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Longer moment arm results in smaller joint moment development, power and work outputs in fast motions
ARTICLE IN PRESS
Journal of Biomechanics 36 (2003) 1675–1681
Longer moment arm results in smaller joint moment development, power and work outputs in fast motions Akinori Naganoa,b,*, Taku Komurac b
a Center for BioDynamics, Boston University, Boston, MA, USA Computer and Information Division, Advanced Computing Center, RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan c Department of Computer Engineering & Information Technology, City University of Hong Kong, Kowloon, Hong Kong
Accepted 7 April 2003
Abstract Effects of moment arm length on kinetic outputs of a musculoskeletal system (muscle force development, joint moment development, joint power output and joint work output) were evaluated using computer simulation. A skeletal system of the human ankle joint was constructed: a lower leg segment and a foot segment were connected with a hinge joint. A Hill-type model of the musculus soleus (m. soleus), consisting of a contractile element and a series elastic element, was attached to the skeletal system. The model of the m. soleus was maximally activated, while the ankle joint was plantarflexed/dorsiflexed at a variation of constant angular velocities, simulating isokinetic exercises on a muscle testing machine. Profiles of the kinetic outputs (muscle force development, joint moment development, joint power output and joint work output) were obtained. Thereafter, the location of the insertion of the m. soleus was shifted toward the dorsal/ventral direction by 1 cm, which had an effect of lengthening/shortening the moment arm length, respectively. The kinetic outputs of the musculoskeletal system during the simulated isokinetic exercises were evaluated with these longer/shorter moment arm lengths. It was found that longer moment arm resulted in smaller joint moment development, smaller joint power output and smaller joint work output in the larger plantarflexion angular velocity region (>120 / s). This is because larger muscle shortening velocity was required with longer moment arm to achieve a certain joint angular velocity. Larger muscle shortening velocity resulted in smaller muscle force development because of the force–velocity relation of the muscle. It was suggested that this phenomenon should be taken into consideration when investigating the joint moment–joint angle and/or joint moment–joint angular velocity characteristics of experimental data. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Modeling; Simulation; Musculoskeletal system; Muscle force
1. Introduction In a musculoskeletal system, forces developed by skeletal muscles produce motions of the skeletal system (Delp et al., 1990; Yamaguchi, 2001). Muscles are attached to bones at a distance from the joint center (center of rotation), as shown in Fig. 1. Therefore linear tensions developed by muscles are transformed into rotational joint moments. The relation between the muscle force development and the joint moment development is described as ~ joint ¼ ~ ~mus ; M rjc-ins F
ð1Þ
*Corresponding author. Computer and Information Division, Advanced Computing Center, RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan. Tel.: +81-48-467-8345; fax: +81-48-467-4389. E-mail address: [email protected] (A. Nagano).
~ joint is the joint moment produced by the muscle where M force, ~ rjc-ins is the vector from the joint center to the muscle insertion, and F~mus is the muscle force vector. The distance noted as MA in Fig. 1 is the moment arm length of the muscle force vector (F~mus ). Assuming that the joint motion occurs in a two-dimensional plane, the relation expressed in Eq. (1) can be simplified into a scalar formula Mjoint ¼ MA Fmus
ð2Þ
where Mjoint is the magnitude of the joint moment, Fmus is the magnitude of the muscle force, and MA is the moment arm length. In human and animal musculoskeletal systems, almost all forces developed by skeletal muscles are transmitted to the external environment in the form of joint moments (Lieber, 1992; Nigg and Herzog, 1994;
0021-9290/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0021-9290(03)00171-4
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M joint = MA . Fmus mu
Lower Leg
Lower Leg Fmus L mus
Fmus
θ
MA
Foot
Mjoint
Foot
Fig. 1. Joint moment (Mjoint ) is produced around a joint, as a function of MA (moment arm length) and Fmus (muscle force).
Winter, 1990). Therefore, it is crucially important to investigate the mechanical structure of joints in terms of moment arm length. Many researchers have reported numerous research results on this issue to date (An et al., 1983; Fowler et al., 2001; Klein et al., 1996; Spoor et al., 1990; Spoor and van Leeuwen, 1992; van Eijden et al., 1997; van Spronsen et al., 1997). The relation described in Eq. (2) may yield a statement such as ‘‘with a longer moment arm, a greater joint moment is (always) developed’’, which is usually true for typical mechanical systems. However, this statement is not always true for biomechanical musculoskeletal systems, especially when the musculoskeletal system is undergoing movements. This is because the force development ability of a muscle is greatly affected by its length and rate of length change, i.e., A.V. Hill’s force–length and force–velocity relations (Hill, 1938) (Fig. 2). In preceding studies, many researchers have highlighted the importance of these relations when considering biomechanical systems (for example, van Zuylen et al., 1988). Generally, when the joint angle changes by Dy (rad) (absolute value), the muscle length changes as much as DLmus (absolute value), such as: DLmus ¼ MA Dy
ð3Þ
The definition of the variables is consistent with Fig. 1. An example of the complete description on this issue can be found in Feynman et al. (1965). This formula suggests that when the moment arm (MA) is longer, the range of joint angle in which active joint moment is developed by the muscle (Dy) is narrower, assuming that the maximal lengthening/shortening range of the muscle is constant (DLmus ). (Note: as the maximal lengthening/shortening range of a sarcomere is approximately 755% (Allinger et al., 1996), similar
Fig. 2. The location of the insertion of the m. soleus was shifted, toward the dorsal direction by 1 cm (Dins =+1 cm) or toward the ventral direction by 1 cm (Dins =1 cm). The manipulation of Dins =+1 cm had an effect of lengthening the moment arm, whereas the manipulation of Dins =1 cm had an effect of shortening the moment arm. The circles represent the origin of the ‘‘tibia’’ coordinate system and the ‘‘calcaneus’’ coordinate system (Delp, 1990). F–L: force–length relation. F–V: force–velocity relation.
