l describe our model of the eye 's mechanics, which is oriented to s imulate normal binocul ar coordi nation, strabismic misa li gnments of various kinds, and eye muscle s urgey . Fin a l ly , I will di s cuss the clinic al us es of the mode l and its implementat i on under UNI X.

Anal ys is by Krewson [6] and Boeder [7] addressed the geometric al problem of a s phere rotated by s tri ngs wrapped around it. They assumed that the muscles t ook great~circle paths across the globe; because of i nt er muscul ar connections, however, this assumption is incorre ct. A s~de from this problem, the muscl e forces and their dependance on length and i nnervat ion were unknown, so that a full solution to the mechanical problem could not be attempt ed.

THE EYE, MUSC LES, AN D ORB IT Each of the s i x extraocul ar musc l es att aches at one end to the globe (it s i nsert ion) and at the other end to the bony orbit (i ts or ig in ) . Fi g . 1 shows the origin and insertion of each of the s i x muscles with the eye looking s traight ahead (~­ mary position) . As the br ai n s ee s it, that i s, in

The latter gap was closed when length-tensioninnervation data were collected by Robinson, et al. [8] and Collins , et al. [9] during s trabismus s ur ger~ unde r lo cal anesthes i a. Us i ng the s e dat a, Robins on [ 10] deve loped a model eye for s trabi smu s ,

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which served as a starting PQint fo:c -the .present model.

ing,.length. For AL =0 the mus£1e exet;t H.n~, force (slack region}. F then rises, roughly linearly, imtil the. muscle i~ stretched about SO..\ of its resting ~ength, at which point its stiffness starts increasing markedly (leash region).

SfRABISMUS Defects in ocular positioning -- strabismus- ~ result from errors ilL innervation to the muscles, and mechanical abnormalities in the eye, muscles, and orbit. The main treatment for strabismus, regardless of its etiology, is eye muscle surge+y; various alterations of the muscles and their insertions are made to restore proper eye alignment.

The developed force (Fig. 28) is a function of innervation I and~L, consistent with the slidingfilament model of muscle action. By definition, Fd = 0 for I = 0; the units of I are arbitrary. Currently, I is a dependant variable in the model, and all simulations must initially solve for the set of innervations.

In the United States some 80,000 eye operations are performed each year for strabismus [11]. Acceptable results are achieved in, perhaps, half of these cases, and many patients must return for repeated surgical or other proceedures. In at least some cases, surgical failure is due to a less than optimal choice of the type or amount of surgery. Given the complexity of orbital mechanics, it is not surprising that the surgeon's intuition is a fallible guide.

Figure 2 shows forces of the lateral rectus. Although each muscle may have unique force surfaces, there are currently no ~ata to require this, and we assume that forces for each of the other 5 muscles can be calculated by scaling the forces in Fig. 2 by the ratio of that muscle's cross-sectional area ·to that of the lateral rectus. Unit Moment Vector

THE "SQUINT" OCULAR MODEL

The center of rotation of the eye, the muscle origin 0, and the point of tangency T determine the muscle plane. The normal to the muscle plane is the unit moment vector For any position of the eye, muscle force acts to rot ate the eye about (Fig . 3).

For the eye to be at rest in any p~sition of gaze, the sum of all rotational forces M must equal 0: -. M

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Nonuniform Muscle Force. The muscles are "bands" about 10mm wide and are tan~ent to the globe along line segments. To calculate m, we need to find the point at which F may be considered to be applied to the eye. Consrder the muscle to consist of many identical independant parallel fibers, each extending the length of the muscle. As the muscl e bends sideways, fibers near the outer edge stretch more, and so exert more force than fibers near the inner edge; we must determine the point along the muscle's width -- the centroid -- at which the total force of the muscle may be considered to act. The centroid is found in the usual way [13) by equating moments to yield:

(1)

where: Pis the rotational force of the passive orbital tissues (fascia, optic nerve, etc.) whose stiffness has been estimated to be about 0.25 g/deg for hori zontal, vertical, and torsional rotations; F is the force of the i-th muscle; m . is the unit m. ~

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moment vector, the axis about which contraction of the muscle i tends to rotate the eye . Muscle Force

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We compute muscle force as the s um of independant developed and passi·:ve forces, based on data collected at the Smith-Kettlewe ll Institute [8, 9, 12) (fig. 2) .

z =

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A. PASSIVE FORCE

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Figure 2: Muscle force s urfaces Fm = Fp

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TOTAL FORCE

2 2 k-w ·sin B·cos B 12 · Fm

( 3)

where: k is a linear apnroximation of muscle s tiffness; w is muscle <·1 idth; B is the angle of sideways bending of the muscle; Fm is total muscle force (Eq. 2) . Note that B must reflect only muscle "bending", not "folding" or "twisting", which do not differentially stretch the edges of the muscle (see [13)).

