The energy of muscle contraction. IV. Greater mass of larger muscles decreases contraction efficiency

Abstract

While skeletal muscle mass has been shown to decrease mass-specific mechanical work per cycle, it is not yet known how muscle mass alters contraction efficiency. In this study, we examined the effect of muscle mass on mass-specific metabolic cost and efficiency during cyclic contractions in simulated muscles of different sizes. We additionally explored how tendon and its stiffness alters the effects of muscle mass on mass-specific work, mass-specific metabolic cost and efficiency across different muscle sizes. To examine contraction efficiency, we estimated the metabolic cost of the cycles using established cost models. We found that for motor contractions in which the muscle was primarily active during shortening, greater muscle mass resulted in lower contraction efficiency, primarily due to lower mass-specific mechanical work per cycle. The addition of a tendon in series with the mass-enhanced muscle model improved the mass-specific work and efficiency per cycle with greater mass for motor contractions, particularly with a shorter excitation duty cycle, despite higher predicted metabolic cost. The results of this study indicate that muscle mass is an important determinant of whole muscle contraction efficiency.

1. Introduction

Much of the energetic cost of locomotion can be attributed to skeletal muscles converting chemical energy that arises from metabolism of the food we eat to mechanical work to move the body. However, due to the difficulty in directly measuring these processes in whole muscle in vivo, most of what we know about this behaviour comes from studies on isolated single fibres or small muscles. To estimate the metabolic cost of contraction in larger whole muscles, the results of these studies on small muscles or single fibres are often extrapolated to larger sizes with the assumption that heat and metabolic cost scale with muscle volume and mass, as is typically done with mechanical work and power [1]. Based on these assumptions, two muscles with identical geometric proportions but different mass are typically assumed to produce the same mass-specific mechanical work and metabolic cost, and therefore efficiency.

Inertial resistance due to muscle mass may limit this assumption that work and metabolic cost scale with muscle volume and mass. Experimental studies examining the intrinsic properties of different fibre-type populations within whole muscle [2] or fibre bundles [3] have shown that surrounding tissue can act to slow the maximum shortening velocity of muscle fibres. Recent modelling work has shown that muscle tissue mass acts to slow the rate of force development [4] and maximum shortening speed of whole muscle [5], and this effect is greater for larger muscles [5–7] and submaximal contractions [5]. Inertial resistance due to muscle mass also acts to decrease the mechanical work output of muscle during simulated [8] and in situ [9] cyclic contractions. While muscle mass has been shown to alter the mechanical work of muscle, the metabolic and efficiency consequences of accelerating this internal mass have not yet been described.

In this study, we examine the effects of muscle tissue mass on the metabolic cost and efficiency of contraction. Additionally, we explore the extent to which muscle fibre-type properties and excitation, external loading conditions and series elasticity alter the relationship between muscle mass and muscle metabolic cost and efficiency. To accomplish this, we predicted the mass-specific mechanical work, metabolic cost and efficiency of cyclic contractions using a Hill-type muscle model that we modified to account for the dynamics of muscle size and mass.

2. Methods

2.1. Muscle model formulations

The force of the massless Hill-type model (figure 1a) was composed of a contractile element (CE) and parallel elastic element (PEE). The CE force depended on the muscle's activation, length and velocity, whereas the PEE force depended only on the muscle's length. We modelled the active and passive force–length and force–velocity curves as Bézier curves fitted to experimental data from Winters et al. [10] and Roots et al. [11], respectively. Details of these curves can be found in [12]. As with most Hill-type muscle models, the massless model does not account for the effects of muscle tissue mass. We added a linear damping term in parallel to the CE to stabilize the distributed mass of the mass model described below. We used the minimum non-dimensional damping constant required to stabilize the masses (0.08), which was similar in magnitude to values previously used to mimic viscosity effects [13,14] that occur within muscle [15,16].

Figure 1.

Diagram of the muscle models. We simulated contraction cycles of a typical massless Hill-type model with a contractile element (CE), parallel elastic element (PEE) and damping element (DE) (a), a mass-enhanced Hill-type model with 16 point masses (p₁–p₁₆) in series separated by Hill-type segments (b), and a mass-enhanced model with a series tendon or series elastic element (SEE) (c).

The mass-enhanced Hill-type model (figure 1b) is described in detail elsewhere [4,5,8]. Briefly, the model accounts for muscle mass in a series of 16 point masses distributed along the muscle's length, which is separated by Hill-type actuators that produce force to accelerate the point masses. We modelled the actuator segments between point masses with the same formulation as the massless Hill-type model.

We ran additional simulations with a massless tendon in series with the mass-enhanced model (figure 1c). Our previous work examining mass effects on muscle performance controlled for series elasticity by excluding a series elastic element from the model formulations [5,8], or by minimizing tendon effects during in situ muscle experiments by attaching the muscle to the servomotor at the myotendinous junction [9]. By doing so, we were able to make more direct interpretations of the effects of muscle mass without the confounding effects of series elasticity. However, tendons stretch when loaded, and if a tendon is long or compliant, this stretch can uncouple the muscle belly velocity from that of the muscle–tendon unit (MTU); this process may reduce muscle energy requirements [17,18] and act to amplify muscle power output [19]. Thus, we also examined simulations with a tendon, as series elasticity has important implications for muscle energetics and could modulate the relationship between muscle mass and contraction work, metabolic cost and efficiency. To model the tendon, we fitted the passive force–length Bézier curve to experimental measures of the human Achilles tendon [20].

