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Journal of Biomechanics 40 (2007) 1768–1775 www.elsevier.com/locate/jbiomech www.JBiomech.com
Is Achilles tendon compliance optimised for maximum muscle efficiency during locomotion? G.A. Lichtwarka,, A.M. Wilsona,b a
Structure and Motion Laboratory, Institute of Orthopaedics and Musculoskeletal Sciences, University College London, Royal National Orthopaedic Hospital, Brockley Hill, Stanmore, Middlesex, HA7 4LP, UK b Structure and Motion Laboratory, Veterinary Basic Sciences, The Royal Veterinary College, University of London, Hawkshead Lane, North Mymms, Hatfield, Hertfordshire, AL9 7TA, UK Accepted 31 July 2006
Abstract Tendon elasticity is important for economical locomotion; however it is unknown whether tendon stiffness is appropriate to achieve an optimal efficiency in various muscles. Here we test the hypothesis that the Achilles tendon is of an appropriate stiffness to maximise medial gastrocnemius muscle efficiency during locomotion with different power requirements. To test this hypothesis we used a three element Hill muscle model to determine how muscle fascicles would be required to change length if the series elastic element stiffness is varied, whilst the limb kinematics and muscle properties are held constant. We applied a model of muscle energetics to these data to predict muscle efficiency for a range of stiffness values in both walking and running conditions. We also compared the model results to in vivo data collected using ultrasonography. The muscle model predicted that optimal series elastic element stiffness for maximising efficiency is equal or slightly higher than that of the average Achilles tendon in running and walking, respectively. Although the peak efficiency values for running (26%) and walking (27%) are similar, the range of stiffness values achieving high efficiency in running is much smaller than that during walking. These results suggest that a compliant tendon, such as the Achilles tendon, is required for efficient running. Such a finding is important, because it describes how the stiffness of a tendon may be adapted to achieve optimal efficiency for particular athletic pursuits. The influence of varying tendon stiffness on kinematic performance may, however, play an important role in determining the efficiency of the muscle. r 2006 Elsevier Ltd. All rights reserved. Keywords: Biomechanics; Tendon; Efficiency; Gastrocnemius; Hill muscle model; Elasticity
1. Introduction Elastic tendon recoil has long been suggested as an energy conserving mechanism during locomotion (Cavagna et al., 1977; Alexander and Bennet-Clark, 1977; Roberts et al., 1997; Alexander, 1988). It has also been suggested that tendon recoil may be responsible for increasing the power output of a muscle (Alexander, 1974; Bobbert et al., 1986; Galantis and Woledge, 2003; Wilson et al., 2003). Corresponding author. Structure and Motion Laboratory, Veterinary Basic Sciences, The Royal Veterinary College, University of London, Hawkshead Lane, North Mymms, Hatfield, Hertfordshire, AL9 7TA, UK. Tel.: +44 1707 666259; fax: +44 1707 666371. E-mail address:
[email protected] (G.A. Lichtwark).
0021-9290/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2006.07.025
The elastic properties of tendon are therefore very important for both muscle power production and efficiency, but, what is the optimal design for a tendon and its muscle? Here we suggest that the Achilles tendon (AT) is designed such that it permits the maximum efficiency of medial gastrocnemius (MG) muscle work production during shortening and allows high efficiency under different contraction conditions and power outputs (Fig. 1). Although the material properties of tendons have been found to be relatively consistent across a range of species and muscles (Pollock and Shadwick, 1994; Bennett et al., 1986), the stiffness of a tendon relative to the force generating capacity of the attached muscle varies greatly between muscles (Ker et al., 1988). Roberts (2002) suggests that most of this variation results from differences in
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Fig. 1. Three-element muscle model representing the medial gastrocnemius (contractile element—CE) attaching to the Achilles tendon (series elastic element—SEE), with an elastic component in parallel with the CE (parallel elastic element—PEE). Total muscle tendon unit (MTU) length is calculated as the sum of the CE and SEE lengths. The model assumes an optimum CE length of 55 mm and a SEE slack length of 237 mm; the average values determined in Lichtwark and Wilson (2006).
