ARTICLE IN PRESS. Journal of Biomechanics

Transcription

1 Journal of Biomechanics 43 (2010) Contents lists available at ScienceDirect Journal of Biomechanics journal homepage: A linearized formulation of triphasic mixture theory for articular cartilage, and its application to indentation analysis Xin L. Lu, Leo Q. Wan, X. Edward Guo, Van C. Mow n Department of Biomedical Engineering, Columbia University, 351 Engineering Terrace 500 West 120 th Street, New York, NY 10027, USA article info Article history: Accepted 9 October 2009 Keywords: Proteoglycan Osmotic Pressure Indentation Creep Stress Relaxation Linearization Aspect Ratio abstract The negative charges on proteoglycans significantly affect the mechanical behaviors of articular cartilage. Mixture theories, such as the triphasic theory, can describe quantitatively how this charged nature contributes to the mechano-electrochemical behaviors of such tissue. However, the mathematical complexity of the theory has hindered its application to complicated loading profiles, e.g., indentation or other multi-dimensional configurations. In this study, the governing equations of triphasic mixture theory for soft tissue were linearized and dramatically simplified by using a regular perturbation method and the use of two potential functions. We showed that this new formulation can be used for any axisymmetric problem, such as confined or unconfined compressions, hydraulic perfusion, and indentation. A finite difference numerical program was further developed to calculate the deformational, electrical, and flow behaviors inside the articular cartilage under indentation. The calculated tissue response was highly consistent with the data from indentation experiments (our own and those reported in the literature). It was found that the charged nature of proteoglycans can increase the apparent stiffness of the solid matrix and lessen the viscous effect introduced by fluid flow. The effects of geometric and physical properties of indenter tip, cartilage thickness, and that of the electrochemical properties of cartilage on the resulting deformation and fluid pressure fields across the tissue were also investigated and presented. These results have implications for studying chondrocyte mechanotransduction in different cartilage zones and for tissue engineering designs or in vivo cartilage repair. & 2009 Elsevier Ltd. All rights reserved. 1. Introduction Indentation test is one of the most frequently used methods for studying the biomechanical behaviors of articular cartilage (Sokoloff, 1966, Mow et al., 1989, Bae et al., 2004, Lu et al., 2004, Sweigart and Athanasiou, 2005). It has been used to determine the material properties of cartilage tissue in situ on the bone. The attachment of the cartilage to the bone avoids specimen preparation complications or other disruptive (e.g., scission of native collagen structure) specimen preparation procedures (Athanasiou et al., 1991, Mow et al., 2005, Lu et al., 2009). In contrast with the ease of performing an indentation experiment, however, the mathematics of analyzing the indentation test on a cartilage bone sample is extremely difficult due to its complex boundary conditions and the multi-dimensional deformations in the tissue layer (Hayes et al., 1972, Mak et al., 1987, Mow et al., 2005). The mathematical solution for the indentation problem using linear elasticity to model a cartilage layer was developed by Hayes et al. (1972), and the solution based n Corresponding author. Tel.: ; fax: address: vcm1@columbia.edu (V.C. Mow). on the biphasic constitutive model for cartilage (Mow et al., 1980) was later performed by Mak et al. for a frictionless, rigid, freedraining indenter tip (Mak et al., 1987), and by Spilker et al. with a finite element method for a frictional, rigid, porous indenter tip (Spilker et al., 1992), respectively. Despite the prevalence of biphasic theory and its wide acceptance in recent years, cartilage is not just a homogeneous mixture of an elastic solid and water. The negative charges fixed on the proteoglycans, a major component of the solid matrix, introduce an osmotic pressure within the tissue that generates significant influence on the stress, strain, fluid pressure, and electrical fields within the cartilage (Donnan, 1924, Maroudas, 1976, Lai et al., 1991). The density of the charges fixed on the proteoglycans is commonly known as the fixed charge density (FCD), and this FCD is responsible for a broad spectrum of observed mechano-electrochemical (MEC) phenomena (Frank and Grodzinsky, 1987, Lai et al., 2000). To account for these osmotic effects, a tertiary mixture theory (i.e., the triphasic theory) was developed for articular cartilage (Lai et al., 1991, Gu et al., 1998). It provides a comprehensive description of cartilage MEC behaviors, and it has been shown that the triphasic constitutive laws can be used to accurately calculate the FCD of a cartilage tissue sample by using indentation testing data (Lu et al., 2007). The mathematical /$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:1016/j.jbiomech

