Research Papers

Displacement Propagation in Fibrous Networks Due to Local Contraction

[+] Author and Article Information
Peter Grimmer

Department of Engineering Physics,
University of Wisconsin-Madison,
1500 Engineering Drive,
Madison, WI 53706
e-mail: pgrimmer@wisc.edu

Jacob Notbohm

Department of Engineering Physics,
University of Wisconsin-Madison,
1500 Engineering Drive,
Madison, WI 53706
e-mail: jknotbohm@wisc.edu

1Corresponding author.

Manuscript received June 19, 2017; final manuscript received November 19, 2017; published online February 12, 2018. Assoc. Editor: Thao (Vicky) Nguyen.

J Biomech Eng 140(4), 041011 (Feb 12, 2018) (11 pages) Paper No: BIO-17-1267; doi: 10.1115/1.4038744 History: Received June 19, 2017; Revised November 19, 2017

The extracellular matrix provides macroscale structure to tissues and microscale guidance for cell contraction, adhesion, and migration. The matrix is composed of a network of fibers, which each deform by stretching, bending, and buckling. Whereas the mechanics has been well characterized in uniform shear and extension, the response to more general loading conditions remains less clear, because the associated displacement fields cannot be predicted a priori. Studies simulating contraction, such as due to a cell, have observed displacements that propagate over a long range, suggesting mechanisms such as reorientation of fibers toward directions of tensile force and nonlinearity due to buckling of fibers under compression. It remains unclear which of these two mechanisms produces the long-range displacements and how properties like fiber bending stiffness and fiber length affect the displacement field. Here, we simulate contraction of an inclusion within a fibrous network and fit the resulting radial displacements to ur ∼ rn where the power n quantifies the decay of displacements over distance, and a value of n less than that predicted by classical linear elasticity indicates displacements that propagate over a long range. We observed displacements to propagate over a longer range for greater contraction of the inclusion, for networks having longer fibers, and for networks with lower fiber bending stiffness. Contraction of the inclusion also caused fibers to reorient into the radial direction, but, surprisingly, the reorientation was minimally affected by bending stiffness. We conclude that both reorientation and nonlinearity are responsible for the long-range displacements.

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Grahic Jump Location
Fig. 1

Fiber network architecture. (a) Domain simulated. The domain's radius is 20 inclusion radii; networks are generated with fibers having average length of 0.27, 0.20, or 0.15 inclusion radii (0.27 shown here). (b) Confocal images of a collagen network. At fiber connections (circles), a single fiber branches into multiple others at acute angles. (c) Two different types of network architecture. Annealed networks are precursors to branched networks. (d) Connectivity distribution for annealed and branched networks. The mean connectivity is 3.3. (e) Branched networks are obtained by reducing the branch angles within the annealed networks. (f) Histograms of fiber lengths and branch angles for the different networks.

Grahic Jump Location
Fig. 6

Effect of dimensionless fiber bending stiffness κ. Results shown here are for a fiber length of L = 0.20 inclusion radii. (a) Displacement decay rate n for each bending stiffness κ. The increase in n with increasing κ indicates that as bending stiffness increases, the displacements propagate over a shorter range. (b) Nonaffinity χ̃ for each bending stiffness. For both panels, markers show medians of eight different random networks; error bars show interquartile range.

Grahic Jump Location
Fig. 7

Fiber reorientation. (a) and (b) Fiber reorientation, defined as the change in fiber angle, for the highest contraction with κ = 10−6 (a) and κ = 3 × 10−4 (b) the inner region within 15 inclusion radii is shown. (c) Histograms of fiber reorientation at the highest contraction, for fibers within 15 inclusion radii, averaged over the eight networks. As the fiber bending stiffness decreases, the fiber reorientation increases, but there is minimal change in reorientation below κ = 10−5. (inset) Histograms for fiber reorientation within a smaller inner region (<6 inclusion radii) shown by the solid circles in panels a and b. The same trends with respect to fiber bending stiffness hold closer to the inclusion where the displacements are larger. (d) Histograms for radial alignment (angle between each fiber and the radial direction) at the highest contraction, averaged over the eight networks. The radial alignment shows very little dependence on fiber bending stiffness.

