Papers »

A biphasic fiber-reinforced FE model for cartilage: continuum basis, computational aspects and applications

Cartilage is a mixture material consisting of fluid and electrolytes (68 - 85%), collagen fibers (15 - 25%), proteoglycans (5 - 10%) and chondrocytes (<0.01%), all by wet weight. From the mechanical point of view, cartilage is a porous, incompressible material combined with locally transversely isotropic solid and fluid behaviors. Moreover, the poro-viscosity and osmotic effects of the porous matrix, as well as material viscoelasticity of both the proteoglycan matrix and collagen fibers, should be taken into account.
In order to describe this complex material, and its resulting mechanical behaviors, we suggest a homogenized mixture model based on the theory of porous media which allows a coupled description of the solid-fluid interaction [1,2]. Therein, the interaction between the fluid pressure, strain-dependent pores and solid matrix stress may be considered. Moreover, the transversely isotropic behavior of the solid impacts both the stress response and the internal fluid permeability, which we model using an invariant formulation of the Helmholtz free energy and a transversely isotropic permeability function.
We model the network of collagen fibers using both a local principal orientation distribution within the tissue (superficial, middle and deep zones are commonly identified) and a measure of local dispersion (the fiber network is locally composed of dispersed fibers). We capture the local orientation and dispersion of the collagen network using experimental data from investigations using multiphoton microscopy [3]. This data calibrates the anisotropic, nonlinear response of the dispersed collagen fiber network which is captured phenomenologically by a strain-energy function extended to consider the dispersion of the collagen fiber about the principal orientation.
Here we first present our calculation concept, including the governing weak formulations for the finite element discretization, and then examine a representative numerical example.

REFERENCES
[1] Pierce D.M., Ricken T., Holzapfel G. A.: A hyperelastic biphasic fibre-reinforced model of articular cartilage considering distributed collagen fibre orientations: continuum basis, computational aspects and applications, Computer Methods in Biomechanics and Biomedical Engineering 16 (12), 1344–1361, 2013. DOI: 10.1080/10255842.2012.670854

[2] Pierce D.M., Ricken T., Holzapfel G. A.: Modeling sample/patient-specific structural and diffusional responses of cartilage using DT-MRI, International Journal for Numerical Methods in Biomedical Engineering 29 (8), 807 – 821, 2013. DOI: 10.1002/cnm.2524

[3] Lilledahl, M. B., Pierce, D. M., Ricken, T., Holzapfel, G. A., de Lange Davies, C.: Structural analysis of articular cartilage using multiphoton microscopy: input for biomechanical modeling, IEEE Transactions on Medical Imaging, 30 (9), 1635 – 1648, 2011. DOI: 10.1109/TMI.2011.2139222

Author(s):

Tim Ricken    
TU Dortmund University
Germany

David M. Pierce    
University of Connecticut
United States