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Research Papers

An Agent-Based Discrete Collagen Fiber Network Model of Dynamic Traction Force-Induced Remodeling

[+] Author and Article Information
James W. Reinhardt

Department of Biomedical Engineering,
The Ohio State University,
270 Bevis Hall,
1080 Carmack Road,
Columbus, OH 43210
e-mail: james.reinhardt@nationwidechildrens.org

Keith J. Gooch

Department of Biomedical Engineering,
The Ohio State University,
270 Bevis Hall,
1080 Carmack Road,
Columbus, OH 43210;
Dorothy M. Davis Heart &
Lung Research Institute,
The Ohio State University,
473 W. 12th Avenue,
Columbus, OH 43210
e-mail: gooch.20@osu.edu

1Corresponding author.

Manuscript received May 11, 2017; final manuscript received September 11, 2017; published online February 22, 2018. Assoc. Editor: Carlijn V.C Bouten.

J Biomech Eng 140(5), 051003 (Feb 22, 2018) (13 pages) Paper No: BIO-17-1207; doi: 10.1115/1.4037947 History: Received May 11, 2017; Revised September 11, 2017

Microstructural properties of extracellular matrix (ECM) promote cell and tissue homeostasis as well as contribute to the formation and progression of disease. In order to understand how microstructural properties influence the mechanical properties and traction force-induced remodeling of ECM, we developed an agent-based model that incorporates repetitively applied traction force within a discrete fiber network. An important difference between our model and similar finite element models is that by implementing more biologically realistic dynamic traction, we can explore a greater range of matrix remodeling. Here, we validated our model by reproducing qualitative trends observed in three sets of experimental data reported by others: tensile and shear testing of cell-free collagen gels, collagen remodeling around a single isolated cell, and collagen remodeling between pairs of cells. In response to tensile and shear strain, simulated acellular networks with straight fibrils exhibited biphasic stress–strain curves indicative of strain-stiffening. When fibril curvature was introduced, stress–strain curves shifted to the right, delaying the onset of strain-stiffening. Our data support the notion that strain-stiffening might occur as individual fibrils successively align along the axis of strain and become engaged in tension. In simulations with a single, contractile cell, peak collagen displacement occurred closest to the cell and decreased with increasing distance. In simulations with two cells, compaction of collagen between cells appeared inversely related to the initial distance between cells. These results for cell-populated collagen networks match in vitro findings. A demonstrable benefit of modeling is that it allows for further analysis not feasible with experimentation. Within two-cell simulations, strain energy within the collagen network measured from the final state was relatively uniform around the outer surface of cells separated by 250 μm, but became increasingly nonuniform as the distance between cells decreased. For cells separated by 75 and 100 μm, strain energy peaked in the direction toward the other cell in the region in which fibrils become highly aligned and reached a minimum adjacent to this region, not on the opposite side of the cell as might be expected. This pattern of strain energy was partly attributable to the pattern of collagen compaction, but was still present when mapping strain energy divided by collagen density. Findings like these are of interest because fibril alignment, density, and strain energy may each contribute to contact guidance during tissue morphogenesis.

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Figures

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Fig. 1

Cells are composed of 109 agents (37 nodes and 72 links). Thirty nodes compose a cell membrane 20 μm in diameter (green squares around outer perimeter) and seven nodes were used to create a central nucleus 10 μm in diameter (black dots). Links connect neighboring membrane nodes (yellow lines between green squares), neighboring nuclear nodes (blue lines that forming a segmented hexagon) and between select membrane and nuclear nodes (green lines). Links behave as Hookean springs to allow forces to be transmitted intracellularly.

Grahic Jump Location
Fig. 2

(a) A complete 2.0 mg/mL network with perfectly straight collagen fibrils measuring 105 μm by 105 μm. Nodes along fibrils are colored green and nodes at cross-links are colored red. Fibril segments with dead ends that do not contribute to the mechanical properties of the network (black arrows) were removed to increase computational efficiency. (b) Networks subjected to tensile strain have all dead ends removed except those that terminate at the left and right edges of the network. This is similar for networks subjected to shear strain, except only dead ends that terminate at the top and bottom edges of the network are preserved. (c) 3.0 mg/mL networks were made by adding 1.0 mg/mL collagen (red) to each 2.0 mg/mL network (blue). This process was repeated to generate 4.0 mg/mL networks. (d) Fibril curvature was introduced to a network by moving nodes along fibrils 0.5, 1.0, 1.5, 2.0, and 2.5 μm perpendicular to the fibril orientation while keeping the position of cross-links fixed. This panel shows how the network in panel B looks when nodes are moved 1.5 μm.

