Journal of Electromyography and Kinesiology 14 (2004) 197–203 www.elsevier.com/locate/jelekin

Effects of the length ratio between the contractile element and the series elastic element on an explosive muscular performance Akinori Nagano a,b,∗, Taku Komura c, Senshi Fukashiro d b

a Center for BioDynamics, Boston University, Boston, MA, USA Computer and Information Division, Advanced Computing Center, RIKEN, Hirosawa 2-1, Wako, Saitama-Ken 351-0198, Japan c Department of Computer Engineering and Information Technology, City University of Hong Kong, Kowloon, Hong Kong d Department of Life Sciences (Sports Sciences), University of Tokyo, Tokyo, Japan

Abstract Effects of the length ratio between the contractile element (CE) and the series elastic element (SEE) on the behavior of the muscle tendon complex were investigated during stretch-shortening cycles. A computer simulation model of the Hill-type muscle tendon complex was constructed. The proximal end of the CE was affixed to a point in the gravitational field, and a massless supporting object was affixed to the distal end of the SEE. A mass was held on the supporting object. Initially, the muscle tendon complex was fixed at a certain length, and the CE was activated at 100%. Through this process, the CE contracted as much as the SEE was stretched. Thereafter, the supporting object was released, which caused the muscle tendon complex to propel the mass upward, simulating a stretch-shortening cycle. The length ratio between the CE and the SEE, the size of the mass and the initial length of the CE were sequentially changed. As a result, it was found that a higher performance is obtained with a longer SEE when the mass is small, while with a shorter SEE when the mass is large.  2003 Elsevier Ltd. All rights reserved. Keywords: Computer simulation; Quick release; Stretch shortening

1. Introduction Dimensional properties of the muscle tendon complex have been of great interest in the field of biomechanics. They are for example, cross sectional area, fascicle length, tendon length, and so on. Generally, a large cross sectional area of the muscle tendon complex is related to a high tensile force development capability [4–6,18], and a long contractile element (CE) is related to a high shortening capability (excursion and shortening velocity) [1,9,15]. In the musculoskeletal system, muscles are attached to bones via tendons. Tendons are primarily composed of collagen tissues, which exhibit elastic properties [2,25]. Therefore, tendons have the capability to store elastic energy when they are stretched, and release the energy when they shorten, in what is called a “stretch-shortening cycle” [10,19,20].

Corresponding author. Tel.: +81-48-467-8354; fax: +81-48-4674389. E-mail address: [email protected] (A. Nagano). ∗

1050-6411/$ - see front matter  2003 Elsevier Ltd. All rights reserved. doi:10.1016/S1050-6411(03)00085-3

Behavior of the tendon during a stretch-shortening cycle is determined by its compliance and length. A higher compliance, as well as a longer length, results in more spring-like behavior of the tendon: a larger tendon length change is expected per unit force applied. It has been reported that compliance of the tendon is typically such that approximately 4% of strain is observed when the maximal isometric force is developed by the muscle fibers [7,12]. As for the length of the tendon, there is a large variation in this value between different muscles, both in terms of absolute length and relative length, i.e., relative to the muscle fiber length [11,33,35]. Delp [8], as well as Hoy et al. [16], reported sets of parameter values of the Hill-type [14] muscle tendon complex of the human lower extremity, which included the optimal length of the CE (LCEopt) and the unloaded slack length of the series elastic element (SEE) (Lslack). When investigating those data sets, a large variation (0– 12) is observed in the length ratio between Lslack and LCEopt (Lslack/LCEopt). Generally, it is observed that (A) muscles that span around the hip and the knee joints have smaller Lslack/LCEopt ratio compared to muscles that

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span around the ankle joint, and (B) monoarticular muscles have smaller Lslack/LCEopt ratio compared to biarticular muscles. As discussed in the previous paragraph, it is assumed that the length ratio, i.e., Lslack/LCEopt, has a major effect on the behavior of the muscle tendon complex in a stretch-shortening cycle. Therefore, it is important to quantitatively investigate how this length ratio is related to the behavior of the muscle tendon complex. Computer modeling and simulation is a useful approach to address this type of question. Specifically for this study, it is possible to manipulate only the length ratio Lslack/LCEopt of the muscle tendon complex, while keeping all other components unchanged. In this way, it is possible to derive a focused analysis and discussion regarding the effects of the length ratio Lslack/LCEopt on the behavior of the muscle tendon complex. A similar approach has been utilized by several researchers [22,[29]. The purpose of this study was to determine the effects of the variation of the length ratio Lslack/LCEopt on an explosive performance of the muscle tendon complex. A jumping-like explosive muscular activity was investigated.

