Journal of Theoretical Biology 353 (2014) 78–83
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Initial tension of the human extraocular muscles in the primary eye position Zhipeng Gao, Hongmei Guo, Weiyi Chen n Shanxi key Laboratory of Material Strength & Structural Impact, Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
H I G H L I G H T S
A model was proposed to estimate the initial tension of extraocular muscles. An optimum method was used to determine the active behavior. Mechanical equilibrium was analyzed using the complementary information.
art ic l e i nf o
a b s t r a c t
Article history: Received 15 June 2013 Received in revised form 10 March 2014 Accepted 11 March 2014 Available online 20 March 2014
This study proposes a mathematical model to estimate the initial tension forces of the extraocular muscles (EOMs). These forces are responsible for the mechanical equilibrium of the eye suspended in primary position. The passive contributions were obtained using the corresponding Cauchy stress– stretch relationships based on the previous clinical experimental data; whereas the active contributions were obtained using an optimum method with weakening the effect of innervation. The initial tension forces of the EOMs were estimated to be 48.87 14.2 mN for the lateral rectus, 89.2 731.6 mN for the medial rectus, 50.6 7 17.6 mN for the superior rectus, 46.2 713.4 mN for the inferior rectus, 15.6 78.3 mN for the superior oblique, and 17.1 7 12.1 mN for the inferior oblique. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Passive behavior Active behavior Optimum method Biomechanics
1. Introduction The extraocular muscles (EOMs) are indispensable to the threedimensional (3D) modeling of eye movement, which can potentially be applied in the diagnosis of vertigo (Dong et al., 2014) and in electronystagmograph analysis (Bedell and Stevenson, 2013). Several models have been established to investigate eye movement; these models can be divided into three categories, i.e., the classical two-element model that includes the EOMs and the eyeball (Robinson, 1975; Miller and Robinson, 1984), the finite element model (Luboz et al., 2004; Schutte et al., 2006), and the modern three-element model (Pascolo and Carniel, 2009; Wei et al., 2010) that includes the EOMs, eyeball and pulley. The primary position of eye is generally set as the reference configuration in eye examination and modeling, in which the initial tension forces of the EOMs play an important role in keeping the eye suspended. n Correspondence to: Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, 79 YingZe West Street, Taiyuan 030024, China. Tel.: þ 86 351 601 8864. E-mail addresses: [email protected], [email protected] (W. Chen).
http://dx.doi.org/10.1016/j.jtbi.2014.03.018 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
Initial tension is defined as the difference between the lengths of the primary position and the resting state of a muscle. This difference urges the corresponding muscle to generate force that would keep the eye suspended in primary position. Theoretically, the mechanical equilibrium of the eye in a 3D space is dependent on the contributions of the six EOMs, i.e., lateral rectus (LR), medial rectus (MR), superior rectus (SR), inferior rectus (IR), superior oblique (SO), and inferior oblique (IO). The present work aims to estimate the initial tension forces of EOMs that maintain the eye suspended in primary position. In studies related to the EOMs, and to the twitch and tetanic activation or the mechanical behavior of eye movement, researchers have set a unified value for the initial tension force of the EOMs; this value ranges from 4 to 200 mN (Gernandt, 1968; Collins et al., 1975; Lennerstrand et al., 2010). The initial tension forces of the EOMs, particularly of the LR muscle, have been measured using horizontal rotation from topical anesthetic patients, whose LR muscles were detached from the globe and were attached with a micromanipulator-mounted strain gauge (Collins et al., 1975; Robinson, 1975). Their experimental results suggest that innervation is a predominant factor in the magnitude
Z. Gao et al. / Journal of Theoretical Biology 353 (2014) 78–83
of the initial tension forces of the EOMs. Different innervations trigger different isometric contractions of a muscle and then produce different active forces. At present, no ideal experimental data can be used to determine this complicated constitutive relationship between active force and the stretch of an EOM because of the anatomical difficulty and ethical requirements, although a similar relationship of the skeletal muscle has been determined (Gordon et al., 1966) and has been widely used in the modern finite element modeling of muscles (Böl and Reese, 2008; Ehret et al., 2011). However, the fiber composition of an EOM is different from that of a skeletal muscle (Kushner, 2010); their biomechanical behaviors also differ (Quaia et al., 2009). To use the corresponding research results on the active force of a skeletal muscle in describing the active behavior of an EOM is inappropriate. As the influence of innervations on the active force of an EOM is rarely studied, the present work focuses on the relaxation state of the eye to weaken the effect of innervation. This study develops a mathematical model to calculate the initial tension forces of EOMs. It refers to previous published works that contain clinical experimental data on the passive tension (Scott, 1971; Collins, 1971; Collins et al., 1975; Quaia et al., 2009) of EOMs and corresponding coordinate information (Miller and Robinson, 1984; Clark et al., 2000) on the primary position of the EOMs. The initial tension force of an EOM comprises passive and active contributions. In this work, the passive forces are calculated using the theory of mechanics, whereas the corresponding active forces are obtained using an optimum mathematical method for establishing the weakening of innervation. The investigation results reveal that the initial tension forces of the EOMs are 48.8714.2 mN for LR, 89.2731.6 mN for MR, 50.6717.6 mN for SR, 46.2713.4 mN for IR, 15.678.3 mN for SO, and 17.1712.1 mN for IO.
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The initial stretch λ of an EOM is obtained by substituting Eq. (2) into Eq. (1). In this paper, the resting state of an EOM is defined as the reference configuration, and its primary position shape is defined as the current (deformed) configuration. The cross-sectional areas of the EOMs in the reference configuration are adopted from the work of Pascolo and Carniel (2009). The corresponding geometric parameters of human EOMs are summarized in Table 2. According to continuum theory, the Lagrange or Piola–Kirchhoff stress along the direction of an EOM fiber in the reference configuration is defined as
s~ ¼ F=A0 ;
ð3Þ
where F is the tension force along the EOM fiber, and A0 is the cross-sectional area in the resting state. Cauchy (true) stress along the direction of the same EOM fiber in the current configuration is defined as
s ¼ F=A;
ð4Þ
where A is the deformed cross-sectional area. Assuming that EOMs are incompressible and that their original cross sections A0 are constant, AL ¼ A0 L0 :
ð5Þ
Substituting Eqs. (1), (4), and (5) into Eq. (3), Eq. (3) yields
s ¼ λs~ :
ð6Þ
A second-order formula in the form of E ¼aε þbε þ c was used to describe Young's modulus of the soft tissue (cornea) in the work of Elsheikh and Anderson (2005). Accordingly, the Lagrange stress 2
2. Methods 2.1. Passive force The resting and initial lengths of an EOM are defined as L0 and L, respectively. On the basis of the definition of stretch in the studies on other muscle types (Calvo et al., 2010; Paersch et al., 2012; Takaza et al., 2013), the initial stretch λ of an EOM in primary position is defined as λ ¼ L=L0 :
ð1Þ
The literature (Miller and Robinson, 1984; Clark et al., 2000) provides the resting length L0 and the key site coordinates of the EOMs, which include the insertions (I), pulleys (P), and origins (O). These parameters are summarized in Table 1. The tangency (T) on the globe can be determined by the above three points of the same muscle. Therefore, the initial length L of an EOM in primary position (Fig. 1) can be obtained by summing up the lengths of the lines of OP, PT, and TI; that is, L ¼ OP þ PT þ TI:
ð2Þ
Fig. 1. Schematic illustration of the length of an EOM.