2.2. Contraction conditions and post-processing

In our previous work examining the effects of muscle mass on mass-specific mechanical work [8], we used a framework that coupled the muscle model with a damped harmonic oscillator to simulate cyclic contraction regimes [12]. By doing so we were able to simulate more variable muscle lengths, velocities and forces than if the muscle length change was constrained to follow a pre-determined sinusoidal path; this allowed us to examine cycles that more closely mimicked the range of contractions seen during locomotion. However, because damped harmonic oscillators are only able to store and return or dissipate energy, they cannot add energy to the system to allow the muscle to act as a brake [21]. Therefore, in this study, we examined two contractile conditions to simulate cyclic contractions of the muscle models: (i) sinusoidal contractions in which we constrained one end of the muscle to follow a sinusoidal trajectory and (ii) harmonic oscillator contractions in which we fixed one end of the muscle model to a damped harmonic oscillator.

The damped harmonic oscillator was composed of a point mass, viscous damper and linear spring. We optimized the oscillator properties and excitation duty cycle to maximize the mass-specific mechanical work per cycle W* of the massless muscle model without a tendon that does not account for the effects of distributed tissue mass (figure 1a) for 1 Hz frequency cycles. The optimized excitation duty cycle was 0.46 and the details of the oscillator properties can be found in the electronic supplementary material. The total length of the muscle and harmonic oscillator system was fixed, and to make the muscle drive the harmonic oscillator, we converted a time-varying square wave excitation function to activation using an excitation–activation transfer function [22]. This activation scaled the active muscle force and was constant across the muscle tissue at each point in time.

For the sinusoidal contractions, we constrained the muscle length to follow a sinusoidal trajectory with a maximum strain amplitude of ±10% of optimal muscle length and frequency of 1 Hz. We used the same activation as for the harmonic oscillator simulations in which the excitation started slightly before the start of shortening (−2.6% of cycle or 26 ms). We also shifted the excitation phase by +50% of the cycle duration so that the muscle was primarily active during lengthening, which produced a braking contraction, and varied the maximum excitation u_max between 0.2 and 1.0, which scaled the excitation amplitude. In addition to using the optimal excitation duty cycle of approximately 0.46 for the harmonic oscillator simulations, we also examined a shorter duty cycle of 0.3. We tested fast and slow fibre-type properties for the harmonic oscillator and sinusoidal simulations without a tendon and tested only fast properties for simulations with a tendon. The fast and slow fibres had maximum shortening strain rates v₀ and activation rate constants of 10 s⁻¹ and 25 ms and 5 s⁻¹ and 45 ms, respectively. For simulations with a tendon, we constrained the MTU rather than the muscle length. Because tendons can uncouple muscle and MTU lengths and velocities, we optimized the stimulation phase relative to the MTU length (using the secant method) so that the excitation would start at the desired time relative to the muscle length. We examined a range of tendon stiffnesses by scaling the slope of the linear region of the tendon force–length curve with the tendon stiffness factor k_T (figure 2).

Figure 2.

Tendon force–strain properties across stiffnesses. (a) Tendon force relative to muscle maximum isometric force F₀ versus tendon strain as a percentage of resting or slack length across selected tendon stiffness factors k_T. (b) Tendon strain at F₀ as a function of k_T.

To explore the effects of muscle mass across a range of muscle sizes, we geometrically scaled the models using a length-scale factor that we herein refer to as ‘scale’. This method of scaling, described in detail in Ross et al. [12], resulted in muscle optimal length l₀ and tendon slack length scaling with the scale, muscle maximum isometric force F₀ with scale² and muscle volume and mass m_m scaling with scale³, assuming a constant muscle density of 1060 kg m⁻³ and maximum isometric stress of 225 kN m⁻². The scale 1 geometry was the approximate size of a fibre bundle with optimal and tendon slack lengths of 0.02 m and an aspect ratio (length : width) of 100. As with our previous work [8], we found minimal differences between the scale 1 and 10 simulations and so we report only scale 10 and 100 results in this paper.

