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Research Papers

Agent-Based Modeling Traction Force Mediated Compaction of Cell-Populated Collagen Gels Using Physically Realistic Fibril Mechanics

[+] Author and Article Information
James W. Reinhardt

Department of Biomedical Engineering,
The Ohio State University,
270 Bevis Hall, 1080 Carmack Rd.,
Columbus, OH 43210

Keith J. Gooch

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

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received September 11, 2013; final manuscript received November 26, 2013; accepted manuscript posted December 9, 2013; published online February 5, 2014. Editor: Victor H. Barocas.

J Biomech Eng 136(2), 021024 (Feb 05, 2014) (9 pages) Paper No: BIO-13-1426; doi: 10.1115/1.4026179 History: Received September 11, 2013; Revised November 26, 2013; Accepted December 09, 2013

Agent-based modeling was used to model collagen fibrils, composed of a string of nodes serially connected by links that act as Hookean springs. Bending mechanics are implemented as torsional springs that act upon each set of three serially connected nodes as a linear function of angular deflection about the central node. These fibrils were evaluated under conditions that simulated axial extension, simple three-point bending and an end-loaded cantilever. The deformation of fibrils under axial loading varied <0.001% from the analytical solution for linearly elastic fibrils. For fibrils between 100 μm and 200 μm in length experiencing small deflections, differences between simulated deflections and their analytical solutions were <1% for fibrils experiencing three-point bending and <7% for fibrils experiencing cantilever bending. When these new rules for fibril mechanics were introduced into a model that allowed for cross-linking of fibrils to form a network and the application of cell traction force, the fibrous network underwent macroscopic compaction and aligned between cells. Further, fibril density increased between cells to a greater extent than that observed macroscopically and appeared similar to matrical tracks that have been observed experimentally in cell-populated collagen gels. This behavior is consistent with observations in previous versions of the model that did not allow for the physically realistic simulation of fibril mechanics. The significance of the torsional spring constant value was then explored to determine its impact on remodeling of the simulated fibrous network. Although a stronger torsional spring constant reduced the degree of quantitative remodeling that occurred, the inclusion of torsional springs in the model was not necessary for the model to reproduce key qualitative aspects of remodeling, indicating that the presence of Hookean springs is essential for this behavior. These results suggest that traction force mediated matrix remodeling may be a robust phenomenon not limited to fibrils with a precise set of material properties.

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Figures

Grahic Jump Location
Fig. 1

(a) A simplified representation of the fibrous ECM that allows for discussion of the rules governing fibril mechanics. (b) A pair of nodes that are connected by a link behaves according to Hooke's Law. If the link is stretched or compressed, a force (Fh) proportional to the difference between the link's length and its resting length will act at each node in an effort to restore the resting length of the link. (c) Each set of three serially connected nodes behaves as a torsional spring. The magnitude of the force (Ft) that acts upon the outer nodes is proportional to the angle formed by the three nodes and acts in the direction perpendicular to the link that connects the outer node to the central node. A complementary force (Ftc) also acts upon the central node such that the sum of the forces acting upon the system is zero. (d) A cross-link is treated as multiple torsional springs. Each torsional spring shares the common node as its central node and a unique, nondivided angle formed by two links. In this example there are four torsional springs each identified by its angle, θ1–θ4, and the corresponding forces that act upon its nodes. Complementary forces are not labeled on this figure.

Grahic Jump Location
Fig. 2

(a) A diagram representing how axial forces were applied to a single fibril. Fibril mechanical properties were tested under tensile (b) and compressive (c) loads and agreed with analytically predicted values <0.001%.

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Fig. 3

(a) A diagram representing how three-point bending of a fibril was performed. A single force was applied at the midpoint of a fibril. The ends of the fibril were free to move in the horizontal but not vertical direction. (b) A plot for the deflection of a 100 μm fibril in response to a range of different loads. (c,d) The deflections for fibrils of different lengths were plotted in response to a constant applied load. The shape of a 100 μm fibril under various loads (e) and of fibrils of different lengths in response to a constant load (f) were plotted in comparison to their analytical solutions.

Grahic Jump Location
Fig. 4

(a) A diagram representing how fibrils were treated as end-loaded cantilevers. A single force was applied at one end of a fibril. The other end of the fibril was fixed in the horizontal and vertical directions. (b) A plot for the deflection of a 100 μm fibril in response to a range of different loads. (c,d) The deflection for fibrils of different lengths was plotted in response to a constant applied load. The shape of a 100 μm fibril under various loads (e) and of fibrils of different lengths in response to a constant load (f) were plotted in comparison to their analytical solutions.

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Fig. 5

Simulations were performed with two cells initially 200 μm apart (n = 10). (a) 0 h. (b) 1 h. (c) 2 h. Scale bar, 100 μm. (d) Percent change in the average density of the fibrous network, measured in links per patch. (e) Percent change in the density of the fibrous network between the two cells. Alignment of fibrils between the two cells was measured by both the anisotropy index (f) and overall direction of the fibrils with respect to a line connecting the nuclei of the two cells (g). *p < 0.05 compared to initial conditions.

Grahic Jump Location
Fig. 6

Simulations that demonstrate the impact of the torsional spring constant value on fibrous network remodeling. Simulations were performed using identical initial conditions but either the standard 1X value for the torsional spring constant (a) or a torsional spring constant that has been increased by a factor of 100 (b). Results are shown after 1 h. Scale bar, 20 μm.

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