Table 1 The key site coordinates of the EOMs (the origin of the SO muscle is the trochlea). EOMs
Insertion
Pulley
Origin
LR MR SR IR SO IO Reference
6.50, 10.08, 0.00 8.42, 9.65, 0.00 7.63, 0.00, 10.48 8.02, 0.00, 10.24 4.41, 2.90, 11.05 7.18, 8.70, 0.00 Miller and Robinson (1984)
9.00, 10.10, 0.30 3.00, 14.20, 0.30 7.00, 1.70, 11.80 6.00, 4.30, 12.90
34.00, 13.00, 0.60 30.00, 17.00, 0.60 31.78, 16.00, 3.60 31.70, 16.00, 2.40 8.24, 15.27, 12.25 11.34, 11.10, 15.46 Miller and Robinson (1984)
Clark et al. (2000)
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Table 2 Geometric parameters of human EOMs. EOM
LR
MR
SR
IR
SO
IO
Reference
Cross-sectional area A0 (mm2) Resting length L0 (mm) Initial length L (mm) Initial stretch λ (dimensionless)
16.73 44.60 50.51 1.13
17.39 35.40 39.42 1.11
11.34 39.30 44.70 1.14
15.85 39.80 45.00 1.13
19.34 20.86 22.17 1.06
19.83 30.60 31.21 1.02
Pascolo and Carniel (2009) Miller and Robinson (1984) This study This study
suspended in primary position is written as
can be written as
s~ ¼ aε þ bε þcε; 3
2
ð7Þ
where ε ¼ ΔL/L0 is the strain along the direction of the same EOM fiber with the displacement ΔL; the coefficients a, b, and c are constants. Substituting Eq. (7) into Eq. (3) yields F ¼ aðA0 =L30 ÞΔL3 þ bðA0 =L20 ÞΔL2 þ cðA0 =L0 ÞΔL:
ð8Þ
The constants a, b, and c can be obtained by fitting the experimental passive F ΔL data (Scott, 1971; Collins, 1971; Collins et al., 1975; Quaia et al., 2009). As ΔL¼L L0, ε ¼ ðL L0 Þ=L0 ¼ λ 1:
ð9Þ
By substituting Eqs. (7) and (9) into Eq. (6), the Cauchy stress can be written as
s ¼ aλðλ 1Þ3 þ bλðλ 1Þ2 þcλðλ 1Þ:
ð10Þ
The initial passive force of an EOM is obtained by multiplying the Cauchy stress–stretch relationship in Eq. (10) by the crosssectional area of the current configuration. The in-order Arabic numerals 1–6 are introduced to represent the LR, MR, SR, IR, SO and IO muscle. The passive force can be written as the function of the stretch λ; that is, F pi ðλi Þ ¼ si ðλi ÞAi ;
ð11Þ
where F pi ðλi Þ with i¼1,2,…,6 is the passive force of the i-th EOM. 2.2. Active force Traditionally, muscle force is formulated as the sum of the passive and active contributions (Hill, 1938; Böl and Reese, 2008; Ramírez et al., 2010; Murtada et al., 2013). Accordingly, the total initial tension force of an EOM can be written as F i ¼ F pi ðλi Þ þ F ai ;
ð12Þ
where F ai represents the active contribution. The mechanical equilibrium of the eye depends on the properties of both the orbital tissue and the EOMs (Luboz et al., 2004). Mechanical equilibrium can be described by the equilibrium equations in a 3D space. The restrictive effect of the orbital tissues surrounding the eye globe should be represented by a resistance torque Mt. Only three valid torque equilibrium equations satisfy the eye globe balance, i.e., 6
6
i¼1
i¼1
∑ Mi þ Mt ¼ ∑ Fi Ri þMt ¼ 0;
ð13Þ
where Mi ¼Fi Ri with i¼1,2,…,6 represents the effective torque generated by the i-th EOM; and Ri and Fi are the corresponding moment arm and total force, respectively (Suzuki et al., 2012; Ferrier et al., 2013). Theoretically, the initial tension forces of the EOMs in primary position should keep the eye at equilibrium. This condition is depicted in Eq. (13) with Mt of zero. Therefore, the torque equilibrium equation used to describe the condition of the eye
6
∑ Mi ¼ Fi Ri ¼ 0:
i¼1
ð14Þ
By dividing Eq. (14) into 3D Cartesian coordinates, we obtain three equilibrium equations; however, the initial tension forces Fi of all six EOMs need to be solved. The initial tension force Fi of an EOM is equal to the sum of the superposition of its passive and active contributions, and the passive contribution F pi can be obtained by the Cauchy stress–stretch relationship (Section 2.1). Once the magnitude of the active force F ai of each EOM is known, the initial tension forces of all six EOMs can be obtained. Referring to the principle of minimum potential energy, we complementally provide an optimized mathematical method to solve the active contribution of each EOM. In the proposed method, the quadratic sum of the stresses contributed by the initial tension forces Fi is considered as the minimum, and the active force of an EOM must take on a non-negative value (Chang et al., 2000). Accordingly, this optimized mathematical equation can be written as 8 6 > > Φ ¼ ∑ðs~ i Þ2 ¼ ∑ ½F i =ðλi Ai Þ2 > > > i¼1 > < 6 : ð15Þ p ¼ ∑ ½ðF i þF ai Þ=ðλi Ai Þ2 > > > > i¼1 > > : a F i Z 0; i ¼ 1; 2; …; 6 Using Eqs. (14) and (15), we can solve the initial active force of each EOM. Finally, substituting the solved passive and active forces of an EOM into Eq. (12) yields the initial tension force Fi of the same EOM. 2.3. Statistics The ANOVA test was applied in the corresponding statistical analysis. A probability value (p) of less than 0.05 was considered as statistically significant. Furthermore, mN was used as the unit of force in the present study, whereas gram force (gf) was commonly used in the previous similar studies of the EOMs. Note that the conversion relationship between gf and the international unit mN is 1 gf ¼9.81 mN (Schutte et al., 2006).