We estimated cycle efficiency η as W* divided by the mass-specific metabolic cost $E_{\mathrm{met}}^{\ast }$. A number of different models have been used to estimate $E_{\mathrm{met}}^{\ast }$ [23–28], and there is currently no consensus on which model best reflects muscle metabolic cost in vivo. Therefore, we chose to estimate η across the results of six cost models. Four of the six models estimate the rate of $E_{\mathrm{met}}^{\ast }$, ${\dot{E}}_{\mathrm{met}}^{\ast }$, as the sum of the mass-specific heat rate ${\dot{H}}_{\mathrm{total}}^{\ast }$ and mechanical power ${\dot{W}}^{\ast }$ [24–27]. Because ${\dot{W}}^{\ast }$ is defined as negative during lengthening, if ${\dot{H}}_{\mathrm{total}}^{\ast }$ is low and ${\dot{W}}^{\ast }$ is higher and negative, ${\dot{E}}_{\mathrm{met}}^{\ast }$ and $E_{\mathrm{met}}^{\ast }$ can be negative, suggesting the muscle can produce metabolic energy through contraction. Therefore, we additionally calculated the metabolic rate as the sum of ${\dot{H}}_{\mathrm{total}}^{\ast }$ and $|{\dot{W}}^{\ast }|$, which we denoted as ${\dot{E}}_{\mathrm{met},\mathrm{abs}}^{\ast }$, to constrain the muscle to only consume and not generate metabolic energy during contraction. To calculate the efficiency η_abs, we divided W* by $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ which we determined by integrating ${\dot{E}}_{\mathrm{met},\mathrm{abs}}^{\ast }$ over time for one cycle. Details of these calculations and the cost model formulations can be found in the electronic supplementary material, as can details of the muscle model equations and the implementation of different contractile conditions.

3. Results

3.1. Mass effects in motor and brake contractions

For the harmonic oscillator simulations, we excited the muscle with a frequency of 1 Hz and for approximately 46% of the total cycle (duty cycle of 0.46). The muscle developed force when it was activated; this caused the muscle to shorten and the harmonic oscillator to stretch to lengths longer than its resting length. The muscle force decreased with deactivation, and because the harmonic oscillator was at a length longer than its resting, the harmonic oscillator force dominated that of the muscle which caused the muscle to stretch. This behaviour repeated every cycle so that the muscle cyclically contracted in time with the activation. Greater u_max resulted in larger muscle shortening strains (figure 3), as did faster fibre-type properties, as these conditions resulted in larger muscle forces to counteract the harmonic oscillator force. In contrast, when the muscle lengths were constrained to follow a sinusoidal trajectory, u_max and the muscle fibre-type properties did not affect the muscle strains.

Figure 3.

Harmonic oscillator versus sinusoidal simulations. Harmonic oscillator versus sinusoidal simulations for maximum excitation u_max of 1.0 and fast fibre-type properties (a), u_max of 0.2 and fast fibre-type properties (b), u_max of 1.0 and slow fibre-type properties (c) and u_max of 0.2 and slow fibre-type properties (d). The top plot in each panel shows muscle strain as a percentage of muscle optimal length and the middle plot shows muscle force normalized to maximum isometric force F₀ over the time of five contraction cycles for harmonic oscillator (dashed) and sinusoidal (solid) simulations. Massless model simulations are shown in the darkest, scale 10 simulations are shown in the intermediate, and scale 100 simulations are shown in the lightest colour. The bar chart in each panel shows the mass-specific mechanical work per cycle W* (striped), mean metabolic cost $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ (solid) and efficiency η_abs (bottom; polka dots) relative to the massless condition for harmonic oscillator scale 10 (H10) and scale 100 (H10), and sinusoidal scale 10 (S10) and scale 100 (S100) simulations.

Despite these differences in muscle strains, both the sinusoidal and harmonic oscillator simulations showed lower W* with greater muscle mass (figures 3 and 4a), except that there was no difference between the harmonic oscillator massless and scale 10 simulations (figure 3). On average across u_max of 0.2 and 1.0 and both fast and slow fibre types, we found a 9.2% reduction in W* for sinusoidal scale 10 simulations relative to massless, and a 4.5 and 13.2% reduction in W* for harmonic oscillator and sinusoidal scale 100 simulations relative to massless, respectively. While higher u_max and fast fibre-type properties resulted in a greater mean $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ across all cost models (figure 4b,c), when u_max and fibre-type properties were controlled for, the effect of muscle mass on $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ was small, particularly for the harmonic oscillator contractions (figure 3). Note that in figure 3 the harmonic oscillator scale 10 and 100 simulations are denoted by ‘H10’ and ‘H100’, respectively, and the sinusoidal scale 10 and 100 simulations are denoted by ‘S10’ and ‘S100’.

Figure 4.

Effects of muscle mass across fibre types and excitations for motor (phase +0%) contractions. Effects of muscle mass on mass-specific work per cycle W*, mean metabolic cost $E{\ast }_{\mathrm{met},\mathrm{abs}}$, and efficiency η_abs during sinusoidal contractions as a percentage of massless at the same maximum excitation u_max and fibre type (a), at u_max of 1.0 and the same fibre type (b) and at the same u_max and fast fibre type (c). Fast fibres are shown as blue squares, slow fibres as red circles, and scale 10 and 100 are shown in dark and light colours, respectively.

