The influence of in vivo mechanical behaviour of the Achilles tendon on the mechanics, energetics and apparent efficiency of bouncing gaits Free

During locomotion, the human body can be considered as a biological engine that utilizes chemical energy to perform mechanical work. A third parameter, the locomotion efficiency (i.e. the ratio between the mechanical work performed and the energy cost), can then be calculated and used to describe the interaction between energy expended and work produced in a given locomotion task (Margaria, 1968).

Mammalian skeletal muscle performs (positive) mechanical work with a muscle efficiency of about 0.25 (0.10–0.34) (e.g. Hill, 1964; Margaria, 1968; Woledge et al., 1985), but in locomotion studies (when mechanical work is calculated at the whole-body level), efficiency is reported to be lower than, equal to or larger than that. Locomotion (apparent) efficiency approximates muscle efficiency values in the forms of locomotion where elastic recoil is nil or negligible (e.g. cycling and swimming; Minetti et al., 2001; Zamparo et al., 2002). When tendons are used as energy savers, ‘apparent’ efficiency can be larger than muscle efficiency, as in running (e.g. 0.60–0.70; Cavagna and Kaneko, 1977; Farris and Sawicki, 2012a), hopping (e.g. 0.50–0.60; Thys et al., 1975; Voigt et al., 1995) or bouncing (e.g. 0.45; Dean and Kuo, 2011). These ‘higher than expected’ efficiency values have been taken as evidence that some of the work generated at the whole-body level is ‘provided’ by tendons (e.g. Alexander and Bennet-Clark, 1977; Alexander and Vernon, 2009; Roberts et al., 1997; Taylor, 1985). In other words, by comparing muscular and apparent efficiency values, it is possible to get insight into the relative role of muscle fibre shortening versus elastic tendon recoil in overall muscle–tendon positive work (e.g. Sawicki and Ferris, 2008).

The underpinning mechanism of the effects of tendon behaviour on energy expenditure is twofold. In energy-saving tasks, during the first half of the stance phase, tendons stretch, storing elastic strain energy in their structure (Alexander and Bennet-Clark, 1977; Roberts and Azizi, 2011; Roberts et al., 1997); during the propulsive phase, part of this energy can be released to propel the body, reducing the need to obtain mechanical energy from the working muscles (e.g. Biewener et al., 1998; Roberts, 2002). Moreover, when the elastic structures accommodate the largest part of the muscle–tendon unit (MTU) displacement during contraction, the muscle fascicles can operate under more isometric conditions, thus reducing their energy expenditure (de Tombe and ter Keurs, 1990; Fletcher and MacIntosh, 2017; Gordon et al., 1966; Smith et al., 2005; Umberger and Martin, 2007).

The role of the elastic structures in determining energy expenditure has been studied extensively in animal locomotion (e.g. Biewener et al., 1998; Roberts, 2002), in modelling studies (e.g. Wakeling et al., 2020; Umberger and Rubenson, 2011; Lichtwark and Wilson, 2007) or in vitro (e.g. MTU preparations; de Tombe and ter Keurs, 1990; Gordon et al., 1966; Robertson and Sawicki, 2015). In humans, hopping and bouncing have mainly been used as experimental models as these are (primarily vertical) motor tasks that involve the plantar-flexors as primary propulsive muscles (Farris and Sawicki, 2012b; Farris et al., 2013; Dean and Kuo, 2011) and where changes in frequency provide different mechanical stimuli for the Achilles tendon (Robertson and Sawicki, 2015; Farris et al., 2013).

To our knowledge, only a few studies have specifically investigated the role of the elastic structures in determining apparent efficiency in humans. Dean and Kuo (2011) used a mechanical muscle–tendon model to quantify elastic and muscle work contribution during a cyclical bouncing task; they observed that the gap between the values of (measured) apparent efficiency and (estimated) muscle efficiency (assumed to be equal to 0.25) could indeed be explained by mechanical tendon behaviour (estimated using a mechanical model). Sawicki and Ferris (2008, 2009) estimated the mechanical power delivered by exoskeletons (in level and inclined walking) from measures of metabolic power by assuming a value of muscle efficiency of 0.25; they then estimated the apparent efficiency of ankle–muscle tendon work based on these data. The gain between apparent and muscular efficiency can be utilized to obtain insight into underlying muscle–tendon function. As suggested by these authors, non-invasive in vivo techniques are necessary to further investigate the relationship between the mechanics and energetics of the lower limb muscle–tendon mechanical function during locomotion.