3. Results 3.1. Coefficients a, b, and c To establish a unified F ΔL coordinate system, Quaia et al. (2009) analyzed three sets of experimental data on the passive tension of human EOMs provided by the same group of investigators (Scott, 1971; Collins, 1971; Collins et al., 1975). In the present work, the coefficients a, b, and c are obtained by fitting the three sets of experimental data on passive F ΔL. The fitting results shown in Fig. 2 reveal that the passive behavior of the EOMs is nonlinear. The corresponding coefficients a, b, and c
Z. Gao et al. / Journal of Theoretical Biology 353 (2014) 78–83
Fig. 2. Fitting results of relevant experimental data in a unified coordinate system.
Table 3 Magnitude of coefficients a, b, and c (SD: standard deviation). Experiment
a
b
c
Scott (1971) Collins (1971) Collins et al. (1975)
16.44 25.45 29.69
10.32 9.25 5.73
1.443 0.725 0.193
Mean SD
23.86 5.52
8.43 1.96
0.787 0.51
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greater than that of SR (38.0 7 10.5 mN; p¼ 0.331), and 6% greater than that of IR (46.2 713.4 mN; p ¼0.817). The p values between MR and SR, between MR and IR, and between SR and IR are 0.978, 0.437, and 0.452, respectively. The passive initial tension force of SO (15.6 78.3 mN) is also not statistically different from that of IO (3.73 72.13 mN), and the corresponding p value is 0.288. However, the passive contributions of the rectus muscles are statistically different from those of the oblique muscles, e.g., the passive initial tension force of LR is approximately twice that of SO (p ¼0.09) and 12 times that of IO (p ¼0.001). Fig. 5 shows the optimization results for the corresponding active contributions. In a pair of agonist–antagonistic muscles, one muscle provides active contribution while another one contributes no active force, i.e., the active force values of the LR, IR, and SO muscles are zero. The active contribution of MR is 51.6 716.6 mN, which is statistically different from that of SR (12.6 7 4.9 mN; p¼ 0.06) and that of IO (13.4 79.4 mN; p ¼0.06). However, no statistical difference exists between the initial active values of the SR and IO muscles (p ¼0.933). The initial tension forces of the EOMs are obtained by the superposition of the passive and active contributions. Therefore, the initial tension forces of the LR, IR, and SO muscles are equal to their passive contributions. The corresponding results shown in Fig. 6 reveal that the initial tension force of MR (89.2 731.6 mN) is statistically different from those of the other rectus muscles, i.e., the initial tension force of MR is 83% greater than that of LR (p ¼0.021), 76% greater than that of SR (50.6 717.6 mN; p¼ 0.026), and 93% greater than that of IR (p ¼0.016). However, no statistical differences exist among the values of the other three rectus muscles except for MR (p 40.78). In addition, the value of SO is not statistically different from that of IO (17.17 12.1 mN; p¼ 0.923).
4. Discussion
Fig. 3. Relevant Cauchy stress–stretch relationship.
related to the three sets of experimental data are summarized in Table 3. 3.2. Cauchy stress–stretch relationship By substituting the coefficients a, b, and c into Eq. (10), we obtain the Cauchy stress–stretch relationship to describe the fitted passive F ΔL curves that are correlated with the above three experiments. The corresponding results are shown in Fig. 3, in which the average Cauchy stress–stretch relationship curve is between the curves obtained from the first two experimental data (Scott, 1971; Collins et al., 1975); this curve is considerably close to that from the third experiment (Collins, 1971). 3.3. Initial tension force The passive contributions of the initial tension of the EOMs are shown in Fig. 4. The passive contributions of the rectus muscles are greater than that of the oblique muscles, with the former showing no statistically significant difference (p 40.05). The passive initial tension force of LR is 48.8 714.2 mN, which is 30% greater than that of MR (37.67 12.2 mN; p¼ 0.319), 28%
The objective of this work is to estimate the initial tension forces of the EOMs through a mathematical model by weakening the effect of innervation. The corresponding information is responsible for the mechanical equilibrium of the eye suspended in primary position. This equilibrium is primarily used to establish a 3D model for assisting the diagnosis of vertigo. Three assumptions were considered in this work. First, the real resting state of a muscle is difficult to determine. In this work, the resting state was assumed as a critical state, in which a muscle is triggered to generate force. Second, from a continuum viewpoint, the microstructural alternation of the internal kinematic state of the properties of a muscle fiber leads to changes in the macroscopic behavior (Ehret et al., 2011). In other words, the elongation of a muscle at the macroscopic muscle level can be considered as equal to its stretch at the single microscopic fiber level. Consequently, the mechanically passive behavior of a muscle fiber could be described by the obtained Cauchy stress–stretch relationship, which was used to calculate the initial tension forces of the EOMs. Third, the orbital system is capable of controlling eye movement by consuming the least energy. Therefore, the optimum results of the corresponding active contributions reveal that not every EOM needs to provide an active force to keep the eye suspended in primary position. The initial tension force values of human EOMs set in the original published report (Gernandt, 1968) were very small, i.e., less than 9.81 mN. The value of the initial tension force of the LR muscle was 40 mN without the effect of innervation, whereas that of the horizontal recti, vertical recti, and obliques were 78 mN, 59 mN, and 39 mN, respectively, with consideration of the effect of innervation (Robinson, 1975). In addition, an EOM
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Fig. 4. Initial passive forces of the EOMs: the left graph shows the calculated result related to the three sets of experimental data, and the right one shows the average result (error bars denote the standard deviations).
Fig. 5. Initial active forces of the EOMs: the left graph shows the result related to the three sets of experimental data, and the right one shows the average result (error bars denote the standard deviations).
Fig. 6. Initial tension forces of the EOMs: the left graph shows the result related to the three sets of experimental data, and the right one shows the average result (error bars denote the standard deviations).
