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Physics, Mechanics & Astronomy
• Article •
May 2014 Vol.57 No.5: 983–987
doi: 10.1007/s11433-013-5227-x
Using the traditional model to evaluate the active force of the
human lateral rectus muscle
GAO ZhiPeng1, CHEN WeiYi1*, JING Lin2, FENG PengFei1, WU XiaoGang1
& GUO HongMei1
1
2
Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China;
Impact Mechanics Laboratory, Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore
Received December 25, 2012; accepted February 18, 2013; published online March 13, 2014
The information on the force of extraocular muscles (EOMs) is beneficial for strabismus diagnosis and surgical planning, and a
direct and simple method is important for surgeons to obtain these forces. Based on the traditional model, a numerical simulation method was proposed to achieve this aim, and then the active force of the lateral rectus (LR) muscle was successfully simulated when the eye rotated every angle from 0° to 30° in the horizontal plane from the nasal to the temporal side. In order to
verify these simulations, the results were compared with the previous experimental data. The comparison shows that the simulation results diverged much more than the experimental data in the range of 0°–10°. The errors were corrected to make the
simulation results closer to the experimental data. Finally, a general empirical equation was proposed to evaluate the active
force of the LR muscle by fitting these data, which represent the relationship between the simulation forces and the contractive
amounts of the LR muscle.
force, extraocular muscles, lateral rectus muscle, eye movement model
PACS number(s): 07.10.Pz, 87.19.Ff, 87.15.Aa
Citation:
Gao Z P, Chen W Y, Jing L, et al. Using the traditional model to evaluate the active force of the human lateral rectus muscle. Sci China-Phys Mech
Astron, 2014, 57: 983987, doi: 10.1007/s11433-013-5227-x
1 Introduction
Eye movement is controlled by extraocular muscles
(EOMs), which are also related to some eye diseases, such
as nystagmus, saccade and strabismus. Especially for strabismus, the information on the force of EOMs is beneficial
for strabismus diagnosis and surgical planning [1]. Based on
the mechanics equilibrium theory, the traditional eye
movement model was proposed [2,3].
Recently, with the discovery of pulley of the rectus muscles [4], several novel models [5–7] were proposed. Finite
element method (FEM) was used to simulate the force of
four rectus muscles [5]; a method [6] was proposed to cal*Corresponding author (email: [email protected])
© Science China Press and Springer-Verlag Berlin Heidelberg 2014
culate the activation of EOMs; and the more detailed situation of eye three-dimensional movement was simulated by
using the OrbitTM software [7]. The Hill-muscle constitutive
model [8] was used in these novel models [6,7] to derive the
force of EOMs. Moreover, the motor unit of EOMs was
measured by chronically implanting the muscle force transducer in alert [9,10]; and in the microscopic neurophysiology and biochemistry, the action of calcium related to the
force of EOMs was also studied [11].
Although these advanced invasive techniques [9,11] and
the novel models [6,7] can effectively help understand the
detailed microscopic structure of EOMs and obtain the more
accurate force of EOMs, they are too complicated and sophisticated for surgeons in clinical diagnosis and surgical
operation [6]. Therefore, this study aims to propose a direct
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Gao Z P, et al.
Sci China-Phys Mech Astron
and simple method to evaluate the active force of EOMs by
gathering the existing anatomical and experimental documents excluding the above invasive techniques.
All of the coordinates in the primary position [3] and the
cross-section areas [6] of every EOM have been proposed;
the credible parameter of the stiffness of the constrained
tissues surrounding the eyeball has been extensively used
[1,5,12]; and the general empirical equation of the passive
tensile test [13] of EOMs also has been proposed [14]. In
this study, these materials will be used with the traditional
model to evaluate the active force of EOMs.
2 Method
Eye movements are controlled by EOMs, four rectus muscles (lateral rectus (LR), medial rectus (MR), superior rectus
(SR) and inferior rectus (IR) muscle) and the two oblique
muscles (superior oblique (SO) and inferior oblique (IO)).
All of the six muscles are hosted within the orbit to which
they are connected on one side, which they inserted onto the
globe on the other. In the primary position (looking straight
ahead), the LR and MR muscle makes the globe rotate towards the temporal and nasal direction, respectively. The
SR and SO muscle’s primary function is elevation and intorsion, respectively; and the IR and IO muscles function
contrary to the above two [6]. The LR and MR muscles play
an important role as an agonist-antagonist pair of muscles
when eye moves in the horizontal plane. In this paper, the
rotation of eye from the nasal to the temporal side in the
horizontal plane (Figure 1(a)) is simulated by the traditional
model, in which the EOMs are treated as elastic strands
[3,7], the eyeball is represented by a center-fixed rigid
sphere [6,7], and the restrictive function of the other tissues
around the eyeball is treated as a moment [7].
Figure 1(a) shows the projection in the horizontal plane
of a point in the globe, in which the OX axis is the line of
sight, OA is the projective vector, and after rotation an angle
θ around the global center to obtain the vector OA'. Ac-
May (2014) Vol. 57 No. 5
cording to the geometric, the relationship between OA and
OA' is
 x  x 2  y 2 cos     ,