2 674 X.L. Lu et al. / Journal of Biomechanics 43 (2010) complexity of the theory, however, has hindered its adaptation for analyzing the indentation experiment. This study will show that, by using a regular perturbation method and potential functions similar to those commonly employed in 3D elasticity theory (Neuber, 1934), the triphasic governing equations can be significantly simplified to four coupled linear partial differential equations (PDEs). A mathematical solution for an indentation test, incorporating the electrolyte concentration in the external bathing solution, was developed based on this new triphasic formulation. The effects of FCD on the indentation responses were determined numerically and compared with the experimental data. The deformational and flow behaviors of articular cartilage, such as MEC fields, were parametrically studied using various material properties defined in the triphasic constitutive equations (Lai et al., 1991), PG contents, indenter tips, and aspect ratios (ratio of the indenter diameter to the tissue thickness). the boundary of the triphasic mixture is defined by the boundary of the solid matrix, and the interstitial free water and ions can freely exchange with those in the external solution; (3) the solid matrix is bonded to the rigid calcified cartilage/ subchondral bone layer that is impervious to fluid phase and ion species; (4) the indenter tip is rigid and its interface with cartilage surface is frictionless; (5) the porous indenter tip is free-draining to both fluid and ionic phases; the latter condition is expressed by the electrochemical potentials at the indenter cartilage interface to be equal to those in the external solution; (6) at the interface of impervious indenter tip and cartilage, all phases, including solid, fluid, and ions, have zero velocity in the axial direction of indenter tip. These six basic assumptions are widely accepted in literature (Mak et al., 1987, Spilker et al., 1992, Julkunen et al., 2007). Based on these assumptions, the indentation boundary conditions can be expressed with the four variables in the new governing equations (see online complementary materials). Due to the existence of FCD, articular cartilage is in a swollen state at physiological condition (Maroudas, 1976). In present study, this free-swollen state will be chosen as reference state for theoretical simulations, and the perturbation on all physical parameters is based on this reference state (Flahiff et al., 2002, Bae et al., 2006, Yao and Gu, 2007) Numerical methods 2. Materials and methods 2.1. Potential representations of the triphasic governing equations In this study, articular cartilage was treated as a triphasic material which consists of a charged incompressible, porous-permeable solid phase, an incompressible interstitial fluid phase (water), and two solute species (monovalent cation and anion) representing the third phase. The details of triphasic theory can be found in Lai et al. (1991) and Gu et al. (1998). The cartilage tissue was further assumed to be homogeneous and isotropic, the porous-permeable extracellular solid matrix to be linearly elastic and to be under infinitesimal deformation. The strain-dependent permeability effect and tension-compression nonlinearity of solid matrix were also ignored in this formulation so that we can focus on the effects of FCD on the tissue s mechanical behaviors. By using a regular perturbation method, the governing equations of the triphasic theory explicitly related to the FCD can be simplified into two parabolic PDEs (Lu et al., 2007) (see online supplementary materials for ¼ A 1r 2 e A 2 r ¼ A 4r 2 g A 5 r 2 e; ð2þ where t is the time; A s are coefficients which depend on FCD (s=1,2,4,5); e the elastic dilatation of solid matrix; g a dimensionless variable related with the ionic concentration. In cartilage indentation testing, the tissue is ideally loaded perpendicular to the end surface of a cylindrical or spherical indenter tip. Thus this is an axial-symmetric problem. In biphasic theory, the governing equations can be shown to be identically satisfied by expressing the fluid pressure, the radial and axial components of the displacement of the solid matrix as functions of two potential functions, j and c (Mak et al., 1987). u s u s @f ; p ¼ 2m þr2 c : ð5þ In the triphasic theory, all other governing equations with implicit reference to the FCD (Eqs. (A1) & (A6) in supplementary materials), can also be transformed into the following equations by expressing u s r, us z, and p as functions of j and c r 2 f ¼ 0; r 2 c ¼ e; ð7þ (see online supplementary materials for mathematical details). Combining Eqs. (1) (2) and Eqs. (6) (7), a new set of triphasic governing equations for axialsymmetric problems was obtained Boundary conditions and initial conditions To perform theoretical simulation for cartilage indentation tests using this new formulation of triphasic theory, some idealizations of the geometric and physical conditions have to be made. We assume: (1) in a small neighborhood of the indentation site, articular cartilage is a uniform flat layer of triphasic