This was recently done by Monte et al. (2020a), who quantified the mechanical work done by the series elastic components of the gastrocnemius medialis MTU in vivo by means of ultrasound analysis; in their study, a significant positive correlation was observed between tendon work and apparent efficiency during running at increasing speeds (the larger the tendon work, the larger the apparent efficiency). Monte et al.’s (2020a) study thus provided strong evidence that the gap between apparent efficiency and muscle efficiency can be explained by the presence of an energy-saving mechanism or, in other words, that some of the work generated at the whole-body level is ‘provided’ by tendons. These authors estimated that, in running, apparent efficiency is reduced to 0.33 when tendon work is nil, a value somewhat larger than ‘expected’ (0.25). This ‘residual gap’ on the one hand could indicate that care should be taken when muscle efficiency is assumed to be equal to 0.25 (as done in previous studies), but on the other could be explained by methodological issues. One of these relates to the calculations of whole-body mechanical work from the sum of internal and external components; this approach has been utilized in many studies of human locomotion (e.g. Cavagna and Kaneko, 1977; Minetti et al., 2001; Zamparo et al., 2002) to which the reader is referred for further details. Another issue relates to the computation of the elastic component itself; indeed, human running is a complex 3D movement task where elastic energy could be provided by elastic structures other than the Achilles tendon, such as the patellar tendon (e.g. Monte et al., 2020b) and the arch of the foot (e.g. Kelly et al., 2018; Ker et al., 1987).

To better understand the role of the elastic structures in the apparent efficiency of human locomotion, a simpler energy-saving task should be investigated, possibly involving a single elastic structure (the Achilles tendon), a single joint (the ankle) and a single direction of movement (vertical). In this regard, bouncing or hopping are also good experimental models because the internal work (associated with the reciprocal movements of the limbs with respect to the centre of mass and to their rotational energy) is negligible in these tasks and thus mechanical work at the whole-body level (and hence apparent efficiency) could be calculated more accurately than in running.

Finally, recent studies of Bohm et al. (2019) and Monte et al. (2020b) suggest a further mechanism that is expected to explain the energy demands (and thus apparent efficiency behaviour) in running. In particular, Bohm and co-workers (2019) suggest that the main mechanism for the reduction of fascicle shortening velocity during the stance phase in running is not tendon displacement per se but tendon gearing (Gt). Gt can be calculated as the ratio between MTU length changes and belly length changes (Gt=ΔMTU length/Δmuscle belly length) and is a measure of the uncoupling between MTU and muscle belly behaviour as, for a given MTU length change, the muscle belly length change depends on the length change (and compliance) of the external tendon (Wakeling et al., 2011). Tendon compliance allows the muscle belly length to change at rates close to the optimal muscle fibre velocity even though the MTU may be changing its length at a different rate (Lichtwark and Barclay, 2010; Wakeling et al., 2011). Therefore, a high Gt indicates that the muscle in that MTU is contracting more isometrically.

Therefore, the purpose of this study was to combine in vivo ultrasound measurements of the gastrocnemius medialis MTU with metabolic, kinematic and kinetic measurements during hopping at different frequencies (as this provides different mechanical stimuli for the Achilles tendon) to better investigate the relationship between tendon mechanics and apparent efficiency. Our hypotheses were as follows: (1) there is a significant positive correlation between tendon power and apparent efficiency (as previously found in running by Monte et al., 2020b) across frequencies; (2) the hopping frequency at which apparent efficiency is maximized corresponds to that at which the mechanical power released by the Achilles tendon is highest (as suggested by Dean and Kuo, 2011, and Robertson and Sawicki, 2015); (3) there is a positive relationship between tendon kinematics (e.g. tendon recoil and Gt) and tendon kinetics (e.g. mechanical power released by the tendon); and (4) Gt is a stronger determinant of tendon power than tendon recoil, as the latter does not take into account the speed of movement.

Whereas investigating the gap between apparent efficiency and muscle efficiency provides insight into tendon behaviour, investigating the gap between total mechanical power and tendon power should give insight into muscle power and muscle efficiency (i.e. muscle behaviour) during locomotion in vivo. Therefore, a further aim of this study was to estimate the mechanical power generated by the contractile components based on values of mechanical power at the whole-body level and tendon power: we expected that the hopping frequency at which muscle power is minimized corresponds to that at which the mechanical power provided by the Achilles tendon is highest (as suggested by Dean and Kuo, 2011, and Robertson and Sawicki, 2015).

Twenty healthy males (25±3 years of age, 75±7 kg body mass, 1.78±0.06 m tall) participated in this study. All participants were free from any type of muscular injury (the only exclusion criteria). The sample size was calculated to observe an effect size of 1 with an error probability of 0.05 and power of 0.8 (G*Power v. 3.1.9.3.); the effect size was estimated based on a previous study (Monte et al., 2020a). The study was performed in agreement with the Declaration of Helsinki for the study of human subjects. The local ethical committee approved the experimental protocol (protocol number 2020-UNVRCLE-0142370) and all subjects gave their written informed consent.