lengthening/shortening range is assumed for muscle fibers as well, relative to the optimal fiber length.) From Eq. (3), the relation between the muscle lengthening/shortening velocity and the joint angular velocity is expressed as ’ ð4Þ L’ mus ¼ MA y; where L’ mus represents the muscle lengthening/shortening velocity (absolute value) and y’ represents the joint angular velocity (absolute value). Again, the definition of the variables is consistent with Fig. 1. The relation expressed in Eq. (4) suggests that the muscle lengthening/shortening velocity is linearly related to the joint angular velocity by the factor of moment arm length (MA). From this, it is derived that when the moment arm is longer, larger muscle lengthening/shortening velocity is required in order to achieve a certain angular velocity of the joint. As Hill (1938) reported, an increase in the shortening velocity results in a non-linear, rapid drop of the force development ability of the muscle (force–velocity relation) (Fig. 2). Therefore, it is important to take into consideration the relation expressed in Eq. (4) on the force development ability of the muscle, when investigating the effects of the moment arm length on the kinetic outputs around the joint (joint moment development, joint power and work outputs). The purpose of this study was to investigate the effects of the moment arm length on the kinetic outputs of a musculoskeletal system using the
ARTICLE IN PRESS A. Nagano, T. Komura / Journal of Biomechanics 36 (2003) 1675–1681
methodology of computer simulation (Anderson and Pandy, 1993; Bobbert, 2001; van den Bogert et al., 1998; Zajac et al., 1984).
2. Methods A computer simulation model of the human ankle musculoskeletal system, which consisted of two rigid bodies connected with a frictionless hinge joint and a Hill-type musculotendon (Hill, 1938), was constructed (Fig. 2). The musculotendon model consisted of two elements, i.e., a contractile element (CE) and a series elastic element (SEE). The effect of pennation angle (apen ) was also considered. All the model development procedures, as well as the numerical integration of ordinary differential equations, were performed using MATLAB (The MathWorks, Inc., Natick, MA, USA). The mathematical representation of the properties of the Hill-type musculotendon (e.g., force–length and force– velocity relations) (Hill, 1938) was adopted from Nagano and Gerritsen (2001). The ankle joint and the musculus soleus (m. soleus) were modeled, as the muscle path of the m. soleus can be represented well with two points (an origin on the tibia and an insertion on the calcaneus) (Delp, 1990). Parameter values of the musculotendon, i.e., the maximal isometric contraction force (Fmax ), the optimal length (LCEopt ) and the pennation angle of the CE, the slack length of the SEE (Lslack ), and the coordinates of the origin and insertion, were adopted from Delp (1990). This resulted in the following parameter values: Fmax ¼ 2839 N, LCEopt ¼ 0:030 m, apen ¼ 25 ; Lslack ¼ 0:268 m. The XYZ coordinates of the origin were (0.0024, 0.1533, 0.0071) on the tibia coordinate system, and the XYZ coordinates of the insertion were (0.0044, 0.0310, 0.0053) on the calcaneus coordinate system. The location of the ankle joint center relative to the tibia coordinate system was (0.000, 0.430, 0.000), and the location of the ankle joint center relative to the calcaneus coordinate system was (0.0488, 0.0412, 0.008) (Delp, 1990) (Fig. 2). It was assumed that dorsiflexion/ plantarflexion motions occur around the Z-axis. With the configuration described in Figs. 1 and 2, a positive angle represents dorsiflexion and a negative angle represents plantar flexion. The muscle (m. soleus) was maximally activated, as the ankle joint was rotated at constant angular velocities. This corresponds to isokinetic exercises on a muscle testing machine (Holmback et al., 1999; Kawakami et al., 2002). Kinetic outputs around the joint (joint moment development, joint power and work outputs) were evaluated for a variation of angular velocities, i.e., from 210 /s (plantarflexion) to +210 /s (dorsiflexion). Plantarflexion motions were initiated from the ankle joint angle=0 (neutral position),
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whereas dorsiflexion motions were initiated from the ankle joint angle=60 (60 plantarflexed), as the range of 60 to 0 covered the maximal range of active force development of this muscle with the parameter values described before. The isometric force and moment development capabilities of the muscle in this range were also evaluated. Thereafter, the same evaluation was repeated, with the X coordinate of the muscle insertion shifted as much as 1 cm, toward the dorsal direction or the ventral direction (Fig. 2). With the configuration described in Fig. 2, shifting the insertion toward the dorsal direction (Dins ¼ þ1 cm) had an effect of increasing the moment arm length, whereas shifting the insertion toward the ventral direction (Dins ¼ 1 cm) had an effect of decreasing the moment arm length.
