physically-based modeling and simulation of extraocular muscles

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Progress in Biophysics and Molecular Biology 103 (2010) 273e283

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Progress in Biophysics and Molecular Biology

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Original research

Physically-based modeling and simulation of extraocular muscles

Qi Wei a,b, Shinjiro Sueda b, Dinesh K. Pai b,*aDepartment of Physiology, Feinberg Medical School, Northwestern University, 303 E. Chicago Ave., Chicago, IL, USAb Sensorimotor Systems Laboratory, Department of Computer Science, The University of British Columbia, 2366 Main Mall, Vancouver, BC, Canada

a r t i c l e i n f o

Article history:Available online 22 September 2010

Keywords:Physically-based modelingExtraocular muscleInteractive simulationSoft tissue motionEye movementBiomechanics

* Corresponding author. Tel.: þ1 604 822 8197; faxE-mail address: [email protected] (D.K. Pai).

0079-6107/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.pbiomolbio.2010.09.002

a b s t r a c t

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 intreatment of visuomotor disorders. In this paper we describe the first three dimensional (3D) biome-chanical model which can simulate the dynamics of ocular motility at interactive rates. We representEOMs using “strands”, which are physical primitives that can model an EOM’s complex nonlinearanatomical and physiological properties. Contact between the EOMs, the globe, and orbital structures canbe explicitly modeled.

Several studies were performed to assess the validity and utility of the model. EOM deformationduring smooth pursuit was simulated and compared with published experimental data; the modelreproduces qualitative features of the observed nonuniformity. The model is able to reproduce realisticsaccadic trajectories when the lateral rectus muscle was driven by published measurements of abducensneuron discharge. Finally, acute superior oblique palsy, a pathological condition, was simulated to furtherevaluate the system behavior; the predicted deviation patterns agree qualitatively with experimentalobservations. This example also demonstrates potential clinical applications of such a model.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Vision is one of our most important senses. Rapid and accurateeye movements are crucial for coordinated direction of gaze.Studying human eye movement has significant implications forimproving our understanding of the oculomotor system andtreating visuomotor disorders. Computational simulation hasplayed an important role in analyzing the mechanisms of ocularmotility. In particular, having a realistic model of the extraocularmuscles (EOMs) and simulating their biomechanics in threedimensions (3D) are necessary for investigating the function andneural control of EOMs. In this paper, we present a 3D biome-chanical model for realistically simulating extraocular musclemotion at interactive rates.

Various computational models of the extraocular muscle andorbital mechanics have been proposed, which provide insight andscientific bases for oculomotor biomechanics, control of eyemovement, and binocular misalignment. Early one-dimensionalbiomechanical models were based on physiological data and laidthe groundwork for the subsequent studies (Clark and Stark, 1974;

: þ1 604 822 0848.

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Collins, 1971; Collins et al., 1975; Robinson, 1964, 1981). Thesemodels characterize the viscoelastic properties of the EOMs, basedon experimental measurements such as the tension-length rela-tionship of the EOMs (Robinson, 1964). They focus on the realism ofmuscle behaviors in the horizontal direction, where the data wereacquired. Extension to three dimensions is not trivial. As a matter offact, these dynamic models have only been simulated in 1D.However, to study the kinematics and neural control of the orbitalplant, it is indispensable to analyze its rotation in 3D. The abovemodels are not easy to apply in more general situations.

The first 3D biomechanical model was developed by Robinson(1975), who simplified the formulation by only considering the elas-ticity of the EOMs and ignoring the dynamics. Equilibrium of the netforce on the eyeball was analytically calculated, and static fixations in3Dwere studied. Themodel incorporates anatomically realisticmusclepaths and empirical EOM innervation-length-tension relationships tostudy normal binocular alignment, strabismus, and surgical correc-tions. Two models were developed that improved Robinson’s model:Simonsz’s model (Simonsz and Spekreijse, 1996) and the SQUINTmodel (Miller and Robinson, 1984). The latter became an iterativesoftware tool e the OrbitTM gaze mechanics simulation (Miller et al.,1995). Recently, Quaia et al. (2008) presented simulation of superioroblique palsy using a model refined from their previous work (Quaiaand Optican, 2003). All of the above models in 1D or 3D are lumped

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Fig. 1. 3D model of a human orbit reconstructed from MRI (Wei et al., 2009). The sixextraocular muscles are: LR-lateral rectus, MR-medial rectus, SR-superior rectus, IR-inferior rectus, SO-superior oblique, IO-inferior oblique.

Q. Wei et al. / Progress in Biophysics and Molecular Biology 103 (2010) 273e283274

biomechanicalmodels: the action of each EOM is simplified as a singleforce vector without taking into account the distribution of materialproperties and forces along the EOMs. These lumped models also donot account formusclemassproperly (Pai, 2010). Furthermore, contactbetween the EOMs and the globe cannot be dealt with. Schutte et al.(2006) proposed an orbital model based on 3D finite element model(FEM). The EOMs were assigned constant stiffness and contact ishandled to a limited extent by approximating the interface as a layerwith low elasticity.

