Reconstituted collagen gels are a widely used environment to study cell migration in three dimensions. Collagen gels show strain stiffening and a strong lateral contraction during extension (Poisson ratio > 0.5). This behavior is not explained by linear elastic theory. To make quantitative estimates about the cellular forces and local mechanical properties of the matrix that the migrating cell encounters, a constitutive model of the gel is needed. On a microscopic scale, collagen gels consist of a network of mechanically coupled collagen fibrils that show buckling under compression and tautening and alignment under extension. These effects give rise to non-affine behavior of the network, which is virtually impossible to model or simulate numerically on a larger scale. Instead, we take advantage of the fact that the effects of a non-affine deformation is well captured by an affine network with non-linear elements. We extract the nonlinear force-length relationship of the network elements from two types of rheological measurements. First, we stretch and compress a collagen gel in the horizontal direction and measure its vertical contraction and dilation. From this we extract the asymmetry of the force-length relationship between extension and compression. Second, we shear a collagen gel in a cone plate rheometer. From this we extract the force-length relationship under extension. Finally, with a finite element analysis we compute the deformation field of a collagen gel around a contracting ellipsoid. We find that our model recapitulates the complex gel deformations typically measured around invasive tumor cells, such as a very large contraction of the gel surface above the cell.
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