As the effect of mass on $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ was minimal for simulations without a tendon, the effect of mass on η_abs primarily followed changes in W*. For both the harmonic oscillator and sinusoidal contractions, greater mass resulted in lower η_abs relative to massless across values of u_max and both fibre types, except for harmonic oscillator contractions at scale 10 relative to massless in which there was no difference in W*, $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ or η_abs (figures 3 and 4a). The reduction in η_abs due to larger mass was greater on average for sinusoidal (9.7% for scale 10 and 13.5% for scale 100 relative to massless) compared to harmonic oscillator contractions (0.0% for scale 10 and 3.7% for scale 100 relative to massless) across u_max of 0.2 and 1.0 and both fibre types (figure 3). Both η_abs and η were lower for greater mass across all six metabolic cost models.

Because harmonic oscillators can only store and return or dissipate energy and cannot inject energy into the system, as is required for eccentric braking contractions, we only simulated braking contraction cycles with sinusoidal muscle length changes. W* was negative for these simulations, so a reduction in W* as a percentage of that of the massless condition indicates negative W* that is lower in magnitude relative to massless. All scale 10 and 100 simulations showed reductions in W* relative to massless, indicating less energy was absorbed or dissipated by the muscle (figure 5). Across u_max of 0.2 and 1.0 and both fibre types, we found a 5.7 and 7.4% reduction in W* for sinusoidal scale 10 and 100 simulations relative to massless, respectively. As with motor contractions, differences in η_abs relative to massless primarily followed changes in W* with both scale 10 and 100 simulations showing lower η_abs relative to massless. These trends in η_abs were consistent across all six cost models; however, η depended on the cost model used and its treatment of negative ${\dot{W}}^{\ast }$.

Figure 5.

Effects of muscle mass across fibre types and excitations for brake (phase +50%) contractions. Effects of muscle mass on mass-specific work per cycle W*, mean metabolic cost $E_{\mathrm{met},\mathrm{abs}}^{\ast }$, and mean efficiency η_abs during sinusoidal contractions as a percentage of massless at the same maximum excitation u_max and fibre type. Fast fibres are shown as blue squares, slow fibres as red circles, and scale 10 and 100 are shown in dark and light colours, respectively.

3.2. Effects of series elasticity and muscle mass

For the motor simulations (figure 6a,b), the tendon (with k_T of 1) mitigated some of the reduction in W* due to greater mass (figure 7a). Averaged across u_max of 0.2 and 1.0, W* was 1.7% higher for scale 10 and 3.4% higher for scale 100 simulations with a duty cycle of 0.46, and 10.8% higher for scale 10 and 9.4% higher for scale 100 simulations with a duty cycle of 0.3 with a tendon compared to without. Across both duty cycles, scales and u_maxs, adding a tendon in series to the muscle increased $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ by 3.1% (figure 7a). The added tendon had little effect on η_abs for simulations with a duty cycle of 0.46, whereas the tendon increased η_abs by 7.3% on average across both u_maxs and scales for duty cycle of 0.3. These results were the same for mean $E_{\mathrm{met}}^{\ast }$ and η, and LW05 was the only cost model that predicted lower $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ and $E_{\mathrm{met}}^{\ast }$ with a tendon and duty cycle of 0.46. For duty cycle of 0.3, the added tendon resulted in higher $E_{\mathrm{met},\mathrm{abs}}^{\ast }$, $E_{\mathrm{met}}^{\ast }$, η_abs and η across all cost models.

Figure 6.

Effects of the added tendon and excitation phase on muscle strain and force. Muscle strain as a percentage of optimal length over time (top row), muscle force normalized to maximum isometric force F₀ over time (middle row), and work-loops of normalized muscle force versus strain (bottom) for the scale 10 mass model. (a) Phase +0% (motor) contractions without a tendon, (b) phase +0% (motor) contractions with a tendon, (c) phase +50% (brake) contractions without a tendon and (d) phase +50% (brake) contractions with a tendon. The muscle is shown in dark blue, the muscle–tendon unit is teal, and the tendon is orange. Solid lines show simulations with a duty cycle of approximately 0.46 and dashed lines show simulations with duty cycle of 0.3.

Figure 7.

Effects of series elasticity and muscle mass. Effects of muscle mass on mass-specific work per cycle W*, mean metabolic cost $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ and mean efficiency η_abs during sinusoidal contractions as a percentage of massless for simulations with excitation phase of +0% (a) and +50% (b), duty cycle of 0.46 and 0.3, maximum excitations u_max of 0.2 and 1.0, and fast fibre type properties. Simulations with a tendon of base stiffness (k_T of 1) are shown as triangles and those without a tendon as circles, and scale 10 and 100 are shown as dark blue and light blue, respectively.

For the brake simulations (figure 6c,d), the added tendon resulted in 1.6% higher W* relative to mass simulations without a tendon (more negative absolute W*) on average across both scales and u_maxs with 0.46 duty cycle. The effect of series elasticity was less clear for simulations with a duty cycle of 0.3 (figure 7b). As with the motor contractions, the addition of a tendon increased the $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ for both scales, duty cycles and u_max values, and this was consistent across all six cost models. Brake simulations with a tendon resulted in 13.0% lower η_abs on average (less negative absolute η_abs) relative to mass simulations without a tendon for both scales, duty cycles and u_maxs. The effect of a tendon on $E_{\mathrm{met}}^{\ast }$ and η during braking contractions depended on the cost model used. Note that η_abs is given by W* divided by $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ and so can be negative if W* is negative.