The participants were instructed to hop in place with both feet (with their hands placed on their hips) over two force platforms at four different frequencies: 2, 2.5, 3 and 3.5 Hz (imposed by a metronome). These frequencies were selected based on previous studies (e.g. in the same range as those investigated by Farris and Sawicki, 2012b) and their order was randomized. During these experiments, kinematic, kinetic, ultrasound and metabolic data were collected. The experiments were performed on two different days (two trials per day) and at least 30 min of rest was observed between consecutive trials on the same day to avoid the effects of fatigue. Before the beginning of the experiments, the subjects were familiarized with the equipment and the procedures.

A 3D motion capture system (8 cameras; Vicon, Oxford, UK) was used to record the three-dimensional trajectories of 49 markers (a customized full-body Plug-in-Gait, Vicon) at 250 Hz; a static standing trial was also captured and used to calibrate the individual subject models using established inertial parameters (Winter, 1979).

The 3D ground reaction forces (GRFs) were synchronously collected at 1500 Hz using two force platforms (AMTI and Kistler ) embedded in the floor. Participants hopped with one foot on each force plate, and thus the GRFs were determined separately for the left and right foot. Kinematic and dynamic data were recorded by means of Nexus software (Oxford Metrix, v. 2.6). The muscle–tendon junction (MTJ) of the gastrocnemius medialis (GM, right leg) was synchronously recorded at 50 Hz using B-mode ultrasonography and associated software (Health care S7 Pro, GE Healthcare). This sample frequency was selected in accordance with Waugh et al. (2017) and represents a good approximation between frame rate and image quality. The position of the MTJ was visualized 1–2 cm medial to the inter-muscular septum separating the two gastrocnemius heads (Lichtwark and Willson, 2005; Waugh et al., 2017) using an ultrasound probe that was secured to the skin surface using a custom-made foam cast and elastic bandages. This position was selected to have a good representation of the MTJ displacement.

The 3D position of the probe was identified (by means of motion analysis) using three markers placed on the probe involucre, in order to take into account the probe movements in the calculations of tendon length (Lichtwark and Willson, 2005; Waugh et al., 2017).

Oxygen uptake (V̇_O2) during each hopping trial was determined by means of a breath by breath metabolimeter (K5, Cosmed, Albano Laziale, Italy). Baseline data collection in a standing position was performed for 5 min before the tests, and data were collected for 4 min during exercise; data collected in the last minute of rest/exercise were averaged and used in further analysis.

Blood samples were obtained from the earlobe 2 min before test onset, and at the 3rd, 5th and 7th minute after exercise, while the subject was seated, for blood lactate determination (Biosen, 5030, EKF, Barleben, Germany).

All equipment, with the exception of the metabolimeter, were synchronized by an external trigger.

Kinematic, kinetic and ultrasound data were recorded in three different time windows: after 1 min, after 2 min and in the middle of the last minute of exercise. Data were analysed by means of a customized Matlab program (v.2019a), available by reasonable request from the corresponding author. Ten hops (contact+flight time, see Fig. 1) were analysed for each time window; the mean values of these 30 hops were then calculated for each parameter. As indicated above, metabolic data were analysed in the last minute of exercise.

Marker trajectories were filtered with a forward and reverse low-pass, 2nd order Butterworth filter with a cut-off frequency of 10 Hz, whereas the GRFs were filtered through a forward and reverse low-pass, 4th order Butterworth filter with a cut-off frequency of 30 Hz (consistent with the Nyquist theorem).

Angular position of the ankle, knee and hip joints at touch down and take off, as well as the maximum angular displacement (from touch down to maximum flexion) were calculated using inverse kinematics; joint power (right leg) was calculated using inverse dynamics (the mean values during the contact phase were calculated and reported in further analysis) (Farris and Sawicki, 2012b; Farris et al., 2013; Lai et al., 2016).

The mechanical power, at the whole-body level (P_MEC), to raise and accelerate the body centre of mass (BCoM) with respect to the environment was calculated based on the summation of the increases in total mechanical energy (E_T) during the hop: P_MEC=ΔE_T/Δt (where Δt is the duration of contact and flight phases). The time course of E_T was calculated as the sum of the time course of potential (E_P) and kinetic (E_K) energy based on the 3D BCoM trajectory and speed, respectively. BCoM position, in turn, was calculated by a double integration of the GRF signal (both legs) according to Cavagna and Kaneko (1977) and by taking into account, as the integration constant, the average 3D speed of the BCoM (see Saibene and Minetti, 2003). Integration constants were set as the average speed over a complete hop, in each direction. In this way, the trajectory of the BCoM in space will return, after one complete hop, to the starting position (for more details, see Saibene and Minetti, 2003; Pavei et al., 2017).

