Robinson's Computerized Strabismus Model Comes of Age

Finite Element Model of Ocular Adduction by Active Extraocular Muscle Contraction

Purpose

In order to clarify the role of the optic nerve (ON) as a load on ocular rotation, we developed a finite element model (FEM) of incremental adduction induced by active contractility of extraocular muscles (EOMs), with and without tethering by the ON.

Methods

Three-dimensional (3-D) horizontal rectus EOM geometries were obtained from magnetic resonance imaging of five healthy adults, and measured constitutive tissue properties were used. Active and passive strain energies of EOMs were defined using ABAQUS (Dassault Systemes) software. All deformations were assumed to be caused by EOM twitch activation that rotated the eye about a fixed center. The medial rectus (MR) muscle was commanded to additionally contract starting from 26 degrees adducted position, and the lateral rectus (LR) to relax, further adducting the eye either with or without loading by the ON. Tridimensional heat maps were generated to represent the stress and strain distributions.

Results

Tensions in the EOMs were physiologically plausible during incremental adduction. Force in the MR increased from 10 gm at 26 degrees adduction to approximately 28 gm at 32 degrees adduction. Under identical MR contraction, adduction with ON loading reached 32 degrees but 36 degrees without it. Maximum and minimum principal strains within the MR were 16% and 22%, respectively, but when ON loading was included, resulting stress and strain were concentrated at the optic disc.

Conclusions

This physiologically plausible method of simulating EOM activation can provide realistic input to model biomechanical behavior of active and passive tissues in the orbit to clarify biomechanical consequences of ON traction during adduction.

Instead of pulley bands, does retrobulbar fat keep the eye muscle bellies in place and thereby act like a pulley?

Extraocular muscle pulley bands were described by Tenon in 1805 as "faisceaux tendineux" acting as "poulies de renvoi." The Passive and Active Pulley Hypotheses propose that these connective-tissue bands between muscle and bony orbital rim limit vertical shift of the horizontal rectus muscle belly in up- and downgaze, caused by the muscle's tendency to assume the shortest path from origin to insertion. The band's attachment to the muscle moves 20 mm sagittally when the eye looks from 50° left to 50° right, however, impeding vertical muscle stabilization. Sliding of the muscle in a sleeve would permit sagittal movement, but four anatomical studies could not confirm that. The band would have to be elastic: We measured it after orbital exenteration and found it to be slack, however, and once extended, very stiff. Our research group in Amsterdam suggested in 1984 that the retrobulbar fat and its enveloping connective-tissue sheets including the intermuscular membrane keep muscle bellies in place. We compared horizontal-rectus-muscle positions in up- and down-gaze using frontal CTs through the posterior pole of the eye. The bellies stayed in place while, anteriorly, the tendons bent up- and downward. We also found that the paths of horizontal rectus muscles were curved outwards in horizontal CTs. We surmised that retrobulbar pressure in the fat, resulting from four rectus muscles pulling the eyeball into the orbit, is contained by rectus muscles and connective-tissue sheets and that the resulting tension in the sheets keeps the muscles in place. Years later we repeated the CT study in a Crouzon patient whose bony orbital rim was displaced 2cm posteriorly, preventing pulley-band fixation to the bone: No vertical shift of horizontal rectus muscle bellies occurred in up- and down-gaze. Finally, we developed a mathematical finite-element model of orbit, muscles, fat and eyeball to study whether fat with enveloping connective-tissue sheets could keep eye muscles in place. In simulated eye movements, the retrobulbar fat, with low elasticity as found in vivo, not only kept the eyeball in place but also horizontal rectus muscle bellies in up- and down-gaze and vertical recti in left- and right-gaze.