2
2
 y   x  y sin     ,
 z   z,

(1)
where,   tan 1 ( y / x).
Euler’s theorem points out any two orientations of a rigid
body are related by a single rotation, and Listing’s law
holds all of these equivalent shafts of eye movement lie in
the same plane [15]. In this study, the angle  is the equivalent angle, OZ is the equivalent shaft and the plane of YOZ
is the Listing plane.
In geometric (Figure 1(b)), the relationship among the
insertion C(xc, yc, zc) and the origin D(xd, yd, zd) of an EOM
as well as the tangency T(xt, yt, zt) between this EOM and
the eyeball is
 x t2  yt2  zt2  R 2 ,

 xt  xd  yt  yd  zt  zd  R 2 ,

 yd zc  zd yc   xt   zd xc  xd zc 

 yt   xd yc  yd xc   zt  0,

(2)
where, R is the radius of eyeball. Thus, the coordinates of
the insertion, origin and tangency of every EOM can be
obtained by eqs. (1) and (2), and then the initial (L0) and
new (L) length as well as the variation (L) between them
of every EOM can be calculated by the coordinates of the
above three points.
When eye moves, theoretically, the contractive EOM is
agonist while the elongate EOM is antagonist. For the antagonist muscles, the relationship between the force T and
the variation of the muscle length ∆L can be represented by
a general exponential function [6]. Quaia et al. [14] proposed an empirical equation to describe this relationship,
Figure 1 (Color online) (a) Eye moves in the horizontal plane; (b) (view from above) relationship among the insertion, origin and tangency of the LR
muscle. Note that the Z axis points to the reader (this drawing is created by modifying Figure 2A of ref. [3]).
Gao Z P, et al.
Sci China-Phys Mech Astron
T(L)=0.95(L)+0.74exp(L/2.91)+3.13, based on the test
data in ref. [13], which was obtained by dragging the LR
muscle of five deeply anesthetic patients. EOMs are treated
as the same material in this paper, and then the antagonist
(passive) force of each EOM is represented by the above
equation multiplying a coefficient k, that is,
T  L   k  0.95  L   0.74 exp  L 2.91  3.13 ,
(3)
where, k=Ai/A1, i=1, 2, 3, …, 6 denotes the LR, MR, SR, IR,
SO and IO muscles, respectively; A denotes the cross- section area of EOMs.
The restraint stiffness of the other tissues around eyeball
Kt is 0.1×radius mN/° [5], but the unit of force is gf in the
traditional literature [1,12] about the force of EOMs. Because 1 gf≈9.8 mN, Kt≈0.12 gf/° in this paper. When
eyeball rotates an angle θ around the equivalent axis OZ
from the initial position to a new position, the moment of
the restrictive tissues is Mt=KtR2 (nxi+nyj+nzk), in which
nx=ny=0. Thus, the static equilibrium equation can be represented
 6
2
  mix   K t R  nx  0,
i 1

