The Energy of Muscle Contraction. I. Tissue Force and Deformation During Fixed-End Contractions

Abstract

During contraction the energy of muscle tissue increases due to energy from the hydrolysis of ATP. This energy is distributed across the tissue as strain-energy potentials in the contractile elements, strain-energy potential from the 3D deformation of the base-material tissue (containing cellular and extracellular matrix effects), energy related to changes in the muscle's nearly incompressible volume and external work done at the muscle surface. Thus, energy is redistributed through the muscle's tissue as it contracts, with only a component of this energy being used to do mechanical work and develop forces in the muscle's longitudinal direction. Understanding how the strain-energy potentials are redistributed through the muscle tissue will help enlighten why the mechanical performance of whole muscle in its longitudinal direction does not match the performance that would be expected from the contractile elements alone. Here we demonstrate these physical effects using a 3D muscle model based on the finite element method. The tissue deformations within contracting muscle are large, and so the mechanics of contraction were explained using the principles of continuum mechanics for large deformations. We present simulations of a contracting medial gastrocnemius muscle, showing tissue deformations that mirror observations from magnetic resonance imaging. This paper tracks the redistribution of strain-energy potentials through the muscle tissue during fixed-end contractions, and shows how fibre shortening, pennation angle, transverse bulging and anisotropy in the stress and strain of the muscle tissue are all related to the interaction between the material properties of the muscle and the action of the contractile elements.

The Energy of Muscle Contraction. II. Transverse Compression and Work

“…Strain-energy is the energy stored by a system undergoing deformation. We previously showed that the redistribution of strain-energy potentials within contracting muscle (Wakeling et al, 2020) changes with the pennation angle, and it is likely that work done on and by the muscle generated by forces in the transverse direction would also affect the strain-energy potentials within the muscle (Wakeling et al, 2020). Thus, we would expect that the external transverse loads affect the strain-energy potentials within the muscle, that in turn could explain the changes in force in the longitudinal direction.…”

“…Not all muscles increase in thickness during fixed-end contractions, and thus we should not expect that every muscle will squeeze into neighbouring muscles when they activate. Muscles with lower pennation angles (<15 • ) tend to bulge, whereas more pennate muscle may thin as they activate Wakeling et al, 2020).…”

“…When muscles contract, they increase in their pennation angle both for shortening (Kawakami et al, 1998;Héroux et al, 2016) and for fixed-end contractions (Wakeling et al, 2020). Internal deformations additionally occur within muscle when it is compressed: external transverse loads cause a reduction in the mean fibre pennation angle (Wakeling et al, 2013), and a reduction to the extent the pennation angle increases when the muscle contracts (Ryan et al, 2019).…”

“…Thus, we would expect that the external transverse loads affect the strain-energy potentials within the muscle, that in turn could explain the changes in force in the longitudinal direction. The redistribution between the forms of energy also depends on the muscle length (Wakeling et al, 2020), and may drive an interaction between the muscle length and the force reduction that occurs with external load. However, the relation between the strain energy potentials and a possible compressionrelated reduction in longitudinal force has not previously been examined.…”

“…However, these studies have not explained how the deformation of muscle tissues due to external transverse loads is affected by the internal geometry or by the direction of the applied load relative to the muscle fibres. Our previous description of muscle (Wakeling et al, 2020), that quantifies how strain-energy potentials in the contractile elements are redistributed throughout the muscle volume and the contractile force is redirected across all three dimensions of the muscle tissue, provides a framework that is well suited to understand these mechanics of muscle compression. The purpose of this study was to identify whether the altered muscle forces that occur with compression can be explained in terms of the strain-energy potentials within the muscle, and in particular to account for the role of muscle length, pennation angle and the direction of the external load on the changes to muscle force.…”