One major limitation of the 3D biomechanical models describedabove is that only static eye positions can be simulated. To study theneural control of rapid saccadicmovement, models using simplifiedanatomical and mechanical properties of EOMs have been devel-oped (Quaia and Optican, 1998; Raphan, 1998; Schnablok andRaphan, 1994). These models assume that three pairs of EOMs actin three planes that are orthogonal to each other. They do not takeinto account the anatomical variations of different EOMs, such asmuscle lengths and cross sectional areas. Another criticalassumption made to simplify the analytical solution is on thenature of the EOM force e the active force is assumed to be simplyproportional to the muscle innervation, whereas the actual EOMforce is a complex nonlinear function of the muscle length, velocity,and innervation. Such models have the advantage of supporting 3Ddynamic simulations and have been used to analyze neuralcontrollers and the pulley hypothesis (Quaia and Optican, 1998;Raphan, 1998). These simplifications, however, limit the models’accuracy and plausibility, as has been illustrated by (Quaia andOptican, 2003).

In this paper, we describe a novel 3D biomechanical simulator ofthe extraocular muscles that addresses the drawbacks of theexisting models. We use the spline strand (Sueda et al., 2008) tomodel EOM mechanics. Our proposed model has the followingdesirable properties:

� Strand-based EOMs are associated with realistic subject-specific muscle anatomy, such as muscle paths and crosssectional areas determined frommagnetic resonance images ofhuman orbits.

� Complex nonlinear EOM physiological properties are incorpo-rated in the computation by explicitly representing theexperimental EOM constitutive models.

� Extraocular connective tissues that stabilize the EOMs in theorbit, called pulleys, can be implemented.

� The model is computationally efficient. It provides a goodbalance between simulation accuracy and speed. EOMmotionscan be simulated at interactive rates (w16frames/second) onpersonal computers.

� Contact between EOMs and the globe is physically handledthrough sliding constraints, not only for visualization but alsofor realistically simulating force interaction and lateral slidingmovement.

� Our biomechanical model can simulate extraocular muscledynamics, which have not been supported by other 3Dbiomechanical models.

In a previous report (Wei et al., 2010), we have presented thestructures of our biomechanical model of the orbital plant. In thispaper, we focus on describing the detailed methodology and resultsof the extraocular muscles e the most important components ofthe whole orbital plant. In the following, we first briefly introducethe anatomy and biomechanics of EOMs in Section 2.1. We thenpresent an in-depth description of our strand-based EOM model inSection 2. To assess the accuracy of our extraocular muscle model,we design several validation experiments. We analyze simulatedEOM deformation during smooth pursuit and compare it to

motion-encoded magnetic resonance imaging (MRI) data (seeSection 3.1). We use empirical neural discharges as the controlsignal to reproduce saccadic movement (see Section 3.2). Finally,we present simulation of a pathological case to further evaluate thesystem behavior against experimental data in Section 3.3.

2. Methods

2.1. Anatomy and biomechanics of extraocular muscles

The orbital plant consists of the globe (eyeball), three pairs ofextraocular muscles, and connective tissues. Fig. 1 shows a 3Dmodel of the human orbit reconstructed from magnetic resonanceimaging (MRI) (Wei et al., 2009). Table 1 summarizes the acronymsused throughout the paper. The six EOMs, including four rectusmuscles and two oblique muscles, are controlled by the cranialnerves to generate force, to rotate the globe to track a visual target,and to stabilize the image of the object on the retina.

The four rectus EOMs originate from the annulus of Zinn; theycourse anteriorly, pass through Tenon’s capsule, and insert on thesclera. The lateral rectus (LR) and medial rectus (MR) muscles sharea common horizontal plane. Contractions of these two musclesproduce horizontal eye movement. The superior rectus (SR) andinferior rectus (IR) muscles form the vertical agonisteantagonistpair, which mainly controls vertical eye movement and also affectsrotation about the line of sight (visual axis). The superior oblique(SO) muscle passes through the cartilaginous trochlea attached tothe orbital wall, which reflects the SO path by 54� (von Noordenand Campos, 2001). The inferior oblique (IO) muscle originatesfrom the orbital wall anteroinferior to the globe center and insertson the sclera posterior to the globe equator. The primary actions ofSO and IO cause rotation of the globe around the visual axis andvertical movement. SO and IO muscles also abduct the eye assecondary actions. More details of the orbit anatomy can be foundin (von Noorden and Campos, 2001).

Different from skeletal muscles, EOMs are bilaminar and consistof two layers with different fiber types (Porter et al., 1995). Thiscomplicated and special architecture contributes to the realizationof both fixations and rapid eye movements up to 900�/sec, andrequires cooperative control of the EOMs involving very complexneural circuits.

Based on evidence from MRI and computerized 3D reconstruc-tion, Miller (1989) first proposed that the rectus muscle connectivetissues posterior to the globe equator, called “pulleys”, couplerectus EOMs to the orbital wall and constrain the transverse shiftsof muscle paths. Pulleys are believed to have significant implica-tions on the kinematics and neural control of the orbital plant

Table 1Acronym table.