2. Methods A Hill-type [14] computer simulation model of the muscle tendon complex was constructed (Fig. 1). The model consisted of three elements: i.e., a CE, an SEE

and a parallel elastic element (PEE). 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). Mathematical representation of the Hill-type CE, i.e., standard force–length and force–velocity relations [14], was adopted from Ref. [21]. Quadratic elastic property of the SEE was modeled such that an SEE strain of 4% is observed when the maximal isometric contraction force of the CE is applied to the SEE [7,12]. Mathematical representation of elastic properties of the PEE was derived according to Ref. [8]. In this representation, the PEE develops a passive elastic force when the length of the CE becomes larger than its optimal length. Parameter values of the muscle tendon complex, i.e., maximal isometric contraction force of the CE (Fmax), optimal length of the CE (LCEopt) and pennation angle of the CE (apen) were derived from Ref. [8]. Mean value of the 43 muscles investigated in Ref. [8] was calculated for each parameter. This procedure was performed in order to yield a reasonable combination of parameter values. This resulted in the following values: Fmax = 550 N, LCEopt = 0.1 m, apen = 7ⴰ. Delp [8] reported a large variation of the length ratio between Lslack and LCEopt (0–12). Therefore, a range of Lslack/LCEopt, i.e., between 0.5 and 16 (0.5, 1.0, 2.0, 4.0, 8.0 and 16.0), were investigated in this study. The proximal end of the CE was affixed to a point in the gravitational field. A supporting object was affixed to the distal end of the SEE. A mass was held on the supporting object (Fig. 1). A variation of sizes of the mass were used, such that the weight ranging between 0.05 and 0.40 of Fmax was imposed on the muscle tendon complex in the gravitational field: mass ⫽ Fmax / g ⫻ (mass factor)

(1)

where g is equal to the acceleration due to gravity (9.81 m/s2), and the mass factor ranged between 0.05 and 0.40. Initially, the length of the muscle tendon complex (L0MTC) was adjusted so that the initial length of the CE (L0CE) was set at a certain value relative to LCEopt: L0MTC ⫽ L0CE ⫹ Lslack

(2)

The length of the CE was set to an initial value in the range between 60% and 140% of LCEopt by adjusting the length of the muscle tendon complex according to the equation: L0CE ⫽ L0CE fac ⫻ LCEopt 0 CE fac

Fig. 1. The Hill-type muscle tendon complex model utilized in this study. The muscle tendon complex model consisted of three elements: a CE, an SEE and a PEE.

(3)

where L ranged between 0.6 and 1.4 (Tables 1 and 2). Thereafter, the CE was activated at 100%. Through this process, the CE shortened and the SEE was stretched, until the contractile force of the CE and the elastic force of the PEE balanced the elastic force

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Table 1 The gain in height experienced by the mass (in meters). Four different sizes of the mass (mass factor), three different initial lengths of the CE (L0CE = L0CE fac × LCEopt) and six different length ratios between the SEE and the CE (Lslack/LCEopt) were investigated. “Max. short” stands for the maximal shortening capability of the CE from the corresponding initial length. Values in bold represent the results in which the highest performance was observed for the combination of L0CE fac and mass factor L0CE fac

Max. short (m)