3.3. Effects of mass and series elasticity on instantaneous mechanical power

For motor contractions at u_max of 1.0 without a tendon, ${\dot{W}}^{\ast }$ increased when the muscle was activated and reached its peak approximately halfway through the shortening phase when the muscle shortening velocity was greatest (figure 8a). $\dot{W}$* became negative when the muscle started to lengthen, and the magnitude of this negative $\dot{W}$* was greater for 0.46 duty cycle simulations compared to 0.3, as the muscle was still active for a short period of time at the start of lengthening. Greater muscle mass decreased the maximum $\dot{W}$* during shortening (figure 8b). Across both duty cycles, we found a 4.4 and 10.0 J s⁻¹ kg⁻¹ (4.7 and 10.7%) decrease in peak $\dot{W}$* for scale 10 and 100 relative to massless, respectively.

Figure 8.

The effects of muscle mass and series elasticity on mechanical power. Mass-specific mechanical power ${\dot{W}}^{\ast }$ in J s⁻¹ kg⁻¹ for mass model simulations (a) and difference in ${\dot{W}}^{\ast }$ between mass model and otherwise equivalent massless model simulations $\mathrm{\Delta }{\dot{W}}^{\ast }$ (b) for scale 10 (dark blue) and 100 (light blue) simulations without a tendon over time. (c) Mass-specific mechanical power ${\dot{W}}^{\ast }$ of the muscle (blue), tendon (orange) and muscle–tendon unit (green) for scale 10 (dark colours) and 100 (light colours) mass simulations with a tendon over time. (d) Difference in muscle–tendon unit ${\dot{W}}^{\ast }$ for mass model simulations with a tendon and otherwise equivalent simulations of the mass model without a tendon $\mathrm{\Delta }{\dot{W}}^{\ast }$. The left panel shows simulations with an excitation duty cycle of approximately 0.46 and the right shows simulations with an excitation duty cycle of 0.3. All simulations are for fast fibre type properties and maximum excitation of 1.0.

We activated the muscle 2.6% of the cycle duration prior to the start of muscle shortening, which allowed the tendon to stretch and store energy when the muscle shortened and the MTU length was nearly constant (figures 6b and 8c). The tendon then stayed at a longer length until the muscle was deactivated, where the tendon shortened and released stored elastic energy to increase $\dot{W}$* of the MTU (figure 8d). Because this tendon energy release occurred after peak MTU $\dot{W}$*, the tendon did not act to amplify peak ${\dot{W}}^{\ast }$ of the MTU beyond what the muscle alone could generate (figure 8c).

3.4. Effects of altering tendon stiffness

For the motor contractions at u_max of 1.0, increasing the tendon stiffness decreased W* for both scale 10 and 100 simulations (figure 9). This decrease was much greater with a duty cycle of 0.3 compared to 0.46. The maximum W* for simulations with a tendon occurred at a tendon stiffness that was more compliant than the base stiffness (k_T of 1). The maximum W* also occurred at a lower stiffness when the muscle mass was larger, with the maximum occurring at a k_T of 0.6 for scale 10 and 0.3 for scale 100 with a duty cycle of 0.46 and 0.35 for scale 10 and 0.3 for scale 100 with a duty cycle of 0.3. Lower k_T resulted in higher $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ for both duty cycles, and this was the case across all six cost models. With a duty cycle of 0.46, the highest η_abs occurred at the highest tendon stiffnesses, due to large increases in $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ at low stiffnesses (figure 9a). For a duty cycle of 0.3, maximum η_abs occurred at k_T of 0.5 for scale 10 and 0.7 for scale 100 (figure 9b). These results did not vary when $E_{\mathrm{met}}^{\ast }$ was used to estimate η.

Figure 9.

Effects of tendon stiffness across muscle masses. Mass-specific work per cycle W*, mean metabolic cost $E_{\mathrm{met},\mathrm{abs}}^{\ast }$ and mean efficiency η_abs during sinusoidal contractions expressed as a percentage of massless simulations without a tendon across tendon stiffness factors k_T for simulations with a duty cycle of 0.46 (a) and 0.3 (b). Scale 10 simulations are shown in dark blue, scale 100 are shown in light blue, and mass simulations with and without a tendon are shown as triangles and circles, respectively. All simulations are with fast fibre type properties and maximum excitation of 1.0.

4. Discussion

4.1. Mass effects across excitations and fibre types for motor contractions

We found that for motor contractions where the muscle was primarily active during shortening, greater mass resulted in lower mass-specific mechanical work per cycle. This is consistent with our previous modelling studies using this mass model [8,9], and with our experimental results showing that increasing muscle effective mass in situ decreases mechanical work per cycle [9]. Because altering the mass of muscle had little effect on metabolic cost, greater mass also resulted in lower efficiency per cycle for motor contractions due to the lower mechanical work (figures 3 and 4a). In our previous simulation work [8], and for the simulations here in which the muscle contracted against a damped harmonic oscillator (figure 3), we found that lower excitation resulted in larger reductions in mass-specific mechanical work with greater muscle mass for both fast and slow fibre type properties. This is consistent with experimental [2] and simulation [5] findings in which submaximal activation of either fast or slow fibres within whole muscle resulted in slower maximum contraction speeds compared to maximal activation of the entire muscle tissue, due to the greater inertia of inactive tissue during the submaximal contractions.

