Simulation of Muscle-Tendon Complex During Human Movements

Transcription

1 Review : Biomechanics Simulation in Human Movements Simulation of Muscle-Tendon Complex During Human Movements Senshi Fukashiro *, Dean C. Hay **, Shinsuke Yoshioka *** and Akinori Nagano **** * Graduate School of Interdisciplinary Information Studies, University of Tokyo Komaba, Meguro-ku, Tokyo Japan fukashiro@idaten.c.u-tokyo.ac.jp ** Department of Life Sciences, University of Tokyo Komaba, Meguro-ku, Tokyo Japan *** Department of Life Sciences, University of Tokyo/Computational Biomechanics Unit, RIKEN 2-1 Hirosawa, Wako, Saitama Japan **** Computational Biomechanics Unit, RIKEN 2-1 Hirosawa, Wako, Saitama Japan [Received March 23, 2005 ; Accepted May 14, 2005] The purpose of this paper was to review the research findings regarding the biomechanical behavior of the muscle-tendon complex (MTC) during dynamic human movements, especially those obtained using computer simulation. Specific studies conducted by the authors using the free vibration technique and MTC modeling have been discussed in detail. For determining individual viscoelastic characteristics of the human triceps surae MTC groups, a race difference between Black and White college athletes was investigated using the free vibration technique. It was found that the muscle stiffness was greater among Black athletes. Through computer simulation using a Hill-type MTC model, the benefit of making a countermovement was investigated in relation to the length ratio between the contractile element (CE) and the series elastic element (SEE) and the compliance of the MTC. The integral roles of the SEE were simulated in a cyclic human heel-raise exercise. It was revealed that it is beneficial to make a countermovement for explosive activities like vertical jumping, and the benefit of making a countermovement increases as the compliance of the MTC increases. Also, using a MTC model, the effects of moment arm length on kinetic outputs of the musculoskeletal system were evaluated. 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. It can be said that computer simulation is a powerful tool for determining and evaluating MTC behavior during dynamic human movements. Keywords: Computer simulation, Muscle-tendon complex, Stretch-shortening cycle, viscoelasticity [] 1. Introduction In a stretch-shortening cycle (Norman & Komi, 1979; Fukashiro & Komi, 1987) caused by a countermovement, viscoelastic characteristics of the muscle-tendon complex (MTC) play an important role in enhancing both the effectiveness and the efficiency of human performance. Of particular importance is the ability of these tissues to store energy when deformed (stretched) by external force and to recoil after being stretched (Komi, 1984; Huijing, 1992). In simulated vertical jumping motions, the jumping height was changed associated with the change in the Achilles tendon stiffness (Bobbert, 2001). While simulation studies have discussed the importance of a stretch shortening cycle, a more detailed quantification of the tendon and muscle characteristics in vivo is necessary to estimate the contribution of the stored elastic energy during dynamic movements (Bobbert, et al., 1986a, b). Neither muscle nor tendon behaves as a perfect spring, but both possess mechanical properties that can be described by relatively simple elastic models. Elasticity in an isolated muscle or tendon has been estimated mainly by the following methods: (1) quick release method (Huxley & Simmons, 1971), (2) α method (Morgan, 1977) and (3) spindle null-point 152

2 Fukashiro, S., et al. method (Rack & Westbury, 1984). A general stress-strain relationship for tendons has been usually modeled using a quadratic function (Woo, 1981). While much of the existing research has used animal models or data from cadavers, these conditions make it difficult to estimate living human skeletal muscle and tendon elasticity. The studies that have endeavored to characterize the viscoelastic properties of in vivo MTC in humans have used several methods: (1) quick release method (Pousson, et al., 1990), (2) impulse response or free vibration technique (Cavagna, 1970; Hunter & Kearney, 1982; 1983; Aruin & Zatiorsky, 1984; Shorten, 1987; Lafortune, et al., 1996; Fukashiro, et al., 2001) and (3) ultrasound method (Fukashiro, et al., 1995a). Mechanical parameters such as tendon compliance, estimated from mechanical models and human cadavers and/or animal experiments have provided slightly different values compared to live human studies (Kurokawa, et al., 2001, 2003) To evaluate the behavior of the MTC during human movements, it is important to determine the force as well as length changes in the MTC itself. One of our recent challenges has been to quantitatively determine the viscoelasticity of the MTC by using the free vibration method (Fukashiro, et al. 2001, 2002). Another challenge is to determine the force of the MTC during movements. Regarding