Equation 3 shows that the centroid shift z is a function of muscle force. Shifting the effective insertions, however, alters muscle forces. Consequent ly, calculation of z is an iterative process in which each estimate of alters the muscle path, requiring recalculation

(2)

The passive force of a muscle (Fig . 2A) is a function of AL, the percent stretch from its rest-

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taken by the muscle as it runs along the surface of the globe, from its point of insertion to its point of tangency. Recall (F ig . 1) that in -primary position all of the muscles insert on the globe beyond the center of rotation; as the globe rotates, we expect the muscles to slip sideways. The horizontal recti, for instance, should slip upwards as the eye elevates and downwards as the eye depresses. If sideslip were unrestrained, a muscle would simply take the shortest path on the globe (a great circle) from insertion point to tangency. However, elastic connective tissue surrounds the muscles and joins them to each other, forming a sort of cap anterior to the equator of the globe, and reducing sideslip. To model sideslip we represent the intermuscular membrane by a row of non- linear incremental springs, coupling the muscle to the globe along the arc of contact. These springs are stretched by Fs, the component of F that acts sideways on the muscle along the s~rface of the globe . By constraining the muscle path to be circular, this potentially difficult problem can be easily solved. (see [13]). The springs are characterized by 2 constants (an asympto tic stretch and a compliance) that are deter mined by experimental data such as those collected by Miller, et . al.[14], in which sideslip was visualized by means of X-ray-opaque markers implanted in the l atera l recti of monkeys trained to fixate. Preliminary results from CT scans of human orbits [15 ] s upport the results of the X-ray s tudy.

Figure 3: Unit moment vector of Fm, which leads to another estimate of

z.

To see if calculating the centroid shift is worthwhile, we find the innervation sets (more below about how this is done) necessary to drive a normal eye to each of 81 points evenly spaced over an 80 deg by 80 deg field of gaze . For each of these innervation sets we then find the position a normal eye would take, (1) using the centroid calculation and (2) with the effective insertion fixed at the midpoint of the actual insertion line. The first calculation simply reproduces the original 81 eye positions (a check of the program's calculations). The second method shows the errors of ignoring centroid shift as deviations of eye positions from the 81 fixation points (shown as crosses in Fig. 4A). It can be seen from Fig. 4A that these etrors are small for the normal eye, tending to increase with eccentricity. The centroid ca lculation becomes more important in modeling abnormal eyes in which large muscle forces exist, due to co contraction or inability of ant ago ni stic mus c le s to len gthen. Cl

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Following the method used above to eva luat e the centroid ca lculat ion , we show i n Fig. 4B the sub st ancial errors that res u l t fro m minimizing the sideslip that the model allows . (A llowing "shortest path" sidesli p result s in much larger errorsthan these. ) Having des cribed the muscle' s path, we can f ind the vector abo ut which the eye tends to rot at e: i t is perpendicul a r to the pl ane de t ermine d by the eye ' s ce nter of rot ation C, the mus c le origin 0 , anJ the point T at which the musc le is tangent t o the eye (F i g . 3) .

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Figure 4: Effect of omitt i ng parts of the ca lcul at ion 1-!uscle Sidesl ip.

The eye's cent er of r ot ation C i s not fixeJ; the eye is s urrounded by ~ resi l ient pads of fa t , and can transl at e several t t mm, as one can ve r ify bv ; t ; ; ; ; ; 1 pus hi ng on the eyel i d . ; ; ; ; ; ; Ce rt a in diso rde r s a r e ~ ~ ~ char ac t e ri :ed by l a r ge + + ; ~ f. ~ ~ tr ansla t ions of t he g lo be (usually re tr ~ction - 40 -20 0 20 40 AddUCIIOfl A t>OUCIIOO in t o t he o r bit ) . Thi ,; tt !Degr ee sJ may occur be cause an ~ ta go nis t mus cl e i s un :!b le t o leng t he n ~he n i t s agonist contrac t s , or becau ~c of co - co ntrac t i on o f anta go nis t ic musc l es .