P_MEC thus corresponds to the positive external mechanical power, as calculated in many studies on human locomotion (Cavagna and Kaneko, 1977; Monte et al., 2020b; Pavei and Minetti, 2016; Zamparo et al., 2016). The internal mechanical power (also calculated in these studies) takes into account the reciprocal movement of the upper and lower limbs and the rotational kinetic energy of the body segments. This component was found to be negligible (less than 0.05 W kg⁻¹ at all frequencies) and was not accounted for in further analysis.

As shown in Fig. 1, the overall (positive) changes in total energy during a single hop correspond to the overall (positive) changes in potential energy; thus, P_MEC could also be more simply calculated as m·g·h/t, where m is the subject's mass, g is acceleration due to gravity, h is the total vertical displacement of the BCoM (see Table 1) and t is the sum of contact and flight time.

Data collected in the last minute of rest/exercise were averaged and used in further analysis. A steady state in V̇_O2 was observed after 2 min of exercise in all tests. The percentage difference in V̇_O2 between the 2nd and 4th minute was: 2.9%, 0.4%, 4.3% and 3.8% at 2, 2.5, 3 and 3.5 Hz, respectively. V̇_O2 at rest was then subtracted from V̇_O2 obtained during exercise, in order to obtain net oxygen consumption (V̇_O2,net, expressed in ml O₂ min⁻¹ kg⁻¹). The net metabolic power derived from aerobic energy sources (P_MET,Aer) was then expressed in W kg⁻¹ by using an energy equivalent (J ml⁻¹ O₂) that takes into account the respiratory exchange ratio (Garby and Astrup, 1987).

Net blood lactate accumulation was calculated from the difference between the highest value of lactate recorded after exercise and lactate concentration at rest, determined immediately before the exercise (di Prampero and Ferretti, 1999). The metabolic power derived from anaerobic lactic energy sources (P_MET,AnL) was calculated by multiplying net lactate for an energy equivalent of 3.3 ml O₂ kg⁻¹ (mmol l⁻¹)⁻¹ (di Prampero and Ferretti, 1999) and by dividing it by total exercise duration (e.g. 4 min); P_MET,AnL is thus expressed in ml O₂ min⁻¹ kg⁻¹. Net P_MET,AnL was then expressed in W kg⁻¹ (as for P_MET,Aer). Total net metabolic power (P_MET) was then calculated as P_MET,Aer+P_MET,AnL.

Apparent efficiency (AE) was finally calculated as the ratio P_MEC/P_MET (both expressed in W kg⁻¹).

The Achilles tendon length was defined as the distance between tendon insertion (on the calcaneus) and the MTJ of the gastrocnemius medialis. The 3D position of tendon insertion was recorded by attaching a marker in this position (identified using ultrasound image).

As indicated above, MTJ displacement was investigated by B-mode ultrasound imaging (Health care S7 Pro, GE Healthcare) using a linear array probe (4.5 cm) collecting at 50 Hz; the probe was positioned over the MTJ and oriented to clearly display both the separation point between the aponeuroses and the GM MTJ; the probe was secured to the skin surface using a custom-made foam cast and elastic bandages. The 2D position of the GM MTJ was manually identified, frame by frame, using open source software (Tracker 4.95; www.physlets.org/tracker/). The position of the MTJ was then reconstructed into the 3D coordinate system of the laboratory as proposed by Waugh et al. (2017) and Lichtwark and Wilson (2005).

Briefly, video and ultrasound data were collected synchronously through the motion capture system using an analog trigger. Using the coordinates of three reflective markers positioned over the ultrasound probe (proximal, distal and in the middle of the field of view), it was possible to extrapolate a 3D coordinate representing the mid-point of the probe scanning interface (with the assumption that the markers and probe were a rigid body). As this coordinate is analogous to the mid-point of the ultrasound image x-axis, and the angle of the ultrasound image relative to the 3D motion capture reference frame is represented by the markers on the ultrasound probe, the ultrasound imaging plane could then be transposed into the 3D motion capture reference system. This transformation allowed the position of the GM MTJ to be estimated within the 3D motion capture reference frame.

The instantaneous MTU length of GM during the contact phase was computed based on data of joint angles and segment lengths as proposed by Hawkins and Hull (1990), whereas the muscle belly length (which includes the proximal GM tendon) was calculated by subtracting the Achilles tendon length from the MTU length. Based on these data, the peak values of MTU, Achilles tendon and muscle belly displacement were calculated. Tendon recoil was calculated as the difference between its maximum strain and the Achilles tendon length at take-off (Monte et al., 2020a). Gt was calculated as the ratio between the MTU length changes and the length changes of the muscle belly (Gt=ΔMTU length/Δmuscle belly length) (Dick and Wakeling, 2017; Wakeling et al., 2011). The average value of Gt during the contact phase was finally calculated.