 6
2
 miy   K t R  n y  0,
 i 1
 6
2
  miz   K t R  nz  0.
 i 1
(4)
Substituting eq. (3) into eq. (4), the agonist (active) force
can be obtained. When eye rotates every angle in the horizontal plane from 0° to 30° [16], the force of EOMs is simulated by using the MATLAB software according to this
method, and all of the parameters used in this method are
shown in Table 1.
3
Result and discussion
Table 2 shows the detailed numerical simulation results
including the angles of eye rotation and the related force
and length variation of EOMs. It is shown that the increments of the force and length variation of the LR and MR
muscles with eye movement are obviously much larger than
the other EOMs. This result also demonstrates that the LR
and MR muscles play an important role in controlling eye
movement in the horizontal plane. The relationship between
the length variation L of the LR and MR muscles and the
eye rotation angles is showed in Figure 2. The agonistTable 1
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985
antagonist function of LR-MR muscles can be clearly seen
in Figure 2, that is, the contractive amount of the LR muscle
nearly equals the elongate amount of the MR muscle when
eye rotates every angle from the nasal to the temporal side
in the horizontal plane.
In order to verify the feasibility of this method, the simulation result of the LR muscle was compared with the previous experimental data [1], which give the active force of
LR muscle and the stiffness of the tissues surrounding the
eyeball of twenty-nine normal human subjects when eye
rotated in the horizontal plane in the range of ±50°. The
data of segment (Figure 3(c)) of 0°–30° were selected for
comparison with the simulation results and the comparison
is showed in Figure 3(a).
As shown in Figure 3(a), the simulation data are very
close to the experimental result when eye rotates in the
range of 10°–30°, but the error of the range of 0°–10° is
large. The reason for this phenomenon is as follows. When
eye rotates an angle from the nasal to the temporal side in
the horizontal plane, the LR muscle actively contracts and
the MR muscle is passively elongated at the same time, and
thus the agonist force of the LR muscle must be larger than
the antagonist force of the MR muscle. Table 2, however,
shows the simulation force of the LR muscle is less than the
MR muscle in the range of 0°–10°. The simulation results of
LR muscle in the range of 0°–10° are wrong and must be
corrected.
When eye rotated in the horizontal plane from the nasal
to the temporal side, the difference (∆F=FL-agFM-antag) between the active force of the LR muscle and the passive
force of the MR muscle was measured nearly equal to the
load of the temporal tissues intact eye (Kt×) (For detailed
result see Figure 10 of the ref. [1]). Thus, the simulation
data of the range of 0°–10° were corrected by the following
equation:
FLR  FMR  K t .
(5)
Figure 3(b) shows the corrected result. As shown in Figure
3(b), the error (gray region in Figure 3(b)) between the corrected result and the experimental data is much less than the
previous one (gray region in Figure 3(a)). The reason for the
error remaining between the corrected simulation result and
the experimental data is that the coordinate parameters of
ocular motion plant used in this study are from ref. [3], but
the experimental data are given from ref. [1]. Although both
of the subjects in the previous two studies are human, the
individual differences of the same species cannot be ignored.
The parameters used in this study. Radius of eyeball R is 12.00 mm [3]; stiffness of constrained tissues Kt is 0.12 gf/° [5]
LR
SR
SO
MR
IR
IO
Origin [3]
34.00, 13.00, 0.60 31.78, 16.00, 3.60 8.24, 15.27, 12.25 30.00, 17.00, 0.60 31.70, 16.00, 2.40 11.34, 11.10, 15.46
Insertion [3]
6.50, 10.08, 0.00
7.33, 0.00, 10.48
8.42, 9.65, 0.00
8.02, 0.00, 10.24
7.18, 8.70, 0.00
4.41, 2.90, 11.05
16.73
11.34
19.34
17.39
15.85
19.83
C-S area (mm2) [6]
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Gao Z P, et al.
Table 2
(°)
Sci China-Phys Mech Astron