For the sinusoidal contractions, we found that lower excitations only resulted in larger reductions in work with greater mass when the muscle had slow fibre type properties, whereas muscles with fast fibre type properties showed the greatest reductions in mass-specific work per cycle at the highest excitations (figure 4a). This difference in mass effects across excitations for fast muscle between the sinusoidal and harmonic oscillator simulations may be due to differences in muscle strain during the cycle shortening phase. Because muscle length was not constrained in the harmonic oscillator simulations, higher muscle forces in simulations with higher excitations and fast fibre type properties resulted in larger shortening strains, whereas strains in the sinusoidal simulations were unaffected by both excitation and muscle fibre type. For sinusoidal contractions, we also found far greater reductions in muscle force with greater muscle mass, primarily near the start of shortening, compared to harmonic oscillator contractions. During locomotion, muscle lengths are not directly controlled, and rather emerge due to muscle forces relative to the external loads [21], similar to the harmonic oscillator simulations. However, unlike the harmonic oscillator simulations, muscle lengths can be indirectly controlled by altering the neural drive using sensory feedback. Therefore, whether the sinusoidal or harmonic oscillator simulations better reflect in vivo mass effects for faster muscle is not clear.

Fast fibre type properties resulted in lower mass-specific work with greater mass compared to slow properties across all excitations. When muscle is activated, smaller, slower motor units are typically recruited before larger, faster ones in accordance with the size principle [29]. However, strictly following the size principle may pose a mechanical paradox under certain conditions (for review, see [30]), such as contractions faster than the maximum intrinsic shortening speed of slow fibres [31]. Thus, under these conditions, slow fibres cannot contribute to force production or work output but still incur a metabolic cost. To address this paradox, faster motor units may be recruited first when slower motor units would be mechanically ineffective, and this has been supported by experimental studies [32,33]. The extent to which this alternative recruitment strategy occurs may also depend on muscle mass in addition to contraction speed. In this study, we found that faster fibres led to greater reductions in mass-specific work and efficiency with greater mass compared to slower fibres (figure 4a). While the slow fibres were still more efficient than fast fibres due to the relatively slow sinusoidal velocities (figure 4c), the contraction speed at which fast fibres become more efficient may be faster for larger muscles with greater mass. Thus, there may be less mechanical benefit of recruiting faster fibres before slower fibres during faster contractions for larger muscles.

4.2. Effects of series elasticity and muscle mass

Compliant tendons have been shown to increase MTU work and efficiency per cycle by decreasing muscle shortening velocity and increasing muscle force during contraction cycles with prescribed excitation [18,34], or to decrease muscle work requirements and increase efficiency by allowing the muscle to contract over a shorter length range during locomotion [35–37]. We found that the addition of a tendon with the approximate stiffness of a human Achilles tendon [20] increased work per cycle for motor contractions and mitigated some of the work reductions due to greater mass (figure 7a), and this work increase was greater with the shorter excitation duty cycle of 0.3. With a duty cycle of 0.3, the tendon increased efficiency as it allowed the muscle to contract nearly isometrically during tendon recoil (figure 6b). In contrast, with a duty cycle of 0.46, the muscle underwent greater shortening than the MTU, which increased both work and metabolic cost (figure 7a). This is consistent with a study by Lichtwark & Wilson [17] in which peak average power occurred at a duty cycle between 0.3 and 0.4 and peak efficiency occurred at a much lower duty cycle of around 0.1. Thus, while the work increases due to series elasticity with a duty cycle of 0.3 may be close to maximal, greater increases in efficiency may have been possible with a shorter duty cycle at the expense of high work output.

The effect of a tendon on MTU work also depends on the excitation phase relative to the start of MTU shortening [17,18,34,38,39]. Earlier activation of the muscle while the MTU is still lengthening can allow for greater storage of elastic energy, as the tendon can store energy from external work as well as work done by the shortening fibres [34]. This additional elastic energy can then be returned later during the shortening phase to increase MTU work output and efficiency. In this study, we used an excitation phase of −2.6% relative to the start of muscle shortening to match the phase that emerged from the harmonic oscillator simulations. For the base tendon with a stiffness factor of 1, this resulted in a phase of −3.8% relative to the start of MTU shortening. Because we selected the harmonic oscillator properties to maximize mechanical work per cycle in the massless model without a tendon, this phase may not be optimal for maximizing work and efficiency in the mass-enhanced model with a tendon. Thus, an earlier phase may have led to greater energy storage in the tendon during MTU lengthening. We also used this same excitation phase and duty cycle across all simulations, but because greater muscle mass decreases the rate of force development [4], larger muscles may require earlier and optimally long phases to maximize work per cycle so the tendon can fully stretch and store energy before the start of MTU shortening.