Lslack/LCeopt

(A) Mass factor = 0.5 0.6 1.0 1.4

0.015 0.055 0.094

0.5 0.013 0.070 0.122

1.0 0.014 0.073 0.130

2.0 0.017 0.086 0.137

4.0 0.017 0.127 0.158

8.0 0.011 0.166 0.257

16.0 – 0.132 0.390

(B) Mass factor = 0.10 0.6 1.0 1.4

0.015 0.055 0.094

0.5 0.012 0.059 0.109

1.0 0.012 0.060 0.109

2.0 0.011 0.064 0.109

4.0 0.010 0.076 0.112

8.0 0.003 0.091 0.148

16.0 – 0.069 0.207

(C) Mass factor = 0.20 0.6 1.0 1.4

0.015 0.055 0.094

0.5 0.009 0.051 0.093

1.0 0.008 0.051 0.093

2.0 0.006 0.051 0.093

4.0 0.003 0.052 0.092

8.0 – 0.053 0.098

16.0 – 0.036 0.113

(D) Mass factor = 0.40 0.6 1.0 1.4

0.015 0.055 0.094

0.5 0.002 0.043 0.083

1.0 – 0.042 0.082

2.0 – 0.040 0.080

4.0 – 0.036 0.076

8.0 – 0.029 0.070

16.0 – 0.003 0.063

developed by the SEE (Fig. 1). Note that the total length of the muscle tendon complex was kept constant throughout this process. Elastic energy was stored in the SEE and the PEE through this process as a function of the length of the SEE and the CE, respectively. After the system reached its equilibrium, the supporting object was released. This resulted in an explosive upward acceleration of the mass. This motion is equivalent to an activity of the muscle tendon complex called “stretch-shortening cycle” [10,19,20]. The gain in height achieved by the mass as a result of this muscular propulsion was evaluated. It should be noted that when enough momentum is provided to the mass before the maximal shortening capability of the CE (55% of LCEopt; [3]) is reached, the mass can jump up. Work outputs of the three elements of the muscle tendon complex, i.e., the CE, the SEE and the PEE, were evaluated.

3. Results There was a large variation in the gain in height achieved by the mass. The gain in height achieved by the mass ranged between 0 and 0.390 m. Generally, the gain in height achieved by the mass was larger with a smaller mass, and smaller with a larger mass (Table 1). The gain in height was larger when the simulation was initiated with a longer CE than with a shorter CE. When the size of the mass was relatively small (e.g., mass factor = 0.05), the gain in height was larger with a longer SEE (Lslack). On the other hand, when the size

of the mass was relatively large (e.g., mass factor = 0.4), the gain in height was larger with a shorter SEE (Table 1). An example of time course of the work output rate of the CE, the SEE and the PEE are shown in Fig. 2 (an example, with mass factor = 0.10, L0CE = 1.4 × LCEopt, Lslack / L0CEopt = 4.0). In most cases, the amount of the work performed by the SEE increased as Lslack increased relative to LCEopt. On the other hand, generally, the amount of the work output performed by the CE decreased as Lslack increased relative to LCEopt. The work output performed by the PEE also decreased as Lslack increased relative to LCEopt. 4. Discussion In typical Hill-type [14] representations of the muscle tendon complex, it is widely accepted that the cross sectional area of the CE represents its maximal isometric force [4,5,18], the length of the CE represents its shortening capability [1,9,15,23] and the length of the SEE represents its spring-like property [27,31,32]. In muscle tendon parameter data sets reported in preceding studies, it is observed that there is a large variation in the length ratio between the SEE and the CE (Lslack/LCEopt) among different muscles [8,16,35]. As the length of the SEE is a primary determinant of the spring-like behavior of the muscle tendon complex, it is important to understand the effects of the length variation of the SEE on the behavior of the muscle tendon complex.

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Table 2 Results of the work output analysis (in joules). Work outputs from the CE, the PEE and the SEE were analyzed for combinations of four different sizes of the mass (mass factor), three different initial lengths of the CE (L0CE = L0CE fac × LCEopt) and six different length ratios between the SEE and the CE (Lslack/LCE). Values in bold represent the results in which the highest performance was observed, for the combination of L0CE fac and mass factor L0CEfac