the force measurement of the MTC, methodologies utilizing buckle type transducers (Fukashiro, et al., 1995b; Komi, 2000) and optic fibers (Finne, et al., 2000) have been reported to date. Since there are ethical and technical difficulties associated with these methods, we have tried to investigate the force and length characteristics of the MTC using computer simulation. A Hill-type MTC model, which includes the force-length-velocity relations of the contractile element (CE) and the force-length relation of the elastic elements, was used to evaluate the force development and length change of the MTC (Nagano, et al., 2004a,b). Furthermore, the Hill type model was applied to a human hopping-like motion (Nagano, et al., 2003). The use of computer simulations in conjunction with non-invasive in vivo experiments has helped expand our understanding of the MTC behaviour during movements. Therefore, the purpose of this paper was to review recent simulation studies of the biomechanical behavior of fascicle and tendinous structures during dynamic human movements, with special focus on research conducted by the authors. 2. Determination of viscoelastic properties in human triceps surae MTC groups using free vibration technique F u k a s h i r o, e t a l. ( ) i n v e s t i g a t e d t h e methodological validity of the free vibration technique (Shorten 1987) for determining individual viscoelastic characteristics of the human triceps surae MTC in vivo. Subjects sat with first phalangeal joint of the forefoot on the edge of a force-plate. A special frame placed over the knee was loaded with weight (0±40 kg) for testing. Oscillations of the triceps surae MTC system were initiated with a hand-held hammer by tapping the weight. In order to keep the same posture, the output of the force plate was displayed on the oscilloscope and subjects were asked to maintain the beam on the oscilloscope at a particular location in relation to a reference line. The damped oscillations in conjunction with the equation of motion of a damped mass- spring model were used to calculate the viscosity of muscle (b) and the elasticity of muscle fibers and tendon (k) in each subject, considering moment arm of the ankle joint (Figure 1). With this arrangement, we obtained high reproducibility in this method. The coefficient of variations (CVs) of b and k in five trials at each weight were quite small (range: 0.5±18.7% in b and 1.0±15.1% in k). There were no significant differences in viscoelastic coefficients between right and left legs. Therefore, it appears that the free vibration technique, used here, is adequate for determining the viscoelastic characteristics of the triceps surae in vivo in humans. Using this technique, Fukashiro, et al. (2002) investigated race differences Weight Force-plate A Figure 1 Schematic presentation (A) and model (B) of the free vibration technique. M: mass, B: viscosity, and K: elasticity in this system X K M B B 153

3 Simulation in Human Movements in viscoelastic characteristics of the triceps surae MTC groups. Black and white college athletes (n=44) participated in this study. There were little significant differences in most of physical characteristic variables between black and white athletes. It was found that black athletes have significantly greater muscle viscosity and elasticity than white athletes while tendon elasticity is equivalent. Thus, muscle stiffness is greater among black athletes. Greater muscle stiffness could contribute to greater sprint / jump performance among black athletes compared to white athletes, through faster alteration of foot-ground contact and take-off phases during sprinting / jumping. 3. Mechanical effects of the length ratio between the CE and the SEE on stretch-shortening cycle Dimensional parameters of the muscle-tendon complex, such as cross sectional area, fascicle length and tendon length, have attracted the interests of many biomechanists (Burkholder & Lieber, 2001; Koo, et al., 2002; Lieber, 1992; Lieber & Friden, 2000; Zuurbier & Huijing, 1992). This is because many of the biomechanical functions of the MTC are determined by these parameters. Generally, a greater cross sectional area of the CE is related to a higher force development ability (Brown, et al., 1998; Brown & Loeb, 2000a, b; Kawakami, et al., 2000; Friederich & Brand, 1990). A greater length of the CE is related to a greater contraction (shortening) ability (excursion and shortening velocity) (Abe, et al., 1999; Edman, 1988; Hof, et al., 2002; Pandy, et al., 1990). Zajac (1989) has published more detailed descriptions on dimensional properties of the MTC that determine its functional capability. Biomechanical action of the tendon during the energy storage and release phases in stretch-shortening cycles is mostly determined by its compliance and length. Higher compliance and / or greater length results in more elastic action of the tendon. Based on experimental studies that utilized dissected