muscle and tendon architecture. For instance, muscles with short fibres, high pennation angles and relatively long free tendons are often more compliant than those with long muscle fibres and short tendons. This is obviously related to the role that a muscle performs, however it has yet to be shown whether the elastic properties of a muscle tendon unit (MTU) is tuned to optimise its efficiency. During MG force production in both walking and running, muscle fascicles remain almost isometric and thus produce very little work (Fukunaga et al., 2001a; Hof et al., 2002; Lichtwark et al., 2005). During force decline the muscle fascicles shorten and hence produce work during the period of elastic recoil of the tendon and aponeurosis. Therefore, the periods of activation are accompanied by little shortening whilst the periods of fastest shortening occur during deactivation. A similar finding has been found in other anti-gravity muscles of a variety of animals (Griffiths, 1991; Roberts et al., 1997). This pattern of work production is efficient because shortening occurs with lower activations (Woledge et al., 1985). If the compliance of the elastic tissues is varied, however, one must also change the way a muscle activates and subsequently shortens so that the required force output can be achieved (Lichtwark and Wilson, 2005b; Ettema, 2001). We hypothesise that the series elastic tissues of the MG muscle have a compliance which allows the muscle to operate with optimal efficiency during normal walking and running. To test this hypothesis we applied a muscle model to experimental data to determine how varying the compliance of the elastic tissues (AT and aponeurosis)
would influence the efficiency of the MG muscle during walking and running. We also compare the model results to previously reported experimental results of human locomotion that measure the actual MTU interaction. 2. Methods 2.1. Experimental data Average MG muscle fascicle length, AT length, whole MTU length changes and enveloped electromyographic (EMG) signals during an average stride of level treadmill walking (1.4 m/s) and running (2.8 m/s) were taken from previous work (Lichtwark and Wilson, 2006). These data were collected using motion analysis and ultrasound techniques. The average data were taken across three strides from each of six male participants for each condition. 2.2. Modelling the influence of SEE compliance We assumed that walking and running kinematics would not change between SEE (series elastic element) compliance levels (see discussion for justification). Hence, the MTU length and force over time would not change. The contractile element (CE) and SEE length and velocity over time would however change with SEE compliance. We changed SEE compliance and determined the muscle fascicle length changes required to maintain the force and MTU length trajectory over time for walking and for running. We then calculated the required activation levels
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et al. (2002) for the same muscle. Those data suggests a maximum shortening velocity of approximately 8 Lo/s (0.45 m/s). The data also demonstrate that the PEE force is low across most of the length range of the muscle fascicle; however, it increases steeply at the higher end of the length range. MTU strain during the walking and running stance phase were used to determine the MTU length change in the model. This assumes zero strain in a neutral anatomical position (Grieve et al., 1978). The AT force throughout the stance phase was estimated from the measured length changes of the AT during walking and running (Lichtwark and Wilson, 2006). To calculate force, we assumed a linear tendon stiffness of 180 N mm 1 (Lichtwark and Wilson, 2005c; Maganaris and Paul, 2002). The MTU strain and AT force inputs used for walking and running are shown in Fig. 3, with respect to time. The estimated AT force was then used to calculate SEE length changes over time. CE length was calculated as the
Force (Po)
for each compliance condition using a Hill-type muscle model. We used the muscle model to predict muscle energetics under each condition and demonstrated the relationship between SEE stiffness and efficiency. The Hill-type muscle model consisted of a CE, a SEE and a parallel elastic element (PEE). The PEE was assumed to act in parallel with the CE only. The CE of the model was assigned force–length and force–velocity relationships as shown in Fig. 2a and b. Optimum muscle fascicle length (Lo) was assumed to be equal to 55 mm (the average length of the CE during the stance phase of running at 2.8 m/s), (Lichtwark and Wilson, 2006). The muscle fascicle was assumed to be at optimum length in the neutral anatomical position (Lieber et al., 1994). A SEE slack length of 237 mm was used, based on the previously calculated average tendon slack length (Lichtwark and Wilson, 2006). The SEE was modelled as having a linear stiffness which we varied. Both the force–velocity and force–length relationships were chosen to resemble the data of Hof 1.5
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Fig. 3. Muscle-tendon unit (MTU) strain (A) and approximate Achilles tendon (AT) force (B) during the stance phase for walking at 1.4 m/s (solid lines) and running at 2.8 m/s (dashed lines).