Grahic Jump Location
Fig. 2

Displacement field for a network with fiber length L = 0.27 inclusion radii and fiber bending stiffness κ = 10−4. (a) Magnitude of displacements due to radially inward contraction of a circular inclusion by 32%. (b) Radial component of the displacements versus the radial position. The radial displacements are fit to Eq. (1) (solid line). The best fit is n = 0.52, indicating displacements propagate over a longer range than predicted by linear elasticity. For comparison, the 2D linear elastic solution n = 1 is shown by the dashed line. (c) When plotted on logarithmic axes, the fitted value of n matches the slope of the displacements in the regime near the center of the domain, indicating the power n quantifies the rate at which displacements decay over distance. (d) Nonaffine displacements over distance (dots) are fit to the right-hand side of Eq. (1) giving n = 0.46 (solid line). Inset shows the same data on semilogarithmic axes with a fit to a decaying exponential (dash-dot line). (e) Dimensionless nonaffinity χ at each node. Results are shown for only the positions within 75% of the outer radius (shown by thesolid circle in panel a), where the displacements are minimally affected by the boundary.

Grahic Jump Location
Fig. 3

Magnitude of contraction affects propagation of displacements and nonaffinity. (a) The displacement decay rate n versus contraction of the inclusion for networks with fiber length L = 0.27 inclusion radii and fiber bending stiffness κ = 10−4. As contraction increases, the displacements propagate over longer range (decay rate n decreases). (b) Average nonaffinity χ̃. As contraction increases the displacements become more affine with less variation across the networks. For both panels, the different markers correspond to each of eight different random networks having branched geometry.

Grahic Jump Location
Fig. 4

Comparison of different network types. (a) Displacement decay rate n for the two network generation methods, with fiber bending stiffness κ = 10−4. Markers show medians of eight different random networks of each type; error bars give interquartile range. (b) and (c) Network at 32% contraction with fibers above 1% tensile strain darkened. The dots indicate the nodes within the inclusion radius that the displacement boundary condition is applied to. Highly tensed fibers form aligned force chains. A given annealed network (b) exhibits fewer force chains than the associated branched network (c), because the reduced branch angles give pre-aligned load paths for force chains to develop.

Grahic Jump Location
Fig. 5

Effect of fiber length: (a) networks with three different fiber lengths, 0.27, 0.20, and 0.15 inclusion radii, are pictured from left to right, (b) displacement decay rate n for the three fiber lengths, using a fiber bending stiffness of κ = 10−4, and (c) average nonaffinity χ̃ for the three fiber lengths. For panels b and c, markers show medians over eight different random networks of each length; error bars show interquartile range.

Grahic Jump Location
Fig. 8

Results from both contraction and expansion for κ = 10−4 and L = 0.20: (a) displacement field resulting from an expanding inclusion, (b) displacement field in the same network for a contracting inclusion, and (c) displacement decay rate n for both expansion and contraction. For small contraction or expansion n ≈ 1, with higher expansion or contraction giving a greater or smaller decay rate as compared to a classically linear material. Markers show medians; error bars show interquartile range. (d) Nonaffinity for the eight networks for expansion and contraction. Note the logarithmic vertical axis. For panels c and d, the different markers correspond to each of the eight different networks.

Grahic Jump Location
Fig. 9

3D model of contracting spherical inclusion. (a) Undeformed 3D domain, with a horizontal midplane section highlighted. The domain's radius is 8 inclusion radii and the mean fiber length is 0.20 inclusion radii. (b) Themodel is solved using all fibers in panel a; the displacements shown correspond to the midplane section at 32% contraction of the spherical inclusion. (c) Displacement decay rate n. In three dimensions, the analytical solution from classical elasticity gives n = 2, whereas here we see n decreasing as the magnitude of contraction increases. (d) Nonaffinity for the 3D networks. For panels c and d, the different markers correspond to each of four different networks.




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