animal specimens, it has been reported that the compliance of a tendon is typically such that approximately a 4% strain is observed when the maximal isometric force is developed by the muscle fibers (Caldwell, 1995; Gerritsen, et al., 1998). Regarding the length of the tendon, it is observed that there is a large variation in this parameter between different muscles, both in terms of absolute length and relative length, i.e., relative to the muscle fiber length (Friederich & Brand, 1990; Wickiewicz, 1983; Yamaguchi, et al., 1990). More specifically, a MTC with a relatively long SEE will exhibit more elastic action as a whole, whereas the action of a MTC with a relatively short SEE will be less elastic. In Nagano, et al. (2004b), effects of the length ratio between the SEE and the CE on the biomechanical behavior of the MTC were examined. A computer simulation model of the Hill-type muscle tendon complex was developed (Figure 2A). The A Height Power (W) 300 B Height (m) 0.15 CE PEE MTC Height g 200 SEE 0.10 SEE Body Massless Supporting Object Time (s) Figure 2 The Hill-type muscle tendon complex model (A) and Power-Vertical Displacement-Time histories in each element (B) (Nagano, et al.,2004b). The muscle tendon complex model consisted of three elements: a contractile element (CE), a series elastic element (SEE) and a parallel elastic element (PEE). Effects of pennation angle were also considered. 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 body was placed on the supporting object. CE PEE

4 Fukashiro, S., et al. proximal endpoint of the CE was affixed to a point in the space, and a massless supporting object was affixed to the distal endpoint of the SEE. A body was placed on the supporting object. Initially, the MTC was fixed at a certain initial length, and the CE was activated maximally. Through this process, the CE shortened as much as the SEE was stretched, whereas the total length stayed constant. Thereafter, the supporting object was suddenly released. This caused the MTC to thrust the body upwards, which simulated a stretch-shortening cycle (Figure 2B). Values of such parameters as length ratio between the SEE and the CE, mass of the body, and the initial length of the MTC, were modified systematically. As a result, the following were found: (1) When the load imposed on the MTC was small, a higher performance (the maximal height reached by the body) was obtained with a longer SEE. (2) When the load imposed on the MTC was large, a higher performance was obtained with a shorter SEE. These findings were consistent with the insights obtained by investigating the muscle-tendon parameter data sets of preceding experimental measurements (Yamaguchi, et al., 1990). 4. Effects of the stiffness of the SEE on the MT during a countermovement activity Bobbert (2001) investigated the influences of the series elasticity compliance of the mm. triceps surae on a SQJ performance in a computer simulation study by systematically modifying the SEE compliance. It was found that the jumping height was the highest when the SEE compliance was the greatest. This result was basically in accordance with the findings of a preceding investigation reported by Pandy (1990). Considering the insights obtained through these studies, it seems meaningful to investigate the action of the MTC in explosive activities that involve a countermovement, with a special emphasis on the contribution of series elasticity. In Bobbert, et al. (1996), the biomechanics of two types of jumping motions (SQJ and CMJ) were compared. In that setup, the computer simulation model was directed to perform these two distinct types of motions independently. It was found that the jumping height was greater in the CMJ than in the SQJ by 3.4 cm. This difference was attributed to the development of higher muscular force in the CMJ before the start of the shortening of muscle fibers. An effect of storage and reutilization of elastic energy Activation Inputs L CEini F SEEini Simulation Numerical Optimization Bremermann (1970) Maximum Range of Motion (Cyclic Motion) Figure 3 Outline of the numerical optimization approach used in Nagano, et al. (2003). Optimal combination of the initial length of the contractile element, initial force of the series elastic element and the neural activation inputs were searched using numerical optimization. The goal was to find the combination of those variables that maximized the vertical range of motion experienced by the body at the same time as making the motion cyclic. was ruled out as a potential explanation. This finding suggested the same basic mechanism of the action of the MTC reported by Avis, et al. (1986) as well as by Anderson and Pandy (1993). However, in normal activities, humans first set up a goal of a motion (e.g., to reach as high as possible), implicitly choose the best strategy (e.g., CMJ), and then execute the most suitable motion. From this perspective, in computer modeling and simulation