difference between the MTU length and the SEE length. The required activation level over time was estimated by dividing the instantaneous muscle force by the maximum possible force at the estimated instantaneous velocity and length (assuming force–length and force–velocity relationships as in Fig. 2a and b). Instantaneous muscle force was estimated as 16% of the force of the AT, (Fukunaga et al., 1992), and a maximum isometric force of 1290 N was assumed (Fukunaga et al., 1992; Narici et al., 1996). Detailed equations for calculating the required SEE and CE length changes and the required activation are described in the Supplementary Information (detailed equations). The modelled activation levels and length changes were then used to calculate the heat output and efficiency of muscle contraction for each stiffness in both level walking and running, as will be described further below. 2.3. Model of muscle efficiency Previously, we have developed a model of muscle contraction that predicts energy expenditure and efficiency of muscle. This is a phenomenological Hill-type muscle model that predicts heat and work outputs of muscle (Lichtwark and Wilson, 2005a). Muscle efficiency is defined as the fraction of chemical energy used that is converted into mechanical work (Barclay and Weber, 2004), therefore the ratio of work to the total cost of work (heat+work) is the efficiency. The heat produced by a muscle is dependent on the active state of the muscle and also on the shortening
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velocity of the muscle (Fenn, 1924; Hill, 1938; Woledge, 1998). The rate of heat production increases relatively linearly with CE shortening velocity, while the active state of the muscle scales heat rate at any given velocity (Woledge et al., 1985). Heat is also produced while the CE is actively lengthening, partly due to the cost of activating the muscle and due to work being done on the active CE. Approximately 50% of the work done on the CE is seen as heat during lengthening (Constable et al., 1997). The relationship between velocity and force, power, heat rate and instantaneous efficiency at different activation levels (and optimum muscle length) from the model are shown in Fig. 2b–e. During lengthening (negative shortening velocity), muscle power output and efficiency are both negative. The current model also incorporates an active force–length relationship (Fig. 2E) which is implemented into the energetic model of muscle contraction as described in Lichtwark and Wilson (2005b). Detailed equations and model parameters are provided in the Supplementary Information (detailed equations). Throughout the course of the stance phase, the instantaneous power and energy output will change depending on the activation level, CE length and CE velocity. Total muscle fascicle efficiency across the stance phase was calculated as the total work done by the muscle fascicle in this period divided by the sum of the total heat produced in this period and the total work done.
2.4. Data analysis The required CE length changes and corresponding activation level required to achieve the MG force during walking and running were assessed across a range of stiffness values (45–720 N mm 1) for both walking and running. This range corresponds to 0.25 to 4 times the measured stiffness of the AT (Lichtwark and Wilson, 2005c). These results were then implemented into the model of energetics to determine stance phase efficiency of the muscle. We then compared the modelled length changes of the muscle AT and MG fascicles to experimentally determined results, and the modelled activation patterns with the average enveloped EMG signal (Lichtwark and Wilson, 2006). To determine the influence of where the muscle fascicles operate on the force–length curve, the length of the MG fascicles in the anatomically neutral position was altered to 720% of the optimum length and the model rerun to determine the influence on efficiency. To account for the difference in operating range of the muscle fascicles on the MTU operating range, the SEE was shortened or lengthened to maintain a constant MTU length in both conditions. This will have an effect on the required activation level, because it may change the maximum possible number of bound crossbridges or operate on a different part of the PEE force–length curve.
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3. Results 3.1. Muscle efficiency with varying series elastic stiffness The model predictions for fascicle length trajectory, fascicle velocity, activation level and heat output for four different SEE stiffness values (0.5, 1, 2 and 4 times the measured AT stiffness) in both the walking and running stance phase are shown in Fig. 4A and B. At high SEE stiffness values the muscle fascicle trajectory is similar to that of the MTU, where it lengthens throughout most of the stance phase until it shortens rapidly (compare Figs. 3 and 4). In contrast, the muscle fascicles are predicted to shorten throughout the stance phase and then lengthen rapidly for low SEE stiffness values. The model predicts that the GM muscle would have to activate to levels above 1 (the maximum activation level) to achieve the required forces for some stiffness values during running (0.5 and 4 AT stiffness). The stiffness values that most closely resembled the fascicle length trajectory range measured experimentally and the muscle activations (Fig. 4A and B)
was the same as close to the average measured AT stiffness reported in Lichtwark and Wilson (2005c). An animation of the required fascicle length and activation levels for each stiffness value in both walking and running is presented in Supplementary Information (videos 1 and 2). The energetic model predicted that the least amount of heat will be produced at a SEE stiffness the same as the average measured stiffness of the AT. This is the case during both walking and running. It is apparent that the most heat is produced during periods of high activation in combination with high shortening velocity. The predicted relationship between SEE stiffness and efficiency is shown in Fig. 5. During walking, the most efficient tendon stiffness was predicted to be slightly higher than the value measured experimentally in Lichtwark and Wilson (2005c). A more compliant SEE (below 0.5 AT stiffness) would reduce the efficiency substantially, and would also require an activation greater than 140% to achieve the required forces. At higher values of stiffness (above 2 AT stiffness), the efficiency only reduces from 26% to 23%. During running, the optimal stiffness for
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Fig. 4. Modelled CE length change (A), shortening velocity (B), activation level (C) and heat output (D) required for the stance phase of walking and running with a series elastic element stiffness of 0.5, 1, 2 and 4 times the Achilles tendon stiffness (as labelled in muscle fascicle length panels). Heat output was estimated from the energetic model of Lichtwark and Wilson (2005a), see Appendix 1 for details. The average muscle fascicle length changes and enveloped EMG signals (+/ two standard errors) measured for each condition (Lichtwark and Wilson, 2006) are also shown for comparison (shaded areas) to the modelled results.