The history of the Ostschweizerische Pleoptik und Orthoptik Schule in St. Gallen

In judging the achievements of Alfred Bangerter in treatment and research of amblyopia it is easy to conclude that his pleoptic exercises have been forgotten because occlusion therapy is more effective and cheaper. However, Bangerter introduced the visuscope to determine the point of fixation directly on the retina, he started the first "school" (exercise treatment facility) for pleoptics and orthoptics in St. Gallen only 18 years after Mary Maddox did so in London and he started a training program for orthoptists. In 1957 the Genossenschaft Ostschweizerische Pleoptik-und Orthoptik-Schule, the OPOS Society, was founded, that in the following years built a clinic especially for the treatment of amblyopia. The idea was to treat children not in a clinic but in a home for children that offered optimal treatment but also adequate lodging and care for the children with amblyopia. The Cantonal government contributed by donating a right to build on the premises of the Cantonal Hospital. The new OPOS Clinic measured more than 500 square meters, had 4 floors and a cellar, and contained outpatient treatment facilities, two operating theatres, patient bedrooms, pleoptic and orthoptic exercise rooms with many devices and classrooms for orthoptic students. There were 56 beds for children. After Bangerter retired as chief physician of the Eye Clinic in 1974, he continued and expanded his clinical and surgical activity in the OPOS Clinic next to the Eye Clinic. After his successor in the OPOS Clinic retired in 1987, the OPOS Foundation sold the OPOS Clinic to the Canton that reintegrated it into the Eye Clinic. In the meantime, Bangerter had continued to pursue his ideal of amblyopia treatment and built a new clinic in Heiden in the neighbouring Canton Appenzell Ausserrhoden, for pleoptics, orthoptics, strabismus surgery, plastic eye surgery, but also for controversial treatments for macular degeneration and other retinal disorders. This Rosenberg Clinic opened in 1982 but Bangerter already stepped down in April 1983 and opened a day clinic in the Rosenbergstrasse in St. Gallen some years later instead. Strangely enough, one of the reasons he had moved to the Rosenberg Clinic was that he insisted on lengthy clinical stays for the treatment of amblyopia, but exactly that was one of the main causes of financial problems.

In vivo experimental study on the resistance and stiffness of orbital suspension tissues with/without the extraocular muscles

BACKGROUND

The accuracy of the surgical amount of extraocular muscle (EOM) is key to the success of strabismus surgery. To establish an accurate eye movement model, it is of great theoretical value and clinical significance to determine the surgical amount of EOM. At present, only resistance and stiffness data of orbital suspension tissues with EOMs exist, while those of orbital suspension tissues without EOMs, which is critical information for eye movement modeling, have not been reported. The aim of this research is to study the resistance and stiffness of orbital suspension tissues with/without EOMs.

METHODS

Fifteen healthy New Zealand white rabbits with body weights of 2.41 ± 0.13 kg were used in the study. Two recti (two horizontal recti of the left eye or two vertical recti of the right eye) or all EOMs were detached from each eye under general anesthesia. Then, a 5-0 silk suture was attached to the stump of the detached rectus insertion (two horizontal recti insertions of the left eye and two vertical recti insertions of the right eye) on the isolated eyeball. The 5-0 silk suture was connected to the INSTRON 5544 tester to facilitate the horizontal rotations of the left eyes and the vertical rotations of the right eyes, respectively.

RESULTS

The resistance and stiffness of orbital suspension tissues with superior rectus, inferior rectus, superior oblique, and inferior oblique EOMs were obtained during horizontal eye movement. Similarly, the resistance and stiffness of orbital suspension tissues with lateral rectus, medial rectus, superior oblique, and inferior oblique EOMs were obtained during vertical eye movement. Then, the resistance and stiffness of orbital suspension tissues without EOMs were obtained during horizontal and vertical eye movements. The resistance and stiffness data of orbital suspension tissues with EOMs were compared with those of orbital suspension tissues without EOMs. The comparison results showed no significant difference in the resistance values between these two cases. In addition, the stiffness values of these two cases statistically differed.

CONCLUSIONS

The two horizontal recti play a major role in passive horizontal eye movement. In addition, when the eye is passively moved vertically, the two vertical recti play major roles. The stiffness of orbital suspension tissues with EOMs, which has been used in eye movement modeling, is not accurate. The results of this work may serve as a reference for improving the accuracy in eye movement modeling, and then it will be beneficial for determining the surgical amount of EOMs in clinical surgery.