was found to slack when its tension force was less than 98.1 mN (Collins et al., 1975). Accordingly, the initial tension force (196.2 mN) was set to be higher than 98.1 mN in the recent investigation into the activation of the EOMs (Lennerstrand et al., 2010). The calculation results in Figs. 4–6 suggest that the initial tension forces of the EOMs estimated in this work are acceptable than those in previous data. Several studies (Demer et al., 1997; Kono et al., 2002; Miller, 2007) have found that the functional origin of a rectus muscle is a pulley. However, a controversial theme exists in the field of orbital motility, i.e., which function determines the ocular motility between the neural and the mechanical factor (Demer, 2006). Current studies on ocular biomechanics (Demer, 2006) tend to support the notion that ocular kinematics is mechanically determined. This viewpoint is also supported by pulley theory
(Miller, 2007), i.e., pulley locations can change depending on instantaneous eye position and eye movement type (Kono et al., 2002; Miller, 2007; Thurtell et al., 2012). The calculation results obtained in the present work, wherein the effect of innervations is weakened, agree with modern orbital biomechanical theory. However, EOMs are programmed by the brain to maintain the stability of the eye in fixation positions (Jampel, 2009). An eye suspended in primary position is at a specific stable fixation. Thus, the initial active forces of the EOMs identified here may be controlled by the corresponding innervations. Studies on skeletal muscles have demonstrated that the activation of a muscle is correlated with the Ca–calsequestrin association in the sarcoplasmatic reticulum (Cannell and Allen, 1984; Ramírez et al., 2010). Therefore, the active force of a muscle depends to a certain extent on the variation of the free Ca2 þ concentration in cytosolic
Z. Gao et al. / Journal of Theoretical Biology 353 (2014) 78–83
(Berchtold et al., 2000). Nevertheless, Feng et al. (2012) provided an evidence for the abundant spontaneous activities in the EOMs by examining the calcium transients in the EOMs of chickens. This discovery suggests that the active force of an EOM might also be generated by the adaptation of the EOM itself. With the mechanical equilibrium of the eye suspended in primary position, the initial active force of an EOM might be related to its adaptation of spontaneous calcium transients, to the innervation programmed by the brain, or to both factors. However, the calculation results as shown in Figs. 4 and 5 suggest that the active contributions of the EOMs have a lower participation in keeping the eye suspended than the corresponding passive contributions do. In addition, innervation influences the initial tension forces of the EOMs in the form of agonist–antagonistic muscle pairs, i.e., one muscle of the horizontal recti, vertical recti, and obliques contracts and generates active contribution by innervation while another one is in slack and provides no active force to keep the eye suspended (Fig. 5). Although the initial tension of the EOMs in primary position is more or less controlled by innervation, the mechanical equilibrium of the eye suspended in primary position is mainly determined by the biomechanical behavior of the EOMs themselves. This conclusion coincides with modern orbital biomechanical theory (Demer, 2006). 5. Conclusion A mathematical model was developed in this study to estimate the initial tension forces of the EOMs with weakening innervation. The corresponding passive contribution was described using the Cauchy stress–stretch relationship, whereas the active contribution was calculated using an optimized mathematical method. In the analysis of the calculation results, the mechanical equilibrium of the eye suspended in primary position was mainly determined by the mechanical behavior of the EOMs. The initial tension forces of the EOMs calculated in this study are 48.8 714.2 mN for LR, 89.2 731.6 mN for MR, 50.6 717.6 mN for SR, 46.2 713.4 mN for IR, 15.6 78.3 mN for SO, and 17.1 712.1 mN for IO. Acknowledgment The authors thank Prof. An Meiwen and Dr. Jing Lin for some helpful advice. This study was supported by the National Natural Science Foundation of China (No. 11032008). References Bedell, H.E., Stevenson, S.B., 2013. Eye movement testing in clinical examination. Vis. Res. 90, 32–37. Berchtold, M.W., Brinkmeier, H., Müntener, M., 2000. Calcium ion in skeletal muscle: its crucial role for muscle function, plasticity, and disease. Physiol. Rev. 80, 1215–1265. Böl, M., Reese, S., 2008. Micromechanical modeling of skeletal muscles based on the finite element method. Comput. Method Biomech. 11, 489–504. Calvo, B., Ramírez, A., Alonso, A., Grasa, J., Soteras, F., Osta, R., et al., 2010. Passive nonlinear elastic behavior of skeletal muscle: experimental results and model formulation. J. Biomech. 43, 318–325. Cannell, M.B., Allen, D.G., 1984. Model of calcium movements during activation in the sarcomere of frog skeletal muscle. Biophys. J. 45, 913–925. Chang, Y.W., Hughes, R.E., Su, F.C., Itoi, E., An, K.N., 2000. Prediction of muscle force involved in shoulder internal rotation. J. Shoulder Elb. Surg. 9, 188–195. Clark, R.A., Miller, J.M., Demer, J.L., 2000. Three-dimensional location of human rectus pulleys by path inflections in secondary gaze positions. Investig. Ophthalmol. Vis. Sci. 41, 3787–3797.
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Collins, C.C., 1971. Orbital mechanics. In: Bach-Y-Rita, P., Collins, C.C., Hyde, J.E. (Eds.), The Control of Eye Movements, Academic Press, New York, pp. 283–325 Collins, C.C., O'Meara, D., Scott, A.B., 1975. Muscle tension during unrestrained human eye movements. J. Physiol. 245, 351–369. Demer, J.L., Poukens, V., Miller, J.M., Micevych, P., 1997. Innervation of extraocular pulley smooth muscle in monkeys and humans. Investig. Ophthalmol. Vis. Sci. 38, 1774–1785. Demer, J.L., 2006. Current concepts of mechanical and neural factor in ocular motility. Curr. Opin. Neurol. 19, 4–13. Dong, C., Wang, Y., Zhang, Q., Wang, N., 2014. The methodology of dynamic gausality graph for intelligent diagnosis of vertigo. Comput. Methods Progr. Biomed. 113, 162–174. Ehret, A.E., Böl, M., Itskov, M., 2011. A continuum constitutive for the active behavior of skeletal muscle. J. Mech. Phys. Solids 59, 625–636. Elsheikh, A., Anderson, K., 2005. Comparative study of corneal strip extensometry and inflation tests. J. R. Soc. Interface 2, 177–185. Feng, C.Y., Hennig, G.W., Corrigan, R.D., Smith, T.K., Von Bartheld, C.S., 2012. Analysis of spontaneous and nerve-evoked calcium transients in intact extraocular muscle in vitro. Exp. Eye Res. 100, 73–85. Ferrier, A., Larrea, X., Camine, V.M., Pioletti, D.P., Farron, A., 2013. Importance of the subscapularis muscle after total shoulder arthroplasty. Clin. Biomech. 28, 146–150. Gernandt, B.E., 1968. Interactions between extraocular myotatic and ascending vestibular activities. Exp. Neurol. 20, 120–134. Gordon, A.M., Huxley, A.F., Julian, F.J., 1966. The variation in isometric tension with sarcomere length in vertebrate muscle fibres. J. Physiol. 184, 170–192. Hill, A.V., 1938. The heat of shortening and the dynamic constants of muscle. Proc. R. Soc. 126, 136–195. Jampel, R.S., 2009. The function of the extraocular muscles, the theory of the coplanarity of the fixation planes. J. Neurol. Sci. 280, 1–9. Kono, R., Clark, R.A., Demer, J.L., 2002. Active pulleys: magnetic resonance imaging of rectus muscle paths in tertiary gazes. Investig. Ophthalmol. Vis. Sci. 43, 2179–2188. Kushner, B.J., 2010. Incomitant strabismus: does extraocular muscle form denote function? Arch. Ophthalmol. 128, 1604–1609. Lennerstrand, G., Bolzani, R., Tian, S., Benassi, M., Fusari, M., Campos, E., et al., 2010. Succinylcholine activation of human horizontal eye movement. Acta Ophthalmol. 88, 872–876. Luboz, V., Pedrono, A., Ambard, D., Boutault, F., Payan, Y., Smider, P., 2004. Prediction of tissue decompression in orbital surgery. Clin. Biomech. 