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Stiffness (Relative to Achilles tendon) Fig. 5. Modelled muscle fascicle efficiency for walking (black line and crosses) and running (black line and circles) on 0% incline across a range of SEE stiffness values (relative to the Achilles tendon stiffness measured experimentally from Lichtwark and Wilson (2005c)). To demonstrate the influence of where the muscle fascicles operate on the force–length curve, the same relationship is shown when the length of the muscle fascicles in the anatomically neutral position is changed to +20% of the optimum length (solid grey line) and 20% of the optimum length (dashed grey line) for both walking (crosses) and running (circles). Efficiency values are not shown for stiffness values where the maximum activation exceeded 140% of maximum activation, as these were deemed impossible results.
achieving maximum efficiency is equal to the experimentally measured value. However, at higher stiffness values, the efficiency of the muscle fascicles drops from 26% to 16%. For stiffness values higher than three times the AT stiffness, the muscle cannot achieve the required forces without an activation greater than 140% of its maximum. Peak efficiency is similar in walking (27%) compared to running (26%), despite running requiring a greater SEE strain. The effect of operating length range on the model, i.e. the length of the muscle fascicle length in the anatomically neutral position, is also shown in Fig. 5. This demonstrates that the model is sensitive to the operating range in terms of maximum capable efficiency; however, the maximum efficiency values for both walking and running still occur at similar tendon stiffness values. 4. Discussion We have demonstrated that an optimal value of SEE stiffness exists to maximise muscle fascicle efficiency in different gait conditions. Varying the tendon compliance in the model changes the required shortening pattern of the muscle fascicles (assuming that the same kinematics are required), and hence the energetics are greatly affected. The optimal value for tendon stiffness in our model was very similar to that measured experimentally for the AT during both walking and running. The model suggests that periods of high activation, coupled with periods of high shortening
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velocity are detrimental to the muscle’s efficiency. Differences in the optimal value for SEE stiffness depending on the gait suggests that the SEE stiffness may affect the conditions under which an individual performs best. High MG efficiency could be achieved across a broad range of SEE stiffness values during walking, but not in running. In the walking gait, only very low stiffness values were found to be detrimental to the efficiency and the muscle’s capability to produce the required forces. This is due to the high fascicle shortening velocity required to stretch the tendon to longer lengths during force production with less stiff tendons. In contrast, stiff tendons were found to be unfavourable for achieving maximum efficiency during running. During running, the MTU is required to shorten a long distance in a short-time period. With a stiff tendon, all of the shortening must occur in the CE; therefore high CE shortening velocities are required. The model showed that for stiffness values above three times the AT stiffness, these shortening velocities cannot be achieved at the given force requirement (activation levels in excess of 140% of the maximum were required). The energetic model predicts that high activations coupled with high shortening velocities will increase the rate of heat production; therefore, stiff tendons are not energetically efficient during running because the muscle fascicles must operate with high shortening velocities whilst producing force. The muscle model used to make these predictions did not include any additional source of series elasticity such as aponeurosis. Previously, it has been shown that the aponeurosis is required to strain during locomotion in order to achieve the measured muscle fascicle shortening (Lichtwark and Wilson, 2006). Aponeurosis in the MG has also been shown to contribute substantially to elastic strain energy storage, although the aponeurosis is much stiffer than the AT, particularly at low strains (Magnusson et al., 2001, 2003; Maganaris et al., 2001). The aponeurosis acts to increase the length and reduce the stiffness of the SEE. If the CE length and the MTU strain pattern remain constant, then the overall length change of the MTU will increase. Therefore, to achieve the required length changes the CE or the aponeurosis must perform work. The aponeurosis can be viewed as a second spring acting in series with the tendon which acts to reduce the SEE stiffness. This reduction in SEE stiffness enables the SEE to stretch further for the same force and store more of the work of the CE, thus ensuring that the efficiency of the CE remains high. The structural properties of the AT can vary substantially between individuals (Lichtwark and Wilson, 2005c; Hof, 1998; Maganaris and Paul, 2002), as can the force generating capacity and the architecture of the muscle (Fukunaga et al., 2001b; Ichinose et al., 1998; Kawakami et al., 1998). Therefore, the way a muscle interacts with the tendon will also change depending on architecture. Recent studies have suggested that AT may be stiffer for people with greater muscle strength (Muraoka et al., 2005) and