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Note that, as wit h and F , s lid eslip and F are mutually dependant, so thaf each iterative m es t imate of F requires a reestimate of muscle sideslip. m

ll'e now consider the path

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Because of its diagnostic value, globe translation is of interest as an output variable in our model. Moreover, translation is important as it affects eye rotation: Globe translation alters muscle stretch and, so the muscle force F • Translation also alters the position of tWe globe relative to the muscle origins and, so, alters the unit moment vectors ~-

the desired positions. The~ program (Fig. 5) reads eye parameters and innervations to produce the positions that the eye would asslime under the given innervations. In either case, the unknowns (either 3 innervations or 3 components of eye position) are estimated in an iterative process until M(Eq. 1) is acceptably close to zero. The programs that iteratively §olve innerv ation and position problems differ in the method of guessing the next innervation set or eye position. In the innervation program, a good guess at the next innervation can be obtained by making a linear expansion of Eq. 1 around the current I, and then finding ~I to bring M to 0. (see [10); Eqs . 46-

Accounting for translation in the model is straightforward. First, we resolve and sum the muscle forces in an appropriate rectangular coordinate system, and calculate translation of the globe center along each axis according to linear estimates of the orbital fat stiffness. Then, we recalculate muscle forces with the new globe center.

54).

The significance of the effect of translation on rotation can be assessed as be fore. Fig . 4C shows the errors in rotation that would result from ignoring globe translation in a normal eye. These errors are ~all for a normal eye. In abnormal eyes, especially those with restrictive or ~ ~ontrac­ tive syndromes (in which globe retraction is observed clinically), very large errors can result from ignoring translation.

In the position program, we see no way to arrive at a quick, a ccurate guess at the next position. Our present method involves taking small steps in each coordinate around the current eye position, to estimate the stiffness of the eye. This yeilds a 3 by 3 stiffness matrix, which is inverted to give a compliance matrix, and used to predict the change in eye Eosition necessary to bring the residual moment ~to zero.

Solution of the Force-Balance Equation

Binocularity

Currently, our computer program is optimaized to s olve the system of equations described here and in [10) and [13) for either innervations or the coordinates of eye position. We call these innervation problem and the position problem, respectively .

According to Hering's law of equal innervation [1, 2, 3), the two eyes are not independantly controlled by the brain. With normal eyes, if one is covered while the other looks about, the covered eye will move in synchrony, pointing to the same places as the s eeing eye; if the eyes are abnormal, the covered eye will point elsewhere. To model the binocular case, we first find the innerv at i ons to the fi xing eye for the positions of interest. Then, a llowing ·for the mirror symmet ry of the two eye s , we supply the same innervations t o the other, foll owing eye, and determine i ts posit ions . There are two complications in thi s approach . Fi r s t, to solve the innervati on problem we must know the 3 component s o f eye pos ition for the fixi ng eye . Phys iologi ca ll y , the eye has onl y 2 degr ees o f f reed om, with the t or s ional component -- rot ati on o f the eye a bout it s line of s i ght -- determined by the br ain fo r each hori zont a l and verti ca l coordin at e pair (see [1 6) ) . If the fi xing eye is norm a l , t or sion is de scr i bed by Lis t ing ' s Law (a f un cti on o f t he hor izont a l and vert i cal component s ; see [ 10, 13) ) . But if the fi xing eye is not normal , t or s i on is unknown. Lis t i ng ' s l aw i s r ea ll y a co ns tra int on the pat te r n of i nne r vations and not on the mech ani cs of the orbi t i t se lf [16) . If i t is onl y the fixin g eye that is a bno rm a l, and not the i nnervations , the n the f att er are s t i l l Lis t i ng i nnerv ations, th at is, i nne r vat ions whi ch woul d dri ve a no r ma l eye so that i t obe yed Li s tin g ' s l aw. We then : ( l ) Gene r at e a f i e ld of norm a l e ye posit i ons, (t ors i on ac cordin g t o Lis tin g ' s law) . (2) So l ve for t he i nn er vati on s et s which woul d brin g a norm a l eye t o eac h posi t ion in t he fie l d . The s e a re Li s t i ng i nner vat i on s . (3) Find the posi t i on t he abnorma l f i xi ng eye woul d move t o under e ac h List in g inner va t i on set . Th e~e poi nt s fo rm a f i xab l e f ie ld, and defi ne a s urface gi vin g t o r sion o f the abnorm a l fi xin g eye f or ~1ny

8 Fi gure 5 :

Mo nocul ar mode l

The innervation pr ob lem is so l ved by a pr og ram ca ll ed inn (Fi g . 5) . The i nn pr ogram t a ke s as i nput a s~o f eye paramet e r~desc ribin g the me chanic a l char a ct e ri s tics of an eye , and a set of des ired positi ons. It produce s as output the innervatio n sets that would drive the de s cribed eye to each of

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DIAGNOSIS AND PREDICTION OF SURGICAL OUTCOME

horizontal-vertical eye position pair. Algorithms for interpolating on this surface are available [17]. The proceedure is shown schematically in Fig. 6A.

