Realizations of highly heterogeneous collagen networks via stochastic reconstruction for micromechanical analysis of tumor cell invasion

Three-dimensional (3D) collective cell migration in a collagen-based extracellular matrix (ECM) is among one of the most significant topics in developmental biology, cancer progression, tissue regeneration, and immune response. Recent studies have suggested that collagen-fiber mediated force transmission in cellularized ECM plays an important role in stress homeostasis and regulation of collective cellular behaviors. Motivated by the recent in vitro observation that oriented collagen can significantly enhance the penetration of migrating breast cancer cells into dense Matrigel which mimics the intravasation process in vivo [Han et al. Proc. Natl. Acad. Sci. USA 113, 11208 (2016)], we devise a procedure for generating realizations of highly heterogeneous 3D collagen networks with prescribed microstructural statistics via stochastic optimization. Specifically, a collagen network is represented via the graph (node-bond) model and the microstructural statistics considered include the cross-link (node) density, valence distribution, fiber (bond) length distribution, as well as fiber orientation distribution. An optimization problem is formulated in which the objective function is defined as the squared difference between a set of target microstructural statistics and the corresponding statistics for the simulated network. Simulated annealing is employed to solve the optimization problem by evolving an initial network via random perturbations to generate realizations of homogeneous networks with randomly oriented fibers, homogeneous networks with aligned fibers, heterogeneous networks with a continuous variation of fiber orientation along a prescribed direction, as well as a binary system containing a collagen region with aligned fibers and a dense Matrigel region with randomly oriented fibers. The generation and propagation of active forces in the simulated networks due to polarized contraction of an embedded ellipsoidal cell and a small group of cells are analyzed by considering a nonlinear fiber model incorporating strain hardening upon large stretching and buckling upon compression. Our analysis shows that oriented fibers can significantly enhance long-range force transmission in the network. Moreover, in the oriented-collagen-Matrigel system, the forces generated by a polarized cell in collagen can penetrate deeply into the Matrigel region. The stressed Matrigel fibers could provide contact guidance for the migrating cell cells, and thus enhance their penetration into Matrigel. This suggests a possible mechanism for the observed enhanced intravasation by oriented collagen.

Figure 1

SEM (scanning electron microscopy) image showing the microstructure of the model heterogeneous ECM which contains a collagen region (upper part) attached to a layer of dense Matrigel (lower part).

Figure 2

Schematic illustration of the bond-node mode (b) for a collagen network derived from its confocal image (a).

Figure 3

Schematic illustration of the overall network reconstruction procedure based on stochastic optimization.

Figure 4

Randomly displacing an arbitrary node in the network to generate a new configuration (b) from an old configuration (a). The fibers affected by this perturbation are highlighted using red color (or dark gray in the print version).

Figure 5

Schematic illustration of the polarized contraction of an ellipsoidal cell embedded in the collagen network. The nodes within a cone region defined by the polar angle $\theta$ are pulled towards the cell center by an affine transformation.

Figure 6

Target and simulated network statistics including the valence number distribution (a) and the fiber length distribution (b) for a homogeneous network with randomly oriented fibers.

Figure 7

(a) Power-law cooling schedule used for the reconstruction. (b) Evolution of the average energy of each stage during the reconstruction.

Figure 8

3D visualization of the reconstructed homogeneous network with randomly oriented fibers that is statistically identical to the target collagen network as obtained using 3D confocal microscopy imaging.

Figure 9

3D visualization of the reconstructed homogeneous network with fibers aligned with the $z$ axis.

Figure 10

Target and simulated network statistics including the valence number distribution (a) and the fiber length distribution (b) for a homogeneous network with prealigned fibers.

Figure 11

3D visualization of the reconstructed heterogeneous collagen network with continuous variation of fiber orientation along the $z$ axis.

Figure 12

The target and simulated ${\mathrm{\Omega }}^{*}\left(z\right)$ (i.e., the average cosine value of the angle between the fibers and $z$ axis) as a function of $z$ position (normalized with respect to the box length) in the network.

Figure 13

Target and simulated network statistics including the valence number distribution (a) and the fiber length distribution (b) for a heterogeneous network with continuously varying fiber orientations.

Figure 14

3D visualization of the reconstructed heterogeneous collagen network with a collagen region composed of aligned fibers and a Matrigel region composed of randomly oriented fibers.

Figure 15

The target and simulated ${\mathrm{\Omega }}^{*}\left(z\right)$ (i.e., the average cosine value of the angle between the fibers and $z$ axis) as a function of $z$ position (normalized with respect to the box length) in the network.

Figure 16

Target and simulated network statistics including the valence number distribution (a) and the fiber length distribution (b) for the binary heterogeneous network.

Figure 17

Propagation of active forces generated by a polarized contractile cell in a homogeneous network with randomly oriented fibers (a), a homogeneous network with prealigned fibers (b), and a heterogeneous network with continuously varying fiber orientations along the $z$ axis (c). The fibers carrying large forces $(>5.0\times {10}^{-7}$ N) are highlighted using red color (dark gray in the print version).

Figure 18

Propagation of active forces generated by a polarized contractile cell in a highly heterogeneous ECM composed a collagen region (upper) with prealigned fibers and a Matrigel region (lower) with randomly oriented fibers. The fibers carrying large forces $(>5.0\times {10}^{-7}$ N) are highlighted using red color (dark gray in the print version).

Figure 19

Comparison of the angle between a fiber in the Matrigel and the $z$ axis before and after the contraction of the cell. The size of the circle indicates the force magnitude in the fiber. Reorientation of the fibers carrying relatively large forces along the $z$ axis can be observed.

Figure 20

Propagation of active forces generated by a group of polarized contractile cells mimicking a tumor invasion front in a highly heterogeneous ECM composed a collagen region (upper) with prealigned fibers and a Matrigel region (lower) with randomly oriented fibers. The fibers carrying large forces $(>5.0\times {10}^{-7}$ N) are highlighted using red color (dark gray in the print version).

Figure 21

Comparison of the angle between a fiber in the Matrigel and the $z$ axis before and after the collective contraction of the group of tumor cells. The size of the circle indicates the force magnitude in the fiber. Reorientation of the fibers carrying relatively large forces along the $z$ axis can be observed.