Integrating a 3D vertex model of a cell spheroid with a fiber-network model to study cell-ECM interactions.

Left: Full system at the final simulation time, i.e., time t = tf and Right: Zoom in of the spheroid. The black denote active linker springs coupling the cells to the fibers.

Distributions of cellular shear stress, cell shape, and stress-shape correlation for solid-like and fluid-like spheroids at fiber-network occupation probability p = 0.8.

(a) Histogram of the maximum shear stress for the cells for two different target s0s. Gamma fit parameters: s0 = 5.2: α = 3.06, θ = 9.34 × 10−4; s0 = 5.8: α = 1.19, θ = 7.70 × 10−4. (b) Histogram of the cell shape anisotropy for the cells for two different target s0s. Gamma fit parameters: s0 = 5.2: θ = 3.20, 0 = 1.66 × 10−2; s0 = 5.8: α = 2.78, θ = 6.21 × 10−2. (c) The overlap, or dot product, between the eigenvector associated with the largest maximum shear stress and the eigenvector associated with the largest gyration eigenvalue.

Spatial distribution of cellular stresses and cellular shape in solid-like spheroids.

(a) Sample cross-section of the cellular maximum shear stress. (b) Sample cross-section of the cellular shape anisotropy. (c) Maximum shear stress for both fluid-like and solid-like spheroids as a function of radial distance from the center of mass of the spheroid. The inset shows the result for p = 0, i.e., the non-embedded spheroid with no surrounding fiber network.

Cellular strain stiffening for volume-preserving deformations.

(a) Initial cell configuration and a strained cell configuration for cells from a fluid-like spheroid. (b) Distribution of maximum bulk shear stress for the initial cells (dark blue) and for the strained cells (light blue) for s0 = 5.8. (c) The average maximum shear stress versus strain demonstrating a nonlinear relationship for both types of the spheroids and, thus, exhibiting strain stiffening phenomena. The inset shows a deviation from linear behavior around 0.1 strain, yielding an estimate of the onset of strain stiffening. By linearly fitting the low and high strain parts of the stress-strain curve, the intersection of the two fits yields a second estimate for the crossover between low and high strain behavior at approximately 0.4 (dashed vertical line). (d) The average cell shape anisotropy versus strain.

Different spheroid cell breakout modes.

(a) Experiment with MEF spheroid in 1.5 mg/ml collagen I. matrix. (b) Single-cell breakout as the target spring length of the four fibers attached to the deep yellow cell decreases. The thicker purple cell-cell adhesions are weaker than the thinner red ones. (c) Two-cell breakout as the target spring length of the four fibers attached to the deep yellow cell decreases. The cyan denotes strong anisotropic cell-cell adhesion springs between the leader boundary dark yellow cell and the center blue cell that is becoming elongated in the dark yellow cell extension to ultimately follow it.

Table of the dimensionless parameters used in the embedded spheroid simulations.

Histogram for maximum shear stress with p = 0.75.

Gamma fit parameters: s0 = 5.2: α = 3.03, θ = 9.53 × 10−4; s0 = 5.8: α = 1.24, θ = 5.32 × 10−4.

Histogram for maximum shear stress with no surrounding fiber network.

Gamma fit parameters: s0 = 5.2: α = 4.44, θ = 1.21 × 10−3; s0 = 5.8: α= 1.71, θ = 4.25 × 10−4

Spatial distribution of cellular stresses and cellular shape in fluid-like spheroids.

(a) Sample cross-section of the cellular maximum shear stress. (b) Sample cross-section of the cellular shape anisotropy.

Non-volume preserving cellular deformation.

(a) Initial cell configuration and a strained cell configuration. (b) Distribution of maximum shear stress for the initial cells (dark orange) and for the uni-axial strained cells (light orange) for s0 = 5.2. (c) Overlap of the stress with the shape for the initial cells and for the strained cells for s0 = 5.2. The inset is the same analysis as performed in the inset for Fig. 3(c) indicating the onset of strain stiffening at approximately a strain of 0.15. In addition, the vertical dashed line represents a crossover strain behavior from lower to higher strain. (d) Distribution of maximum shear stress for the initial cells (blue) and for the uni-axial strained cells (pink) for s0 = 5.2. (e) Overlap of the stress with the shape for the initial cells and for the strained cells for s0 = 5.8.