Research Papers

A Coupled Fiber-Matrix Model Demonstrates Highly Inhomogeneous Microstructural Interactions in Soft Tissues Under Tensile Load

[+] Author and Article Information
Lijuan Zhang

Scientific Computation Research Center,
Rensselaer Polytechnic Institute,
Low Center for Industrial Innovation, CII-4011,
110 8th Street,
Troy, NY 12180

Spencer P. Lake, Victor H. Barocas

Department of Biomedical Engineering,
University of Minnesota,
7-105 Nils Hasselmo Hall,
312 Church Street SE,
Minneapolis, MN 55455

Victor K. Lai

Department of Chemical Engineering
and Materials Science,
University of Minnesota,
421 Washington Ave SE,
Minneapolis, MN 55455

Catalin R. Picu

Scientific Computation Research Center,
Rensselaer Polytechnic Institute,
Low Center for Industrial Innovation,
CII-4011, 110 8th Street,
Troy, NY 12180;
Department of Mechanical,
Aerospace and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Jonsson Engineering Center,
Rm. 2049, 110 8th Street,
Troy, NY 12180

Mark S. Shephard

Scientific Computation Research Center,
Rensselaer Polytechnic Institute,
Low Center for Industrial Innovation,
CII-4011, 110 8th Street,
Troy, NY 12180
e-mail: shephard@rpi.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received July 26, 2012; final manuscript received November 16, 2012; accepted manuscript posted December 8, 2012; published online December 27, 2012. Assoc. Editor: James C. Iatridis.

J Biomech Eng 135(1), 011008 (Dec 27, 2012) (9 pages) Paper No: BIO-12-1323; doi: 10.1115/1.4023136 History: Received July 26, 2012; Revised November 16, 2012; Accepted December 08, 2012

A soft tissue's macroscopic behavior is largely determined by its microstructural components (often a collagen fiber network surrounded by a nonfibrillar matrix (NFM)). In the present study, a coupled fiber-matrix model was developed to fully quantify the internal stress field within such a tissue and to explore interactions between the collagen fiber network and nonfibrillar matrix (NFM). Voronoi tessellations (representing collagen networks) were embedded in a continuous three-dimensional NFM. Fibers were represented as one-dimensional nonlinear springs and the NFM, meshed via tetrahedra, was modeled as a compressible neo-Hookean solid. Multidimensional finite element modeling was employed in order to couple the two tissue components and uniaxial tension was applied to the composite representative volume element (RVE). In terms of the overall RVE response (average stress, fiber orientation, and Poisson's ratio), the coupled fiber-matrix model yielded results consistent with those obtained using a previously developed parallel model based upon superposition. The detailed stress field in the composite RVE demonstrated the high degree of inhomogeneity in NFM mechanics, which cannot be addressed by a parallel model. Distributions of maximum/minimum principal stresses in the NFM showed a transition from fiber-dominated to matrix-dominated behavior as the matrix shear modulus increased. The matrix-dominated behavior also included a shift in the fiber kinematics toward the affine limit. We conclude that if only gross averaged parameters are of interest, parallel-type models are suitable. If, however, one is concerned with phenomena, such as individual cell-fiber interactions or tissue failure that could be altered by local variations in the stress field, then the detailed model is necessary in spite of its higher computational cost.

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Fig. 4

The magnitude of stress contributions differed depending on whether simulations were evaluated at constant total strain (solid lines) or constant total stress (dashed lines); (a) while matrix stress values increased in both cases, (b) fiber stresses show opposite trends for the two cases, where decreasing values for the constant-stress case demonstrates stress-shielding (by the matrix) at high shear moduli

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Fig. 5

Average (a) matrix stress, (b) fiber stress, and (c) fraction of total stress at 10% strain and with νNFM = 0.1 show good agreement between the parallel and coupled models; stress values at a larger Poisson's ratio (i.e., νNFM = 0.45) at G = 110 Pa show a small shift of stress from the matrix to fibers (mean ± 95% CI; n = 5)

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Fig. 3

(a) Fiber stress, (b) matrix stress, (c) total stress, and (d) RVE Poisson's ratio versus engineering strain for the coupled fiber-matrix model at varying values of the NFM shear modulus; the stress values increased with increasing G, particularly for the matrix, while Poisson's ratio decreased (data are representative results from a single network)

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Fig. 2

Development of the coupled fiber-matrix model: (a) work-flow demonstrating construction protocol for the coupled model; (b) illustration of the conforming multidimensional mesh and the mesh classification to the geometric model; (c) schematic showing the interior of the multidimensional mesh showing fibers (black lines) and meshed matrix (yellow elements with blue borders)

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Fig. 1

Computational representations of collagen fiber networks were constructed by randomly placing a set of seed points in a representative volume element, constructing a Voronoi tessellation about these points, defining fibers as the edges of the Voronoi elements, removing seed points, and placing pin-joint nodes at each edge-edge and edge-boundary intersection

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Fig. 6

Average (a) fiber stretch, (b) RVE Poisson's ratio, and (c) Ω11 (representing fiber orientation in the loading direction) at 10% strain and with νNFM = 0.1 show decreased values for the coupled model compared to the parallel model, but similar qualitative changes as a function of increasing G; for the case where νNFM = 0.1 and G = 110 Pa, the fiber stretch and Poisson's ratio increased and decreased, respectively, with no change in Ω11 (mean ± 95% CI; n = 5)

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Fig. 7

Interior normal and shear stress fields at 10% strain on the mid-section slice for a representative network (G = 720 Pa; νNFM = 0.1) demonstrates a highly inhomogeneous distribution for all six independent stress components; slices were cut normal to the loading (1-) direction in the 2-3 plane (represented by the dashed lines in the RVE schematic) and black dots indicate locations where fibers intersect the cutting plane; examples of two regions of high stress concentrations are indicated by arrows in the σ23 plot

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Fig. 8

(a) Maximum, and (b) minimum principal stress (normalized by the shear modulus of the matrix material) distributions at 10% strain over all mesh elements of the matrix material; the mean value increased and the standard deviation decreased as G increased, with curves moving more towards the affine model (the values from five networks are lumped and plotted)

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Fig. 9

Distribution of fiber stretch values at 10% strain and with νNFM = 0.1; values concentrated near ∼1 when G was low (i.e., G = 10 Pa and 110 Pa), but fibers were stretched to a greater extent at higher G values, similar to what was shown in the affine model (the values from five networks are lumped and plotted)




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