Acronym Term

EOM Extraocular muscleLR Lateral rectusMR Medial rectusSR Superior rectusIR Inferior rectusSO Superior obliqueIO Inferior obliqueCE Contractile elementSE Series elasticPE Passive elasticFL ForceelengthFV ForceevelocityABN Abducens nucleusSOP Superior oblique palsy

Q. Wei et al. / Progress in Biophysics and Molecular Biology 103 (2010) 273e283 275

(Demer, 2004, 2007; Haslwanter, 2002; Miller, 2007; Quaia andOptican, 1998). In order to understand their specific functions,biomechanical simulation of the orbit incorporating the pulleyconnective tissues are needed (Haslwanter, 2002; Miller, 2007).

The Hill-type muscle constitutive model (Hill, 1938) widelyapplied in describing the force generation mechanism of skeletalmuscles has been adopted to describe extraocular muscle mechanics(Clark and Stark, 1974; Collins et al., 1975; Robinson, 1964, 1981). Itconsists of an active contractile element (CE), a series elastic element(SE), and a parallel elastic element (PE). See Fig. 2 for a representativemodel. CE, the active muscle force, depends on the muscle activationdynamics (a), the active force-length (FL) relationship fl, and theactive force-velocity (FV) relationship fv. PE models the passivemuscle force as a function of muscle length, characterized by thepassive FL curve. The active and passive FL relationships have beenmeasured experimentally (Robinson et al., 1969). Models of fv havebeen proposed based on Hill’s FV curve of skeletal muscles, experi-mental data on rat EOMs, and maximum saccadic velocity of humaneyes (Robinson, 1981). SE represents the passive muscle force fromelastic elements in series with CE, contributed by tendons andconnective tissues within muscles.

2.2. Strand-based EOM model

As mentioned in Section 1, previous 3D biomechanical EOMmodels do not simulate dynamics. In addition, the OrbitTM simu-lator does not consider the local deformation of EOMs, which isnonuniform caused by the varying cross sectional areas along themuscle path and fast dynamic motions. The finite element orbitalmodel (Schutte et al., 2006), similar to many other FEM studies,faces numerical difficulties while modeling thin structures like theEOM tendons. The range of deformation that can be simulated islimited and model parameters are difficult to choose. Furthermore,these are expensive to simulate, which restricts their interactiveanalysis and applications.

To simulate both static and dynamic eyemovements realisticallyand interactively, we use the strand model (Sueda et al., 2008) torepresent EOMs. A strand is a musculotendon modeling primitivebased on a spline with inertia. Each strand represents a part of the

Fig. 2. Three-element Hill-type muscle constitutive model. CE: contractile element;PE: parallel elastic element; SE: series elastic element.

musculotendon aligned with the fibers; depending on the level ofdetail needed, a strand can be as fine as a single fascicle or as coarseas an entiremuscle.Wemodel EOMs, consisting of parallel fibers, asa collection of strands. EOM orbital layer (OL) and global layer (GL)are explicitly modeled. An EOM strand incorporates the curvedmuscle path, which provides geometric realism and handlescontact. Compared to other biomechanical models that treat eachEOM as a single unit, the strand-based EOM model computes theforce and motion continuously distributed along the EOM path. Inother words, it models EOM mechanical properties locally andpossesses a higher level of complexity. Our EOM model is efficientto compute, benefiting from the parametric representation.

In the following, we first review the strand simulator in Section2.2.1 and then describe the strand-based EOMmodel in Section 2.2.2.

2.2.1. Strand musculoskeletal simulatorThe simulator consists of two dynamic primitives: rigid bodies

for the globe and orbit, and strands for the EOMs.The path of a strand is described by a cubic B-spline curve,

whose control points are the generalized coordinates, (position,qiðtÞ and velocity, _qiðtÞ), which define the strand dynamics. Giventhe generalized coordinates, a point on a strand at a parameter s iscomputed as pðs; tÞ ¼ P3

i¼0 biðsÞqiðtÞ, where biðsÞ are the B-splineblending functions. With these generalized coordinates of thestrand, we derive the kinetic and potential energies, and then theEulereLagrange equations of motions. Although a strand can havean arbitrary number of control points, a point on a strand onlydepends on four control points, due to the local support of the cubicB-spline basis, making the summation a function of only four terms(i ¼ 0.3). Because of this, the mass matrix of a strand (Qin andTerzopoulos, 1996) also depends on only four generalized coordi-nates, making the system mass matrix sparse. The B-spline basedstrand gives C2 continuity on the geometry and the dynamics.

The infinitesimal strain can be computed at any point along thestrand. eðs; tÞ ¼ ðP3

i¼0 b0iðsÞqiðtÞÞ=LðsÞ � 1, where b0iðsÞ are the

spline derivatives of the blending functions, and LðsÞ is the initialstrain. Based on these quantities, the passive (fp) and active (fa)forces acting on the strand control points can be computed, to getthe equations of motion of a single strand:

M _qðkþ1Þ ¼ M _qðkÞ þ h�f p þ af a

�; (1)

where M is the generalized mass matrix of the strand, h is the stepsize, and a is the activation level.