Prediction of muscle activation for an eye movement with finite element modeling

In this paper, a 3D finite element (FE) modeling is employed in order to predict extraocular muscles' activation and investigate force coordination in various motions of the eye orbit. A continuum constitutive hyperelastic model is employed for material description in dynamic modeling of the extraocular muscles (EOMs). Two significant features of this model are accurate mass modeling with FE method and stimulating EOMs for motion through muscle activation parameter. In order to validate the eye model, a forward dynamics simulation of the eye motion is carried out by variation of the muscle activation. Furthermore, to realize muscle activation prediction in various eye motions, two different tracking-based inverse controllers are proposed. The performance of these two inverse controllers is investigated according to their resulted muscle force magnitude and muscle force coordination. The simulation results are compared with the available experimental data and the well-known existing neurological laws. The comparison authenticates both the validation and the prediction results.

The nonlinearity of passive extraocular muscles

Passive extraocular muscles (EOMs), like most biological tissues, are hyperelastic, that is, their stiffness increases as they are stretched. It has always been assumed, and in a few occasions argued, that this is their only nonlinearity and that it can be ignored in central gaze. However, using novel measurement techniques in anesthetized paralyzed monkeys, we have recently demonstrated that EOMs are characterized by another prominent nonlinearity: the forces induced by sequences of stretches do not sum. Thus, superposition, a central tenet of linear and quasi-linear models, does not hold in passive EOMs. Here, we outline the implications of this finding, especially in light of the common assumption that it is easier for the brain to control a linear than a nonlinear plant. We argue against this common belief: the specific nonlinearity of passive EOMs may actually make it easier for the brain to control the plant than if muscles were linear.

Physically-based modeling and simulation of extraocular muscles

Dynamic simulation of human eye movements, with realistic physical models of extraocular muscles (EOMs), may greatly advance our understanding of the complexities of the oculomotor system and aid in treatment of visuomotor disorders. In this paper we describe the first three dimensional (3D) biomechanical model which can simulate the dynamics of ocular motility at interactive rates. We represent EOMs using "strands", which are physical primitives that can model an EOM's complex nonlinear anatomical and physiological properties. Contact between the EOMs, the globe, and orbital structures can be explicitly modeled. Several studies were performed to assess the validity and utility of the model. EOM deformation during smooth pursuit was simulated and compared with published experimental data; the model reproduces qualitative features of the observed nonuniformity. The model is able to reproduce realistic saccadic trajectories when the lateral rectus muscle was driven by published measurements of abducens neuron discharge. Finally, acute superior oblique palsy, a pathological condition, was simulated to further evaluate the system behavior; the predicted deviation patterns agree qualitatively with experimental observations. This example also demonstrates potential clinical applications of such a model.

Simulation of oculomotility in craniosynostosis patients

Craniosynostosis syndromes can be associated with missing extraocular muscles, or muscles with abnormal insertions, and so provide useful test cases for assessing our understanding of the mechanics of the extraocular muscles. Patient with craniosynostosis syndromes often show eye movements in which a horizontal movement by one eye is accompanied by upshoot or downshoot in the other eye. An hypothesis which has been put forward to explain these movements is that the muscles in the patients are excyclorotated and that the upshoots and downshoots follow directly from the application of Hering's law of equal innervation. We modelled the mechanics of the excyclorotated muscles and verified this hypothesis. However, excyclorotation of the orbit often occurs in combination with anomalous muscle anatomy in craniosynostosis syndromes. In keeping with this finding, we have found that surgical transposition of the rectus muscles is insufficient by itself to correct the anomalous eye movements, but that transposition in combination with weakening of the obliques is effective.