The mechanical power released by the Achilles tendon (P_TEN) was calculated as the area under the force–elongation curve divided by contact time (Lichtwark and Wilson, 2005b). The values of P_TEN utilized in further calculations were multiplied by 2 to take into account the contribution of both legs. We thus implicitly assumed that muscle–tendon behaviour is the same on both limbs, as their movement is synchronized during this motor task (i.e. we assumed as negligible the eventual right/left asymmetries).

The Achilles tendon force was calculated as the ratio between the ankle joint moment and the tendon lever arm. The latter was measured by taking into account the Achilles tendon curvature, as the distance from the tendon's line of action to the centre of rotation of the ankle, as suggested by Rasske et al. (2017). Briefly, four markers were positioned on the medial malleolus (representing the ankle centre of rotation), insertion of the Achilles tendon, and at 5 and 10 cm proximal to the calcaneus (to represent the Achilles tendon line of action) and their location was determined by ultrasound scanning.

More specifically, the energy recovered from the tendon is represented by the area under the descending limb of the force–elongation curve and the positive mechanical power released by the tendon corresponds to this energy in the unit of time. Finally, Achilles tendon hysteresis was calculated by dividing the difference between the area under the loading and the unloading curves by the area under the loading curve alone (Lichtwark and Willson, 2005).

The mechanical power generated by the contractile components (P_MUS) was estimated by subtracting the Achilles tendon mechanical power from the mechanical power at the whole-body level: P_MUS=P_MEC−P_TEN. As calculated, P_MUS does not represent the GM contribution alone but is an estimate of the mechanical power provided by all contractile components. However, in hopping, the GM MTU is prevalent in determining both contractile and elastic behaviour and it is thus reasonable to assume that the contractile components of the GM MTU contribute most to P_MUS (this is argued in more detail in the ‘Limitations’ section at the end of the Discussion). Muscle efficiency was finally calculated as the ratio between the mechanical power done by the contractile components and metabolic power (P_MUS/P_MET).

The values are presented as means±s.d. All data were tested for normality using the Shapiro–Wilk test. To test for statistical differences in dependent variables among hopping frequencies, a one-way ANOVA with repeated measures was performed using SPSS software (v. 23; IBM). The independent variable for the ANOVA was the hopping frequency: 2, 2.5, 3, 3.5 Hz. If a significant main effect was found, a Bonferroni post hoc test was utilized. Finally, Pearson's correlation coefficient was used to check for possible correlations between apparent efficiency and tendon power as well as between P_TEN and tendon gearing. The level of significance was selected as P<0.05.

The effect of frequency is reported throughout Results as a main effect, whereas post hoc tests are reported in tables and figures.

The (positive) mechanical power produced at the joint level decreased as a function of hopping frequency (P=0.023) in all lower limb joints (Table 1). The relative contribution of the ankle joint to overall joint power (sum of ankle, knee and hip) increased as a function of hopping frequency (from 57% at 2 Hz to 76% at 3.5 Hz); the contribution of the knee joint decreased from 30% at 2 Hz to 11% at 3.5 Hz, whereas that of the hip joint was about 13% at all frequencies.

Maximum joint flexion decreased as a function of hopping frequency (P<0.001), whereas no effects of frequency were observed at touch-down or at take-off in the ankle, knee and hip angles (see Table 1).

MTU, muscle and tendon behaviour are reported in Fig. 2 (left): muscle belly length changes were the lowest at the intermediate frequencies and tendon length changes accounted for the larger part of the MTU length changes in all conditions. Peak values of MTU, muscle and tendon strain (Fig. 2, right) were all affected by hopping frequency (P<0.001). The relative contribution of Achilles tendon length changes to MTU length changes increased from 86% at 2 Hz to 94% at 3 Hz to further drop to 60% at 3.5 Hz; Gt followed the same trend: it increased from 5.5 at 2 Hz to 6.6 at 3 Hz and then decreased to 4.6 at 3.5 Hz (Table 2). Tendon recoil decreased steadily with hopping frequency (P<0.001), whereas Achilles tendon hysteresis was unaffected by it (Table 2).

The positive mechanical power released by the Achilles tendon (P_TEN) changed as a function of hopping frequency (P<0.001) and accounted for more than 45% of the mechanical power at the whole-body level (P_MEC); the highest value of tendon power was observed at 3 Hz, where Gt is the highest, but not tendon recoil (which was highest at the lowest frequency).