The equations of motion for the rigid body (globe) are based onthe standard body-local formulation with Coriolis term (Murrayet al., 1994). The sliding surface, on which the EOM strands slide,is modeled as a bicubic B-spline patch that is kinematicallyattached to the globe.

Constraints are required for modeling EOM origins/insertionsand for dealing with EOM contact with the globe. Velocity-levelconstraints on the relative velocities of the globe and the EOMstrands are used. Origins/insertions aremodeled as a constraint thatallows no relative velocity between a point on a strand and a pointon the globe. For sliding constraints, the relative velocity of a pointon a strand and a point on the sliding surface on the globe is con-strained to have no normal component. This way, only tangentialrelative velocities are permitted, allowing the strand to slide freelyon the surface patch. Since strands are based on parametric curves,it is computationally inexpensive to keep track of the contact points.At each time step, the closest points on another strand or a rigidbody B-spline surface are updated using NewtoneRaphson search.The combined velocity vector of rigid body and strands is denotedby F. The equality constraints with a stabilization term is formu-lated as GF ¼ mg. G is the constraint Jacobian for the fixed and

0.8 1 1.2 1.40

50

100

150

normalized length (L/L0)

mus

cle

forc

e (g

)

PassiveActiveTotal

−1−0.500.510

0.5

1

1.5

V (m/sec)

f v

b

a

Fig. 3. (a) Isometric Force-length curves of the LR muscle, based on Orbit 1.8TM. (b)Force-velocity relationship.


sliding constraints outlined above; g is the vector of positionconstraint functions; and m is the stabilizer weight (Baumgarte,1972).

Overall, a linear Karush-Kuhn-Tucker (KKT) system (Boyd andVandenberghe, 2004), is solved at each time step to obtain thegeneralizedvelocities of the rigid bodyand the strands at thenext step.

�M GT

G 0

��Fðkþ1Þ

l

�¼

�MFðkÞ þ hf

�mg

�; (2)

where M is the generalized mass matrix of the globe and the EOMstrands, and l is the vector of Lagrange multipliers for theconstraints. A direct method based on Gaussian Elimination is usedto solve this matrix. Once the new velocities, Fðkþ1Þ, are computed,the new positions are computed trivially (using Rodrigues’ formula(Murray et al., 1994) for the rigid body.)

2.2.2. EOM strandsWe have two types of strand elements, contractile elastic strands

and non-contractile elastic strands. A contractile elastic strandmodels muscle fibers; its total force is simplified to be the sum ofthe active force (FCE) and the passive force (FPE):

Fm ¼ FCE þ FPE ¼ a$fl$fv þ FPE (3)

The muscle activation a is assumed to be a unitless variablebetween 0 and 1, with a ¼ 1 meaning the muscle is fully inner-vated. The active (passive) force-length function fl (FPE) models thegenerated (passive) muscle force as a function of muscle length.The forceevelocity function fv relates the generated muscle force tothe muscle velocity. Both fl and fv have complex nonlinear charac-teristics, as does FPE The SE element of the Hill muscle modelillustrated in Fig. 2 is not included in our EOM constitutive model.As Robinson (1981) pointed out, the force mechanism of the EOMseries element has not been adequately defined due to the lack ofphysiological data. Therefore, it is difficult to derive a reasonablemodel for the SE element. None of the existing 3D biomechanicalmodels of the orbit incorporates SE because of its indeterminacy,and we follow this common choice.

We adopt the the active force-length curve of a fully innervatedEOM and the passive FL curve used in the OrbitTM 1.8 simulator,which best approximate the published empirical data. Cubic B-spline curves are fitted to the original FL data to convenientlyevaluate Fm at any given muscle length. Fig. 3(a) shows the force-length relationships of the LR muscle. In our simulation, otherEOMs’ strengths are scaled by their cross sectional areas. We usethe force-velocity model in (Robinson, 1981) for shortened EOMs,and function fv ¼ 2� ð0:18þ 0:5VÞ=ðV þ 0:18Þ for lengthenedEOMs, where V is the EOM lengthening rate. Fig. 3(b) shows the FVcurve used in simulating dynamics.

Noncontractile elastic strands are used to model EOM tendonsand the intermuscular coupling between IR and IO. These strandscan be passively stretched because of the attachments; they do notactively shorten to generate force. EOM tendons are very stiff andthus we use a high stiffness to model them. The IReIO strandstiffness is chosen such that the anteroposterior shift of IO is halfthe movement that the IR insertion travels, which has beenobserved from MRI studies (Demer et al., 2003).