3. Results Smaller muscle force was developed with longer moment arm in the plantar flexion motions (concentric action) (Fig. 3A, Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater muscle force was developed with longer moment arm (Fig. 3B, Table 1). When the magnitude of the angular velocity (oankle ) was relatively small (30 /s to 90 /s), greater plantarflexion moment was developed with longer moment arm, in the plantar flexion motions (concentric action). However, when the magnitude of oankle was relatively large (120 /s to 210 /s), smaller plantarflexion moment was developed with longer moment arm (Fig. 4A, Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater plantarflexion moment was developed with longer moment arm for all angular velocities (Fig. 4B, Table 1). When the magnitude of oankle was relatively small (–30 /s to –90 /s), greater positive joint power output was observed with longer moment arm, in the plantarflexion motions (concentric action). However, when the magnitude of oankle was relatively large (120 /s to 210 /s), smaller positive joint power output was observed with longer moment arm (Fig. 5A, Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater negative joint power output was observed with longer moment arm for all angular velocities (Fig. 5B, Table 1). Smaller positive joint work output was observed with longer moment arm in the plantar flexion motions (concentric action) for all angular velocities (Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater negative joint work output was observed with longer moment arm for all angular velocities (Table 1).
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Fmus (N) 2500
∆ins = +1cm ∆ins = 0 cm ∆ins = -1 cm
2000 1500 1000 500 0 -60
θ ankle
-40
) (deg
-20
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-210
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3000 2000 1000 0 -60
θ ankle
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0
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60
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Fig. 3. Profiles of the muscle force (Fmus ) developed by the m. soleus, as a function of the ankle joint angle (yankle ) and the ankle joint angular velocity (oankle ) (isometric=0 /s). Circle, triangle and square stand for Dins =+1 cm, Dins =0 cm and Dins =1 cm, respectively (Fig. 2). (A) concentric action and (B) eccentric action.
4. Discussion In a musculoskeletal system, forces developed by skeletal muscles are transmitted to the external environment through skeletal structures. Formulae (1) and (2) suggest that a greater joint moment development (thus joint power and work outputs) is associated with a longer moment arm, assuming that the magnitude of the muscle force is constant. However, in biomechanical musculoskeletal systems, the relation expressed in Eqs. (1) and (2) has to be considered together with the force–length and force–velocity relations of the muscle (Hill, 1938). The importance of these relations for biomechanical systems has been highlighted in many preceding studies (for example, van Zuylen et al., 1988). The purpose of this study was to investigate the effects
of the moment arm length on the kinetic outputs around the joint. Smaller muscle force was developed with longer moment arm in the concentric action (Fig. 3A, Table 1). When the moment arm is longer, larger muscle shortening velocity is required to achieve a certain joint angular velocity, as expressed in Eq. (4). Because of the force–velocity relation reported by Hill (1938), smaller muscle force development was observed when the moment arm was longer. On the other hand, in the eccentric action, Eq. (4) suggests that larger lengthening velocity is imposed on the muscle with longer moment arm. Hill’s force–velocity relation suggests the enhancement of the muscle force development ability with the increase of the muscle lengthening velocity. This is the reason why greater muscle force was developed with longer moment arm in the eccentric action (Fig. 3B, Table 1). In the concentric action, greater plantarflexion moment was developed with longer moment arm, when the magnitude of the joint angular velocity (oankle ) was relatively small (Fig. 4A, Table 1). Although the muscle force development was smaller with longer moment arm (Fig. 3A, Table 1), longer moment arm multiplied to the muscle force made up for that difference. However, when the magnitude of oankle was relatively large, further decrease in the muscle force development was observed (Fig. 3A, Table 1). This is why smaller plantarflexion moment was developed with longer moment arm in this joint angular velocity region (Fig. 4A, Table 1). In the eccentric action, greater plantarflexion moment was consistently developed with longer moment arm (Fig. 4B, Table 1). This is because both moment arm length and muscle force (Fig. 3B, Table 1) were greater with longer moment arm in the eccentric action. Similar characteristics were observed for the joint power output. In the concentric action, the (positive) power output was greater with longer moment arm when the magnitude of oankle was relatively small, and the (positive) power output was smaller with longer moment arm when the magnitude of oankle was relatively large (Fig. 5A, Table 1). In the eccentric action, the power output was negative in all cases, as the m. soleus exerted ankle plantar flexion moment while the ankle joint underwent dorsiflexion. Greater (negative) power output was observed with longer moment arm in all cases (Fig. 5B, Table 1). The (positive) joint work output in the concentric action was consistently smaller with longer moment arm (Table 1). Joint work output is determined as joint power output integrated over time. The characteristics of the power output (Fig. 5A, Table 1) and the characteristics of the range of active joint moment development (narrower range with longer moment arm and wider range with shorter moment arm, as derived
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Table 1 Results of the kinetic analysis oankle ( /s) Dins =+1 cm 210 180 150 120 90 60 30 0 30 60 90 120 150 180 210
282 439 634 882 1206 1637 2252 2839 4009 4252 4371 4435 4485 4530 4571
oankle ( /s) Dins =+1 cm 210 180 150 120 90 60 30 0 30 60 90 120 150 180 210
69 92 110 122 124 111 75 — 129 273 420 568 717 865 1027
Peak Fmus ðNÞ Dins =0 cm 416 577 783 1035 1351 1771 2349 2839 3956 4214 4332 4406 4448 4483 4539 Peak Pankle ðW Þ Dins =0 cm 90 107 119 125 122 105 69 — 110 234 360 488 617 747 882
Peak Mankle ðN mÞ Dins =0 cm
Dins =1 cm
Dins =+1 cm
565 736 939 1188 1507 1913 2451 2839 3890 4149 4279 4361 4415 4442 4486
19 29 42 58 79 106 144 178 246 260 267 271 274 275 280
25 34 46 60 78 101 131 155 211 223 229 233 236 238 241
29 38 48 60 75 94 118 133 177 188 195 197 200 201 203
Dins =1 cm
Dins =+1 cm
Wankle ðJÞ Dins =0 cm
Dins =1 cm
106 118 125 125 118 98 62 — 93 197 306 413 523 633 743
4 7 10 15 21 28 38 — 84 90 93 95 96 99 100
7 9 13 18 23 30 40 — 83 88 92 93 95 97 98
Dins =1 cm
9 12 16 20 26 33 42 — 81 87 90 92 94 95 97
Peak muscle force of the m. soleus (Fmus ; Fig. 3), peak joint moment around the ankle joint (Mankle, Fig. 4), peak joint power output around the ankle joint (Pankle, Fig. 5) and work output around the ankle joint (Wankle), as a function of the ankle joint angular velocity (oankle ) (isometric=0 /s). Since the coordinate system was defined as shown in Fig. 2, dorsiflexion is positive and plantarflexion is negative. Insertion of the m. soleus was shifted by 1 cm toward the dorsal direction (Dins ¼ þ1 cm) and the ventral direction (Dins ¼ 1 cm) (Fig. 2).