Lslack/LCEopt

(A) Mass factor = 0.05 0.6

1.0

1.4

CE SEE PEE CE SEE PEE CE SEE PEE

0.5 0.25 0.10 0.00 1.55 0.36 0.00 0.32 0.42 3.23

1.0 0.20 0.19 0.00 1.26 0.73 0.00 0.38 0.81 2.62

2.0 0.16 0.29 0.00 0.93 1.43 0.00 0.53 1.53 1.69

CE SEE PEE CE SEE PEE CE SEE PEE

0.5 0.56 0.10 0.00 2.89 0.37 0.00 2.29 0.42 3.23

1.0 0.46 0.17 0.00 2.59 0.73 0.00 2.50 0.82 2.62

CE SEE PEE CE SEE PEE CE SEE PEE

0.5 0.89 0.08 0.00 5.27 0.36 0.00 6.55 0.43 3.23

CE SEE PEE CE SEE PEE CE SEE PEE

0.5 0.31 0.01 0.00 9.12 0.29 0.00 14.64 0.36 3.23

4.0 0.11 0.35 0.00 0.87 2.61 0.00 0.87 2.86 0.60

8.0 0.05 0.26 0.00 0.76 3.79 0.00 1.27 5.79 0.01

16.0 – – – 0.40 3.22 0.00 1.25 9.46 0.00

2.0 0.34 0.28 0.00 2.06 1.43 0.00 2.75 1.54 1.69

4.0 0.21 0.33 0.00 1.56 2.61 0.00 2.70 2.86 0.60

8.0 0.05 0.13 0.00 1.19 3.79 0.00 2.31 5.79 0.01

16.0 – – – 0.59 3.22 0.00 1.89 9.46 0.00

1.0 0.74 0.13 0.00 4.89 0.72 0.00 6.75 0.82 2.62

2.0 0.51 0.19 0.00 4.18 1.42 0.00 6.95 1.54 1.69

4.0 0.17 0.12 0.00 3.10 2.61 0.00 6.69 2.86 0.60

8.0 – – – 2.04 3.79 0.00 4.93 5.80 0.01

16.0 – – – 0.93 3.04 0.00 2.99 9.47 0.00

1.0 – – – 8.59 0.59 0.00 14.69 0.68 2.62

2.0 – – – 7.57 1.16 0.00 14.59 1.28 1.69

4.0 – – – 5.80 2.15 0.00 13.72 2.42 0.60

8.0 – – – 3.37 3.01 0.00 10.27 5.11 0.01

16.0 – – – 0.24 0.42 0.00 5.08 8.74 0.00

(B) Mass factor = 0.10 0.6

(C) Mass factor = 0.10 0.6

1.0

1.4

(D) Mass factor = 0.10 0.6

1.0

1.4

As a result of this simulation study, a large variation was observed in the gain in height achieved by the mass (Table 1). Generally, the gain in height was larger when a smaller mass was used, and smaller when a larger mass was used, for obvious reasons. In several cases, when the initial length of the CE (L0CE) was set at a too short length, it was impossible for the muscle tendon complex model to lift up the mass (Table 1, noted with “–”). This

is because the size of the mass was too large compared to the force development capability of the CE determined by its Fmax, as well as the force–length and force– velocity relations [14]. On the other hand, in several cases, the gain in height achieved by the mass was larger than the maximal shortening capability of the CE (55% of LCEopt; [3]) (Table 1). This is because enough momentum had been provided to the mass before the maximal

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Fig. 2. Time course of the work output rate (power) from the muscle tendon complex (circle), the CE (triangle), the SEE (square) and the PEE (cross). The duration of 0.20 s after the release is shown. Solid line stands for the time course of the position of the mass, the highest point noted with a circle. This data set is an example obtained with mass factor = 0.10, L0CE = 1.4 × LCEopt and Lslack / LCEopt = 4.0.

shortening capability of the CE was reached. Generally, the gain in height achieved by the mass was larger when the initial length of the CE (L0CE) was set longer (Table 1), which suggests that it is beneficial to utilize a larger excursion of the CE in order to maximize its mechanical work output. When a small mass was imposed on the muscle tendon complex, a larger gain in height was observed with a larger LSEE/LCEopt ratio (Table 1). This suggests that when the goal of an activity is to maximize the work output with a relatively small mass imposed on the muscle tendon complex, it is beneficial to have a relatively long SEE in stretch-shortening cycle. The work output of the SEE was generally larger than the work output of the CE (Table 2, Fig. 3). This result is consistent with the finding reported in Ref. [26], in which “muscle shortening work” was calculated to be smaller than (~25%) “tendon energy recovery” in turkeys during running. As a longer SEE implies an enhanced spring-like behavior of the SEE, the work output of the SEE increased as its length increased (Table 2, Fig. 3). When a large mass was imposed on the muscle tendon complex, a larger gain in height was observed with a smaller LSEE/LCEopt ratio (Table 1). This suggests that when the goal of an activity is to maximize work output with a relatively large mass imposed on the muscle tendon complex, it is beneficial to have a relatively short SEE in stretchshortening cycles. As the length of the SEE increased, the work output of the CE as well as that of the PEE decreased (Table 2, Fig. 3). Although the work output of the SEE increased with the increase of the SEE length (Table 2, Fig. 3), the decrease in the work output of the CE and the PEE was larger than the increase of the work