Each rectus muscle has two contractile elastic strands in parallelmodeling the orbital and global layers of the muscle segment, anda tendon strand in series. The SO muscle has a muscle strand anda long tendon strand (30 mm). The IO muscle has one musclestrand, assuming its short tendon is negligible. Subject-specificEOM anatomy, including the EOM path and strength determined bycross section, is incorporated based on 3D geometric models of the

EOMs reconstructed from MRI (Wei et al., 2009). Fig. 4(a) isa representative LR muscle mesh. The medial axis depicting theaverage longitudinal muscle path is represented by a B-spline curvein blue; the control points are shown as green crosses. We partitionthe medial spline into two connected curves and fit two strands tomodel the muscle (in blue) and tendon (in gray) segmentsrespectively (see Fig. 4(b)). The cross sectional area A is a variablealong the strand, which allows simulation of realistic EOM localdeformation. We use the radius r ¼ ffiffiffiffiffiffiffiffiffi

A=pp

associated with eachcontrol point to approximate the area A. Fig. 4(c) renders the strandas a cylinder to demonstrate the nonuniform cross sections.

The globe is modeled as a spherical rigid body with a ball-and-socket joint allowing eye rotation in 3D. EOM origins and insertionsare implemented as attachment constraints (cyan spheres in Fig. 4(a)), which couple their positions to the attachment sites. Contactbetween the extraocular tendonand the globe surface is detected onthe fly to prevent penetration. Surface sliding constraints (red violetspheres) are pre-specified on the EOM tendon strands such thattendons always slide on the globe. 3e5 evenly spaced sliding pointsare sufficient because of the parametric representation of strands.

Rectus pulleys are approximated by their discrete functionallocations in 3D that have been estimated from MRI (Clark et al.,1997; Kono et al., 2002). Our goal is to develop a model thatcaptures the primary characteristics of pulleys while introducinga minimal number of undetermined parameters. We use prismaticjoints to abstract the mechanical functions of the rectus pulleys.These joints allow the pulleys to slide along the one-dimensionaljoint axis with zero movement in the transverse direction (see theorange double ended arrows in Fig. 4(d)). In other words, by usingprismatic joints, we apply zero longitudinal stiffness and infinitely

Fig. 4. Strand-based modeling of an EOM.


large transverse stiffness on the pulleys to stabilize the rectusmuscle paths. Small (<1 mm) transverse sideslips observed fromnormal subjects under physiological conditions (Kono et al., 2002)are neglected. Because of its simplicity, the strand-based orbitalplant model with pulleys is particularly suitable for studying thedynamics and neural control of ocular motility.

Our simulator is sufficiently general to support other imple-mentations of pulleys, for instance using elastic suspensions toincorporate transverse shifts (see simulation of coordinated pulleysin (Wei, 2010)).

2.3. Innervation solver

As pointed out in (Robinson, 1975), simulating orbital mechanicsinvolves twoparts. Forward simulation computes EOMforces and eye

movement, given the EOM innervations (activation levels). Inversecontrol simulation solves innervations of the six EOMs to hold the eyein fixation or to move the eye to target positions. Previous 3Dbiomechanical models of the orbit deal with static eye movementonly. In steady states, the net force of the six EOMs balances thepassive force of the orbital connective tissues. Utilizing this forceequilibrium relationship, innervations required to stabilize the eye atany position can be found (Miller et al., 1995; Robinson, 1975). Toaddress the redundancy problem (three degrees of freedom of theeye rotation vs. six EOMs), most existing models apply Sherrington’sLaw of Reciprocal Innervation by using a symmetric hyperbolafunction to explicitly define the innervation relationship of an agonistand antagonist EOM pair. The deficiency of this symmetry assump-tion has been noted recently (Quaia et al., 2008).

To simulate dynamic eye movement, we cannot use equilibriumequations but need a more complicated solver taking into accounteye velocities.We extend the activation solver in (Sueda et al., 2008)to compute EOM innervations. At any time step, given the desiredvelocity of the globe vx, the activation levels a˛½0;1� of the six EOMsare solved by optimizing a constrained quadratic objective function:

mina

wxjj�Hxaþ vf

�� vxjj2þwajjajj2

s:t: 1 � ai > a0i cv�veye

�cp�peye

�:

(4)

The first term minimizes the difference between the simulatedglobe velocity and the target motion. The matrix Hx can be thoughtof as the effective inverse inertia experienced by the muscle acti-vation levels in order to produce the target motion. Hxa determinesthe velocity contributed by the active EOM forces, and is derived byre-ordering the KKT system (Equation (2)) to solve for the activa-tion levels, a. The vf vector is the globe velocity due to the nonactiveforces. The second term in the objective function minimizes thetotal activation from all EOMs such that among the numerouspossible solutions, the set of innervations with the minimal sum-med activation energy are preferred. It also serves as a regulariza-tion of the quadratic problem. wx and wa are scalar weightsblending the two terms.

In order to incorporate the characteristics of ocular motorneuron activities (Collins et al., 1975; Sylvestre and Cullen, 1999),we add the constraint terms in the optimization in Equation (4).Each inequality specifies the lower innervation bound of anextraocular muscle EOMi as a function of the minimal activationlevel a0i , the velocity factor cv, and the position factor cp. Wedefine

cv�veye

� ¼ max�0;

vmax �veye

vmax

�; (5)

cp�peye

�¼ max

�0;

pmax �peye

pmax

�: (6)

peye and veye are the position and velocity of the globe in EOMi’saction dimension: horizontal for LR and MR, and vertical for theother EOMs. vmax is the velocity defining the onset and offset ofa saccade, which is set to 20�/s (Sylvestre and Cullen, 1999). Theposition factor cp linearly decreases the lower bound with eccentriceye position. pmax is the maximum OFF direction (contralaterallydirected) eye position, at which an antagonist EOM’s innervation iscompletely off; we assume pmax ¼ 30� in our experiments.