from Eq. (3)) determined this profile of the (positive) work output. In the eccentric action, consistently greater (negative) work output was observed with longer moment arm. This result agrees well with the finding that the (negative) power output was consistently greater with longer moment arm in the eccentric action (Fig. 5B, Table 1). The results of this study have important implications for a category of experimental studies, performed using human subjects in order to investigate the muscle force– length and force–velocity relations in vivo (de Koning et al., 1985; Funato et al., 1997; Kanehisa and Fukunaga, 1999; Uchiyama and Akazawa, 1999; van Bolhuis and Gielen, 1997). In these studies, the joint moment–joint angle and/or joint moment–joint angular velocity profiles have been measured in order to discuss the force–length and/or force–velocity relations of
muscles. It is suggested that individual variations in the moment arm length and their effects on the kinetic outputs around the joint should be taken into consideration in this type of studies. Bobbert and van Ingen Schenau (1988) studied human squat jumping motion, in which the peak plantar flexion angular velocity was calculated to be approximately 900 /s. Winter (1990) reported a set of kinematic data of human walking, in which the peak plantar flexion angular velocity was calculated to be approximately 250 /s. Jacobs et al. (1996) studied human sprinting push-off, in which the peak plantar flexion angular velocity was calculated to be approximately 950 /s. Therefore it is suggested that the non-linear effect of the moment arm length on the joint kinetic outputs investigated in this study may come into play in many cases of daily human locomotion.
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-Mankle (N (Nm) m)
Pank le (W) 120
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-120
-Pankle (W) 1000
250
∆ins = +1cm ∆ins = 0 cm ∆ins = -1 cm
200 150
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800 600
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50
200
0
0
-60
-60
-20
0
(B)
0
30
60
90
120
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180
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ωankle (deg / s)
Fig. 4. Profiles of the ankle joint moment development (Mankle ), as a function of the ankle joint angle (yankle ) and the ankle joint angular velocity (oankle ) (isometric=0 /s). Circle, triangle and square stand for Dins =+1 cm, Dins =0 cm and Dins = cm, respectively (Fig. 2). (A) concentric action and (B) eccentric action. As the coordinate system was defined as shown in Fig. 2, plantarflexion moment is negative (dorsiflexion moment is positive).
The X (dorsal–ventral component) coordinate of the muscle insertion was shifted by 71 cm in this study (Fig. 2). This amount of manipulation, i.e., 1 cm, is within the range of values found in preceding studies. Rugg et al. (1990) performed a detailed examination of the moment arm length of the Achilles tendon determined using magnetic resonance imaging (MRI), and reported that up to 12% of variation is to be expected associated with that procedure. Similarly, Klein et al. (1996) examined the ankle joint musculature of 10 human cadaver legs, and reported substantial individual variations in the Achilles tendon moment arm length (approximately 4.3–6.3 cm). In reality, the ankle joint is a complex joint that performs three-dimensional motions
-40
) (deg
) (deg
θ ankle
θ ankle
-40
-20 0
(B)
0
30
60
90
120
150
ωankle (deg / s)
Fig. 5. Profiles of the ankle joint power output (Pankle), as a function of the ankle joint (yankle ) and the ankle joint angular velocity (oankle ). The power output was equal to zero during the isometric exercise (i.e., no motion; oankle ¼ 0 /s). Circle, triangle and square stand for Dins =+1 cm, Dins =0 cm and Dins =1 cm, respectively (Fig. 2). (A) concentric action and (B) eccentric action. Power output during the concentric action is positive and power output during the eccentric action is negative.
(Leardini et al., 1999; van den Bogert et al., 1994). A simplification was adopted in this study, i.e., the ankle joint was modeled as a frictionless hinge joint with its axis parallel to the Z-axis (Fig. 2). This assumption was relevant, as by far the most dominant function of the m. soleus is ankle plantar flexion (Delp et al., 1990). With this simplification, clear messages were obtained: it is important to consider the effects of the moment arm length on the joint kinetic outputs, together with the force–velocity relation of the muscle. Depending on the magnitude of the joint angular velocity, this effect results in a smaller joint moment development with a longer moment arm. Basically the
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same behavior will be observed in any other Hill-type skeletal muscles as well, as the findings of this study are derived from the force–velocity relation of Hill-type muscles.