material; (2) ð1þ ð4þ ð6þ A finite difference program was developed to solve Eqs. (1), (2) and (6), (7) for the indentation configuration as shown in Fig. 1. The numerical method of lines (Schiesser, 1991) was employed and central-difference approximation was chosen for the spatial derivatives. The numerical integration over time scale was performed by a custom-modified commercial solver (MATLAB R13, The MathWorks, Inc., Natick, MA) based on backward differentiation formulas (i.e., Gear s method). In simulation, the following baseline parameters were used: temperature=293 k, external solution concentration c n =5 M, indenter tip radius a=5 mm, cartilage thickness h=1.471 mm, water content f w =0.74, intrinsic aggregate modulus (defined as that modulus without the osmotic effect introduced by fixed charge density) H a =0.24 MPa, Poisson s ratio u=5, permeability k= m 4 /Ns, FCD=5 meq/ml, D + = 10 9 m 2 /s, and D = m 2 /s. In numerical analysis, these parameters were necessary to calculate the five coefficients A 1 5 in governing equations, Eqs. (1) and (2), and to transform the results (j, c, e, and g) back into original physical parameters (e.g., stress and strain) (see supplementary materials). 3. Results 3.1. Results for creep test The theoretical triphasic prediction of cartilage creep response under a constant loading (Heaviside loading) was compared with data from an indentation test (Fig. 2). This mathematical prediction was obtained by using the above triphasic formulation for the indentation creep test. The input values of intrinsic Young s modulus and Poisson s ratio were determined by curve-fitting the indentation data of samples bathed in a hypertonic solution with the biphasic indentation solution (Mow, Gibbs et al., 1989, Lu et al., 2004). The hypertonic solution was used to remove the Donnan osmotic pressure -z a Articular Cartilage Subchondral Bone 5a Indenter Tip Fig. 1. A schematic diagram of indentation test showing a charged-hydrated cartilage attached to a rigid bone block indented by a circular, rigid, and frictionless indenter tip. The indenter tip may be free-draining or impervious. Due to the symmetry with respect to r=0, a half part of the sample was numerically analyzed. r

3 X.L. Lu et al. / Journal of Biomechanics 43 (2010) Creep Displacement (mm) Indentation Experiment Triphasic Simulation Fig. 2. A typical indentation creep data of cartilage bathed in a physiological environment (5 M PBS) and the triphasic theoretical prediction at the same condition using measured material coefficients and FCD. Axial Strain E zz z / h Creep Displacement (m) FCD (meq/ml) Dilatation e k z / h Fig. 3. A group of indentation creep curves of articular cartilage with various FCD values for numerical simulation. The loading profile and all the mechanical properties are kept identical. Reaction Force (N) FCD (meq/ml) 5 0 Porous-free draining 5 Porous-free draining Time (second) Fig. 4. Reaction force from articular cartilage surface during displacement control tests (i.e., stress-relaxation experiments) were calculated for cartilage with different FCD values. Both porous free-draining and impervious indenter tips were employed for these calculations. inside the tissue. According to previous experimental study, strains in the deformed configuration are small relative to the hypertonic configuration, as well as to the isotonic state of the Fluid Pressure (Pa) 120k 110k 100k 90k z / h Fig. 5. The transient history of (A) axial strain (E zz ) in the loading direction, (B) solid matrix dilatation, and (C) fluid pressure. All variables are calculated at three different depths across the cartilage during stress-relaxation tests. tissue (Narmoneva et al., 2001, Flahiff et al., 2002, Lu et al., 2007). For verification, the FCD value was obtained using biochemical GAG assay (Lu et al., 2004). By comparison with the triphasic simulation curve, Fig. 2 shows an accurate theoretical prediction. To study the effect of FCD on the creep response of cartilage, five creep tests were simulated using the same mechanical properties and loading conditions but with different FCD values (Fig. 3). The differences between equilibrium values illustrate the effect of FCD on tissue s apparent stiffness. Higher fixed charge

4 676 X.L. Lu et al. / Journal of Biomechanics 43 (2010) Axial Strain E zz z/h = Porous z/h = r / a Radial Displacement (Ur / h) Aspect Ratio (a/h) Normalized depth (z / h) 10k 0-3 Axial Stress σ zz (Pa) -10k -20k -30k -40k.5 z/h = Porous.5 z/h = -50k r / a Axial Strain E zz -6-9 Aspect Ratio Normalized depth (z / h) Fluid Pressure (Pa) 20k 18k 16k 14k 12k 10k.5 z/h = Porous.5 z/h = Dilatation e Aspect Ratio 8k r / a Fig. 6. Effect of indenter tip porosity (free-draining versus impervious) on the mechanical responses at the end of loading phase (t 0 =400 s): (A) axial strain versus radial position at three depths, (B) axial stress versus radial position, and (C) fluid pressure versus radial position Normalized depth (z / h) Fig. 7. Effect of aspect ratio (cartilage thickness/indenter radius) on cartilage mechanical behaviors at the