Our constraints model the following properties of ocularmotoneuron activities that have been observed experimentally. AllEOMs are innervated at primary position (Collins et al., 1975). Infixations, ocular motor neuronal discharges decrease with contra-lateral eye position (Collins et al., 1975). Many of the antagonist

Fig. 5. Inversely computed innervations of LR and MR plotted as a function of hori-zontal (ABD e abduction, ADD e adduction) and vertical (DOWN e downward, UP e

upward) eye positions between �30� and 30� .


motoneurons cease firing during contralaterally directed saccades(Sylvestre and Cullen, 1999).

Fig. 5 shows computed innervations of LR and MR as a functionof horizontal and vertical eye positions. The EOM innervations areinterpolated and plotted as iso-innervation curves. The LR and MRactivation levels depend on the horizontal eye positions, and arenearly independent of the vertical eye positions, as is consistentwith the physiology of the horizontal EOMs (von Noorden andCampos, 2001). Note that our innervation solver solves for theinnervations of all six EOMs simultaneously without enforcing thereciprocal relationship. However, the reciprocal innervation char-acteristics is obtained automatically which can be seen from Fig. 5,due to the fact that we have a realistic orbital plant with nonlineargeometry and mechanics.

3. Results

Our simulator is programmed in Java. Experiments were per-formed on a PC with an Intel Core2 Duo 2.4GHz CPU and 3GB ofRAM. Interactive simulation of the extraocular muscle dynamicsand the orbital plant is achieved at about 16 frames per second.

Fig. 6 shows simulated horizontal eyemovements. The influenceof the pulleys on the rectus muscle paths is clearly observed. As the

Fig. 6. Simulated horizontal eye movements. Rectus muscle pulleys are represented as prismthe anteroposterior movement of the LR and MR pulleys. (For the interpretation of the refere

eye rotates horizontally, the LR and MR pulleys move along theirrespective joint axis anteroposteriorly while the SR and IR pulleysremain stable in the mediolateral direction. The SR and LR pulleyscause sharp muscle path inflections e the SR and IR muscle belliesposterior to the pulleys are fixed relative to the orbit and theiranterior paths follow rotation of the globe in eccentric gazes.

3.1. EOM deformation during smooth pursuit

Extraocular muscle deformation as a function of gaze and time isan important parameter in studying extraocular mechanics (Milleret al., 2006). We validate the dynamic EOM deformation of ourmodel by comparing simulation to the in vivo EOM motion data.Motion-encoded MRI was recently proposed to assess EOMmotionduring smooth pursuit eyemovement (Piccirelli et al., 2007). Sparsetissue points on the transverse imaging planes of tagged MRI weretracked as the subject was instructed to visually pursue a slowlymoving stimulus. EOM deformation was inferred by analyzing therecorded positions of the tissue points. Stretch ratio, defined as theratio between the deformed muscle length and its initial length,was used to quantify the deformation. Along the longitudinal axisin the primary eye position, the EOM path was divided into threesegments of equal length. Stretch ratios of the segments in the twohorizontal rectus muscles were analyzed.

We simulate the same smooth pursuit movement as in (Piccirelliet al., 2007). The eyemoves horizontally from20� in abduction to 20�

in adductionwith zeroverticalmovement. The velocity is a sinusoidalfunctionwith a 2 s period. Themaximumspeed is 6.4�/s. Applying thesame analysis scheme, we present the simulated EOM strains in solidcurves in Fig. 7. For comparison, simulated EOM strains are overlaidon top of the in vivo EOM motion data reported in (Piccirelli et al.,2007). Simulation of this smooth pursuit movement is demon-strated in the Supplementary video “MoviePursuit”

Our simulated deformation trends of the LR and MR musclesfollow the published MRI data. All EOM segments have sinusoidaldeformation pattern. Nonuniform shortening and elongation areobserved in the two horizontal EOMs. The middle segment, whichhas larger cross sectional areas, deforms more than the posteriorsegment. The anterior part contains more tendinous tissue anddeforms the least. Simulated strains of all segments except for the

atic joints, shown as green arrows. Note the inflection of the SR and IR muscle paths andnce to color in this figure legend the reader is referred to the web version of this article.)

a b

c d

e f

Fig. 7. Comparisonof the lateral rectus andmedial rectusmuscle strains fromsimulation (in solid curves) to publishedMRI data. The shaded regions show theaverage stretch ratios (mean�1S.D.) from 7 subjects over time (reproduced from (Piccirelli et al., 2007)). Values larger than one indicate muscle lengthening; values smaller than one indicate muscle shortening.


middle LR fall in the 95% confidence interval. The mean percentagedifference (MPD) between the simulated strain Ss and the averageexperimental strain Se is computed as the percentage difference%Diff ¼ jSs � Sej=Se � 100 averaged over all the time frames inwhich tagged MR images were acquired. MPDs of LR and MR aresummarized in Table 2. The main disparity occurs at the middlesegmentse simulated LR andMR show larger deformation than theempirical data by less than 10%. It is known that due to low contrast,

Table 2Computed mean percentage difference (MPD) between the simulated strain and theaverage experimental strain of the LR and MR muscle segments.