studies, it would be closer to reality to "tell" the computer simulation model the final goal to be accomplished, and let the model choose the best strategy to accomplish that goal. This way, it may become possible to investigate why the existence (or non-existence) of a countermovement is advantageous for the purpose of enhancing the performance of explosive muscular activities. Nagano, et al. (2004a) performed a computer simulation investigation of this question. The two goals of the study were: (1) to evaluate the effects of the SEE on an explosive activity that allows for making a countermovement, and (2) to examine whether or not a countermovement is spontaneously generated in the optimal explosive activity. A Hill-type MTC model that is composed of a CE and a SEE was constructed (Figure 2A). The proximal end of the CE was affixed to a point in the space, and a massless supporting object was affixed to the distal end of the SEE. A body was placed on the supporting object. The goal of this explosive activity was to raise the body as high as possible. A variety of SEE compliance was investigated within the physiological range. The best strategy was searched using numerical optimization for each value of SEE compliance (Figure 3). Two major findings 155

5 Simulation in Human Movements were obtained: (1) As the SEE elasticity increased, the maximal height reached by the body also increased, primarily because of the enhanced force production of the CE. (2) A countermovement was spontaneously generated for all values of SEE compliance. It was found that it is beneficial to make a countermovement for this type of explosive activity, and the benefit of making a countermovement increases as the compliance of the MTC increases. 5. Simulation of human cyclic heel-raise exercise with a focus on the contribution of the SEE Biomechanical properties of individual components of the MTC have been translated into elegant mathematical equations by many researchers (Gielen, et al., 2000; van den Bogert, et al., 1998; Wu & Herzog, 1999; Zahalak & Ma, 1990). Craib, et al. (1996) investigated the relation between joint flexibility and running economy of athletes. Anderson & Pandy (1993) evaluated the mechanism of storage and utilization of elastic energy during jumping. There are numerous other preceding studies that addressed this issue (Kubo, et al., 1999; 2000; Lieber, 1992). Properties of the SEE have great impacts on the generation of cyclic activities of musculoskeletal systems (Winters, 1990; Yamaguchi, 2001). As a clear example, the oscillation frequency of a system composed of a rigid body and a linear spring is determined by the inertia of the rigid body and the stiffness of the spring. Therefore it is important to consider the properties of the SEE when studying cyclic activities of the musculoskeletal system. This has important implications for researchers in biomechanics, as many human activities, such as walking and running are cyclic. Several researchers have investigated the behavior of the CE and the SEE during cyclic activities of animals in vivo (Biewener, et al., 1998; Roberts, et al., 1997; Wilson, et al., 2001). Biewener, et al. (1998) measured the muscle fiber length change and tendon force in hopping wallabies. They found that an increasing amount of strain energy stored within the hindlimb tendons is usefully recovered at faster steady hopping speeds without being dissipated by increased stretch of the muscle fibers. Biewener, et al. (1998) concluded that tendon elastic saving of energy is an important function by which wallabies are able to hop at faster speeds with little increase in metabolic energy expenditure. Roberts, et al. (1997) measured the muscle force and length change in running turkeys, from which work output of the muscle and the tendon was individually calculated. As a result, it was found that as running speed increased, contribution of the tendon in terms of positive work output increased. The authors of the study concluded that muscles that act as active struts improve running economy. These in vivo studies have contributed greatly to the understanding of the biomechanics of animal movements. However, as discussed previously, the oscillation frequency of a mass-spring system is determined by the elastic property of the spring and the inertia of the rigid body. This suggests that the behavior of the system that contains elasticity depends on its dimensional properties. Therefore it is not self-evidently correct to simply extrapolate the mechanisms of animal movements to those of humans. Although it is difficult to perform the same type of invasive measurements using human subjects, the methodology of computer simulation can provide a great advantage for solving this problem. In a computer simulation, muscle and tendon force, length change, power and work outputs, and other biomechanical variables can be isolated and manipulated in ways simply not possible in in