Acknowledgements This study was partially supported by the Integrated Rehabilitation Engineering Program, National Institute of Disabled and Rehabilitation Research. Akinori Nagano would like to thank Professor J.J. Collins (Center for BioDynamics) for his support.
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Journal of Biomechanics 36 (2003) 1675–1681
Longer moment arm results in smaller joint moment development, power and work outputs in fast motions Akinori Naganoa,b,*, Taku Komurac b
a Center for BioDynamics, Boston University, Boston, MA, USA Computer and Information Division, Advanced Computing Center, RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan c Department of Computer Engineering & Information Technology, City University of Hong Kong, Kowloon, Hong Kong
Accepted 7 April 2003
Abstract Effects of moment arm length on kinetic outputs of a musculoskeletal system (muscle force development, joint moment development, joint power output and joint work output) were evaluated using computer simulation. A skeletal system of the human ankle joint was constructed: a lower leg segment and a foot segment were connected with a hinge joint. A Hill-type model of the musculus soleus (m. soleus), consisting of a contractile element and a series elastic element, was attached to the skeletal system. The model of the m. soleus was maximally activated, while the ankle joint was plantarflexed/dorsiflexed at a variation of constant angular velocities, simulating isokinetic exercises on a muscle testing machine. Profiles of the kinetic outputs (muscle force development, joint moment development, joint power output and joint work output) were obtained. Thereafter, the location of the insertion of the m. soleus was shifted toward the dorsal/ventral direction by 1 cm, which had an effect of lengthening/shortening the moment arm length, respectively. The kinetic outputs of the musculoskeletal system during the simulated isokinetic exercises were evaluated with these longer/shorter moment arm lengths. It was found that longer moment arm resulted in smaller joint moment development, smaller joint power output and smaller joint work output in the larger plantarflexion angular velocity region (>120 / s). This is because larger muscle shortening velocity was required with longer moment arm to achieve a certain joint angular velocity. Larger muscle shortening velocity resulted in smaller muscle force development because of the force–velocity relation of the muscle. It was suggested that this phenomenon should be taken into consideration when investigating the joint moment–joint angle and/or joint moment–joint angular velocity characteristics of experimental data. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Modeling; Simulation; Musculoskeletal system; Muscle force
1. Introduction In a musculoskeletal system, forces developed by skeletal muscles produce motions of the skeletal system (Delp et al., 1990; Yamaguchi, 2001). Muscles are attached to bones at a distance from the joint center (center of rotation), as shown in Fig. 1. Therefore linear tensions developed by muscles are transformed into rotational joint moments. The relation between the muscle force development and the joint moment development is described as ~ joint ¼ ~ ~mus ; M rjc-ins F
ð1Þ
*Corresponding author. Computer and Information Division, Advanced Computing Center, RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan. Tel.: +81-48-467-8345; fax: +81-48-467-4389. E-mail address: [email protected] (A. Nagano).
~ joint is the joint moment produced by the muscle where M force, ~ rjc-ins is the vector from the joint center to the muscle insertion, and F~mus is the muscle force vector. The distance noted as MA in Fig. 1 is the moment arm length of the muscle force vector (F~mus ). Assuming that the joint motion occurs in a two-dimensional plane, the relation expressed in Eq. (1) can be simplified into a scalar formula Mjoint ¼ MA Fmus
ð2Þ
where Mjoint is the magnitude of the joint moment, Fmus is the magnitude of the muscle force, and MA is the moment arm length. In human and animal musculoskeletal systems, almost all forces developed by skeletal muscles are transmitted to the external environment in the form of joint moments (Lieber, 1992; Nigg and Herzog, 1994;
0021-9290/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0021-9290(03)00171-4
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M joint = MA . Fmus mu
Lower Leg
Lower Leg Fmus L mus
Fmus
θ
MA
Foot
Mjoint
Foot
Fig. 1. Joint moment (Mjoint ) is produced around a joint, as a function of MA (moment arm length) and Fmus (muscle force).
Winter, 1990). Therefore, it is crucially important to investigate the mechanical structure of joints in terms of moment arm length. Many researchers have reported numerous research results on this issue to date (An et al., 1983; Fowler et al., 2001; Klein et al., 1996; Spoor et al., 1990; Spoor and van Leeuwen, 1992; van Eijden et al., 1997; van Spronsen et al., 1997). The relation described in Eq. (2) may yield a statement such as ‘‘with a longer moment arm, a greater joint moment is (always) developed’’, which is usually true for typical mechanical systems. However, this statement is not always true for biomechanical musculoskeletal systems, especially when the musculoskeletal system is undergoing movements. This is because the force development ability of a muscle is greatly affected by its length and rate of length change, i.e., A.V. Hill’s force–length and force–velocity relations (Hill, 1938) (Fig. 2). In preceding studies, many researchers have highlighted the importance of these relations when considering biomechanical systems (for example, van Zuylen et al., 1988). Generally, when the joint angle changes by Dy (rad) (absolute value), the muscle length changes as much as DLmus (absolute value), such as: DLmus ¼ MA Dy
ð3Þ
The definition of the variables is consistent with Fig. 1. An example of the complete description on this issue can be found in Feynman et al. (1965). This formula suggests that when the moment arm (MA) is longer, the range of joint angle in which active joint moment is developed by the muscle (Dy) is narrower, assuming that the maximal lengthening/shortening range of the muscle is constant (DLmus ). (Note: as the maximal lengthening/shortening range of a sarcomere is approximately 755% (Allinger et al., 1996), similar
Fig. 2. The location of the insertion of the m. soleus was shifted, toward the dorsal direction by 1 cm (Dins =+1 cm) or toward the ventral direction by 1 cm (Dins =1 cm). The manipulation of Dins =+1 cm had an effect of lengthening the moment arm, whereas the manipulation of Dins =1 cm had an effect of shortening the moment arm. The circles represent the origin of the ‘‘tibia’’ coordinate system and the ‘‘calcaneus’’ coordinate system (Delp, 1990). F–L: force–length relation. F–V: force–velocity relation.
