A hyperelastic biphasic fiber reinforced model of articular cartilage incorporating the influences of osmotic pressure and damage (original) (raw)
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On the numerical simulation of the mechanical behaviour of articular cartilage
International Journal for Numerical Methods in Engineering, 2006
In this paper, a fibre-reinforced porohyperelastic model is extensively presented and used for the simulation of soft hydrated tissues, like cartilage. The derivation of the classical governing equations for biphasic tissues is obtained from a complete and robust description for multiphasic systems. An Augmented Lagrangian formulation has been used to enforce the incompressibility condition in the whole tissue. Porohyperelasticity has been included in order to simulate the biphasic nature of the tissue associated to a porous solid matrix and interstitial fluid. Anisotropy in the hyperelastic behaviour has been formulated by means of a strain energy function that takes into account the influence of collagen fibres. Several numerical examples are shown in order to validate the proposed model. The confined compression problem has been solved to demonstrate the improvement that an Augmented Lagrangian Method provides with respect to a penalty approach. On the other hand, the unconfined compression problem allowed us to study the load distribution in each phase and its dependence on the strain rate. Finally, one realistic application of this model is briefly discussed.
Investigation of mechanical behavior of articular cartilage by fibril reinforced poroelastic models
Biorheology, 2003
The fibril reinforced poroelastic models have been found successful in describing some mechanical behaviors of articular cartilage in unconfined compression that were not understood previously, including the strong and nonlinear transient response, the strain-magnitude and strain-rate dependent cartilage stiffness and the depth-varying stresses and strains. It has been demonstrated that a better description for the mechanical behavior of cartilage is obtained by introducing a fibrillar matrix into a poroelastic model, in addition to the nonfibrillar matrix and water. This paper reports the development of the nonlinear fibril reinforced homogeneous and nonhomogeneous models and further explores the potentials of the models for investigation of cartilage mechanical response. Some comments are made in regard to further applications of the models and improved accuracy of the material representation.
Journal of Biomechanics, 2000
The depth dependence of material properties of articular cartilage, known as the zonal di!erences, is incorporated into a nonlinear "bril-reinforced poroelastic model developed previously in order to explore the signi"cance of material heterogeneity in the mechanical behavior of cartilage. The material variations proposed are based on extensive observations. The collagen "brils are modeled as a distinct constituent which reinforces the other two constituents representing proteoglycans and water. The Young's modulus and Poisson's ratio of the drained non"brillar matrix are so determined that the aggregate compressive modulus for con"ned geometry "ts the experimental data. Three nonlinear factors are considered, i.e. the e!ect of "nite deformation, the dependence of permeability on dilatation and the "bril sti!ening with its tensile strain. Solutions are extracted using a "nite element procedure to simulate uncon"ned compression tests. The features of the model are then demonstrated with an emphasis on the results obtainable only with a nonhomogeneous model, showing reasonable agreement with experiments. The model suggests mechanical behaviors signi"cantly di!erent from those revealed by homogeneous models: not only the depth variations of the strains which are expected by qualitative analyses, but also, for instance, the relaxation-time dependence of the axial strain which is normally not expected in a relaxation test. Therefore, such a nonhomogeneous model is necessary for better understanding of the mechanical behavior of cartilage.
Finite Element Modeling and Simulation of the Multiphysic Behavior of Articular Cartilage
2018
The finite element method (FEM) is the most widely used numerical method for solving complex problems mathematically represented by one or several coupled field equations in continuum physics. The main power of the FEM appears in coupled, nonlinear, and heterogeneous problems where several physical fields interact pointwise in a nonlinear way. This is the case of the behavior of articular cartilage in animals and humans, where the stress distribution is coupled with the fluid flow around the solid matrix and the constitutive behavior is controlled by the complex interaction of the fluid pressure, the Donnan pressure induced by the internal electrical charges in the fluid and matrix, and, finally, the fiber collagen distribution in the solid matrix. These three problems (electrochemistry, solid mechanics, and fluid diffusion) are strongly coupled and contribute together to the characteristic swelling and viscoelastic behavior of this biological tissue, which, in turn, is critical in ...
Study on biphasic material model and mechanical analysis of knee joint cartilage
Journal of Physics: Conference Series, 2008
A material model of articular cartilage is formulated, and fundamental problems are analyzed. The soft tissue is assumed to comprise two phases: solid and fluid. The biphasic theory proposed by to deal with such materials is reviewed, and some new additional analyses are carried out on the basis of this theory. Assuming the elasticity for the solid phase and introducing the pressure, which is defined by the product of the volume change and penalty coefficient, it is shown that the viscoelastic property of the soft tissue can be reproduced. A preferable solution is obtained for the solid phase by using the reduction integral, even if a high-order interpolation function is used. However, the high-order element cannot satisfactorily capture the velocity distribution of fluids. The pressure distribution is studied by assuming the change in the surface characteristics of the cartilage tissue with the progress of osteoarthritis. The pressure is strongly related to the lubrication conditions, i.e., perfect lubrication, perfect adhesion, and partial adhesion.