19, 202–208. Miller, J.M., Robinson, D.A., 1984. A model of the mechanics of binocular alignment. Comput. Biomed. Res. 17, 436–470. Miller, J.M., 2007. Understanding and misunderstanding extraocular muscle pulleys. J. Vis. 10, 1–15. Murtada, S., Arner, A., Holzapfel, G.A., 2013. Experiments and mechanochemical modeling of smooth muscle contraction: significance of filament overlap. J. Theor. Biol. 297, 176–186. Paersch, C., Trimmer, B.A., Dorfmann, A., 2012. A constitutive model for active– passive transition of muscle fibers. Int. J. Nonlin. Mech. 47, 377–387. Pascolo, P., Carniel, R., 2009. From time series analysis to a biomechanical multibody model of the human eye. Chaos Solitons Fractals 40, 966–974. Quaia, C., Ying, H.S., Nichols, A.M., Optican, L.M., 2009. The viscoelastic properties of passive eye muscle in primates. I: Static forces and step responses. PloS One 4, e4850. Ramírez, A., Grasa, J., Alonso, A., Soteras, F., Osta, R., Muńoz, M.J., et al., 2010. Active response of skeletal muscle: in vivo experimental results and model formulation. J. Theor. Biol. 267, 546–553. Robinson, D.A., 1975. A quantitative analysis of extraocular muscle cooperation and squint. Investig. Ophthalmol. 14, 801–825. Schutte, S., Van den Bedem, S.P., Keulen, F.V., Van der Helm, F.C., Simonsz, H.J., 2006. A finite-element analysis model of orbital biomechanics. Vis. Res. 46, 1724–1731. Scott, A.B., 1971. Extraocular muscle forces in strabismus. In: Bach-Y-Rita, P., Collins, C.C., Hyde, J.E. (Eds.), The Control of Eye Movements, Academic Press, New York, pp. 327–342 Suzuki, Y., Nomura, T., Casadio, M., Morasso, P., 2012. Intermittent control with ankle, hip and mixed strategies during quiet standing: a theoretical proposal based on a double inverted pendulum model. J. Theor. Biol. 310, 55–79. Takaza, M., Moerman, K.M., Gindre, J., Lyons, G., Simms, C.K., 2013. The anisotropic mechanical behavior of passive skeletal muscle tissue subjected to large tensile strain. J. Mech. Behav. Biomed. 17, 209–220. Thurtell, M.J., Joshi, A.C., Walker, M.F., 2012. Three-dimensional kinematics of saccadic and pursuit eye movements in humans: relationship between Donder's and Listing's law. Vis. Res. 60, 7–15. Wei, Q., Sueda, S., Pai, D.K., 2010. Physically-based modeling and simulation of extraocular muscles. Prog. Biophys. Mol. Biol. 103, 273–283.
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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Initial tension of the human extraocular muscles in the primary eye position Zhipeng Gao, Hongmei Guo, Weiyi Chen n Shanxi key Laboratory of Material Strength & Structural Impact, Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
H I G H L I G H T S
A model was proposed to estimate the initial tension of extraocular muscles. An optimum method was used to determine the active behavior. Mechanical equilibrium was analyzed using the complementary information.
art ic l e i nf o
a b s t r a c t
Article history: Received 15 June 2013 Received in revised form 10 March 2014 Accepted 11 March 2014 Available online 20 March 2014
This study proposes a mathematical model to estimate the initial tension forces of the extraocular muscles (EOMs). These forces are responsible for the mechanical equilibrium of the eye suspended in primary position. The passive contributions were obtained using the corresponding Cauchy stress– stretch relationships based on the previous clinical experimental data; whereas the active contributions were obtained using an optimum method with weakening the effect of innervation. The initial tension forces of the EOMs were estimated to be 48.87 14.2 mN for the lateral rectus, 89.2 731.6 mN for the medial rectus, 50.6 7 17.6 mN for the superior rectus, 46.2 713.4 mN for the inferior rectus, 15.6 78.3 mN for the superior oblique, and 17.1 7 12.1 mN for the inferior oblique. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Passive behavior Active behavior Optimum method Biomechanics
1. Introduction The extraocular muscles (EOMs) are indispensable to the threedimensional (3D) modeling of eye movement, which can potentially be applied in the diagnosis of vertigo (Dong et al., 2014) and in electronystagmograph analysis (Bedell and Stevenson, 2013). Several models have been established to investigate eye movement; these models can be divided into three categories, i.e., the classical two-element model that includes the EOMs and the eyeball (Robinson, 1975; Miller and Robinson, 1984), the finite element model (Luboz et al., 2004; Schutte et al., 2006), and the modern three-element model (Pascolo and Carniel, 2009; Wei et al., 2010) that includes the EOMs, eyeball and pulley. The primary position of eye is generally set as the reference configuration in eye examination and modeling, in which the initial tension forces of the EOMs play an important role in keeping the eye suspended. n Correspondence to: Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, 79 YingZe West Street, Taiyuan 030024, China. Tel.: þ 86 351 601 8864. E-mail addresses: [email protected], [email protected] (W. Chen).
http://dx.doi.org/10.1016/j.jtbi.2014.03.018 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
Initial tension is defined as the difference between the lengths of the primary position and the resting state of a muscle. This difference urges the corresponding muscle to generate force that would keep the eye suspended in primary position. Theoretically, the mechanical equilibrium of the eye in a 3D space is dependent on the contributions of the six EOMs, i.e., lateral rectus (LR), medial rectus (MR), superior rectus (SR), inferior rectus (IR), superior oblique (SO), and inferior oblique (IO). The present work aims to estimate the initial tension forces of EOMs that maintain the eye suspended in primary position. In studies related to the EOMs, and to the twitch and tetanic activation or the mechanical behavior of eye movement, researchers have set a unified value for the initial tension force of the EOMs; this value ranges from 4 to 200 mN (Gernandt, 1968; Collins et al., 1975; Lennerstrand et al., 2010). The initial tension forces of the EOMs, particularly of the LR muscle, have been measured using horizontal rotation from topical anesthetic patients, whose LR muscles were detached from the globe and were attached with a micromanipulator-mounted strain gauge (Collins et al., 1975; Robinson, 1975). Their experimental results suggest that innervation is a predominant factor in the magnitude
Z. Gao et al. / Journal of Theoretical Biology 353 (2014) 78–83
of the initial tension forces of the EOMs. Different innervations trigger different isometric contractions of a muscle and then produce different active forces. At present, no ideal experimental data can be used to determine this complicated constitutive relationship between active force and the stretch of an EOM because of the anatomical difficulty and ethical requirements, although a similar relationship of the skeletal muscle has been determined (Gordon et al., 1966) and has been widely used in the modern finite element modeling of muscles (Böl and Reese, 2008; Ehret et al., 2011). However, the fiber composition of an EOM is different from that of a skeletal muscle (Kushner, 2010); their biomechanical behaviors also differ (Quaia et al., 2009). To use the corresponding research results on the active force of a skeletal muscle in describing the active behavior of an EOM is inappropriate. As the influence of innervations on the active force of an EOM is rarely studied, the present work focuses on the relaxation state of the eye to weaken the effect of innervation. This study develops a mathematical model to calculate the initial tension forces of EOMs. It refers to previous published works that contain clinical experimental data on the passive tension (Scott, 1971; Collins, 1971; Collins et al., 1975; Quaia et al., 2009) of EOMs and corresponding coordinate information (Miller and Robinson, 1984; Clark et al., 2000) on the primary position of the EOMs. The initial tension force of an EOM comprises passive and active contributions. In this work, the passive forces are calculated using the theory of mechanics, whereas the corresponding active forces are obtained using an optimum mathematical method for establishing the weakening of innervation. The investigation results reveal that the initial tension forces of the EOMs are 48.8714.2 mN for LR, 89.2731.6 mN for MR, 50.6717.6 mN for SR, 46.2713.4 mN for IR, 15.678.3 mN for SO, and 17.1712.1 mN for IO.