lengthening/shortening range is assumed for muscle fibers as well, relative to the optimal fiber length.) From Eq. (3), the relation between the muscle lengthening/shortening velocity and the joint angular velocity is expressed as ’ ð4Þ L’ mus ¼ MA y; where L’ mus represents the muscle lengthening/shortening velocity (absolute value) and y’ represents the joint angular velocity (absolute value). Again, the definition of the variables is consistent with Fig. 1. The relation expressed in Eq. (4) suggests that the muscle lengthening/shortening velocity is linearly related to the joint angular velocity by the factor of moment arm length (MA). From this, it is derived that when the moment arm is longer, larger muscle lengthening/shortening velocity is required in order to achieve a certain angular velocity of the joint. As Hill (1938) reported, an increase in the shortening velocity results in a non-linear, rapid drop of the force development ability of the muscle (force–velocity relation) (Fig. 2). Therefore, it is important to take into consideration the relation expressed in Eq. (4) on the force development ability of the muscle, when investigating the effects of the moment arm length on the kinetic outputs around the joint (joint moment development, joint power and work outputs). The purpose of this study was to investigate the effects of the moment arm length on the kinetic outputs of a musculoskeletal system using the
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methodology of computer simulation (Anderson and Pandy, 1993; Bobbert, 2001; van den Bogert et al., 1998; Zajac et al., 1984).
2. Methods A computer simulation model of the human ankle musculoskeletal system, which consisted of two rigid bodies connected with a frictionless hinge joint and a Hill-type musculotendon (Hill, 1938), was constructed (Fig. 2). The musculotendon model consisted of two elements, i.e., a contractile element (CE) and a series elastic element (SEE). The effect of pennation angle (apen ) was also considered. All the model development procedures, as well as the numerical integration of ordinary differential equations, were performed using MATLAB (The MathWorks, Inc., Natick, MA, USA). The mathematical representation of the properties of the Hill-type musculotendon (e.g., force–length and force– velocity relations) (Hill, 1938) was adopted from Nagano and Gerritsen (2001). The ankle joint and the musculus soleus (m. soleus) were modeled, as the muscle path of the m. soleus can be represented well with two points (an origin on the tibia and an insertion on the calcaneus) (Delp, 1990). Parameter values of the musculotendon, i.e., the maximal isometric contraction force (Fmax ), the optimal length (LCEopt ) and the pennation angle of the CE, the slack length of the SEE (Lslack ), and the coordinates of the origin and insertion, were adopted from Delp (1990). This resulted in the following parameter values: Fmax ¼ 2839 N, LCEopt ¼ 0:030 m, apen ¼ 25 ; Lslack ¼ 0:268 m. The XYZ coordinates of the origin were (0.0024, 0.1533, 0.0071) on the tibia coordinate system, and the XYZ coordinates of the insertion were (0.0044, 0.0310, 0.0053) on the calcaneus coordinate system. The location of the ankle joint center relative to the tibia coordinate system was (0.000, 0.430, 0.000), and the location of the ankle joint center relative to the calcaneus coordinate system was (0.0488, 0.0412, 0.008) (Delp, 1990) (Fig. 2). It was assumed that dorsiflexion/ plantarflexion motions occur around the Z-axis. With the configuration described in Figs. 1 and 2, a positive angle represents dorsiflexion and a negative angle represents plantar flexion. The muscle (m. soleus) was maximally activated, as the ankle joint was rotated at constant angular velocities. This corresponds to isokinetic exercises on a muscle testing machine (Holmback et al., 1999; Kawakami et al., 2002). Kinetic outputs around the joint (joint moment development, joint power and work outputs) were evaluated for a variation of angular velocities, i.e., from 210 /s (plantarflexion) to +210 /s (dorsiflexion). Plantarflexion motions were initiated from the ankle joint angle=0 (neutral position),
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whereas dorsiflexion motions were initiated from the ankle joint angle=60 (60 plantarflexed), as the range of 60 to 0 covered the maximal range of active force development of this muscle with the parameter values described before. The isometric force and moment development capabilities of the muscle in this range were also evaluated. Thereafter, the same evaluation was repeated, with the X coordinate of the muscle insertion shifted as much as 1 cm, toward the dorsal direction or the ventral direction (Fig. 2). With the configuration described in Fig. 2, shifting the insertion toward the dorsal direction (Dins ¼ þ1 cm) had an effect of increasing the moment arm length, whereas shifting the insertion toward the ventral direction (Dins ¼ 1 cm) had an effect of decreasing the moment arm length.