EOM Anterior Middle Posterior

LR 0.84% 9.50% 1.46%MR 1.71% 8.48% 4.29%

accurate identification of EOM insertions from MR images is diffi-cult. The tracking curves manually placed on the images (seePiccirelli et al., 2007; Fig. 2) show more posterior insertional endsthan the actual insertions. We conjecture that this variation indelineating muscle paths may partially contribute to thediscrepancy.

3.2. Simulation of saccades

In order to test the plausibility of our biomechanical model, wesimulated saccadic movement using experimental EOM neuralactivity data as the neural input of the LRmuscle to drive the orbitalplant. We then compared the simulated motion to the desiredmotion to evaluate the system behavior.

The neural drive of a saccadic eye movement can be character-ized by a pulse component to overcome the viscoelasticity of the


orbital plant, a step component to stabilize the eye in the newposition, and a slide component that models the gradual transitionbetween the pulse and step (Robinson, 1964). The neural control ofsaccades has been studied extensively (Fuchs and Luschei, 1970;Horn and Cullen, 2009; Robinson, 1964; Sylvestre and Cullen,1999). The activities of neurons are characterized by thedischarge rates recorded at the neuronal sites. The movement ofthe LR muscle is controlled by abducens neurons (ABNs) whosedischarges during saccades can be expressed as a first orderequation (Sylvestre and Cullen, 1999):

FR ¼ bþ kE þ r _E: (7)

Themodel approximates the ABN firing rates as a linear functionof the eye position E and velocity _E. b is a bias constant, which is theneural firing rate at the stationary central eye position. k and r areconstants and their optimal values have been estimated by fittingthe above model to the actual neuron discharge recordings frommonkeys (Sylvestre and Cullen, 1999).

Fig. 8 shows the analysis of a 20� abduction saccade while theeye is elevated by 20�. This movement is generated by contractionof the LR muscle, relaxation of the MR muscle, and co-contractionof other EOMs. Here we experiment with a saccade froma secondary position to a tertiary position, which is more inter-esting and challenging than a purely horizontal saccade. The

Fig. 8. Simulated 20� abduction saccade in 20� elevation. (

torsional component in the tertiary gaze is a standard test toevaluate the biomechanics of eye movement.

Fig. 8(a) is a snapshot of the simulated gaze trajectory and theorbit configuration when saccade is completed. Muscle activationlevels are shown in red on the EOM strands. Obviously, the LR and SRmuscles are the two active EOMs. Fig. 8(b) shows the approximatedABN firing rate profile for a 20� saccade based on the model inEquation (7). Themodel coefficient values are b ¼ 156, k ¼ 4:2, andr ¼ 0:42. Fig. 8(c) plots our estimated innervations of the LR andMRmuscles e the primary rectus muscles contributing to horizontalsaccades. The other EOMs maintain the vertical elevation of the eyethroughout this movement. Note that the computed LR innervationhas the pulse-slide-step characteristics of saccadic neural control.Also note that the computed MR innervation realistically models thebehavior of antagonistmotoneurons during OFF direction saccadesemost of them completely cease firing after saccade onsets (Sylvestreand Cullen, 1999). The model-based firing rate profile in Fig. 8(b) islinearly scaled such that the starting and ending values match thecomputed innervation. The scaled profile is plotted in Fig. 8(c),overlaying on the estimated innervation for comparison.We observethat the computed LR innervation dynamics (in blue) agreeswith theexperimental ABNs discharge dynamics (in red).

We use the scaled empirical neural discharge profile as the neuraldrive of LR and re-simulate the 20� abduction saccade. The bluecurves in Fig. 8(d) are the desiredposition and velocity profiles, based

See supplementary demonstration file MovieSaccade).

Table 3Root mean squared errors (RMSEs) of the simulated saccadic positions and veloci-ties, and maximum torsional errors.

SaccadeAmplitude (

�)

PositionRMSE (

�)

VelocityRMSE (

�/sec)

Max VelocityDifference (

�/sec)

Max TorsionalError (

�)

10 1.27 34.31 6.31 0.1620 1.27 32.93 24.00 0.1030 1.78 41.27 12.22 0.09


on saccade traces recorded from monkeys (Sylvestre and Cullen,1999). The superimposed red curves are the simulation results. Thesimulated saccade reasonably follows the desired saccadic trajectory.The root mean square errors (RMSEs) are analyzed quantitatively inTable 3. The associated torsion is very small, which further demon-strates the simulation accuracy and realism of the nonlinear EOMmodel in reproducing saccades.We also simulate abduction saccadeswith the amplitudes of 10 and 30�. Simulation results are shown inFig. 9 and Fig. 10. Agreement on the eye position and velocitytrajectories is also achieved. Simulation of the saccade movement isshown in the Supplementary movie “MovieSaccade”.