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output of the SEE. Therefore, a longer SEE resulted in a lower height achieved by the mass (Table 1). Delp [8] reported a set of muscle tendon complex parameter data for 43 lower limb muscles. A large variation is observed in the length ratio between the SEE and the CE. Generally, there is a tendency that (A) muscles that span around the hip and the knee joints have smaller Lslack/LCEopt ratio compared to muscles that span around the ankle joint, and (B) monoarticular muscles have smaller Lslack/LCEopt ratio compared to biarticular muscles. In comparison with the results of this current study, the finding (A) may be explained by the fact that the inertial load imposed on muscles is generally larger for muscles that span around the hip and the knee joints compared to muscles that span around the ankle joint. Therefore, the results of this current study may be consistent with the findings derived from the data set of muscle tendon complex parameter values reported in Ref. [8]. Although practically all natural human motions are generated through interactions between multiple segments and multiple muscles, and individual muscles have different primary functions [34], the result of this study suggests the morphological advantage of individual muscles in explosive activities. Regarding finding (B), many researchers have discussed that the primary function of biarticular muscles is work transfer instead of work generation [13,17,24,28]. This statement implies that the primary determinant of the optimal morphological feature of biarticular muscles is work transfer characteristic instead of work generation characteristic. In order to transfer mechanical energy from proximal segments to distal segments with as small energy expenditure as possible, it may be beneficial to have a shorter CE and a longer SEE [26]. This hypothesis might be consistent with the finding (B). Characteristics similar to findings (A) and (B) are also observed in data sets of muscle tendon complex parameter values reported in other preceding studies [12,16,30]. To summarize, it was found that (1) when the inertial load imposed on the muscle tendon complex during a stretch-shortening cycle is relatively low, it is more beneficial to have a longer SEE ;and (2) when the inertial load imposed on the muscle tendon complex during a stretch-shortening cycle is relatively high, it is more beneficial to have a shorter SEE. These findings might be consistent with the characteristics found in the data sets of muscle tendon complex parameter values reported in previous studies.

Acknowledgements This study was partly supported by the Integrated Rehabilitation Engineering Program, NIDRR. Akinori Nagano would like to thank Professor James J. Collins

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Fig. 3. Results of the work output analysis. The work output of the CE, the SEE and the PEE, with the initial length of the CE set at 1.4 × LCEopt (Table 2). Generally, it was observed that as the length ratio Lslack/LCEopt increased, (1) work output of the CE decreased, (2) work output of the SEE increased and (3) work output of the PEE decreased.

at Boston University for his support. The authors would like to thank Dr. Keitaro Kubo at the University of Tokyo for his insightful discussions.

[11] [12]

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Akinori Nagano received a B.L.A. degree in 1996 from the Department of Natural and Artificial Systems, the University of Tokyo. He received a M.L.A. degree in 1998 from the Division of International and Interdisciplinary Studies, the University of Tokyo. His Ph.D. degree was awarded in 2001 for a dissertation on computer simulation of a human ancestor (Australopithecus afarensis) from the Interdisciplinary Ph.D. program in Exercise Science, Arizona State University. He participated in the Integrated Rehabilitation Engineering Program at the Center for BioDynamics, Boston University, between 2002 and 2003 as a post-doctoral researcher. Currently he is a contract researcher at the Institute of Physical and Chemical Research (RIKEN). Research interests include computer simulation of human movements, as well as experimental investigation of the human balance control system. He is a member of four professional societies. Taku Komura received his B.Sci. (1995), M.Sci. (1997) and D.Sci. (2000) degrees in Information Science from the University of Tokyo. His research interests include automatic motion generation and motion analysis based on the musculoskeletal model for use in computer graphics, robotics, biomechanics and ergonomics. He is currently an Assistant Professor in the Department of Computer Engineering and Information Technology at the City University of Hong Kong. He is a member of IEEE and ACM SIGGRAPH. Senshi Fukashiro received his M.Sci. and Ph.D. degrees in biomechanics (sport science) from the University of Tokyo in 1981 and 1993, respectively. Currently, he is an Associate Professor of Biomechanics in the Department of Life Sciences at the University of Tokyo. His major area of teaching and research is sport biomechanics with a special emphasis on the behavior of muscle tendon complex during human movements. He has contributed over 30 refereed research articles on a broad range of issues in sport biomechanics such as threedimensional film/video analyses, in vivo ultrasonographic investigation of human muscle architecture and function, new approaches to estimate mechanical outputs of individual joints, and so on. He is on the editorial board of three professional journals in Japan. He is a member of five professional societies, including a membership of the Executive Council of the Japanese Society of Biomechanics. He has been a member of the Executive Council of the International Society of Biomechanics since 1999.