3. Results Smaller muscle force was developed with longer moment arm in the plantar flexion motions (concentric action) (Fig. 3A, Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater muscle force was developed with longer moment arm (Fig. 3B, Table 1). When the magnitude of the angular velocity (oankle ) was relatively small (30 /s to 90 /s), greater plantarflexion moment was developed with longer moment arm, in the plantar flexion motions (concentric action). However, when the magnitude of oankle was relatively large (120 /s to 210 /s), smaller plantarflexion moment was developed with longer moment arm (Fig. 4A, Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater plantarflexion moment was developed with longer moment arm for all angular velocities (Fig. 4B, Table 1). When the magnitude of oankle was relatively small (–30 /s to –90 /s), greater positive joint power output was observed with longer moment arm, in the plantarflexion motions (concentric action). However, when the magnitude of oankle was relatively large (120 /s to 210 /s), smaller positive joint power output was observed with longer moment arm (Fig. 5A, Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater negative joint power output was observed with longer moment arm for all angular velocities (Fig. 5B, Table 1). Smaller positive joint work output was observed with longer moment arm in the plantar flexion motions (concentric action) for all angular velocities (Table 1). On the other hand, in the dorsiflexion motions (eccentric action), greater negative joint work output was observed with longer moment arm for all angular velocities (Table 1).
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Fmus (N) 2500
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3000 2000 1000 0 -60
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60
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Fig. 3. Profiles of the muscle force (Fmus ) developed by the m. soleus, as a function of the ankle joint angle (yankle ) and the ankle joint angular velocity (oankle ) (isometric=0 /s). Circle, triangle and square stand for Dins =+1 cm, Dins =0 cm and Dins =1 cm, respectively (Fig. 2). (A) concentric action and (B) eccentric action.
4. Discussion In a musculoskeletal system, forces developed by skeletal muscles are transmitted to the external environment through skeletal structures. Formulae (1) and (2) suggest that a greater joint moment development (thus joint power and work outputs) is associated with a longer moment arm, assuming that the magnitude of the muscle force is constant. However, in biomechanical musculoskeletal systems, the relation expressed in Eqs. (1) and (2) has to be considered together with the force–length and force–velocity relations of the muscle (Hill, 1938). The importance of these relations for biomechanical systems has been highlighted in many preceding studies (for example, van Zuylen et al., 1988). The purpose of this study was to investigate the effects
of the moment arm length on the kinetic outputs around the joint. Smaller muscle force was developed with longer moment arm in the concentric action (Fig. 3A, Table 1). When the moment arm is longer, larger muscle shortening velocity is required to achieve a certain joint angular velocity, as expressed in Eq. (4). Because of the force–velocity relation reported by Hill (1938), smaller muscle force development was observed when the moment arm was longer. On the other hand, in the eccentric action, Eq. (4) suggests that larger lengthening velocity is imposed on the muscle with longer moment arm. Hill’s force–velocity relation suggests the enhancement of the muscle force development ability with the increase of the muscle lengthening velocity. This is the reason why greater muscle force was developed with longer moment arm in the eccentric action (Fig. 3B, Table 1). In the concentric action, greater plantarflexion moment was developed with longer moment arm, when the magnitude of the joint angular velocity (oankle ) was relatively small (Fig. 4A, Table 1). Although the muscle force development was smaller with longer moment arm (Fig. 3A, Table 1), longer moment arm multiplied to the muscle force made up for that difference. However, when the magnitude of oankle was relatively large, further decrease in the muscle force development was observed (Fig. 3A, Table 1). This is why smaller plantarflexion moment was developed with longer moment arm in this joint angular velocity region (Fig. 4A, Table 1). In the eccentric action, greater plantarflexion moment was consistently developed with longer moment arm (Fig. 4B, Table 1). This is because both moment arm length and muscle force (Fig. 3B, Table 1) were greater with longer moment arm in the eccentric action. Similar characteristics were observed for the joint power output. In the concentric action, the (positive) power output was greater with longer moment arm when the magnitude of oankle was relatively small, and the (positive) power output was smaller with longer moment arm when the magnitude of oankle was relatively large (Fig. 5A, Table 1). In the eccentric action, the power output was negative in all cases, as the m. soleus exerted ankle plantar flexion moment while the ankle joint underwent dorsiflexion. Greater (negative) power output was observed with longer moment arm in all cases (Fig. 5B, Table 1). The (positive) joint work output in the concentric action was consistently smaller with longer moment arm (Table 1). Joint work output is determined as joint power output integrated over time. The characteristics of the power output (Fig. 5A, Table 1) and the characteristics of the range of active joint moment development (narrower range with longer moment arm and wider range with shorter moment arm, as derived
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Table 1 Results of the kinetic analysis oankle ( /s) Dins =+1 cm 210 180 150 120 90 60 30 0 30 60 90 120 150 180 210
282 439 634 882 1206 1637 2252 2839 4009 4252 4371 4435 4485 4530 4571
oankle ( /s) Dins =+1 cm 210 180 150 120 90 60 30 0 30 60 90 120 150 180 210
69 92 110 122 124 111 75 — 129 273 420 568 717 865 1027
Peak Fmus ðNÞ Dins =0 cm 416 577 783 1035 1351 1771 2349 2839 3956 4214 4332 4406 4448 4483 4539 Peak Pankle ðW Þ Dins =0 cm 90 107 119 125 122 105 69 — 110 234 360 488 617 747 882
Peak Mankle ðN mÞ Dins =0 cm
Dins =1 cm
Dins =+1 cm
565 736 939 1188 1507 1913 2451 2839 3890 4149 4279 4361 4415 4442 4486
19 29 42 58 79 106 144 178 246 260 267 271 274 275 280
25 34 46 60 78 101 131 155 211 223 229 233 236 238 241
29 38 48 60 75 94 118 133 177 188 195 197 200 201 203
Dins =1 cm
Dins =+1 cm
Wankle ðJÞ Dins =0 cm
Dins =1 cm
106 118 125 125 118 98 62 — 93 197 306 413 523 633 743
4 7 10 15 21 28 38 — 84 90 93 95 96 99 100
7 9 13 18 23 30 40 — 83 88 92 93 95 97 98
Dins =1 cm
9 12 16 20 26 33 42 — 81 87 90 92 94 95 97
Peak muscle force of the m. soleus (Fmus ; Fig. 3), peak joint moment around the ankle joint (Mankle, Fig. 4), peak joint power output around the ankle joint (Pankle, Fig. 5) and work output around the ankle joint (Wankle), as a function of the ankle joint angular velocity (oankle ) (isometric=0 /s). Since the coordinate system was defined as shown in Fig. 2, dorsiflexion is positive and plantarflexion is negative. Insertion of the m. soleus was shifted by 1 cm toward the dorsal direction (Dins ¼ þ1 cm) and the ventral direction (Dins ¼ 1 cm) (Fig. 2).