May (2014) Vol. 57 No. 5
The numerical simulation results in this study
L (mm)(“” is contraction, “+” is elongate)
Force (gf)
LR
MR
SR
IR
SO
IO
LR
MR
SR
IR
SO
IO
0
2.56
4.02
3.22
3.66
4.47
4.44
0.00
0.00
0.00
0.00
0.00
0.00
1
1.45
4.40
3.40
3.75
4.53
4.47
0.21
0.30
0.07
0.07
0.04
0.02
2
0.35
4.78
3.57
3.83
4.58
4.51
0.41
0.59
0.13
0.14
0.08
0.05
3
0.74
5.16
3.74
3.92
4.64
4.54
0.62
0.87
0.20
0.22
0.12
0.08
4
1.84
5.54
3.91
4.00
4.69
4.57
0.83
1.16
0.26
0.28
0.15
0.11
5
2.93
5.93
4.06
4.07
4.74
4.61
1.04
1.43
0.33
0.35
0.19
0.14
6
4.02
6.32
4.22
4.15
4.79
4.64
1.25
1.71
0.38
0.41
0.22
0.18
7
5.10
6.72
4.37
4.22
4.83
4.68
1.46
1.97
0.44
0.47
0.25
0.23
8
6.17
7.12
4.52
4.29
4.88
4.72
1.67
2.23
0.49
0.53
0.28
0.27
9
7.23
7.54
4.65
4.35
4.93
4.76
1.88
2.50
0.54
0.59
0.31
0.32
10
8.28
7.96
4.79
4.41
4.96
4.80
2.09
2.76
0.59
0.64
0.34
0.37
11
9.31
8.40
4.92
4.47
4.99
4.84
2.30
3.01
0.64
0.69
0.37
0.43
12
10.3
8.84
5.04
4.53
5.03
4.88
2.51
3.27
0.68
0.73
0.39
0.49
13
11.3
9.31
5.16
4.58
5.06
4.92
2.72
3.52
0.72
0.78
0.42
0.55
14
12.3
9.79
5.28
4.63
5.10
4.96
2.92
3.77
0.76
0.82
0.44
0.61
15
13.2
10.2
5.39
4.67
5.13
5.00
3.13
4.02
0.79
0.85
0.46
0.68
16
14.2
10.8
5.49
4.72
5.15
5.04
3.34
4.26
0.82
0.89
0.48
0.75
17
15.1
11.3
5.59
4.75
5.18
5.08
3.55
4.51
0.85
0.92
0.50
0.82
18
16.1
11.8
5.68
4.79
5.21
5.13
3.76
4.75
0.88
0.95
0.51
0.89
19
16.9
12.4
5.77
4.82
5.22
5.17
3.97
5.00
0.91
0.97
0.53
0.97
20
17.7
13.1
5.85
4.84
5.24
5.21
4.18
5.24
0.92
0.99
0.54
1.05
21
18.5
13.7
5.92
4.87
5.26
5.26
4.39
5.48
0.94
1.01
0.55
1.13
22
19.3
14.4
5.99
4.89
5.27
5.30
4.60
5.72
0.96
1.03
0.56
1.22
23
20.1
15.1
6.05
4.91
5.29
5.34
4.81
5.96
0.97
1.04
0.57
1.31
24
20.8
15.8
6.11
4.92
5.30
5.38
5.02
6.20
0.98
1.05
0.58
1.39
25
21.4
16.6
6.16
4.93
5.31
5.42
5.23
6.43
0.98
1.06
0.58
1.48
26
22.1
17.4
6.21
4.93
5.31
5.46
5.44
6.67
0.99
1.06
0.59
1.57
27
22.6
18.3
6.24
4.94
5.31
5.51
5.64
6.91
0.99
1.07
0.59
1.67
28
23.1
19.2
6.27
4.93
5.32
5.54
5.85
7.14
0.98
1.06
0.59
1.76
29
23.5
20.2
6.29
4.93
5.32
5.58
6.06
7.38
0.98
1.06
0.59
1.86
30
23.9
21.3
6.30
4.91
5.31
5.62
6.27
7.61
0.97
1.05
0.59
1.96
fitted by the formula T(L)=a(L)b+c, and the result
(R2=0.99, Figure 3(d)) was obtained
T (L)  2.43(L)1.18  3.85.
4
Figure 2
Relationship between θ and ∆L of the LR and MR muscles.
The relationship between the corrected simulation forces
and the relevant contractive amounts of the LR muscle was
(6)
Summary
Using the traditional eye movement model, we have introduced a simple numerical simulation method to evaluate the
active force of human LR muscle. The situation that eye
rotates every angle from 0° to 30° in the horizontal plane
from the nasal to the temporal side has been simulated by
using this method. The error between the simulation results
and the previous experimental data [1] is large in the range
of 0°–10°. Therefore, the simulation data in the range of
0°–10° are corrected by eq. (5), and the corrected simulation
results are much closer to the experimental data [1]. Finally,
Gao Z P, et al.
Sci China-Phys Mech Astron
May (2014) Vol. 57 No. 5
987
Figure 3 (Color online) (a) and (b) represent the comparison between the simulation result and the experimental data [1] of LR muscle and the comparison
among the simulation result and the experimental data [1] as well as the corrected result of the LR muscle (the shadow region is the error of the range of
0°–10°, and the tint region is the corrected amount), respectively; (c) experimental data of the active force of the left LR muscle (Figure 8 of the ref. [1]),
which was measured in the subject K.C., and the data of gray region were selected for comparison with the simulation results; (d) fitting result of the simulation force of the LR muscle.
we propose a general empirical equation (eq. (6)) to represent the relationship between the active force and the contractive amount of the LR muscle by fitting the corresponding data points.
7
This work was supported by the National Natural Science Foundation of
China (Grant No. 11032008).
10
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