vivo experiments. In Nagano, et al. (2003), contribution of series elasticity of the human mm. triceps surae in cyclic heel-raise exercise was evaluated. A two-dimensional skeletal model of the human body was constructed. An upright posture was maintained in the simulation to prevent the model from falling down. A mathematical representation of the mm. triceps surae was attached to the model. The muscle was activated by the neural activation input with a time resolution of 50 ms. Cyclic heel-raise exercise, which was similar to hopping except for that the feet did not leave the ground, over a wide range of cycle durations was generated using an optimization approach. The goal of the numerical optimization was to generate cyclic motions with as large a range of motion as possible. Realistic heel-raise motions were generated as a result. It was found that the contribution of the series elasticity in positive mechanical work output of the MTC during the push-off phase increased as the motion frequency increased. Relatively higher muscle activation was found during the downward moving phase when the motion frequency was higher. It was suggested that series elasticity plays a substantial role 156

6 Fukashiro, S., et al. in the generation of a cyclic human motion. 6. Effects of the moment arm length on the mechanical outputs around joints Y Z X Force - Length In musculoskeletal systems, forces developed by skeletal muscles produce motions of the joints. Muscles are attached to bones at a distance from the center of rotation (joint centers). Linear tensile forces generated by muscles are transformed into rotational joint moments. The relation between muscle forces and joint moments is described as: M = F * MA (Eq. 1) where M is joint moment, F is muscle force and MA is moment arm length. This equation suggests that a greater joint moment development is expected with a longer moment arm, whereas a smaller joint moment development is expected with a shorter moment arm, assuming that the magnitude of the muscle force is constant. In human musculoskeletal systems, almost all forces developed by muscles are transmitted to the external environment in the form of joint moments (Lieber, 1992; Nigg and Herzog, 1994; Winter, 1990). Therefore it is important to consider the mechanical structure of joints in terms of moment arm length. The relation described in (Eq. 1) often yields a misleading statement: "with a longer moment arm, a greater joint moment is always developed". This statement is not necessarily true, especially when the musculoskeletal system is undergoing dynamic movements. This is because force development capability of a muscle is greatly influenced by its length and shortening / lengthening speed, i.e., force-length-velocity relations. Generally speaking, when the joint angle changes by θ, the muscle length changes as much as L, such as: L = MA * θ (Eq. 2) This equation suggests that when the MA is greater, the range of joint angle in which active joint moment is developed by the muscle ( θ) is narrower, with an assumption that the maximal lengthening / shortening range of the muscle stays constant. From (Eq. 2), the relation between muscle lengthening / shortening velocity and joint angular velocity is expressed as V = MA * ω (Eq. 3) where V represents muscle lengthening / shortening velocity and ω represents joint angular velocity. The relation expressed in (Eq. 3) suggests that the muscle lengthening / shortening velocity is linearly +1 cm -1 cm Force - Velocity Figure 4 In Nagano and Komura (2003), the location of the insertion of the m. soleus was shifted dorsally and ventrally by 1 cm. These manipulations had an effect of changing (lengthening and shortening, respectively) the moment arm of the m. soleus. related to the joint angular velocity by the factor of moment arm length (MA). Therefore it is derived that when the moment arm is longer, a larger muscle lengthening / shortening velocity is necessary in order to achieve a certain angular velocity of the joint. As Hill (1938) reported, an increase in shortening velocity results in a non-linear drop of force development ability of the muscle. This effect has to be taken into consideration when evaluating mechanical outputs around joints. Nagano and Komura (2003) evaluated the effects of moment arm length on kinetic outputs of the musculoskeletal system. A skeletal system of the human ankle joint was constructed: a lower leg segment and a foot segment were connected via a hinge joint. A Hill-type muscle model of the m. soleus was attached to the skeletal system (Figure 4). The model of m. soleus was maximally activated while the ankle joint was plantarflexed / dorsiflexed at a variety of angular velocities to simulate conditions used in human ankle isokinetic experiments. Profiles of kinetic outputs such as muscle force development, joint moment development, joint power output and joint work output were calculated. Thereafter, 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 moment arm, respectively. Kinetic outputs of the musculoskeletal system were evaluated with these longer / shorter moment arm lengths. It was found the that longer moment arm resulted in smaller 157

