Affine Versus Non-Affine Fibril Kinematics in Collagen Networks: Theoretical Studies of Network Behavior

[+] Author and Article Information
Preethi L. Chandran

Department of Biomedical Engineering, University of Minnesota, 312 Church St. SE, Minneapolis, MN 55455

Victor H. Barocas1

Department of Biomedical Engineering, University of Minnesota, 312 Church St. SE, Minneapolis, MN 55455baroc001@umn.edu


Corresponding author.

J Biomech Eng 128(2), 259-270 (Oct 21, 2005) (12 pages) doi:10.1115/1.2165699 History: Received January 23, 2004; Revised October 21, 2005

The microstructure of tissues and tissue equivalents (TEs) plays a critical role in determining the mechanical properties thereof. One of the key challenges in constitutive modeling of TEs is incorporating the kinematics at both the macroscopic and the microscopic scale. Models of fibrous microstructure commonly assume fibrils to move homogeneously, that is affine with the macroscopic deformation. While intuitive for situations of fibril-matrix load transfer, the relevance of the affine assumption is less clear when primary load transfer is from fibril to fibril. The microstructure of TEs is a hydrated network of collagen fibrils, making its microstructural kinematics an open question. Numerical simulation of uniaxial extensile behavior in planar TE networks was performed with fibril kinematics dictated by the network model and by the affine model. The average fibril orientation evolved similarly with strain for both models. The individual fibril kinematics, however, were markedly different. There was no correlation between fibril strain and orientation in the network model, and fibril strains were contained by extensive reorientation. As a result, the macroscopic stress given by the network model was roughly threefold lower than the affine model. Also, the network model showed a toe region, where fibril reorientation precluded the development of significant fibril strain. We conclude that network fibril kinematics are not governed by affine principles, an important consideration in the understanding of tissue and TE mechanics, especially when load bearing is primarily by an interconnected fibril network.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Micromesh. A micromesh generated by growing segments from seed points and ending at intersections. Each intersection contains three segments, two of which are collinear. The mesh contains 553 segments.

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Figure 2

Fibril constitutive equations. Four constitutive equations were used. The primary equation was Eq. 1 with A=120nN and B=2, giving a highly nonlinear response in tension (solid line). The same equation was also used with A=480nN and B=0.5 (dashed line), giving a weaker nonlinear response. A bilinear form with stiffness 240nN in tension, and 2nN in compression (dotted line), and a linear form with uniform stiffness 240nN in extension and compression (not shown) were also considered. All four models gave a spring constant k=240nN for small extensional strains of a unit fibril.

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Figure 3

Confocal micrograph of a tissue equivalent. The Image is a 110μm square.

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Figure 4

Cumulative distribution functions TE and micromesh. Both the micromesh and the TE had a nearly uniform distribution of orientations (a). The segment length distributions (b) both possessed a wide uniform region, but the micromesh had many more very short segments and fewer very long segments.

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Figure 5

Micromesh at 30% strain for the network model. The model shows significant rearrangement from the undeformed state (Fig. 1).

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Figure 6

Stress response in the micromeshes. (a) The affine model shows a much more rapid rise and a higher stress than the network model. (b) The network model gives similar stress response for each of the four constitutive equations described in Fig. 2. The toe region, the upward curvature, and the small stress compared to the affine model are present in all cases. The stresses in the linear (circles) and the bilinear (dotted) case were nearly indistinguishable.

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Figure 7

Orientation response in the micromeshes. The network and affine models give similar values for the orientation parameter ΩXX at all strains.

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Figure 8

Fibril kinematics—orientation. (a) Final orientation versus initial orientation for 30% strain. Each point represents a single segment. In the affine model, the final orientation is determined by initial orientation, so a smooth, monotonic curve results. In the network model, interactions among connected segments lead to much more scatter. (b) The cumulative distribution function is shifted farther to the right in the network model than in the affine model, indicating that more segments have been recruited into the direction of stretch.

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Figure 9

Fibril kinematics—stretch. (a) Final segment stretch versus initial orientation for 30% strain. Because of the network’s ability to rearrange, most segments experience less stretch in the network model than in the affine model. Some of the segments in the network model are in compression (λ<1). (b) The probability distribution function for the stretches shows much more stretch in the affine model.

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Figure 10

Fibril Kinematics—stretch versus initial length. The network model (a) shows a negative correlation between initial length and stretch, but the affine model (b) does not.

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Figure 11

Fibril kinematics—correlations. The differences between affine and network behavior, reflected in the scatter plots of Figs.  8910, are quantified as correlation coefficients and plotted against strain. Correlations are shown for (a) initial angle θ0 and final angle θf; (b) initial length L0 and final stretch ratio λf; and (c) initial angle and final stretch ratio. Especially in the first two correlations, the network showed a shift in behavior at the end of the toe region.

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Figure 12

Stress parameters. The standard stress parameter φ does not capture the differences between affine and network reorientation (a), but the modified stress parameter ψ of Eq. (12), as does (b).



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