79
The initial stretch λ of an EOM is obtained by substituting Eq. (2) into Eq. (1). In this paper, the resting state of an EOM is defined as the reference configuration, and its primary position shape is defined as the current (deformed) configuration. The cross-sectional areas of the EOMs in the reference configuration are adopted from the work of Pascolo and Carniel (2009). The corresponding geometric parameters of human EOMs are summarized in Table 2. According to continuum theory, the Lagrange or Piola–Kirchhoff stress along the direction of an EOM fiber in the reference configuration is defined as
s~ ¼ F=A0 ;
ð3Þ
where F is the tension force along the EOM fiber, and A0 is the cross-sectional area in the resting state. Cauchy (true) stress along the direction of the same EOM fiber in the current configuration is defined as
s ¼ F=A;
ð4Þ
where A is the deformed cross-sectional area. Assuming that EOMs are incompressible and that their original cross sections A0 are constant, AL ¼ A0 L0 :
ð5Þ
Substituting Eqs. (1), (4), and (5) into Eq. (3), Eq. (3) yields
s ¼ λs~ :
ð6Þ
A second-order formula in the form of E ¼aε þbε þ c was used to describe Young's modulus of the soft tissue (cornea) in the work of Elsheikh and Anderson (2005). Accordingly, the Lagrange stress 2
2. Methods 2.1. Passive force The resting and initial lengths of an EOM are defined as L0 and L, respectively. On the basis of the definition of stretch in the studies on other muscle types (Calvo et al., 2010; Paersch et al., 2012; Takaza et al., 2013), the initial stretch λ of an EOM in primary position is defined as λ ¼ L=L0 :
ð1Þ
The literature (Miller and Robinson, 1984; Clark et al., 2000) provides the resting length L0 and the key site coordinates of the EOMs, which include the insertions (I), pulleys (P), and origins (O). These parameters are summarized in Table 1. The tangency (T) on the globe can be determined by the above three points of the same muscle. Therefore, the initial length L of an EOM in primary position (Fig. 1) can be obtained by summing up the lengths of the lines of OP, PT, and TI; that is, L ¼ OP þ PT þ TI:
ð2Þ
Fig. 1. Schematic illustration of the length of an EOM.
Table 1 The key site coordinates of the EOMs (the origin of the SO muscle is the trochlea). EOMs
Insertion
Pulley
Origin
LR MR SR IR SO IO Reference
6.50, 10.08, 0.00 8.42, 9.65, 0.00 7.63, 0.00, 10.48 8.02, 0.00, 10.24 4.41, 2.90, 11.05 7.18, 8.70, 0.00 Miller and Robinson (1984)
9.00, 10.10, 0.30 3.00, 14.20, 0.30 7.00, 1.70, 11.80 6.00, 4.30, 12.90
34.00, 13.00, 0.60 30.00, 17.00, 0.60 31.78, 16.00, 3.60 31.70, 16.00, 2.40 8.24, 15.27, 12.25 11.34, 11.10, 15.46 Miller and Robinson (1984)
Clark et al. (2000)
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Table 2 Geometric parameters of human EOMs. EOM
LR
MR
SR
IR
SO
IO
Reference
Cross-sectional area A0 (mm2) Resting length L0 (mm) Initial length L (mm) Initial stretch λ (dimensionless)
16.73 44.60 50.51 1.13
17.39 35.40 39.42 1.11
11.34 39.30 44.70 1.14
15.85 39.80 45.00 1.13
19.34 20.86 22.17 1.06
19.83 30.60 31.21 1.02
Pascolo and Carniel (2009) Miller and Robinson (1984) This study This study
suspended in primary position is written as
can be written as
s~ ¼ aε þ bε þcε; 3
2
ð7Þ
where ε ¼ ΔL/L0 is the strain along the direction of the same EOM fiber with the displacement ΔL; the coefficients a, b, and c are constants. Substituting Eq. (7) into Eq. (3) yields F ¼ aðA0 =L30 ÞΔL3 þ bðA0 =L20 ÞΔL2 þ cðA0 =L0 ÞΔL:
ð8Þ
The constants a, b, and c can be obtained by fitting the experimental passive F ΔL data (Scott, 1971; Collins, 1971; Collins et al., 1975; Quaia et al., 2009). As ΔL¼L L0, ε ¼ ðL L0 Þ=L0 ¼ λ 1:
ð9Þ
By substituting Eqs. (7) and (9) into Eq. (6), the Cauchy stress can be written as
s ¼ aλðλ 1Þ3 þ bλðλ 1Þ2 þcλðλ 1Þ:
ð10Þ
The initial passive force of an EOM is obtained by multiplying the Cauchy stress–stretch relationship in Eq. (10) by the crosssectional area of the current configuration. The in-order Arabic numerals 1–6 are introduced to represent the LR, MR, SR, IR, SO and IO muscle. The passive force can be written as the function of the stretch λ; that is, F pi ðλi Þ ¼ si ðλi ÞAi ;
ð11Þ
where F pi ðλi Þ with i¼1,2,…,6 is the passive force of the i-th EOM. 2.2. Active force Traditionally, muscle force is formulated as the sum of the passive and active contributions (Hill, 1938; Böl and Reese, 2008; Ramírez et al., 2010; Murtada et al., 2013). Accordingly, the total initial tension force of an EOM can be written as F i ¼ F pi ðλi Þ þ F ai ;
ð12Þ
where F ai represents the active contribution. The mechanical equilibrium of the eye depends on the properties of both the orbital tissue and the EOMs (Luboz et al., 2004). Mechanical equilibrium can be described by the equilibrium equations in a 3D space. The restrictive effect of the orbital tissues surrounding the eye globe should be represented by a resistance torque Mt. Only three valid torque equilibrium equations satisfy the eye globe balance, i.e., 6
6
i¼1
i¼1
∑ Mi þ Mt ¼ ∑ Fi Ri þMt ¼ 0;
ð13Þ