7 Simulation in Human Movements Ankle Plantar Flexion Moment (N m) θ (deg) longer MA neutral MA shorter MA ω (deg / s) Figure 5 Profiles of ankle joint plantar flexion moment development, as a function of ankle joint angle and angular velocity. It was observed that the joint moment was greater with a longer moment arm in the low angular velocity region, whereas the joint moment was greater with a shorter moment arm in the high angular velocity region. Nagano and Komura (2003) joint moment development, smaller joint power output and smaller joint work output in the region of greater plantarflexion angular velocity (Figure 5). This is because greater muscle shortening velocity was necessary with a longer moment arm in order to achieve a certain joint angular velocity. Greater muscle shortening velocity resulted in smaller muscle force production because of the non-linear force- velocity relation. It was suggested that this phenomenon should be taken into consideration when investigating joint moment - angle -angular velocity properties of the musculoskeletal system. 7. Concluding Remarks It can be said that the computer simulation is a powerful tool for determining and evaluating MTC behavior during dynamic human movements. This methodology is especially useful when considering mechanical variables that are not easily accessible through in vivo measurements. Additionally, it becomes possible to impose small modifications on the musculoskeletal system and to evaluate the effects of those modifications on the behavior of the system as a whole. However, complex characteristics of the human musculoskeletal system need to be simplified when developing a simulation model so that many of the characteristics unique to individual subjects are omitted in the process. As computing power continues to increase and models become more detailed, it is hoped that this issue will be addressed in future studies. Acknowledgment This research was supported by Ministry of Education, Culture, Sports, Science and Technology in Japan (No: ). References Abe, T., Brown, J.B. & Brechue, W.F. (1999) Architectutal characteristics of skeletal muscle in black & white college football players. Medicine and Science in Sports and Exercise 31, Anderson, F.C. & Pandy, M.G. (1993) Storage and utilization of elastic strain energy during jumping. Journal of Biomechanics 26: Aruin, A.S. and Zatiorsky, V.M. (1984) Biomechanical characteristics of human ankle-joint muscles. European Journal of Applied Physiology 52, Avis, F.J., Toussaint, H.M., Huijing, P.A. & van Ingen Schenau, G.J. (1986) Positive work as a function of eccentric load in maximal leg extension movements. European Journal of Applied Physiology 55, Biewener, A.A., Konieczynski, D.D. & Baudinette, R.V. (1998) In vivo muscle force-length behavior during steady-speed hopping in tammar wallabies. Journal of Experimental Biology 201: Bobbert, M.F., Huijing, P.A. & Ingen Schenau, G.J. van. (1986a) A model of the human triceps surae muscle-tendon complex applied to jumping. Journal of Biomechanics 19, Bobbert, M.F., Huijing, P.A. & Ingen Schenau, G.J. van. (1986b) An estimation of power output and work done by the human 158

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Journal of Biomechanics 25, Name: Senshi Fukashiro Affiliation: Sport Biomechanics, Graduate School of Interdisciplinary Information Studies, University of Tokyo Address: Komaba, Meguro, Tokyo , Japan Brief Biographical History: 1993 PhD, University of Tokyo Executive Council of Japanese Society of Biomechanics Executive Council of International Society of Biomechanics Executive Council of Japan Society of Physical Education, Health and Sport Sciences President of Tokyo branch of Japan Society of Physical Education, Health and Sport Sciences Associate Editor of Journal of Japan Society of Physical Education, Health and Sport Sciences Associate Editor of International Journal of Sport and Health Science Main Works: S. Fukashiro (1990): Science of Jumping. Taishukan-shoten, Tokyo. S. Fukashiro, S. Sakurai, Y. Hirano and M. Ae (2000): Sport Biomechanics. Asakura-shoten, Tokyo. S. Fukashiro and A. Shibayama (2000): Handbook of Mathematics and Physics in Sports. Asakura-shoten, Tokyo. S. Fukashiro, M. Noda and A. Shibayama (2001): In vivo determination of muscle viscoelasticity in the human leg. Acta Physiol. Scand. 172: S, Fukashiro, T. Abe, A. Shibayama and W.F. Brechue(2002): Comparison of viscoelastic characteristics in triceps surae between Black and White athletes. Acta Physiol Scand. 175(3): Membership in Learned Societies: Japanese Society of Biomechanics International Society of Biomechanics Japan Society of Physical Education, Health and Sport Sciences 160