end of loading phase (t 0 =400 s): (A) displacement of solid matrix in radial direction (U r ) versus depth, (B) axial strain (E zz )inz direction versus cartilage depth, and (C) dilatation (e) of solid matrix versus depth. density increases the apparent stiffness of the tissue, which generates a smaller equilibrium deformation. Another important phenomenon revealed in Fig. 3 is that tissues with higher FCD reach equilibrium much faster. The characteristic time for the creep curve from 1 meq/ml FCD is 263 s while the one for meq/ml FCD is 50 s. Therefore, the FCD affects not only the apparent stiffness of cartilage but also the flow-dependent transient responses (i.e., governed by tissue permeability) of the tissue under mechanical loading Results for stress relaxation test The reaction force on indenter tip from the cartilage with various FCDs (5 versus meq/ml) and different indenter tip permeabilities i.e., porous free draining versus solid impervious, in stress relaxation tests are shown in Fig. 4. The tissue is subjected to a displacement e 0 =10% of its initial thickness (i.e., average strain) with a ramp time t 0 =400 s. This loading protocol was previously adopted in the literature to study the mechanical

5 X.L. Lu et al. / Journal of Biomechanics 43 (2010) behaviors of cartilage (Mak et al., 1987, Spilker et al., 1992). Tissue with higher FCD showed a more distinct stiffening effect, consistent with the finding in indentation creep behaviors (Fig. 3) and those reported in previous experimental studies (Holmes et al., 1985, Frank and Grodzinsky, 1987, Lai et al., 1991). The impervious indenter tip (Fig. 4) generated higher reaction force than that of the porous free-draining tip before reaching the same equilibrium value. Fig. 5 showed the transient histories of three important physical parameters during the stress relaxation test: axial strain E zz (strain in loading direction z), dilatation of the solid matrix (tissue volume change), and fluid pressure, at three different depths across the tissue. The axial strain at the surface (z/h=0), middle zone (z/h=), and deep zone (z/h=1) showed significant differences during the relaxation phase. There is a recoil effect for E zz in both the surface and the middle zones (Fig. 5B). Due to the extremely low permeability of the solid matrix, the compression in z direction generated high hydraulic pressure (Fig. 5C), and fluid drag forces inside the solid matrix further forced the matrix to expand in the radial direction. This mechanism endows the tissue a nearly incompressible character in the initial loading phase which is confirmed by the small volume change at the end of loading phase (o1%) (Fig. 5B). As the fluid is being squeezed out, the solid matrix slowly recoiled back along the radial direction, and this process caused a redistribution of axial strain along the cartilage depth (Fig. 5A). These results indicated that, for the in situ situation, the middle zone cartilage is compressed the most in the loading direction (Fig. 5A), but has the lowest volume change (Fig. 5B). The largest tissue compressive dilatation occurs in the superficial zone (Fig. 5B), and the deep zone has the lowest axial strain. The fluid pressure across the tissue thickness direction is relatively uniform (Fig. 5C). Fig. 6 shows the effect of indenter tip porosity on cartilage mechanical behaviors at three different depths under indentation test. Three major mechanical parameters, axial strain and axial stress in z direction and the fluid pressure, are plotted versus radial position at time t 0 =400 s, the end of the ramp phase of compression. First, for all three components, severe gradients exist at cartilage surface due to the sharp edge of cylindrical indenter tip. With increasing depth, smoother distributions are observed for all three variables. Second, significant differences exist in the magnitudes of these three variables for impervious and porous indenters in superficial zone, while such difference vanishes at the deeper zones. Third, the value of all variables diminishes to zero rapidly outside the neighborhood of the indenter tip. All these three phenomena can be simply yet thoroughly explained by St. Venant s principle in elasticity. In cartilage indentation test, the aspect ratio k is an important parameter in experimental design (Bae et al., 2006). This number is defined as the ratio between indenter radius a and cartilage thickness h. Fig. 7 shows the effect of aspect ratio on three major mechanical responses along tissue depth with the tissue thickness kept constant while the aspect ratio varies from to 5. While the aspect ratio affects the radial strain distribution most significantly, it has less effect on the axial strain (Fig. 7A and B). Fig. 7B shows that the highest axial strain under indenter has similar magnitudes, although high aspect ratio generates an axial Err (Peak) Err (Final) Erz (Peak) Erz (Final) Ezz (Peak) Ezz (Final) Fig. 8. Strain fields inside cartilage tissue under indentation tests at the peak-loading time point (t 0 =400 s) and the final steady state (t=12,000 s).