4.3. Effects of mass and series elasticity on instantaneous mechanical power

Achieving high instantaneous muscle power output is required for a variety of locomotor tasks, such as jumping and accelerating the body from rest, and is thought to be limited by the muscle's intrinsic force–velocity properties [40]. In this study, we found that greater muscle mass decreased muscle mass-specific peak instantaneous power (figure 8a,b), indicating that a muscle's mass may be an important consideration for maximizing power in addition to its intrinsic force–velocity properties.

The mechanical power requirements of tasks such as jumping have been shown to exceed the maximum instantaneous power output predicted from the intrinsic force–velocity properties of muscle. MTU instantaneous power can be increased using elastic mechanisms in which energy stored in the elastic structure is rapidly released to amplify the MTU power beyond what muscle alone can generate [19]. For amplification to occur, the tendon must first stretch to store elastic energy, and then later release this energy at the time coinciding with peak muscle power [41]. In this study, we found that tendon energy release occurred later than peak MTU power for both excitation duty cycles (figure 8c,d). For a duty cycle of 0.46, the tendon recoiled and released its stored energy near the start of lengthening, and the elastic energy release was not effective in increasing peak instantaneous MTU power. With a duty cycle of 0.3, tendon energy release occurred during shortening, which enhanced the instantaneous MTU power output but did not amplify the maximum power. As with increasing mechanical work per cycle, power amplification to mitigate reductions in peak instantaneous power with greater mass may be achieved with a shorter duty cycle or shifting the excitation phase earlier relative to the start of MTU shortening.

4.4. Effects of altering tendon stiffness

Measures of tendon mechanical properties have shown that their stiffness relative to maximum muscle force varies widely across mammals [41], and tendon stiffness can change with age [42] and training [43]. Even within the same tendon, like the extensively studied human Achilles tendon, stiffness can vary substantially, even within single studies using the same measurement technique [44,45]. Compliant tendons can increase cyclic work and efficiency to a greater extent than stiff tendons [18,34,38,46,47], but if a tendon is too compliant, the muscle must shorten too quickly to stretch the tendon and store elastic energy before the start of MTU shortening [46], increasing metabolic cost [48] and decreasing efficiency [46]. Our results confirm that an intermediate tendon stiffness results in the highest work and efficiency per cycle compared to overly compliant or stiff tendons (figure 9). Regardless of the duty cycle or muscle mass, optimal work and efficiency occurred at more compliant stiffnesses than the base tendon with human Achilles tendon properties (stiffness factor of 1). This may be because our muscle model properties and geometry did not reflect those of a muscle that attaches to the Achilles tendon in vivo. Modelling the muscle as a human plantarflexor muscle may have resulted in the optimal work and efficiency occurring closer to the base tendon stiffness, as in [46], assuming that the material properties of tendon are optimized to maximize work and efficiency for a given muscle design.

For simulations in which we varied tendon stiffness, we kept both the excitation duty cycle and phase relative to the muscle (not MTU) length constant. As with different scales, the optimal duty cycle and phase for peak work and efficiency per cycle may vary depending on the tendon stiffness. Compliant tendons have been shown to achieve high average power output and efficiency with earlier excitation phases relative to the start of MTU shortening [17,34] and shorter duty cycles or longer periods of relaxation during shortening [18,34] compared to stiffer tendons. Despite the possibly non-optimal work and efficiency across tendon stiffnesses, work reductions due to muscle mass were entirely compensated for by a tendon with optimal stiffness with a duty cycle of 0.3 (figure 9b). While reductions in efficiency were only entirely compensated for with the smaller scale 10 simulations, the largest scale 100 model was able to overcome approximately half of the efficiency reduction due to its mass with an optimal stiffness tendon.

4.5. Implications for whole-body locomotion and comparative biomechanics

In this study, we found that larger muscles are less efficient because they generate less work due to their greater mass compared to smaller muscles or simulated massless models. In living animals, skeletal muscles do work to move the skeleton and also account for a large proportion of metabolic energy expenditure during locomotion [49]. Thus, one could expect that whole-body efficiency during locomotion would be lower for larger animals. However, comparative studies have found that larger animals exhibit lower mass-specific metabolic costs of locomotion relative to smaller animals [50]. Because mass-specific mechanical work required to move a given distance varies little across animals of different sizes [51], the overall efficiency of the body during locomotion increases with body size. Differences in the mechanical design of MTUs in smaller and larger animals, such as differences in speed and muscle fibre type properties, elastic energy storage and muscle geometry, may account for this difference in locomotor efficiency predicted by our muscle-level results compared to measured whole-body values.

