Eye Movements With Pulleys Geometrical Anatomy of the Eye and Eye Movements



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Geometrical Anatomy of the Eye and Eye Movements

In a Model with Restricted Extraocular Muscles

Thomas P. Langer


Introduction


In a companion paper, the geometrical anatomy of the eye and its implications for the eye’s movements were explored under the assumption that the eye muscles were free to follow their insertions as the eye moved. Comparatively recently it has been discovered that the eye’s extrinsic muscles are not free to follow their insertion (Demer, Miller et al. 1995; Clark, Miller et al. 1997; Demer, Poukens et al. 1997; Clark, Miller et al. 2000; Demer, Oh et al. 2000). Rather, they are constrained by fascial slings that act as pulleys. These pulleys are located a short distance posterior to the globe’s equator in neutral gaze, which might have profound consequences for the eye movements that they produce. The location and general anatomy of the relevant fascia has been known for some time (Duke-Elder and Wybar 1961; Williams, Bannister et al. 1995), but the role it plays in eye movements is only fairly recently appreciated. In this paper the same type of analysis will be done as was done in the free muscle model, but with the pulleys as part of the model.

It turns out that the formal differences between the two models are comparatively small. The only substantial difference is that the origins of the rectus muscles are no longer at the annulus of Zinn. While the muscles still take their anatomical origins from that common tendinous ring, their functional origins are a collection of fascial slings embedded in the fascia that encircles the globe near its equator and binds it to the orbital wall. These slings resist displacement of the muscles relative to the orbital wall, but allow the muscle’s tendons to freely follow their insertions as the eye moves. The model defines the origin to mean the functional origin, the point from which the muscle pulls, rather than the anatomical origin. Consequently, the origins move from the back of the orbit and medial to the eyeball to a series of slings near the transverse equator of the eyeball.



There is a great deal of detailed anatomy that relates to the orbital fascia, but the main points that we need to extract for the purposes of computing the eye movements is the locations of the pulleys for each of the recti. These are given in a recent paper that used MRI to monitor the eye muscles (Clark, Miller et al. 2000). As the eye was adducted and abducted the vertical recti flexed at their pulleys, and as the eye was elevated and depressed, the horizontal recti flexed at their pulleys. It was found that the locations of the pulleys were as summarized in the following table.


Normal Rectus Pulley Positions Relative to the Globe’s Center

Muscle

Anterior

Lateral

Superior

Medial Rectus

-3 ± 2 mm.

-14.2 ± 0.2 mm.

-0.3 ± 0.3 mm.

Lateral Rectus

-9 ± 2 mm.

10.1 ± 0.1 mm.

-0.3 ± 0.2 mm.

Superior Rectus

-7 ± 2 mm.

-1.7 ± 0.3 mm.

11.8 ± 0.2 mm.

Inferior Rectus

-6 ± 2 mm.

-4.3 ± 0.2 mm.

-12.9 ± 0.1 mm.

These measurements in millimeters translate into unit vectors originating from the center of the globe as follows.


Normal Rectus Pulley Positions Relative to the Globe’s Center

Muscle

i

j

k

Medial Rectus

-0.25

1.2

-0.03.

Lateral Rectus

-0.25

-0.84

-0.03

Superior Rectus

-0.58

0.07

0.98

Inferior Rectus

-0.50

0.36

-1.1

Once we have computed the locations of the rectus pulleys, we can plug them into the model developed for the freely moving eye muscles as the origins of the muscles. The calculations are otherwise the same as for that model.

Methods


The methods for this analysis are essentially as described in the companion paper. The principal differences are in the changes in the locations of the origins of the rectus muscles.

The calculations were done with the same model as for the analysis of free muscle anatomy, except for the changes in the muscle origins indicted above. All the calculations were done in Mathematica and most of the figures are taken from Mathematica with some labeling added in Canvas 9.


Results

Changes in Muscle Length with Changes in Gaze Direction





Figure 1. The distribution of the change in the medial rectus muscle’s length as a function of horizontal and vertical offset from neutral gaze: Restricted muscle model. The plotted variable is the difference between the muscle’s length in neutral gaze and its length in the offset gaze (ML). The vertical and horizontal offsets may run in different directions in the different plots, to best display the surface’s configuration. See the text for a description of the surface.

Gaze direction and orientation are completely determined by the set of muscle lengths of the six extraocular muscles. We have examined the distribution of the extraocular muscles as a function of gaze direction when the muscles are free the follow their insertions (Langer, 2004). It was found that the surface that represents this relationship is complex, but sections of the surface for each muscle is generally a fairly shallow hyperbolic, or saddle-shaped, surface tilted with respect to the coordinate plane for gaze direction. When the calculations are done with the pulleys acting as functional origins for the muscles, the results are similar.



The surfaces for the superior and inferior oblique muscles are the same because their origins and insertions are the same as in the free muscle model. The general differences for the four rectus muscles are that they are less curved than in the free muscle model. Despite the flatter surfaces the hyperbolic shape is sometimes more apparent.


Figure 2. The distribution of the change in the lateral rectus muscle’s length as a function of horizontal and vertical offset from neutral gaze: Restricted muscle model. The conventions are the same as for the medial rectus muscle figure.



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