from Eq. (3)) determined this profile of the (positive) work output. In the eccentric action, consistently greater (negative) work output was observed with longer moment arm. This result agrees well with the finding that the (negative) power output was consistently greater with longer moment arm in the eccentric action (Fig. 5B, Table 1). The results of this study have important implications for a category of experimental studies, performed using human subjects in order to investigate the muscle force– length and force–velocity relations in vivo (de Koning et al., 1985; Funato et al., 1997; Kanehisa and Fukunaga, 1999; Uchiyama and Akazawa, 1999; van Bolhuis and Gielen, 1997). In these studies, the joint moment–joint angle and/or joint moment–joint angular velocity profiles have been measured in order to discuss the force–length and/or force–velocity relations of
muscles. It is suggested that individual variations in the moment arm length and their effects on the kinetic outputs around the joint should be taken into consideration in this type of studies. Bobbert and van Ingen Schenau (1988) studied human squat jumping motion, in which the peak plantar flexion angular velocity was calculated to be approximately 900 /s. Winter (1990) reported a set of kinematic data of human walking, in which the peak plantar flexion angular velocity was calculated to be approximately 250 /s. Jacobs et al. (1996) studied human sprinting push-off, in which the peak plantar flexion angular velocity was calculated to be approximately 950 /s. Therefore it is suggested that the non-linear effect of the moment arm length on the joint kinetic outputs investigated in this study may come into play in many cases of daily human locomotion.
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-Mankle (N (Nm) m)
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0
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Fig. 4. Profiles of the ankle joint moment development (Mankle ), as a function of the ankle joint angle (yankle ) and the ankle joint angular velocity (oankle ) (isometric=0 /s). Circle, triangle and square stand for Dins =+1 cm, Dins =0 cm and Dins = cm, respectively (Fig. 2). (A) concentric action and (B) eccentric action. As the coordinate system was defined as shown in Fig. 2, plantarflexion moment is negative (dorsiflexion moment is positive).
The X (dorsal–ventral component) coordinate of the muscle insertion was shifted by 71 cm in this study (Fig. 2). This amount of manipulation, i.e., 1 cm, is within the range of values found in preceding studies. Rugg et al. (1990) performed a detailed examination of the moment arm length of the Achilles tendon determined using magnetic resonance imaging (MRI), and reported that up to 12% of variation is to be expected associated with that procedure. Similarly, Klein et al. (1996) examined the ankle joint musculature of 10 human cadaver legs, and reported substantial individual variations in the Achilles tendon moment arm length (approximately 4.3–6.3 cm). In reality, the ankle joint is a complex joint that performs three-dimensional motions
-40
) (deg
) (deg
θ ankle
θ ankle
-40
-20 0
(B)
0
30
60
90
120
150
ωankle (deg / s)
Fig. 5. Profiles of the ankle joint power output (Pankle), as a function of the ankle joint (yankle ) and the ankle joint angular velocity (oankle ). The power output was equal to zero during the isometric exercise (i.e., no motion; oankle ¼ 0 /s). Circle, triangle and square stand for Dins =+1 cm, Dins =0 cm and Dins =1 cm, respectively (Fig. 2). (A) concentric action and (B) eccentric action. Power output during the concentric action is positive and power output during the eccentric action is negative.
(Leardini et al., 1999; van den Bogert et al., 1994). A simplification was adopted in this study, i.e., the ankle joint was modeled as a frictionless hinge joint with its axis parallel to the Z-axis (Fig. 2). This assumption was relevant, as by far the most dominant function of the m. soleus is ankle plantar flexion (Delp et al., 1990). With this simplification, clear messages were obtained: it is important to consider the effects of the moment arm length on the joint kinetic outputs, together with the force–velocity relation of the muscle. Depending on the magnitude of the joint angular velocity, this effect results in a smaller joint moment development with a longer moment arm. Basically the
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same behavior will be observed in any other Hill-type skeletal muscles as well, as the findings of this study are derived from the force–velocity relation of Hill-type muscles.
Acknowledgements This study was partially supported by the Integrated Rehabilitation Engineering Program, National Institute of Disabled and Rehabilitation Research. Akinori Nagano would like to thank Professor J.J. Collins (Center for BioDynamics) for his support.
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