Fig 1.
(A) shows the uniaxial stress-stretch relationship for the intrinsic properties of the homogenization: ECM (blue), cellular (yellow), and averaged whole muscle components (red), along with experimental data from Gillies et al. [39] (ECM, blue dot), Jin et al. [30] (brain grey matter, yellow dot), and Mohammadkhah et al. [44] (transverse muscle response, red dot). The averaged whole muscle component was fit to experimental data and is shown for comparison. Total (yellow), passive (red), and active (blue) stress-stretch relationships are shown for the along-fibre response in (B) with the experimental data obtained by Winters et al. [45] and normalized to σ0 = 2 × 105 Pa. ECM component was scaled by 200 in (A) to account for cross-sectional area calculations.
Table 1.
Summary of parameters used in the model.
List of the values for the aforementioned parameters used in this model. ci,cell/ecm are the Yeoh model parameters shown in Eq 11 and were obtained using nonlinear regression analysis.
Fig 2.
(A) Mesh of the geometry used for the numerical experiments. The geometry had dimensions 20cm × 6cm × 4cm and muscle fibre properties are orientated along the x axis. (B) Shear experiment setup. The −x face was constrained in all directions, while the +x face was constrained in the x direction only. The arrow represents direction of applied shear stress.
Fig 3.
Comparison of model results to experimental data.
Comparison of model passive stress-stretch curves to experimental data for skeletal muscle. (A) Gives the model with a parameter of sECM = 150, while (B) is the model with a parameter of sECM = 250. α was varied between 0.02—0.20, which corresponds to 2—20% volume fraction of ECM. The grey lines represent experimental data from Takaza et al. [48] (circle) and Mohammadkhah et al. [44] (dot). Error bars represent ± standard deviation when available.
Fig 4.
Plot of the shear stress-strain relationship for α = 0.05, 0.10 and sECM = 250, while the muscle is passive (A) and active (B). (C) Shear stress-strain relationship for bulk moduli of 1 × 106, 1 × 107, and 1 × 108 Pa. Wire mesh of muscle model after shear stress was applied then model was activated (D), and after first activation then application of shear stress (E). (D,E) Color represents the dilation seen in the muscle model. (C) Shear stress-strain relationship for bulk moduli of 1 × 106, 1 × 107, and 1 × 108 Pa.
Fig 5.
Volumetric impact on stress-strain relationship.
Stress-strain relationship with κcell = 1 × 106, 1 × 107, 1 × 108 Pa. Stress was applied in the longitudinal direction on the +x face of the muscle model. Increasing values of the bulk moduli result in a stiffer material.
Table 2.
Total volume change and normalized stress on the +x face of the muscle after passive lengthening to a stress of 1 × 105 Pa and fixed length contraction to an activation of 100%.
The stress was normalized to σ0. These values are measured with homogenization parameters of α = 0.05 and sECM = 250.
Fig 6.
Strain energy-density with varying κcell.
Plots of passive fibre, base material, isochoric, volumetric, and total internal strain energy-densities. The energies are plotted over a passive lengthening period, up to a traction of 1 × 105 Pa, from timestep 3 to 13, and a linearly increasing fixed-length activation from timestep 13 to 23. κcell is varied between values 1 × 106 Pa, 1 × 107 Pa, and 1 × 108 Pa. The larger values of κcell demonstrate increasing incompressibility and approach the bulk moduli of water (2.15 × 109 Pa [49]), which is considered to be almost completely incompressible. The total internal strain energy-density is the combination of the volumetric, isochoric, and activation (not shown in figure) energies.
Fig 7.
Strain energy-density with varying ECM volume fraction.
Plots of passive fibre, base material, isochoric, volumetric, and total internal strain energy-densities. The energies are plotted over a passive lengthening period, up to a traction of 1 × 105 Pa, from timestep 3 to 13, and a linearly increasing fixed-length activation from timestep 13 to 23. Volume fractions of the ECM are varied between 2%, 25%, 50%, 75%, and 100% to investigate the physics of the model.