Many kinds of information are considered in arriving at a diagnosis of strabismus. But, regardless of the clues used to reach it, the diagnosis itself is (or should be) a hypothesis about abnormalities of orbital mechanics or innervations. One way in which our model can serve as a diagnostic aid is by determining if the physician's diagnosis can account for the pattern of misalignment found in the patient. Parameters of the model are modified to reflect the diagnosis, and the positions assumed by the model eye are compared with tho se of the patient. If there is a discrepancy, the diagnosis may be a ltered and the process repeated until a satisfactory match is achieved. The SQUINT model is also able to suggest diagnosis, by searchin g for the values of eye parameters that minimize the differences between the posit ions of the patient's eye and those of the model eye. Automatic diagnosis has been implemented in the eye program (Fig. 5) using a st r ai ghtforward iterative method. We are currently appl ying the model in a retrospective study of c lini ca l cases to gauge its usefulness, and de t ermine where improvements are needed. The aut omat ed diagnosis are often quite interesting, although it is usually not pos s ible to verify them wi th mechanical measurements on the patients eyes. Our criterion of success in these cases is agreement with the diagnosi s of an experienced strabismus s urgeon . We feel that the model shows considerable promise as a di agnostic t ool.

8 data

Figure 6: Binocular model flow chart Now that we have the 3 position cordinates of the fixing eye, we find its innervations (Fig. 6 B). We then want to supply "corresponding" innervations, according to Hering's law, to the following eye and calculate its positions. The second complication, then, is the proper treatment of Hering's l aw. Since the normal structure of the two eyes has mirror symmetry a coordinated (conjugat e) movement (say, t o the upper right ) will involve different innervations to the muscles in the t wo eye s . The direct mapping of fixing-eye innervati ons into following-e ye innervations is unknown . li01veve r , we do know the mapping of fixing-eye po s iti ons to following-eye positions for norm~l eyes : a s i mple reflection of the horizont a l coordinate . That is, if we ass ume the neural mechanisms that program innervations t o be normal, we can follow the pro ceedure diagramed in Fig. 6 C. Finally, the binocular simulation is complet ed by s olv ing for the positions of the following-eye (Fig. 6 0).

In light of the cumbersome nature of the full bino cul ar so lution, we should point out th at i f ever ythi ng i s normal except for the fol lo win g- eye mechanics, most of the a bove compl ic at ion s are avoi ded. In this case the abbreviat ed proceedur e indicated as Fig. 6 E can be used.

Whether the und er l yi ng abnormalit y is in the brain or in the orbit, the mos t common treatment for strabismus is s urger y on the extraocul ar muscle s, mainly alterations i n insertions and muscle lengths. Such mechanica l manipul ations can be described as changes in the parameters of our model. Prediction of s urgical outcome is often quite accurate, but in some important cases it is not. Indi vidua l difference s in the response t o surgery (scarring, fo r example) may expl ain some failures of pr edict ion, but we also s uspect t hat our muscl e path ana l ys i s i s not soph i s t icated enough t o dea l with the compl e x pat hs that muscles may t a ke as a r es ult o f s ur gery . I ~1PL H1E'HATIO N

The SQ UI NT s i mul ati on sys t em i s organized around the inn , ~· and eye commands and the three t ype s of f i les they out put (see Fig . 5). The commands a ll run as independant UNIX proces s es so that the U~ I X fac il ities fo r I/0 redirecti on and jo b control may be used . The fi l es ar e all ASCII s trings, a ll owi ng full us e of UNIX te xt manipul a tlon pro gr ams . Sur ge r y commands , s uch as rece ss , resect , and faden, ope r ate on e ye par ame ter files , yiel din g modified eye pa rameter file s that reflect the effect s of s ur gery . Prett y- print pro grams t rp os , tr1nn, and tre ye, trim down the numerical preci s i on and otherwi s e pr epare file s for hum an v i ewin g . Creat ing a new eye positi on file is easily done

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with an editor; normal values for torsion can then be entered in the file with listing, which computes Listing's law. ~ imlements [1 7 ), (see "Bino-:ularity" above and Fig. 6A). Compound commands allow more complex simulations; they are actually UNIX Shell scripts composed of simple commands. hc2 for instance, performs the operations shown in Fig. 9, producing both tabul ar and graphical output. In designing the SQUINT system we sought to make it portable, modifiable, and, of course, correct. Portability was achieved by choosing UN I X as the operating system, Most of the code is written in C and has passed the lint, error-checking complier. A few programs are-written in ANSI Fortran, so that a Fortran 77 complier is als o required. Since it s development on a Digital Equipment PDP- 11/ 45 running a mixture of JHU UNIX (from Johns Hopkins Universit y) and Version 7 UNIX (Bell Laboratory) , it has been ported to a Digital Equipment VAZ -1 1/ 780 under 4.2 BSD UNIX (University of Cal iforni a Berkeley), and to a Masscomp MC-500 under RTU system III UNIX (Masscom).