where Mi ¼Fi Ri with i¼1,2,…,6 represents the effective torque generated by the i-th EOM; and Ri and Fi are the corresponding moment arm and total force, respectively (Suzuki et al., 2012; Ferrier et al., 2013). Theoretically, the initial tension forces of the EOMs in primary position should keep the eye at equilibrium. This condition is depicted in Eq. (13) with Mt of zero. Therefore, the torque equilibrium equation used to describe the condition of the eye
6
∑ Mi ¼ Fi Ri ¼ 0:
i¼1
ð14Þ
By dividing Eq. (14) into 3D Cartesian coordinates, we obtain three equilibrium equations; however, the initial tension forces Fi of all six EOMs need to be solved. The initial tension force Fi of an EOM is equal to the sum of the superposition of its passive and active contributions, and the passive contribution F pi can be obtained by the Cauchy stress–stretch relationship (Section 2.1). Once the magnitude of the active force F ai of each EOM is known, the initial tension forces of all six EOMs can be obtained. Referring to the principle of minimum potential energy, we complementally provide an optimized mathematical method to solve the active contribution of each EOM. In the proposed method, the quadratic sum of the stresses contributed by the initial tension forces Fi is considered as the minimum, and the active force of an EOM must take on a non-negative value (Chang et al., 2000). Accordingly, this optimized mathematical equation can be written as 8 6 > > Φ ¼ ∑ðs~ i Þ2 ¼ ∑ ½F i =ðλi Ai Þ2 > > > i¼1 > < 6 : ð15Þ p ¼ ∑ ½ðF i þF ai Þ=ðλi Ai Þ2 > > > > i¼1 > > : a F i Z 0; i ¼ 1; 2; …; 6 Using Eqs. (14) and (15), we can solve the initial active force of each EOM. Finally, substituting the solved passive and active forces of an EOM into Eq. (12) yields the initial tension force Fi of the same EOM. 2.3. Statistics The ANOVA test was applied in the corresponding statistical analysis. A probability value (p) of less than 0.05 was considered as statistically significant. Furthermore, mN was used as the unit of force in the present study, whereas gram force (gf) was commonly used in the previous similar studies of the EOMs. Note that the conversion relationship between gf and the international unit mN is 1 gf ¼9.81 mN (Schutte et al., 2006).
3. Results 3.1. Coefficients a, b, and c To establish a unified F ΔL coordinate system, Quaia et al. (2009) analyzed three sets of experimental data on the passive tension of human EOMs provided by the same group of investigators (Scott, 1971; Collins, 1971; Collins et al., 1975). In the present work, the coefficients a, b, and c are obtained by fitting the three sets of experimental data on passive F ΔL. The fitting results shown in Fig. 2 reveal that the passive behavior of the EOMs is nonlinear. The corresponding coefficients a, b, and c
Z. Gao et al. / Journal of Theoretical Biology 353 (2014) 78–83
Fig. 2. Fitting results of relevant experimental data in a unified coordinate system.
Table 3 Magnitude of coefficients a, b, and c (SD: standard deviation). Experiment
a
b
c
Scott (1971) Collins (1971) Collins et al. (1975)
16.44 25.45 29.69
10.32 9.25 5.73
1.443 0.725 0.193
Mean SD
23.86 5.52
8.43 1.96
0.787 0.51
81
greater than that of SR (38.0 7 10.5 mN; p¼ 0.331), and 6% greater than that of IR (46.2 713.4 mN; p ¼0.817). The p values between MR and SR, between MR and IR, and between SR and IR are 0.978, 0.437, and 0.452, respectively. The passive initial tension force of SO (15.6 78.3 mN) is also not statistically different from that of IO (3.73 72.13 mN), and the corresponding p value is 0.288. However, the passive contributions of the rectus muscles are statistically different from those of the oblique muscles, e.g., the passive initial tension force of LR is approximately twice that of SO (p ¼0.09) and 12 times that of IO (p ¼0.001). Fig. 5 shows the optimization results for the corresponding active contributions. In a pair of agonist–antagonistic muscles, one muscle provides active contribution while another one contributes no active force, i.e., the active force values of the LR, IR, and SO muscles are zero. The active contribution of MR is 51.6 716.6 mN, which is statistically different from that of SR (12.6 7 4.9 mN; p¼ 0.06) and that of IO (13.4 79.4 mN; p ¼0.06). However, no statistical difference exists between the initial active values of the SR and IO muscles (p ¼0.933). The initial tension forces of the EOMs are obtained by the superposition of the passive and active contributions. Therefore, the initial tension forces of the LR, IR, and SO muscles are equal to their passive contributions. The corresponding results shown in Fig. 6 reveal that the initial tension force of MR (89.2 731.6 mN) is statistically different from those of the other rectus muscles, i.e., the initial tension force of MR is 83% greater than that of LR (p ¼0.021), 76% greater than that of SR (50.6 717.6 mN; p¼ 0.026), and 93% greater than that of IR (p ¼0.016). However, no statistical differences exist among the values of the other three rectus muscles except for MR (p 40.78). In addition, the value of SO is not statistically different from that of IO (17.17 12.1 mN; p¼ 0.923).
4. Discussion
Fig. 3. Relevant Cauchy stress–stretch relationship.
