Fig 1.
Tendon force–strain relationship measured experimentally and used in our simulations.
The model curves (solid black lines) produced tendon strains of 1–10% when the muscles were developing maximum isometric force (Fmax). The curve labeled “E” is one physiologically plausible curve based on the experimental data reported by Maganaris and Paul [31] and Magnusson et al. [30] (shaded regions).
Fig 2.
Average whole-body metabolic power consumed during running, normalized by subject mass.
The experimental data (gray lines) were obtained by indirect calorimetry collected from 5 males of sufficient fitness to run at all speeds aerobically, as reported by Steudel-Numbers and Wall-Scheffler [48]. Simulations of running at 2, 3, 4, and 5 m/s were generated for 10 male long-distance runners using the methods reported by Hamner and Delp [33] with tendon force–strain curves that fit experimental data (see the curve labeled “E” in Fig 1); colored bars indicate mean ±1 standard deviation. A basal rate of 1.13 W/kg [45, 49] was added to the simulation results to form a valid comparison with the gross metabolic power reported by Steudel-Numbers and Wall-Scheffler.
Fig 3.
Average whole-body metabolic power consumed during running as tendon compliance varies.
The mean (line) and standard deviation (vertical bars) predicted by our simulations at each speed and tendon compliance are shown in (a); the mean increase in average metabolic power from the lowest average power at each speed is shown in (b), expressed as a percentage of the lowest average power at each speed. Filled circles in (a) indicate the lowest value at each speed; open circles denote values that are not significantly greater than these minima (p < 0.05, matched pairs t-test). The “experimental range” indicated in (b) is 4.9±1% strain at Fmax, the mean and standard deviation reported by Maganaris and Paul [31]. The optimal tendon compliance was near this range for all running speeds; less compliant tendons were substantially more favorable when running at 2 m/s.
Fig 4.
Average metabolic power consumed by lower extremity muscles during running as tendon compliance varies.
The mean increase in average metabolic power from the lowest average power at each speed is shown for lower extremity muscles crossing the hip, knee, and ankle, expressed as a percentage of the lowest average whole-body metabolic power at each speed (filled circles in Fig 3(a)). The metabolic power associated with biarticular muscles was distributed between each joint in proportion to instantaneous flexion/extension moment arms. At the knee, more compliant tendons were favored when running at 3–5 m/s. At the ankle, very compliant tendons were particularly detrimental when running at 2 m/s.
Fig 5.
Average metabolic power consumed by the gastrocnemius and soleus muscles during running as tendon compliance varies.
The mean increase in average metabolic power from the lowest average power consumed is shown when running at 2 m/s. The gastrocnemius (medial and lateral heads) consumed the greatest power when tendons were very compliant, as did all the other plantarflexors except the soleus, which consumed the greatest power when tendons were least compliant.
Fig 6.
Dynamics and metabolics of the medial gastrocnemius and soleus muscles during running at 2 m/s.
Simulated muscle activations, fiber lengths, and fiber velocities (top row) and outputs from our model of muscle energetics (bottom row) for the right medial gastrocnemius (a) and soleus (b) muscles are shown over the gait cycle. Our model of muscle energetics predicted the rate of heat generation due to sarcoplasmic reticular ion transport and actin–myosin interaction (activation and maintenance heat rate), the rate of heat generation due to shortening and lengthening of the fibers, and the mechanical power of the fibers [37]. The mean (line) and standard deviation (shaded region) are shown for the seven rearfoot-striking subjects when low (2% strain at Fmax; orange) and high (10% strain at Fmax; blue) tendon compliances were used. When tendons were very compliant, the soleus fibers were operating nearly isometrically during stance, thereby reducing the average shortening and lengthening heat rate predicted by the energetics model (from 128 to 61 mW/kg). In contrast, the medial gastrocnemius fibers were operating far from their optimal lengths during stance when tendons were very compliant, thereby requiring greater activation to generate a similar plantarflexion moment and increasing the average activation and maintenance heat rate (from 38 to 57 mW/kg).
Fig 7.
Muscle activations and positive fiber mechanical power during running as tendon compliance varies.
The mean (line) and standard deviation (vertical bars) at each speed and tendon compliance are shown. Filled circles indicate the lowest value at each speed; open circles denote values that are not significantly greater than these minima (p < 0.05, matched pairs t-test). Although the sum of squared activations (a) reveals similar trends as average metabolic power at the whole-body level, activation-based metrics may disagree with metabolics at the muscle level (e.g., see Fig 6(b)). Comparison of Figs 3(a) and 7(b) reveals that average positive fiber mechanical power achieved minima at greater tendon compliances than average whole-body metabolic power consumption, suggesting that positive fiber mechanical power is a poor surrogate for metabolic power.