A microstructurally based continuum model of cartilage viscoelasticity and permeability incorporating measured statistical fiber orientations

• Published in: Biomechanics and modeling in mechanobiology 2016-02, Vol.15 (1), p.229-244
• Main Authors: Pierce, David M., Unterberger, Michael J., Trobin, Werner, Ricken, Tim, Holzapfel, Gerhard A.
• Format: Article
• Language: English
• Subjects: Biological and Medical Physics; Biomedical Engineering and Bioengineering; Biophysics; Cartilage; Cartilage, Articular - anatomy & histology; Collagen; Collagen - chemistry; Collagens; Computer Simulation; Diffusion Tensor Imaging; Elasticity; Engineering; Fibers; Finite Element Analysis; Fluid pressure; Interstitials; Mathematical analysis; Mathematical models; Mechanical properties; Microstructure; Models, Biological; Models, Statistical; Networks; Original Paper; Permeability; Shear stress; Space life sciences; Strain; Stress, Mechanical; Theoretical and Applied Mechanics; Viscosity
• ISSN: 1617-7959; 1617-7940
• DOI: 10.1007/s10237-015-0685-x

The remarkable mechanical properties of cartilage derive from an interplay of isotropically distributed, densely packed and negatively charged proteoglycans; a highly anisotropic and inhomogeneously oriented fiber network of collagens; and an interstitial electrolytic fluid. We propose a new 3D finite strain constitutive model capable of simultaneously addressing both solid (reinforcement) and fluid (permeability) dependence of the tissue’s mechanical response on the patient-specific collagen fiber network. To represent fiber reinforcement, we integrate the strain energies of single collagen fibers—weighted by an orientation distribution function (ODF) defined over a unit sphere—over the distributed fiber orientations in 3D. We define the anisotropic intrinsic permeability of the tissue with a structure tensor based again on the integration of the local ODF over all spatial fiber orientations. By design, our modeling formulation accepts structural data on patient-specific collagen fiber networks as determined via diffusion tensor MRI. We implement our new model in 3D large strain finite elements and study the distributions of interstitial fluid pressure, fluid pressure load support and shear stress within a cartilage sample under indentation. Results show that the fiber network dramatically increases interstitial fluid pressure and focuses it near the surface. Inhomogeneity in the tissue’s composition also increases fluid pressure and reduces shear stress in the solid. Finally, a biphasic neo-Hookean material model, as is available in commercial finite element codes, does not capture important features of the intra-tissue response, e.g., distributions of interstitial fluid pressure and principal shear stress.