The mechanical power at the whole-body level (P_MEC) decreased steadily as a function of hopping frequency (P<0.001), following the trend of BCoM displacement and peak vertical GRF. Total metabolic power (P_MET) changed with hopping frequency following a U-shaped function and was lowest at the intermediate frequencies (see Table 2); the aerobic contribution to P_MET was about 90–92% at all frequencies. Apparent efficiency changed as a function of hopping frequency with an opposite trend compared with P_MET, with the highest values at 2.5–3 Hz (see Table 2). The mechanical power of the contractile components of the GM MTU (estimated as: P_MUS=P_MEC−P_TEN) followed the same trend of P_MET (P<0.001) and was lowest at the intermediate frequencies (2.5–3 Hz, see Table 2). Therefore, from a metabolic point of view, the optimal hopping frequency is at about 2.5–3 Hz, where P_MET and P_MUS are lowest, apparent efficiency is highest and where the Achilles tendon changes explain the largest part of the MTU length changes (i.e. where Gt is highest).

Positive correlations were observed between P_TEN and P_MEC, P_MET and apparent efficiency (AE) (see Fig. 3, where data points are individual values and refer to all hopping frequencies). These correlations indicate that larger values of P_TEN are associated with larger values of P_MEC (Fig. 3A) and that the subjects with the highest tendon work (P_TEN) are those with the lowest P_MET (Fig. 3B) and the largest apparent efficiency (Fig. 3C). In this last case, the intercept with the y-axes indicates the value of apparent efficiency that could be expected were the Achilles tendon not operating as an energy saver: AE=0.22+0.15P_TEN (R²=0.67, N=80, P<0.001). Gt is also positively related to apparent efficiency (Fig. 4B): AE=0.23+0.05Gt (R²=0.50, N=80, P<0.001), the y-intercept being similar to that of the apparent efficiency versus P_TEN relationship.

A positive correlation was also observed between P_TEN and Gt: P_TEN=0.13+0.31Gt (R²=0.73, N=80, P<0.001): the power released by the tendon increases with the fraction of the MTU displacement explained by tendon displacement (Fig. 4A). P_TEN is also related to tendon recoil (TR) but this correlation is less strong than in the case of Gt: P_TEN=0.99+0.48TR (R²=0.34, N=80, P=0.005), suggesting that Gt plays a more important role in tendon mechanical power release. No relationship was observed between P_TEN and tendon hysteresis.

By knowing the mechanical power at the whole-body level and tendon power, it is possible to calculate the mechanical power done by the contractile components in vivo (P_MUS=P_MEC−P_TEN); the changes in P_MUS as a function of hopping frequency mirror the changes in P_TEN (Fig. 5A). Finally, muscle efficiency can (also) be estimated from the ratio between the mechanical power done by the contractile components and metabolic power (P_MUS/P_MET): it amounts to about 0.22, with no significant changes as a function of hopping frequency (Fig. 5B).

This study was designed to investigate the relationship between tendon mechanical behaviour (in vivo) and its energy-saving capacity; more specifically, it was designed to investigate the relationship between tendon power and apparent efficiency (in bouncing gaits). Hopping was utilized as an experimental model because it is a simple movement task involving mainly a single elastic structure (the Achilles tendon), a single joint (the ankle) and a single direction of movement (vertical). Further, changes in hopping frequency can be utilized to provide different mechanical stimuli to the Achilles tendon.

The data indicate that our hypotheses are accepted. The hopping frequency at which apparent efficiency was maximized (2.5–3 Hz) corresponds to the frequency at which the mechanical power provided by the Achilles tendon (P_TEN) was highest and muscle mechanical power was lowest; moreover, at this frequency, Gt was also maximized. P_TEN and Gt were positively correlated with AE: the intercept of this relationship (0.22–0.23) is closer to the expected values of muscle efficiency than previously estimated in running. The mechanical power released by the tendon is related to its kinematical features: the larger the tendon gear (and to a smaller degree the elastic recoil), the larger the mechanical power released.

Data reported in this study, therefore, point to the role of Gt (a measure of the uncoupling between MTU and belly velocity) in determining the tendon’s capability to store and release elastic energy in vivo. Indeed, tendon recoil, per se, cannot explain the U-shaped behaviour of P_MET and apparent efficiency (as it decreases steadily as a function of hopping frequency), whereas Gt and tendon power show a U-shaped behaviour and were highest at the same frequency that optimizes P_MET and apparent efficiency.