Allometry of muscle fibre-type properties and speed of locomotion. The reduction in efficiency of larger muscles may be mitigated in larger animals by them having slower and less costly muscle fibre types. At the fibre level, a faster fibre will have higher work output for an otherwise equivalent contraction cycle than a slower fibre because it can generate more force at a given shortening velocity (or strain rate) and also activate and deactivate more quickly. However, because faster fibres also incur higher metabolic cost compared to slower fibres [52,53], faster fibres are less efficient at slower shortening velocities [53] but more efficient at faster velocities where their force and work output is higher than slower fibres [54]. If muscles with faster fibres experience a greater reduction in mass-specific mechanical work per cycle due to greater mass compared to muscles with slower fibres, as we found in this study (figure 4), the contraction velocity at which faster fibres become more efficient may be higher for larger muscles. Thus, faster fibres may have less value in larger animals with muscles that operate at velocities below those corresponding with this efficiency threshold. Additionally, smaller animals have faster stride frequencies [55] that require rapid rates of force development and relaxation. Faster fibres that activate and deactivate quickly are required to achieve this rapid force development and relaxation, which increases metabolic cost and decreases efficiency. Again, there may be less need for larger animals to operate faster, but more costly, fibres because of their slower stride frequencies.

Muscle fibres in larger animals do in fact have lower maximum intrinsic shortening velocities than equivalent fibre types in smaller animals [56,57], and they also operate at lower frequencies [55] and strain rates [58]. These differences in muscle fibre physiology that are adapted to the slower gait parameters may mitigate the otherwise possible reductions in efficiency that would occur with recruiting the fastest fibres for the largest animals.

Allometry of elastic energy storage and return. Our results show that tendon with appropriate stiffness can increase MTU work and efficiency and offset some of the performance deficits due to greater muscle mass. For larger muscle, this work and efficiency benefit was greatest at a lower tendon stiffness compared to smaller muscle with lower mass. Studies have shown that the elastic modulus of tendon is relatively constant across terrestrial animals of different sizes [59,60]; however, tendon cross-sectional area relative to that of muscle, and therefore their approximate ability to withstand forces, decreases with body size [61,62]. This means that tendons in larger animals will stretch and store more elastic energy for given relative muscle stress, which is mechanically akin to varying tendon stiffness as we did in this study. Although there have been opposing arguments to suggest elastic energy storage is more important in smaller animals [63], greater elastic savings in larger animals could help mitigate the effects of their greater muscle mass.

Allometry of muscle geometry and architecture. Larger muscles may have different geometry and architecture to limit reductions in work and efficiency due to greater muscle mass. The effects of mass on contractile performance are due to a balance between the CE forces within muscle and the inertial loads due to the tissue mass. If muscles were parallel-fibred and scaled isometrically in size, in that the length l, width and thickness increased proportionately, like the muscle models scaled in this study, the muscle force would scale in proportion to the muscle cross-sectional area A and the inertial load (product of mass and acceleration) would scale in proportion to Al², assuming constant density and maximum isometric stress. Thus, in this scenario, the effects of mass on muscle work and efficiency would scale with the muscle length squared, or in simpler terms, longer muscles would experience greater reductions in work and efficiency compared to shorter muscles. Studies comparing muscle architecture across a range of animal sizes have shown that muscles in larger animals have relatively larger cross-sectional areas and shorter fibres [61,62]. While this has been suggested to reduce the high stresses and increase the safety factor of muscles in larger animals, it may also have work and efficiency benefits for larger muscles.

While this is an appealing explanation, only a small proportion of muscles are considered to be entirely parallel-fibred; in reality, most muscles are pennate and non-uniform in geometry. In pennate muscle, fibres are oriented at an angle relative to the force-generating axis of the muscle.

We previously found smaller reductions in mass-specific mechanical work due to greater mass in muscles with higher fibre pennation angles at rest [64]. If we assume that pennation angle has little effect on metabolic cost, which may be reasonable given how little muscle mass altered metabolic cost in this study, muscles with higher pennations may be more efficient for larger muscles in larger animals. Pennation angle has been shown to increase with body mass in monitor lizards ranging in size from 8 g to 40 kg [65], and this may be a mechanism to limit the work and efficiency penalty of greater muscle mass. Aponeurosis may also play a role in limiting the effects of muscle mass in larger animals. We previously showed that larger muscles with greater mass store more energy in aponeurosis during cyclic contractions [64]. However, it is not yet clear what role this energy storage has in mitigating the effects of greater muscle mass, and if this role is altered with changes in aponeurosis properties with body size.

Data accessibility

All relevant data are within the paper and its electronic supplementary material. The data are provided in the electronic supplementary material [66].

Authors' contributions

Conceptualization: S.A.R., J.M.W.; data curation: S.A.R.; formal analysis: S.A.R.; funding acquisition: J.M.W.; investigation: S.A.R.; methodology: S.A.R.; project administration: S.A.R.; resources: J.M.W.; software: S.A.R.; supervision: J.M.W.; visualization: S.A.R.; writing—original draft: S.A.R.; writing—review and editing: S.A.R., J.M.W.

Competing interests

We declare we have no competing interests.

Funding

We gratefully acknowledge support from the Natural Sciences and Engineering Research Council of Canada for a Discovery Grant to J.M.W. and an Alexander Graham Bell Canada Graduate Scholarship-Doctoral to S.A.R.