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TECHNICAL PAPERS: Cell

Deterministic Material-Based Averaging Theory Model of Collagen Gel Micromechanics

[+] 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

In this document, the term “cross-links” refers to connections between fibrils.

Throughout this paper, we use “matrix” in the composite-materials sense of nonfibrillar material, rather than in the biological sense of extracellular matrix.

1

Corresponding author.

J Biomech Eng 129(2), 137-147 (Aug 14, 2006) (11 pages) doi:10.1115/1.2472369 History: Received March 03, 2005; Revised August 14, 2006

Mechanics of collagen gels, like that of many tissues, is governed by events occurring on a length scale much smaller than the functional scale of the material. To deal with the challenge of incorporating deterministic micromechanics into a continuous macroscopic model, we have developed an averaging-theory-based modeling framework for collagen gels. The averaging volume, which is constructed around each integration point in a macroscopic finite-element model, is assumed to experience boundary deformations homogeneous with the macroscopic deformation field, and a micromechanical problem is solved to determine the average stress at the integration point. A two-dimensional version was implemented with the microstructure modeled as a network of nonlinear springs, and 500 segments were found to be sufficient to achieve statistical homogeneity. The method was then used to simulate the experiments of Tower (Ann. Biomed. Eng., 30, pp. 1221–1233) who performed uniaxial extension of prealigned collagen gels. The simulation captured many qualitative features of the experiments, including a toe region and the realignment of the fibril network during extension. Finally, the method was applied to an idealized wound model based on the characterization measurements of Bowes (Wound Repair Regen., 7, pp. 179–186). The model consisted of a strongly aligned “wound” region surrounded by a less strongly aligned “healthy” region. The alignment of the fibrils in the wound region led to reduced axial strains, and the alignment of the fibrils in the healthy region, combined with the greater effective stiffness of the wound region, caused rotation of the wound region during uniaxial stretch. Although the microscopic model in this study was relatively crude, the multiscale framework is general and could be employed in conjunction with any microstructural model.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Inter-fibril rotational force. (a) The rotational force due to changes in angle between neighboring fibrils connected at a node is assumed to arise due to localized bending off the fibril close to the node. The force acts along the chord of the angle subtended by the connected fibrils. (b) Segment stretch ratio versus initial orientation at 4% extension for K=0.01. Few fibrils are recruited into stretch. The corresponding affine prediction for 4% extension is also shown in gray. (c) Segment stretch ratio versus initial orientation at 4% extension for K=10. A large number of fibrils undergo stretch. The affine prediction at 4% extension is shown in gray. The network model fibril strains are comparable to the affine.

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Figure 2

Macroscopic boundary value problem—uniaxial tensile test

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Figure 3

Parameterization of micromesh mechanical response. (a) Stress response at 30% as function of initial orientation state. Stress response is dependent on initial orientation, but differs between each size group. (b) Scaling stress response by fibril density gives consistent response between the different size groups.

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Figure 4

The undeformed (a) and deformed state at 27% extension (b) of a cross-loaded TE with initial alignment 88deg to the stretch axis. The RVEs within each finite element are shown at 10× magnification.

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Figure 5

Comparison of simulated and experimental response of cross-loaded TE in uniaxial extension (a) Simulation result: Force (scaled to fit the plot), retardation and its components as function of sample extension. (b) Experimental result: Force (scaled to fit the plot), retardation and its components as function of sample extension. (c) Simulation result: Force (scaled to fit the plot), components of the orientation angle as a function of sample extension. (d) Experimental result: Force (scaled to fit the plot), components of the orientation angle as a function of sample extension.

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Figure 6

The deformed state of the three TE microstructures—based on normal skin, wounded skin, and normal skin with a central wounded region—at 10%, 20%, and 30% extension. The wounded region is shown in gray. Each column is one TE type, and each row is one extension level.

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Figure 7

Stress response of homogeneous normal, homogenous wound, and non-homogeneous TEs. The wounded TE, highly aligned and oriented along the direction of stretch, gives a stiffer response than the normal TE. The force at 30% extension is 11 times that of the normal. If, however, the wounded region is within a surrounding normal TE, the stiffness of the resulting nonhomogeneous TE is much lower, approximately twice the normal TE response at 30% extension.

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Figure 8

The undeformed micromeshes based on normal and wounded skin, and their deformed state from selected regions of the TE at 30% extension. (a) Input micromeshes for normal and wounded TEs. The normal micromesh has a net orientation of 52.1deg. The wound micromesh has a net orientation of 0deg and is more aligned than the normal. (b) Reference positions for the micromeshes shown in Fig. 9(c). (c) Typical deformation state of micromesh at the locations marked in Fig. 9(b). Each column corresponds to one tissue type. Each row corresponds to one reference location. All micromeshes are of the same scale.

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