Strong triaxial coupling and anomalous Poisson effect in collagen networks

Abstract

While cells within tissues generate and sense 3D states of strain, the current understanding of the mechanics of fibrous extracellular matrices (ECMs) stems mainly from uniaxial, biaxial, and shear tests. Here, we demonstrate that the multiaxial deformations of fiber networks in 3D cannot be inferred solely based on these tests. The interdependence of the three principal strains gives rise to anomalous ratios of biaxial to uniaxial stiffness between 8 and 9 and apparent Poisson's ratios larger than 1. These observations are explained using a microstructural network model and a coarse-grained constitutive framework that predicts the network Poisson effect and stress-strain responses in uniaxial, biaxial, and triaxial modes of deformation as a function of the microstructural properties of the network, including fiber mechanics and pore size of the network. Using this theoretical approach, we found that accounting for the Poisson effect leads to a 100-fold increase in the perceived elastic stiffness of thin collagen samples in extension tests, reconciling the seemingly disparate measurements of the stiffness of collagen networks using different methods. We applied our framework to study the formation of fiber tracts induced by cellular forces. In vitro experiments with low-density networks showed that the anomalous Poisson effect facilitates higher densification of fibrous tracts, associated with the invasion of cancerous acinar cells. The approach developed here can be used to model the evolving mechanics of ECM during cancer invasion and fibrosis.

Keywords: 3D cell traction force microscopy; fibrous matrices; matrix realignment; tissue swelling.

Fig. 1.

The early onset of triaxial and biaxial stiffening of fibrous networks is accompanied by greater stretching of individual fibers compared with bending and rotation. Snapshots of the fiber network model (A) before loading and after (B) uniaxial tension, (C) equibiaxial tension, and (D) triaxial tension. The network snapshots show the microstructure of the network and the anomalous Poisson effect in uniaxial and biaxial deformation. A–D visualize the difference in the microstructural evolution of fiber reorientation, realignment, stretching, and buckling in the various modes of loading, quantified in Figs. 1H and 2D, and SI Appendix, Fig. S2. In A, all fibers are colored green, whereas in B–D fibers are colored by maximum principal strain. (E) Stress versus stretch response of the fiber networks in uniaxial, equibiaxial, and volumetric tension and compression. (F) The stretch ratio in the transverse direction in uniaxial and biaxial tension and compression. (G) The apparent Poisson ratios of the networks are smaller than 0.5 for the isotropic networks but grow larger than 1 in the case of the anisotropically deformed networks. The large apparent Poisson ratio of the networks is associated with a decrease in sample volume during deformation. (H) The stretch energy of fibers normalized by the total strain energy as a function of the applied stretch. The curves in E and F correspond to results from the coarse-grained constitutive model (Eq. 1).

Fig. 2.

The role of fiber reorientation and fiber stretching in the pore size-dependent multiaxial coupling and Poisson effect of fibrous networks. (A) The apparent Poisson ratio of fiber networks versus uniaxial stretch at different concentrations of collagen. The network pore size was tuned by changing the number of random seed points used to generate the network geometry. (B) Apparent Poisson’s ratio of the networks at a stretch of 1.2 for networks having different pore sizes. (C) The normalized fiber stretch energy is larger in networks with smaller pores subjected to uniaxial stretch. (D) The reorientation per fiber as a function of the applied stretch in uniaxial and triaxial deformation and uniaxial deformation of a network with smaller pores. The reorientation of each fiber was calculated as the angle between the vectors marking the orientation of the fiber before and after deformation. In C and D, the original network and the network with smaller pores correspond to values of pore size/fiber diameter of 43 and 23, respectively.

Fig. 3.

The pore size-dependent stiffening of fibrous networks in triaxial and biaxial deformation. (A) Map of the normalized strain energy density of a fiber network in biaxial stretch tests. (B) Stiffening of networks in equibiaxial deformation increases with increasing network pore size. The stiffening in biaxial deformation and strain energy increase in biaxial deformation are displayed using red triangles and blue diamonds, respectively. In these tests, pore size was tuned by changing the number of random seed points used to generate the network geometry. The reported pore size was normalized by the fiber diameter. The maximum possible biaxial-to-uniaxial stiffness ratio for linear elastic isotropic materials corresponds to materials having ν = 0.5 (red dashed line). The biaxial and uniaxial stiffnesses are equal for linear elastic isotropic materials exhibiting no Poisson effect, ν = 0.0 (green dotted line). The Inset displays a snapshot of a fiber network tested in equibiaxial tension. The horizontal axis is plotted in reverse. (C) Variation of network storage modulus against axial stress. Two different samples were used in the tension and compression rheological experiments. The Insets display schematics of the coupled axial–shear rheometry experiments in tension and compression. A G′₀ value of 450 Pa was used in normalizing the rheological data. (D) Snapshot of a fiber network model in shear loading after multiaxial deformation. Colors display the axial strain of fibers. (E) The map of the network shear modulus after biaxial deformation indicates that the networks become stiffer in shear if at least one of the applied axial stretches is larger than the strain–stiffening threshold. The purple asterisk marks the increase in shear modulus of a network under compression in one direction and tension past the stiffening threshold in another direction. (F) The apparent shear stiffness of a fiber network increases with volumetric expansion. The dashed curve is plotted to guide the eye. All panels present results obtained using the computational fiber network model, and C also includes results from the axial–shear rheometry experiments.

Fig. 4.

Influence of the Poisson effect on the apparent stiffness of thin collagen networks in extensional rheology tests. (A) The response of the computational coarse-grained constitutive model in the instantaneous extension of a thin sample. The sample curves inward at the free boundary due to the positive apparent Poisson ratio. The axis of symmetry is marked by a vertical dashed-dotted line. (B) Computational models of isotropic linear elastic materials do not exhibit curving or inhomogeneity of strain in the absence of the Poisson effect, for ν = 0.0. The dark blue color indicates vanishing radial strain in the sample. The color scale of A is also used in plotting the snapshot in B. (C) Snapshot of a computational fiber network model in tension. (D) The ratio of apparent elastic stiffness over the elastic modulus of the gels as a function of the specimen length (or thickness)-to-diameter ratio. The experimental data points were collected from rheometry experiments (23, 50) and tension experiments using dog bone-shaped samples (17). In addition to experiments, results are plotted from the computational network model and the computational coarse-grained model. Both axes are plotted logarithmically. The Insets display gels tested using (Left) extensional rheometry and (Right) uniaxial stretch of dog bone-shaped samples.

Fig. 5.

The density-dependent Poisson effect in collagen ECMs induced by cellular forces. (A) Schematic of the Poisson effect in a collagen tract formed by a pair of mechanically interacting cell clusters. The studied fibrous tracts promote the invasion of cancerous acinar cells. (B) Bright-field image of a pair of interacting acini, showing the two cell clusters atop a gel formed at a collagen concentration of 1 mg/mL. The region of interest (ROI) is marked by the yellow dotted box. (C) Mean apparent Poisson’s ratio at the ROI as a function of stretch in the horizontal direction. (D–F) The ratio of the largest to smallest principal stretch ratios in collagen gels deformed by a pair of mammary acini. Collagen gels polymerized at collagen concentrations of (D) 1, (E) 2, and (F) 3 mg/mL were tested. (G–I) The corresponding results obtained using the coarse-grained model. D–I correspond to a mean λ_max = 1.1 at the ROI. All panels except C correspond to a pair of cell clusters, seeded atop the horizontal surface of a gel, as viewed from the top.