3.3. Simulation of superior oblique palsy

We simulated the pathological case in which the superior obli-que (SO)muscle action is weakened due to paralysis of the trochlear

Fig. 9. Simulated 10� abduction

nerve. Because the contractibility is diminished, a paretic EOM’sgenerated force is inadequate for moving the eye by the desiredamount. The change of the oculomotor plant mechanics caused byparetic SO muscle leads to ocular misalignment, or strabismus.Biomechanical simulation is useful for systematically under-standing the mechanical influences of the paretic EOMs.

To simulate superior oblique palsy (SOP), we first computed themuscle innervations that stabilize a normal eye in various eyepositions. To mimic an eye with paretic SO muscle, we then set theSO innervation to zero while keeping the values of the other EOMsunchanged. The new fixation locations are the predicted eye posi-tions of a pathological eye with SOP.

We compare simulation results with published data froma monkey with acute SOP (Shan et al., 2007). We choose this datafor comparison because the controlled experiment ruled out otherabnormalities possibly affecting alignment. Our goal is to evaluatewhether our model is able to predict deviation patterns that arequalitatively consistent with observed pathology.

The predicted eye positions of a SOP eye are plotted in Fig. 11.Monkey alignment data reproduced from (Quaia et al., 2008) aswell as simulated data are superimposed for comparison. Incom-itant deviations dependent on eye positions are observed. Oursimulation shows a vertical deviation for every fixation; the devi-ation is greatest with the eye adducted and down. These results areconsistent with the role of SO in depressing the globe. Extorsional

saccade in 20� elevation.

Fig. 10. Simulated 30� abduction saccade in 20� elevation.

Fig. 11. Simulated eye positions of a SOP eye shown as black crosses. Gray crosses aremonkey data, reproduced from (Quaia et al. (2008)). Cyclorotation, multiplied by 1.5for better visualization, is shown as the vertical tilt of the crosses. Courtesy of Dr.Christian Quaia for the monkey data.


deviation, presented as the angle between the vertical bar of thecross and the vertical axis, is also observed. We conclude thatsimulated SOP alignment shows static deviation patterns consis-tent with experimental acute SOP data.

The mean absolute difference between the simulation and thedata shown in Fig.11 is 0.98� in the horizontal direction and 2.50� inthe vertical direction. Again we would like to emphasize that ourmain objective is to validate the model qualitatively rather thanquantitatively based on these numbers, because the simulatedmodel is a human orbit and not a monkey orbit, and the degree ofdeviation due to SOP varies significantly due to anatomical varia-tions (Quaia et al., 2008).

4. Conclusions

This paper presents a biomechanical model that simulates 3Dmotion of the extraocular muscles interactively. Our objective is tolay the foundation of a biomechanical and computational frame-work that possesses fewer limitations than previous models and issufficiently general for both scientific research and clinicalapplications.

Our model is the first 3D biomechanical model that simulatesdynamic eye movements. The strand-based EOM model is associ-ated with nonlinear anatomical and constitutive properties.


Realistic EOM paths and cross sectional areas of these EOM strandsare based on geometric models reconstructed from MR images ofhuman orbits. The model implements pulleys with infinite trans-verse stiffness; it generates reasonable EOM paths, gaze positionsand trajectories given EOM innervations.

Before using the biomechanical model to answer scientificquestions, we need to assess the plausibility and accuracy of thesimulator. We performed several validations from different aspects,which not only evaluate the accuracy of the EOM model but alsoprovide guidance for choosing model parameters. EOM deforma-tion during smooth pursuit was examined and compared toexperimental motion-encoded MRI data from human subjects andconsistent deformation patterns were observed. We simulatedsaccadic eye movements that abduct the eye at different ampli-tudes. Abducens motoneuron discharges based on empirical datawere used to drive the lateral rectus muscle, the primary EOM thatis active in these eye movements. We found that the simulatedsaccadic trajectories reasonably follow the desired trajectories,which shows the realism of the EOM dynamics. We also simulatedstrabismus caused by superior oblique muscle palsy. The predicteddeviations in various directions were qualitatively consistent withexperimental data from controlled subjects. Based on the results,we conclude that our EOM model with nonlinear constitutivecharacteristics and MRI-based anatomical properties is able togenerate realistic static and dynamic eye movements.

In future work, we plan to further investigate the force andmotion of the simulated EOMs by comprehensively analyzingvarious eye movements. Efforts on speeding up simulation toenable real-time applications will be made. We would also like tounderstand the roles of pulleys in coordinating EOMs and testpulley hypotheses that have been proposed.

Acknowledgement

We would like to thank Dr. Joel Miller and Dr. Joseph Demer forvery helpful discussions. We also thank Dr. Kathleen Cullen for themonkey saccade data and Dr. Christian Quaia for the monkey SOPdata. This research was supported in part by NIH NINDSR01NS050942, NIH NIAMS R01AR053608, Canada Research ChairsProgram, Peter Wall Institute for Advanced Studies, NSERC, CIHR,Canada Foundation for Innovation, and BC KDF.

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