Non-Linear Model for Compression Tests on Articular Cartilage

Abstract

Hydrated soft tissues, such as articular cartilage, are often modeled as biphasic systems with individually incompressible solid and fluid phases, and biphasic models are employed to fit experimental data in order to determine the mechanical and hydraulic properties of the tissues. Two of the most common experimental setups are confined and unconfined compression. Analytical solutions exist for the unconfined case with the linear, isotropic, homogeneous model of articular cartilage, and for the confined case with the non-linear, isotropic, homogeneous model. The aim of this contribution is to provide an easily implementable numerical tool to determine a solution to the governing differential equations of (homogeneous and isotropic) unconfined and (inhomogeneous and isotropic) confined compression under large deformations. The large-deformation governing equations are reduced to equivalent diffusive equations, which are then solved by means of finite difference (FD) methods. The solution strategy proposed here could be used to generate benchmark tests for validating complex user-defined material models within finite element (FE) implementations, and for determining the tissue's mechanical and hydraulic properties from experimental data.

The Darcy-Forchheimer law for modelling fluid flow in biological tissues

“…plays the role of a non-linear diffusion coefficient depending on J and its derivative J . A similar result has been recently discussed in [25,26], in which, however, only Darcy's law was considered. The main difference between D in ( 55) and the diffusion coefficient obtained in [25,26] is that D depends also on the derivative of the volumetric ratio.…”

“…A similar result has been recently discussed in [25,26], in which, however, only Darcy's law was considered. The main difference between D in ( 55) and the diffusion coefficient obtained in [25,26] is that D depends also on the derivative of the volumetric ratio. This is a consequence of the Forchheimer's correction that does not arise in the Darcian case.…”

A poroplastic model of structural reorganisation in porous media of biomechanical interest

“…In general, if the model of the considered tissue is inhomogeneous, the parameters α 0 , α 1 , α 2 , and β depend on material points. Sometimes, however, for computational simplicity, or because of lack of experimental data, it is assumed that only one of these parameters is variable; for instance, in modelling articular cartilage [46], α 0 was expressed by fitting experimental data taken from the literature as a thirdorder polynomial function of the axial coordinate parameterising the depth of a cylindrical specimen of tissue, whereas all the other material parameters were assumed to be constant. The expressions of σ s and σ f reported in (8a) and (8b) can be found in many works based on Mixture Theory (cf.…”

“…In (46), N is the unit vector normal to ∂C R , i.e. the boundary of the reference configuration C R , and the sets Γ χ D and Γ p D are the Dirichlet portions of ∂C R , on which the deformation, χ , and the pressure, p, are equal to the prescribed data χ b and p b , respectively.…”

Modelling of bone fracture healing: influence of gap size and angiogenesis into bioresorbable bone substitute

“…The presence of different phases with very heterogeneous material properties, like the case under study, can be dealt with by an approach similar to the one proposed in Cecchi and Rizzi [47]. This extension will also require adaptation of different tissue models: for example, the cartilage models proposed in Grillo et al [48] and Tomic et al [49]. This model could be improved by including the poroelastic effect.…”