Journal of Biomechanical Science and Engineering, 2010
Articular cartilage has high water content and biphasic property. The structures of the tissue are inhomogeneous and anisotropic. Furthermore, the mechanical behavior of cartilage shows depth-dependence. Therefore it is necessary to consider not only the average tissue property but also the local one to explain mechanical and functional behavior. Previously, we created two-dimensional biphasic finite element (FE) cartilage tissue models considering the depth-dependence of elastic modulus distribution based on experimental results. As a result, this finding indicates that the depth-dependence of elastic modulus has a remarked influence on the deformed profile. In this study, the effectiveness of collagen fiber network in addition to the depth-dependent elastic modulus of cartilage tissue is evaluated. By creating of cartilage tissue models using axisymmetric biphasic elements and spring elements, we analyzed the unconfined compressive behaviors of articular cartilage specimens and compared the FE analyses to experimental results. Every FE model has depth-dependence of elastic modulus based on our previous formula, while the Poisson's ratio and permeability of solid phase were assumed as constant in literature data. To compare experimental result with finite element analysis (FEA), boundary conditions for FEA were given to correspond to the compression test. As a result, total load capacity and deformed profiles immediately after compression of FEA simulation on eventual model corresponded to experimental results by controlling spring constant. Furthermore, local strain of axial direction in FEA results for eventual model and experimental ones show the same tendency about time-dependent change. Then, we considered intrinsic fluid flow of eventual model.
Clinical Biomechanics, 1999
Objective. To develop a biomechanical model for cartilage which is capable of capturing experimentally observed nonlinear behaviours of cartilage and to investigate eects of collagen ®bril reinforcement in cartilage. Design. A sequence of 10 or 20 steps of ramp compression/relaxation applied to cartilage disks in uniaxial uncon®ned geometry is simulated for comparison with experimental data. Background. Mechanical behaviours of cartilage, such as the compression-oset dependent stiening of the transient response and the strong relaxation component, have been previously dicult to describe using the biphasic model in uncon®ned compression. Methods. Cartilage is modelled as a¯uid-saturated solid reinforced by an elastic ®brillar network. The latter, mainly representing collagen ®brils, is considered as a distinct constituent embedded in a biphasic component made up mainly of proteoglycan macromolecules and a¯uid carrying mobile ions. The YoungÕs modulus of the ®brillar network is taken to vary linearly with its tensile strain but to be zero for compression. Numerical computations are carried out using a ®nite element procedure, for which the ®brillar network is discretized into a system of spring elements. Results. The nonlinear ®bril reinforced poroelastic model is capable of describing the strong relaxation behaviour and compression-oset dependent stiening of cartilage in uncon®ned compression. Computational results are also presented to demonstrate unique features of the model, e.g. the matrix stress in the radial direction is changed from tensile to compressive due to presence of distinct ®brils in the model. Relevance Experimentally observed nonlinear behaviours of cartilage are successfully simulated, and the roles of collagen ®brils are distinguished by using the proposed model. Thus this study may lead to a better understanding of physiological responses of individual constituents of cartilage to external loads, and of the roles of mechanical loading in cartilage remodelling and pathology.
Journal of Biomechanics, 2010
A time-and depth-dependent Poisson's ratio has been observed during unconfined compression experiments on articular cartilage, but existing cartilage models have not fully addressed these phenomena. The goal of this study was to develop a model which is able to predict and explain these phenomena, while also being able to fit other experimental scenarios on full depth cartilage specimens such as confined and unconfined compressions. A biphasic (poroelastic), fiber-embedded cartilage model was developed. The heterogeneous material properties of the cartilage (aggregate modulus, void ratio tensile modulus) were extracted from reported experiments on individual layers of bovine articular cartilage. The nonlinear permeability material constants were found by fitting the overall response to published experimental data from confined compression. The matrix of the cartilage was modelled as an inhomogeneous isotropic biphasic material with nonlinear strain dependent permeability. Orthotropic layers were added as embedded elements to represent collagen fibers. Material parameters for these layers were derived from tensile tests of different layers of cartilage. With these predefined tensile parameters, the model showed a good fit with multi-step confined and unconfined compression experiments (R 2 ¼0.984 and 0.977, respectively) and could also predict the depth-dependent Poisson's ratio (R 2 ¼ 0.981). The highlight of the model is the ability to explain the time-depth dependent Poisson's ratio and, by association, the strong effect of material inhomogeneity on local stress and strain patterns within the cartilage layer. This material model's response may provide valuable new insight into potential initiation of cartilage fibrillation or delamination in wholejoint simulations.
Inhomogeneous Response of Articular Cartilage: A Three-Dimensional Multiphasic Heterogeneous Study
Articular cartilage exhibits complex mechano-electrochemical behaviour due to its anisot-ropy, inhomogeneity and material non-linearity. In this work, the thickness and radial dependence of cartilage properties are incorporated into a 3D mechano-electrochemical model to explore the relevance of heterogeneity in the behaviour of the tissue. The model considers four essential phenomena: (i) osmotic pressure, (ii) convective and diffusive processes, (iii) chemical expansion and (iv) three-dimensional through-the-thickness heterogeneity of the tissue. The need to consider heterogeneity in computational simulations of cartilage behaviour and in manufacturing biomaterials mimicking this tissue is discussed. To this end, healthy tibial plateaus from pigs were mechanically and biochemically tested in-vitro. Heterogeneous properties were included in the mechano-electrochemical computational model to simulate tissue swelling. The simulation results demonstrated that swelling of the heterogeneous samples was significantly lower than swelling under homogeneous and iso-tropic conditions. Furthermore, there was a significant reduction in the flux of water and ions in the former samples. In conclusion, the computational model presented here can be considered as a valuable tool for predicting how the variation of cartilage properties affects its behaviour, opening up possibilities for exploring the requirements of cartilage-mimicking biomaterials for tissue engineering. Besides, the model also allows the establishment of behavioural patterns of swelling and of water and ion fluxes in articular cartilage.