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hence the current model could be used to determine how muscle strength and tendon stiffness variations might affect muscle efficiency. However, to perform such a study, one must also obtain better estimates of the force–length, force–velocity and heat–velocity properties of the muscle. The distribution of activation between the muscles of the triceps surae should also be considered. In the current model, estimates commonly used in musculoskeletal models have been implemented in the model; however, these relationships may vary greatly amongst individuals, particularly when taking into account fascicle-type and architecture differences. We assumed that ankle and knee kinematics would not change with tendon compliance. The effect of AT stiffness on locomotion has been elegantly studied during walking and running by Hof et al.(2002). They showed that the triceps surae muscle and tendon interacted differently depending on tendon stiffness, particularly in the load sharing between the muscles of the triceps surae muscle group. The ankle kinematics of participants with large differences in tendon stiffness were, however, remarkably similar. This is unsurprising because during walking and running, the base of the foot will be in contact with the ground at mid-stance and ankle angle will therefore be dependent on sweep of the distal leg, which is primarily a function of speed. Any effect would therefore be limited to the period of plantar flexion in late stance. Knee angle will have a small influence on leg angle and MTU length but this will have more effect between walking and running as it is hard to envisage running with a straight knee or using a significantly more flexed knee in either gait (Groucho walking or running). Hof (2003) suggests that the time course of force production may also be influenced by the stiffness of the tendon, however this effect also depends on the effective mass being supported by the tendon. There is considerable variation in AT stiffness between individuals (Hof, 1998; Lichtwark and Wilson, 2005c; Maganaris and Paul, 2002) and future studies might concentrate on how tendon stiffness in the biological range might influence kinematics and energetics and how the two might be coupled. The current model neglects some physiological phenomenon that may influence our results. The model assumes a linear tendon stiffness (180 N mm 1), however the AT has been shown to have a significant, non-linear, toe region (Maganaris and Paul, 2002). This will influence our results because, particularly at low forces, the muscle will be able to shorten more against the smaller resistance to stretch. It has also been suggested that the SEE stiffness can be varied by changing the muscle activation level and hence the amount of aponeurosis that is strained (Hof, 1998). Therefore, perhaps during walking and running, where the activation levels vary substantially, the stiffness of the SEE can be adjusted to optimise efficiency of the muscle fascicles further. The effect of pennation angle was also ignored in the model, however the pennation angle range during the stance phase is only 10–25 degrees throughout
the stance phase — (Lichtwark and Wilson, 2006). Therefore the effect of pennation angle on muscle fascicle length calculation is small (2–8% length error). In conclusion, we have found that the AT stiffness is optimal to achieve the highest efficiency in both walking and running. We have demonstrated that muscles act at high values of efficiency by contracting fibres at favourable speeds, often very different from the speed of the whole MTU, and the fibres are deactivating during periods of shortening. This is achieved by utilisation of SEE elasticity. Changing the elastic properties of the SEE, making it either more or less stiff, would reduce the efficiency of the muscle greatly. We suggest that selective recruitment of SEE structures such as aponeurosis with changes in muscle activation can adjust the stiffness of the MTU and perhaps optimise muscle efficiency further under different gait conditions and power requirements. Acknowledgements The authors would like to acknowledge their funding bodies. Glen Lichtwark receives financial support from the British Council Overseas Research Fellowship and the Royal National Orthopaedic Hospital Trust Research and Development Scholarship. Alan Wilson receives support as a BBSRC Research Fellow and holder of a Royal Society Wolfson Research Merit award. We would also like to acknowledge Anna Wilson for her help with the manuscript. Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech. 2006.07.025. Appendix B. Supplementary data
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