Dynamics of primate oculomotor plant revealed by effects of abducens microstimulation

Despite their importance for deciphering oculomotor commands, the mechanics of the extraocular muscles and orbital tissues (oculomotor plant) are poorly understood. In particular, the significance of plant nonlinearities is uncertain. Here primate plant dynamics were investigated by measuring the eye movements produced by stimulating the abducens nucleus with brief pulse trains of varying frequency. Statistical analysis of these movements indicated that the effects of stimulation lasted about 40 ms after the final pulse, after which the eye returned passively toward its position before stimulation. Behavior during the passive phase could be approximated by a linear plant model, corresponding to Voigt elements in series, with properties independent of initial eye position. In contrast, behavior during the stimulation phase revealed a sigmoidal relation between stimulation frequency and estimated steady-state tetanic tension, together with a frequency-dependent rate of tension increase, that appeared very similar to the nonlinearities previously found for isometric-force production in primate lateral rectus muscle. These results suggest that the dynamics of the oculomotor plant have an approximately linear component related to steady-state viscoelasticity and a nonlinear component related to changes in muscle activation. The latter may in part account for the nonlinear relations observed between eye-movement parameters and single-unit firing patterns in the abducens nucleus. These findings point to the importance of recruitment as a simplifying factor for motor control with nonlinear plants.

A finite-element analysis model of orbital biomechanics

To reach a better understanding of the suspension of the eye in the orbit, an orbital mechanics model based upon finite-element analysis (FEA) has been developed. The FEA model developed contains few prior assumptions or constraints (e.g., the position of the eye in the orbit), allowing modeling of complex three-dimensional tissue interactions; unlike most current models of eye motility. Active eye movements and forced ductions were simulated and showed that the supporting action of the orbital fat plays an important role in the suspension of the eye in the orbit and in stabilization of rectus muscle paths.

Evidence for wide range of time scales in oculomotor plant dynamics: implications for models of eye-movement control

Oculomotor-plant dynamics are not well characterised, despite their importance for modelling eye-movement control. We analysed the time course of the globe's return after horizontal displacements in three rhesus monkeys lightly anaesthetised with ketamine. The eye-position traces were well fitted by a sum of four exponentials (time constants 0.012, 0.099, 0.46, 7.8 s). The two long time-constant terms accounted for 25% of plant compliance, and led to a model that accounted for hitherto unexplained features of ocular motoneuron firing such as (i) hysteresis, and (ii) the inability of a 2 time-constant model to fit data for both fast and slow eye-movements.

Reduction of ocular muscle power by splitting of the rectus muscle I: biomechanics

BACKGROUND/AIM

Based on mechanical considerations, the authors have developed a new approach to weakening oculomotor muscles. They present the biomechanical considerations that have encouraged them to explore this approach, and compare it with existing surgical techniques. Results of application to patients are given in the companion paper, and do not require an analytical understanding of the underlying mechanics.

METHODS

Using a simple biomechanical model for the oculomotor system and vector component analysis, the eye position dependent torque exerted by extraocular muscles on the eyeball was investigated. This model was applied to the healthy eye, as well as to different surgical procedures (Cuppers' Fadenoperation, Y-split muscle recessions, botulinum toxin, and simple muscle recessions).

CONCLUSION

These investigations suggest that a Y-split muscle recession is a simple and efficient way to weaken ocular rectus muscles. Compared to alternative surgical procedures, undesired radial forces that can lead to post-surgical complications can be kept to a minimum. The authors further speculate that their good results may in part be because of possible preservation of proprioceptive inputs from the insertion of the extraocular muscle.

Mechanics of eye movements: implications of the "orbital revolution"

Our understanding of the functional structure of extraocular muscles has undergone a profound change: while these muscles used to be represented by strings running straight from their origin in the posterior orbita to their insertion on the globe, we now know that their paths and pulling directions are dominated by fibromuscular pulley structures, keeping them close to the orbital wall for most of their path. An overview is presented of recent models that have been developed to understand the implications of muscle pulleys for the neural control of eye movements and the applications of such models to the interpretation of experimental data.

A simple control law generates Listing's positions in a detailed model of the extraocular muscle system

The neural commands for maintaining static Listing's positions were identified using a detailed model of extraocular muscle based on Miller and Shamaeva (Orbit 1.5 gaze mechanics simulation 1995). The commands were approximately separable, suggesting a simple control law whereby independent horizontal and vertical commands are combined to generate tertiary positions. Tests showed that this control law (i) generated Listing' s positions to reasonable accuracy over+/-30 deg, provided pulleys were represented in the model; (ii) if driven by retinal coordinates, produced errors close to the theoretical minimum for a commutative system. The proposed commands appear consistent with electrophysiological evidence.

