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TECHNICAL PAPERS

Coupled Macroscopic and Microscopic Scale Modeling of Fibrillar Tissues and Tissue Equivalents

[+] Author and Article Information
Balaji Agoram

Department of Chemical Engineering, University of Colorado, Boulder, CO 80309-0424

Victor H. Barocas

Department of Biomedical Engineering, University of Minnesota, Minneapolis, MN 55455

J Biomech Eng 123(4), 362-369 (Mar 31, 2001) (8 pages) doi:10.1115/1.1385843 History: Received March 07, 2000; Revised March 31, 2001
Copyright © 2001 by ASME
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Figures

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Flowchart of the solution strategy. Our goal is to incorporate microscopic information into the solution of a macroscopic problem. Once the macroscopic problem has been posed (Step 1), the macroscopic nodal positions are projected onto the microscopic domain within each element to generate a set of microscopic problems (Step 2). These problems are solved (independently) to determine how the microstructure must move to accommodate the macroscopic deformation and how much elastic force results from the motion (Step 3). The microscopic forces are then used to determine the macroscopic stress at every node (Step 4), allowing the evaluation of the residual equation, Eq. (4), in Step 5.
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Scanning electron micrographs of oriented and unoriented collagen gels: Gel (a) was 1×3 cm, and constrained compaction led to preferential alignment of the collagen fibrils in the horizontal direction. Gel (b) was 1×1 cm, leading to isotropic compaction and thus no alignment of the collagen fibrils. Scale bar in (a) is 1 μm.
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Experimental (a) (Ψ=0.61) and simulated (b) (Ψ=0.62) behavior of isotropic and anisotropic gels. Results for an anisotropic gel tested parallel to (∥) and perpendicular (⊥) to the direction of orientation are shown, as are results for extension of an isotropic gel (Iso). At 8 percent strain, the simulation is two times higher for F, close to the same for FIso, and five times lower for F.
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The macroscopic problem: Although our method could be applied generally to any stress-deformation problem, we focus here on simple extension of a rectangular (two-dimensional) tissue equivalent (a). The sample is assumed to be clamped on the left and right edges (solid lines) but free on the top and bottom edge (dashed lines). The sample is deformed (b) by moving the clamped edges, which allow no y displacement and require a specified x displacement. The free surface then deforms in response to the resulting internal stresses (deformation greatly exaggerated).
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Representation of FEM model: A micro–macro problem with four finite elements is shown. Macroscopic (finite element) nodes are shown by solid circles. The microscopic scale networks are all the same, but this is not a requirement.
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Schematic of fictional segments: Solid lines indicate real fiber segments. Broken lines indicate fictional segments that provide cost to bending of solid segments about a node. S represents a fiber segment, and F1 and F2 represent two fictional segments. Displacement of segment S (shown by S) causes tension in fictional segment F1 and compression in fictional segment F2.
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Accuracy of micro–macro model: To estimate discretization error in the FEM model stresses (in arbitrary units) for the FEM model and a direct microscopic problem for same TDOF are plotted for three different TDOF. Bars to left correspond to microscopic model. An average difference of 11 percent was found between the two methods.
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Error in micro–macro stress conversion: The percentage variation of s about the estimated value at the 95 percent confidence level is plotted against μDOF The error decreases with increasing μDOF and becomes roughly constant beyond μDOF=120.
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Scaling of models: CPU times are plotted for (i) FEM model versus TDOF for increasing finite elements; (ii) FEM model versus TDOF for increasing μDOF; (iii) direct solution versus TDOF. Best scaling is obtained when we increase the number of finite elements each with a constant μDOF.
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Model predictions of typical anisotropic TE behavior (Ψ=0.62): Higher forces are seen parallel to the direction of fiber orientation (∥) than perpendicular to the direction of orientation (⊥). Intermediate forces are seen for an isotropic TE (Iso).

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