Grahic Jump Location
Fig. 3

Mechanical testing was performed on randomly distributed fibril networks that measured 105 μm × 105 μm and represented a thickness of 1.0 μm. A 2.0 mg/mL collagen network that is unstrained (a) and (e) or shown after subjection to tensile (b)–(d) or shear (f)–(h) strain. Fibril segments labeled red have been stretched greater than 1%. For networks subjected to tension, these stretched fibrils are horizontal and appear perfectly straight. For networks subjected to shear, the stretched fibrils are straight between cross-links and generally aligned from bottom left to top right. This is most evident in panel H. (i) The percentage of fibril segments stretched greater than 1% was quantified for networks subjected to 30% strain (n = 100). Error bars represent the standard deviation (**p ≤ 0.005, ****p ≤ 0.0001).

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Fig. 4

Stress–strain curves for collagen networks at 2.0, 3.0, and 4.0 mg/mL collagen subjected to tensile (a) and shear (b) strain. Overlain on panel B is data from Stein et al. [24] (Reproduced with permission from John Wiley and Sons @ 2010), showing a comparison between the stress–strain response of their 3D finite element model and experimental data obtained by rheometry of a collagen hydrogel. (c) and (d) Anisotropy index values corresponding to the data in (a) and (b). (e) Incremental Young's modulus for a simulated 3.0 mg/mL collagen network subjected to tensile strain (n = 1). An identical network was subjected to the same strain, but torsional springs were removed. Removal of torsional springs meant that nodes along fibrils and at cross-links would act as freely rotating hinges that do not provide resistance to movement. (f) Fibril alignment was measured by quantifying the anisotropy index for the entire network. Error bars represent the standard deviation (n = 10).

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Fig. 5

Effect of collagen concentration on modulus (a) and fold increase in modulus (b) as measured in response to low (0.002) tensile and shear strain. Error bars represent the standard deviation.

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Fig. 6

A single 2.0 mg/mL collagen network was modified to introduce varying degrees of fibril curvature. These new networks were then subjected to tensile (a) and shear (b) strain. Increasing the curvature of fibrils in the collagen network caused an increase in the value of strain at which simulated networks strain-stiffened.

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Fig. 7

Remodeling around a single cell was quantified over 432,000 iterations by the inward displacement of nodes on 2.0 mg/mL collagen networks (n = 12 simulations). Nodes that were tracked belonged to concentric rings around the cellat the start of the simulation (i.e., 100 ± 3 μm, 133 ± 3 μm, 167 ± 3 μm, etc.). Also included on this figure is experimental data quantified at 6 h from Mohammadi et al. [46]. Error bars represent the standard deviation except for data from Mohammadi et al. for which error bars represent the standard error of the mean.

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Fig. 8

(a) Confocal reflectance image of two endothelial cells cultured within a collagen network after 4 h. This image was kindly provided by Dr. Alisha Sarang-Sieminski. Experimentally, gathering intensity was quantified between two cells by dividing the average intensity in a rectangular intercellular region (box 1) by the average intensity in a region that had not been remodeled (box 2). Cells are identified by black arrows. (b) Previously published experimental data from McLeod et al. [40] reproduced with permission. (c) Images from computational simulations with cells separated by 75 μm before (top) and after (middle) applying traction force, and cells separated by 150 μm after applying traction force (bottom). Pseudopodia actively contracting the network are shown as short rectangles (green) extending from the periphery of the two cells (solid blue circles). Computationally, gathering intensity was quantified by dividing the average collagen concentration in a rectangular intercellular region by the average initial collagen concentration. (d) Gathering intensity was quantified in simulations after 288,000 iterations. Collagen concentration was changed by varying fibril number. Error bars represent the standard deviation.

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Fig. 9

Average strain energy maps (a)–(e) and strain energy as measured 10 μm from cells was plotted as a function of position around the cell (f)–(j) from simulations with a collagen density of 2.0 mg/mL. Due to symmetry in two planes, these values are the average from 32 data points (n = 8, 2 cells per simulation, top and bottom symmetry). 0 deg corresponds to the direction toward the other cell and 180 deg is away from the other cell. Minimum strain energy is indicated with an arrow. Error bars denote the standard deviation.

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