Simulation of pathological ocular counter-roll and skew-torsion by a 3-D mathematical model

A basic version of a 3-D mathematical model for simulation of otolithic control of binocular static eye position was extended by introducting excitatory commissural fibers between the vestibular nuclei, and physiological non-linearities: the force-response relationship of utricular neurons and a quadratic relationship between eye muscle innervation and force. These modifications appeared to be necessary in order to simulate the gain asymmetry of ocular counter-roll to lateral head tilt in patients with unilateral utricular loss. The current model can adequately simulate skew-torsion in patients with unilateral utricular loss, lesions of the vestibular nuclei, and central graviceptive pathway lesions. The direction of simulated skew-torsion corresponds satisfactorily to data from normals and patients with acute vestibular loss. The relatively low values of predicted eye deviations for peripheral vestibular lesions suggest that part of the effects seen in patients is caused by affection of the semicircular canals.

Optimality of position commands to horizontal eye muscles: A test of the minimum-norm rule

Six muscles control the position of the eye, which has three degrees of freedom. Daunicht proposed an optimization rule for solving this redundancy problem, whereby small changes in eye position are maintained by the minimum possible change in motor commands to the eye (the minimum-norm rule). The present study sought to test this proposal for the simplified one-dimensional case of small changes in conjugate eye position in the horizontal plane. Assuming such changes involve only the horizontal recti, Daunicht's hypothesis predicts reciprocal innervation with the size of the change in command matched to the strength of the recipient muscle at every starting position of the eye. If the motor command to a muscle is interpreted as the summed firing rate of its oculomotor neuron (OMN) pool, the minimum-norm prediction can be tested by comparing OMN firing rates with forces in the horizontal recti. The comparison showed 1) for the OMN firing rates given by Van Gisbergen and Van Opstal and the muscle forces given by Robinson, there was good agreement between the minimum-norm prediction and experimental observation over about a +/-30 degrees range of eye positions. This fit was robust with respect to variations in muscle stiffness and in methods of calculating muscle innervation. 2) Other data sets gave different estimates for the range of eye-positions within which the minimum-norm prediction held. The main sources of variation appeared to be disagreement about the proportion of OMNs with very low firing-rate thresholds (i.e., less than approximately 35 degrees in the OFF direction) and uncertainty about eye-muscle behavior for extreme (>30 degrees ) positions of the eye. 3) For all data sets, the range of eye positions over which the minimum-norm rule applied was determined by the pattern of motor-unit recruitment inferred for those data. It corresponded to the range of eye positions over which the size principle of recruitment was obeyed by both agonist and antagonist muscles. It is argued that the current best estimate of the oculomotor range over which minimum-norm control could be used for conjugate horizontal eye position is approximately +/-30 degrees. The uncertainty associated with this estimate would be reduced by obtaining unbiased samples of OMN firing rates. Minimum-norm control may result from reduction of the image movement produced by noise in OMN firing rates.

Pseudo-inverse control in biological systems: a learning mechanism for fixation stability

The problem of redundancy in motor control is common to both robotics and biology. Pseudo-inverse control has been proposed as a solution in robotics and appears to be used by the oculomotor system for eye position. Learning mechanisms for implementing pseudo-inverse control using a distributed system of ocular motor units were investigated by modelling integrator calibration for horizontal eye movements. Ocular motoneuron (OMN) input weights were adjusted with a gradient-descent learning rule, using a retinal-slip estimate as an error signal. Firing-rate threshold only became related to motor-unit strength when a noise term was added to OMN firing rates. The learning rule suppressed those units making the largest contribution to the noise-related error, causing the strongest units to have the highest thresholds (size principle). Because the size principle and pseudo-inverse control are related, the trained system approximated pseudo-inverse control over the central +/-35 degrees of the oculomotor range.