In agreement with previous studies, our data indicate that tendon length changes accompany the largest part of the MTU length changes during hopping at all the investigated frequencies (Waugh et al., 2017; Lichtwark and Willson, 2005). As a consequence, the tendon strain that occurs during the absorption phase could be recoiled during the propulsive phase, releasing most of the shortening work required for take-off. In contrast, the muscle belly (plus the elastic structures of the proximal tendon) only stretches and shortens by small amounts during hopping at different frequencies, thereby reducing muscle fibre work (Lichtwark and Willson, 2005a; Fletcher and MacIntosh, 2017; Gordon et al., 1966; Roberts and Azizi, 2011). This behaviour allows the muscle fibres to act almost isometrically, and thus to produce more force with a lower active muscle volume (Roberts, 2002; Bohm et al., 2019; Monte et al., 2020a). Indeed, we observed a negative correlation between Achilles tendon mechanical power and metabolic energy expenditure.

Even though, in this study, we did not measure GM fascicle displacement directly, the muscle belly length data reported here are comparable with those reported by Lichtwark and Wilson (2005b) and Waugh et al. (2017), who observed GM fascicle displacements ranging from 10% to 20% of their resting length during hopping at different frequencies, reinforcing the idea that the plantar-flexor muscles operate at the upper region of their force–length and force–velocity relationships in different locomotor tasks. This behaviour allows the efficiency of muscle contraction to be maximized (Hill, 1964). Without tendons, the fascicle length changes would be higher, increasing the cross-bridge turnover and the energy demands for muscle contraction (Woledge et al., 1988). Furthermore, as the force per cross-bridge decreases with increasing fascicle length changes (de Tombe and ter Keurs, 1990), a decrease in the muscle force potential requires upregulation of the muscle activation to maintain the same level of force to support and accelerate the BCoM, thereby increasing the energy cost of contraction and reducing muscle efficiency (Fletcher and MacIntosh, 2017).

The percentage of MTU length changes explained by the Achilles tendon is the highest at about 3 Hz (see Table 2); accordingly, muscle belly excursion is minimized at this frequency, reducing the length changes of the active component and this is expected to reduce the amount of positive power performed by the contractile elements, as we indeed found in our study. At lower or higher frequencies, the percentage of MTU length changes explained by the tendon length changes decreases, and this indicates that the body did not behave like a Hookean mass–spring system, possibly in an attempt to avoid excessive BCoM excursion (Taylor, 1985). Indeed, as suggested by Robertson and Sawicki (2015) and Farley et al. (1991), resonance hopping frequency is somewhere around 2.5 Hz. Hopping at a higher or lower frequency could affect the mechanical behaviour of the ‘spring’, increasing the amount of work produced by the contractile elements and reducing the elastic energy released by the elastic structures. To compensate for the higher mechanical demands imposed by a low hopping frequency (e.g. larger vertical peak GRF and higher BCoM vertical displacement; Table 1), larger values of muscular mechanical work are needed to propel the system, thus increasing the metabolic demands. At the other extreme, at high hopping frequencies, we can assume that the force acting along the MTU is not sizeable enough to strain the tendon, reducing the Achilles tendon contribution and its positive effects on the metabolic demands.

As shown by data reported in Table 2, the underpinning mechanism that explains the role of the tendon in hopping mechanics and energetics is not the amount of recoil per se (e.g. tendon excursion), which decreases steadily as a function of hopping frequency, but (rather) tendon power, which shows an ‘optimization’ pattern, being maximal at about 3 Hz, as is the case for Gt (see Table 2). At this same hopping frequency, the metabolic demand is minimized and the apparent efficiency is highest (Table 2 and Fig. 5B). A strong correlation between P_TEN and apparent efficiency should then be expected, as indeed shown in this study (see Fig. 3C).

A strong correlation was also observed between P_TEN and Gt (see Fig. 4A) as both take into account the time course of tendon excursion; Gt is thus expected to play an important role in determining the physiological demand in energy-saving tasks. Indeed, Gt is a measure of the uncoupling between MTU length changes and muscle belly length changes: a high Gt indicates that the muscle is contracting more isometrically, as the largest part of the MTU displacement is accommodated by the elastic structures, and this could potentially increase the amount of elastic energy stored and released.

The mechanical power generated by the contractile components (P_MUS) was calculated based on values of P_MEC (mechanical power at the whole-body level) and P_TEN. This calculation is based on the assumption (see the ‘Limitations’ section, below) that the elastic components of the plantar-flexor muscles contribute most to P_TEN and that the contractile components of the plantar-flexor MTU contribute most to P_MUS.

