Abstract
In this study we examined how the strain energies within a muscle are related to changes in longitudinal force when the muscle is exposed to an external transverse load. We implemented a three-dimensional (3D) finite element model of contracting muscle using the principle of minimum total energy and allowing the redistribution of energy through different strain energy-densities. This allowed us to determine the importance of the strain energy-densities to the transverse forces developed by the muscle. We ran a series of in silica experiments on muscle blocks varying in initial pennation angle, muscle length, and external transverse load. As muscle contracts it maintains a near constant volume. As such, any changes in muscle length are balanced by deformations in the transverse directions such as muscle thickness or muscle width. Muscle develops transverse forces as it expands. In many situations external forces act to counteract these transverse forces and the muscle responds to external transverse loads while both passive and active. The muscle blocks used in our simulations decreased in thickness and pennation angle when passively compressed and pushed back on the load when they were activated. Activation of the compressed muscle blocks led either to an increase or decrease in muscle thickness depending on whether the initial pennation angle was less than or greater than 15°, respectively. Furthermore, the strain energy increased and redistributed across the different strain-energy potentials during contraction. The volumetric strain energy-density varied with muscle length and pennation angle and was reduced with greater transverse load for most initial muscle lengths and pennation angles. External transverse load reduced the longitudinal muscle force for initial pennation angles of β0 = 0°. Whereas for pennate muscle (β0 > 0°) longitudinal force changed (increase or decrease) depending on the muscle length, pennation angle and the direction of the external load relative to the muscle fibres. For muscle blocks with initial pennation angles β0 ≤ 20° the reduction in longitudinal muscle force coincided with a reduction in volumetric strain energy-density.
Highlights
Muscles can change in length and develop longitudinal force when they contract, and these result in external work done by the muscle
When the external transverse load was applied to the passive muscle blocks on the z-face (“top” loading), the blocks decreased in their thickness in the z-direction (Figure 3A), and in their pennation angle (Figure 4)
Note that the internal pressure in the passive muscle block decreased with increasing external transverse load (Figure 3B): whilst this may seem counter-intuitive this internal pressure is only one of the multiple factors that were balanced through the optimization of energy in these simulations
Summary
Muscles can change in length and develop longitudinal force when they contract, and these result in external work done by the muscle. Muscles expand and develop forces in transverse directions, resulting from internal work done within the muscle. Transverse expansions of contracting muscle have been reported in both animal (Brainerd and Azizi, 2005; Azizi et al, 2008) and human studies (Maganaris et al, 1998; Randhawa et al, 2013; Dick and Wakeling, 2017), and transverse forces generated internally in the muscle can “lift” weights during contraction (Siebert et al, 2014). Transverse loads that compress the muscle in its cross-section should be transferred to forces and changes in length in the longitudinal direction of the muscle
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