related to the three sets of experimental data are summarized in Table 3. 3.2. Cauchy stress–stretch relationship By substituting the coefficients a, b, and c into Eq. (10), we obtain the Cauchy stress–stretch relationship to describe the fitted passive F ΔL curves that are correlated with the above three experiments. The corresponding results are shown in Fig. 3, in which the average Cauchy stress–stretch relationship curve is between the curves obtained from the first two experimental data (Scott, 1971; Collins et al., 1975); this curve is considerably close to that from the third experiment (Collins, 1971). 3.3. Initial tension force The passive contributions of the initial tension of the EOMs are shown in Fig. 4. The passive contributions of the rectus muscles are greater than that of the oblique muscles, with the former showing no statistically significant difference (p 40.05). The passive initial tension force of LR is 48.8 714.2 mN, which is 30% greater than that of MR (37.67 12.2 mN; p¼ 0.319), 28%
The objective of this work is to estimate the initial tension forces of the EOMs through a mathematical model by weakening the effect of innervation. The corresponding information is responsible for the mechanical equilibrium of the eye suspended in primary position. This equilibrium is primarily used to establish a 3D model for assisting the diagnosis of vertigo. Three assumptions were considered in this work. First, the real resting state of a muscle is difficult to determine. In this work, the resting state was assumed as a critical state, in which a muscle is triggered to generate force. Second, from a continuum viewpoint, the microstructural alternation of the internal kinematic state of the properties of a muscle fiber leads to changes in the macroscopic behavior (Ehret et al., 2011). In other words, the elongation of a muscle at the macroscopic muscle level can be considered as equal to its stretch at the single microscopic fiber level. Consequently, the mechanically passive behavior of a muscle fiber could be described by the obtained Cauchy stress–stretch relationship, which was used to calculate the initial tension forces of the EOMs. Third, the orbital system is capable of controlling eye movement by consuming the least energy. Therefore, the optimum results of the corresponding active contributions reveal that not every EOM needs to provide an active force to keep the eye suspended in primary position. The initial tension force values of human EOMs set in the original published report (Gernandt, 1968) were very small, i.e., less than 9.81 mN. The value of the initial tension force of the LR muscle was 40 mN without the effect of innervation, whereas that of the horizontal recti, vertical recti, and obliques were 78 mN, 59 mN, and 39 mN, respectively, with consideration of the effect of innervation (Robinson, 1975). In addition, an EOM
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Z. Gao et al. / Journal of Theoretical Biology 353 (2014) 78–83
Fig. 4. Initial passive forces of the EOMs: the left graph shows the calculated result related to the three sets of experimental data, and the right one shows the average result (error bars denote the standard deviations).
Fig. 5. Initial active forces of the EOMs: the left graph shows the result related to the three sets of experimental data, and the right one shows the average result (error bars denote the standard deviations).
Fig. 6. Initial tension forces of the EOMs: the left graph shows the result related to the three sets of experimental data, and the right one shows the average result (error bars denote the standard deviations).
was found to slack when its tension force was less than 98.1 mN (Collins et al., 1975). Accordingly, the initial tension force (196.2 mN) was set to be higher than 98.1 mN in the recent investigation into the activation of the EOMs (Lennerstrand et al., 2010). The calculation results in Figs. 4–6 suggest that the initial tension forces of the EOMs estimated in this work are acceptable than those in previous data. Several studies (Demer et al., 1997; Kono et al., 2002; Miller, 2007) have found that the functional origin of a rectus muscle is a pulley. However, a controversial theme exists in the field of orbital motility, i.e., which function determines the ocular motility between the neural and the mechanical factor (Demer, 2006). Current studies on ocular biomechanics (Demer, 2006) tend to support the notion that ocular kinematics is mechanically determined. This viewpoint is also supported by pulley theory
(Miller, 2007), i.e., pulley locations can change depending on instantaneous eye position and eye movement type (Kono et al., 2002; Miller, 2007; Thurtell et al., 2012). The calculation results obtained in the present work, wherein the effect of innervations is weakened, agree with modern orbital biomechanical theory. However, EOMs are programmed by the brain to maintain the stability of the eye in fixation positions (Jampel, 2009). An eye suspended in primary position is at a specific stable fixation. Thus, the initial active forces of the EOMs identified here may be controlled by the corresponding innervations. Studies on skeletal muscles have demonstrated that the activation of a muscle is correlated with the Ca–calsequestrin association in the sarcoplasmatic reticulum (Cannell and Allen, 1984; Ramírez et al., 2010). Therefore, the active force of a muscle depends to a certain extent on the variation of the free Ca2 þ concentration in cytosolic
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(Berchtold et al., 2000). Nevertheless, Feng et al. (2012) provided an evidence for the abundant spontaneous activities in the EOMs by examining the calcium transients in the EOMs of chickens. This discovery suggests that the active force of an EOM might also be generated by the adaptation of the EOM itself. With the mechanical equilibrium of the eye suspended in primary position, the initial active force of an EOM might be related to its adaptation of spontaneous calcium transients, to the innervation programmed by the brain, or to both factors. However, the calculation results as shown in Figs. 4 and 5 suggest that the active contributions of the EOMs have a lower participation in keeping the eye suspended than the corresponding passive contributions do. In addition, innervation influences the initial tension forces of the EOMs in the form of agonist–antagonistic muscle pairs, i.e., one muscle of the horizontal recti, vertical recti, and obliques contracts and generates active contribution by innervation while another one is in slack and provides no active force to keep the eye suspended (Fig. 5). Although the initial tension of the EOMs in primary position is more or less controlled by innervation, the mechanical equilibrium of the eye suspended in primary position is mainly determined by the biomechanical behavior of the EOMs themselves. This conclusion coincides with modern orbital biomechanical theory (Demer, 2006). 5. Conclusion A mathematical model was developed in this study to estimate the initial tension forces of the EOMs with weakening innervation. The corresponding passive contribution was described using the Cauchy stress–stretch relationship, whereas the active contribution was calculated using an optimized mathematical method. In the analysis of the calculation results, the mechanical equilibrium of the eye suspended in primary position was mainly determined by the mechanical behavior of the EOMs. The initial tension forces of the EOMs calculated in this study are 48.8 714.2 mN for LR, 89.2 731.6 mN for MR, 50.6 717.6 mN for SR, 46.2 713.4 mN for IR, 15.6 78.3 mN for SO, and 17.1 712.1 mN for IO. Acknowledgment The authors thank Prof. An Meiwen and Dr. Jing Lin for some helpful advice. This study was supported by the National Natural Science Foundation of China (No. 11032008). References Bedell, H.E., Stevenson, S.B., 2013. Eye movement testing in clinical examination. Vis. Res. 90, 32–37. Berchtold, M.W., Brinkmeier, H., Müntener, M., 2000. Calcium ion in skeletal muscle: its crucial role for muscle function, plasticity, and disease. Physiol. Rev. 80, 1215–1265. Böl, M., Reese, S., 2008. Micromechanical modeling of skeletal muscles based on the finite element method. Comput. Method Biomech. 11, 489–504. Calvo, B., Ramírez, A., Alonso, A., Grasa, J., Soteras, F., Osta, R., et al., 2010. Passive nonlinear elastic behavior of skeletal muscle: experimental results and model formulation. J. Biomech. 43, 318–325. Cannell, M.B., Allen, D.G., 1984. Model of calcium movements during activation in the sarcomere of frog skeletal muscle. Biophys. J. 45, 913–925. Chang, Y.W., Hughes, R.E., Su, F.C., Itoi, E., An, K.N., 2000. Prediction of muscle force involved in shoulder internal rotation. J. Shoulder Elb. Surg. 9, 188–195. Clark, R.A., Miller, J.M., Demer, J.L., 2000. Three-dimensional location of human rectus pulleys by path inflections in secondary gaze positions. Investig. Ophthalmol. Vis. Sci. 41, 3787–3797.
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