Indeed, P_MUS behaviour is consistent with muscle belly behaviour (i.e. muscle belly shortening is reduced at intermediate frequencies; see Figs 2 and 5A). Moreover, P_MUS mirrors P_TEN behaviour (Fig. 5B), and this is consistent with data reported by Robertson and Sawicki (2015). These authors observed that, in an in vitro MTU preparation (bullfrog hindlimb), the contributions of the contractile and elastic elements to overall mechanical power exhibit an opposite behaviour as a function of frequency. Similar results (at least in the range of frequencies used in the present study) were reported by Dean and Kuo (2011), who estimated muscle and tendon work in humans using a mechanical model during a bouncing task. Thus, these calculations support the idea of an important role of the elastic structures in reducing metabolic energy expenditure during in vivo energy-saving tasks in humans.

In this study, we provide estimates of muscle efficiency in vivo by means of two complementary methods, both based on the consideration that the gap between apparent efficiency and muscular efficiency is expected to be nil when tendon power is nil. (1) The y-intercept of the relationship between apparent efficiency and tendon mechanical power (or tendon gearing) indicates a condition when tendon mechanical contribution is nil or negligible and thus corresponds to the efficiency of ‘pure’ muscular work (0.22–0.23; Figs 3C and 4B). (2) When the contribution of tendon mechanical behaviour to total mechanical work at the whole-body level is taken into account, the ratio between mechanical and metabolic power (i.e. apparent efficiency) is a measure of the efficiency of the contractile components (about 0.22; Fig. 5B).

The estimates of muscle efficiency reported in this study are closer to the ‘expected’ muscle efficiency values (0.25) than previously estimated in running (0.33; Monte et al., 2020b); thus, by using a simpler motor task (that primarily involves the ankle plantar flexors as propulsive muscles), we were able to better elucidate the role of the elastic structures and to better estimate the contribution of the active muscle in determining mechanical power at the whole-body level.

There are certain limitations to the approach used in the present study that require consideration. The mechanical power developed by the ‘contractile components’ was calculated as P_MUS=P_MEC−P_TEN. In turn, the mechanical power developed by the ‘elastic components’ (P_TEN) was calculated based on measures of the GM MTJ displacement with the assumption that: (i) the behaviour of the elastic components of the GM MTU is representative of Achilles tendon behaviour; (ii) the ankle plantar flexors contribute most to overall joint power in this motor task; and (iii) negligible elastic energy could be provided by elastic structures other than the Achilles tendon.

To our knowledge, no data are reported in the literature regarding the effects of hopping frequency on tendon gearing or tendon work for the other triceps surae muscles. As far as running is concerned, Lai et al. (2014), using a muscle–tendon model where each plantar flexor was modelled as a single MTU, showed that the soleus MTU stores and releases higher values of elastic strain energy compared with the GM at increasing running speeds, and Bhom et al. (2019) investigated tendon gearing of the soleus MTU at a slow running speed. The soleus is the largest of the plantar flexor muscles and its contribution to ankle work is probably larger than that of the GM. Further studies are thus necessary to understand whether our estimates of P_TEN (and hence P_MUS) in hopping would differ when calculated based on data from a different plantar-flexor MTU.

As a function of hopping frequency, the ankle joint contribution increased from 57% to 76%, that of the knee joint decreased from 30% to 5% and that of the hip joint was about 13% (irrespective of frequency). Thus, this assumption ‘holds’ at high frequencies, whereas at low hopping frequencies, the knee joint contribution is probably not negligible. This means that the changes in metabolic energy expenditure (and hence apparent efficiency) across frequencies could partially be attributable to changes in joint work distribution.

During hopping, elastic energy could also be provided by the patellar tendon and the arch of the foot. Regarding the latter, to our knowledge, no data have been reported in the literature about the work/power contribution of these structures (in hopping or in human locomotion) as a function of frequency; it is therefore difficult to estimate their effect on P_TEN. Regarding the former, in a recent study we investigated the joint function of the lower limbs during hopping at different frequencies (Monte et al., 2021; same subjects, equipment and procedures as in this study) and we observed that, at high frequencies, the elastic contribution of the patellar tendon could be considered negligible compared with that of the Achilles tendon: the ankle joint operates mainly as a spring, whereas the knee joint mainly works as a strut to support the body. Thus, the primary source of P_TEN seems indeed attributable to the elastic function of the ankle (but at the slowest frequency).

This study provides further evidence about the role of the Achilles tendon as energy saver in terrestrial locomotion: without the elastic contribution, the energy expenditure would be higher and apparent efficiency would be closer to the value of muscle efficiency. What this paper adds to the body of evidence is that gearing rather than absolute recoil is what matters: it is the influence of tendon elasticity on muscle fibre mechanics that can influence the energy consumed by muscle, not the energy stored and returned in the tendon itself.

We would like to thank Francesca Morra, Raffaele Piras, Stefano Vaiani and Alessandro Vicentini for their help in data collection, and the subjects for participating in the study.