7.

P. Boeder, "Co-operative action of extraocular muscles," Br. J. Ophthalmol. vol: 46, pp. 397403, 1962.

8.

D.A. Robinson, D.M. O'Meara, A.B. Scott, et.al., ''Mechanical components of human eye movements," J. Appl. Physiol. vol . 26, pp. 548-553, 1969.

9.

C.C. Collins, A.B. Scott, and D.M. O'Meara, "Elements of the peripheral oculomotor apparatus." Am. J. Optom. vol. 46, no. 7, pp. 510515, 1969.

10.

D.A. Ro binson , "A quantitative analysis of extraocular muscle cooperation and squint," Invest. Ophtal. vo1. 14, pp. 801-825 , 1975.

11.

Vision Research: A National Pl an 1983-1987 : The 1983 Report of The National Advisory Eye Council, vol. 1, p 43. USDHHS, PHS, NIH , 1983.

12.

C. C. Collins, M. R. Carlson, A.B. Scot t, and A. Jampolsky , "Extraocular muscl e forces in normal human s ubjects," I nvest . Opthal . Vis . Sci . vol . 20 , no. 5, pp 652-664 , 1981.

13.

J .M. Mi ll er and D.A . Ro binson, "A model of the me chanics of binocul ar al i gnment," submi tt ed to Computers and Biomedical Research, 1984

14.

J.M. Miller, D.A. Ro binson, A.B . Scott, D. Robins, "Side-slip and the act ion of ocul ar musc les ," s upplement t o Invest. Vis . Sci . vo l. 25 , no. 3, p . 182, March

15.

H. J . Simonsz, F. Harti ng, B.J . de Waa l, and B. W. J .M. Verbeeten, "Sideways displaceme nt and curved path of recti eye musc les," unpublished manuscript, 1984 .

16.

K. Nakayama, "Coordination of extraocular muscle s," In Basic Mechanisms of Ocul ar Motility and Their Cli ni cal Implications (G . Lenners tr and and P. Bach - y-Rita , Eds . ) , pp . 193-207. Pergamon Press: New Yo rk, 1975 .

17.

H. Akima , "A method of bi varate interpolation and smooth s ur face fit tin g for irregul arly distribut ed dat a point s . " .\01 Trans. ~l ath . Software . vol. 4 no . 2, pp. 14 8- 159, 1978 .

C-p

The system will certainly by modi fied as more is learned about orbital mechanics and i nnerv ation, and to facilitate changes we have made SQUINT highl y modular at both the command and subroutine levels. The SQUINT source code occupies abo ut 0 . 4 Mbytes. About 2 .0 Mbytes is snough disc space for the entire system including the source, object library, load modules, and some user-produced output. Several SQUINT processes are l arger th an 64 KBytes; machines (such as small PDP-l l' s) wi th limited addressing range may be unabl e t o run the full system. We are ple ased to make SQUINT available to any interested investigator. REFERENCES 1.

G. K. von Noorden, Binocular Vision and Ocul ar Motility: Theory of Management of Strabi smus , 2nd ed . St. Loui s: C.V. Mosby, 1980

2.

G. K. vo n Noorden, Atlas of St rabi smus , 3rd ed . St. Lo ui s: C.V. Mosby, 1977 .

3.

M. M. Parks, Ocu l ar Mo tilit y and Strabismus . Hagerstown: Harper &Row, 1975

4.

A.W . Volkman, "On the mechanics of th e eye muscles," Ber. Ve rh. Sachs . Ges . ll'sch . vol. 21 , pp . 28- 69,-r869-.--- - ---- ---

5.

T. Nakagawa , "Topo graphic anat omical s tudies on t he orbit and it s cont ent s, "Act a Soc . ~ . ~ . vol. 69, pp . 2155-2179 , 1965 .

6.

W. E. Krewso n, "The action of the ext raocular muscles; A met hod of vector anal ysis wi t h comput ations , " Trans. Am. Opthalmol. Soc . vo l, 48 , pp . 443-49 6, 1950.

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and extr aOphth al 1984.