6 678 X.L. Lu et al. / Journal of Biomechanics 43 (2010) P (impervious) P (free-draining) P (equilibrium) Dilatation e strain peak on the surface while a lower aspect ratio generates a peak around the middle zone. On the contrary, a small aspect ratio generates a much higher dilatation in solid matrix (i.e., sharper indentation tip), especially in the surface zone (Fig. 7C). Fig. 8 shows the strain fields, E rr, E zz, and E rz, across the tissue under an impervious indenter at peak loading time (t 0 =400 s) and at final steady state (t=12000 s). Evidently the sharp edge of the cylindrical indenter generates a stress concentration axisymmetric ring inside the tissue. Besides, at the singular point (edge), axial strain and radial strain reaches the highest value a little below the cartilage surface and diminishes sharply at the deep zone, which is consistent with previous experimental observations (Bae et al., 2006) and classical 3D elasticity results. Fig. 9 shows the pressure field at the time of peak loading (t 0 =400 s) from impervious and porous free-draining indenter tips. When a porous free-draining tip is used, the fluid pressure reaches peak value at the deep zone. On the contrary, loading from an impervious indenter tip creates the highest pressure right under the tip. Although indenter porosity influences the fluid movement during the entire loading and relaxation process, the fluid pressure fields from two cases share the same profile at equilibrium state (Fig. 9). The osmotic pressure inside the tissue at equilibrium state occupies a unique inhomogeneous pattern due to the non-zero dilatation of the solid matrix, which generates an inhomogeneous FCD across the tissue. This explains the same pattern shared by the dilatation field (e) of solid matrix and osmotic pressure field (Fig. 9). 4. Discussion 9 MPa 9 MPa 9 MPa Fig. 9. Effects of indenter tips on fluid pressure field inside cartilage tissue at peakloading time point (t 0 =400 s): (A) impervious indenter tip and (B) porous freedraining indenter tip. The pressure fields at equilibrium (C) for both setups are identical and similar with the pattern of the dilatation field of solid matrix (D). The governing equations of triphasic mixture theory were linearized and simplified by using a regular perturbation method and by the use of two potential functions (Eqs. (3) (5)). This new formulation is applicable for any axisymmetric loading case, such as confined or unconfined compressions, hydraulic perfusion, and indentation. In the present study, numerical solutions for both creep and stress-relaxation tests were developed for indentation analysis. The results showed that the mechanical behaviors of indented cartilage, both the transient and equilibrium responses, are significantly affected by the proteoglycan endowed charged nature of solid matrix. The effect of geometric and physical properties of the indenter tip, cartilage thickness, and chemical properties of cartilage on the resulting mechanical fields across the tissues were investigated and presented. The results may provide valuable implications for studying chondrocyte mechanotransduction in different cartilage zones and designing appropriate loading profiles for tissue regeneration. Certain theoretical limitations should be emphasized regarding the present study. The solid phase of tissue is simplified as a homogeneous, isotropic, linearly elastic material undergoing infinitesimal strain. In reality, the cartilage solid matrix is an inhomogeneous, anisotropic, nonlinear viscoelastic material, with a well-known nonlinear strain dependent permeability (Huang et al., 2005, Mow et al., 2005, Yao and Gu, 2007), and tension compression nonlinearity (Huang et al., 2005). The strain level found in this study also indicates a need for considering a finite deformation model, especially for the analysis around the stress concentration points. Despite the above limitations, applications of the linearized triphasic formulation to axisymmetric problems undergoing infinitesimal deformation is considerably simple. The major conclusions about cartilage indentation tests reported in the present study, as with all previously reported indentation analyses, appear to be valid and provide additional and valuable insights for cartilage indentation behavior. Conflict of interest statement The authors and collaborators have no conflicts of interest. Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at doi:1016/j.jbimech References Athanasiou, K.A., Rosenwasser, M.P., Buckwalter, J.A., Malinin, T.I., Mow, V.C., Interspecies comparisons of in situ intrinsic mechanical properties of distal femoral cartilage. Journal of Orthopaedic Research 9, Bae, W.C., Law, A.W., Amiel, D., Sah, R.L., Sensitivity of indentation testing to step-off edges and interface integrity in cartilage repair. Annals of Biomedical Engineering 32, Bae, W.C., Lewis, C.W., Levenston, M.E., Sah, R.L., Indentation testing of human articular cartilage: effects of probe tip geometry and indentation depth on intra-tissue strain. Journal of Biomechanics 39, Donnan, F.G., The theory of membrane equilibria. Chemical Reviews 1, Flahiff, C.M., Narmoneva, D.A., Huebner, J.L., Kraus, V.B., Guilak, F., Setton, L.A., Osmotic loading to determine the intrinsic material properties of guinea pig knee cartilage. Journal of Biomechanics 35, Frank, E.H., Grodzinsky, A.J., Cartilage electromechanics-ii. A continuum model of cartilage electrokinetics and correlation with experiments. Journal of Biomechanics 20, Gu, W.Y., Lai, W.M., Mow, V.C., A mixture theory for charged-hydrated soft tissues containing multi-electrolytes: passive transport and swelling behaviors. Journal of Biomechanical Engineering 120, Hayes, W.C., Keer, L.M., Herrmann, G., Mockros, L.F., A mathematical analysis for indentation tests of articular cartilage. Journal of Biomechanics 5, Holmes, M.H., Lai, W.M., Mow, V.C., Singular perturbation analysis of the nonlinear, flow-dependent compressive stress relaxation behavior of articular cartilage. Journal of Biomechanical Engineering 107, Huang, C.Y., Stankiewicz, A., Ateshian, G.A., Mow, V.C., Anisotropy, inhomogeneity, and tension-compression nonlinearity of human glenohumeral cartilage in finite deformation. Journal of Biomechanics 38, Julkunen, P., Korhonen, R.K., Herzog, W., Jurvelin, J.S., Uncertainties in indentation testing of articular cartilage: a fibril-reinforced poroviscoelastic study. Medical Engineering & Physics 30, Lai, W.M., Hou, J.S., Mow, V.C., A triphasic theory for the swelling and deformation behaviors of articular cartilage. Journal of Biomechanical Engineering 113, Lai, W.M., Mow, V.C., Sun, D.D., Ateshian, G.A., On the electric potentials inside a charged soft hydrated biological tissue: streaming potential versus diffusion potential. Journal of Biomechanical Engineering 122,

7 X.L. Lu et al. / Journal of Biomechanics 43 (2010) Lu, X.L., Miller, C., Chen, F.H., Guo, X.E., Mow, V.C., The generalized triphasic correspondence principle for simultaneous determination of the mechanical properties and proteoglycan content of articular cartilage by indentation. Journal of Biomechanics 40, Lu, X.L., Mow, V.C., Guo, X.E., Proteoglycans and mechanical behavior of condylar cartilage. Journal of Dental Research 88, Lu, X.L., Sun, D.D., Guo, X.E., Chen, F.H., Lai, W.M., Mow, V.C., Indentation determined mechanoelectrochemical properties and fixed charge density of articular cartilage. Annals of Biomedical Engineering 32, Mak, A.F., Lai, W.M., Mow, V.C., Biphasic indentation of articular cartilage-i. Theoretical analysis. Journal of Biomechanics 20, Maroudas, A.I., Balance between swelling pressure and collagen tension in normal and degenerate cartilage. Nature 260, Mow, V.C., Gibbs, M.C., Lai, W.M., Zhu, W.B., Athanasiou, K.A., Biphasic indentation of articular cartilage-ii. A numerical algorithm and an experimental study. Journal of Biomechanics 22, Mow, V.C., Gu, W.Y., Chen, F.H., Structure and function of articular cartilage and meniscus. In: Mow, V.C., Huiskes, R. (Eds.), Basic Orthopaedic Biomechanics and Mechano-biology. Lippincott Williams & Wilkins, Philadelphia, pp Mow, V.C., Kuei, S.C., Lai, W.M., Armstrong, C.G., Biphasic creep and stress relaxation of articular cartilage in compression? Theory and experiments. Journal of Biomechanical Engineering 102, Narmoneva, D.A., Wang, J.Y., Setton, L.A., A noncontacting method for material property determination for articular cartilage from osmotic loading. Biophysical Journal 81, Neuber, H., On a new attempt at solving the spatial problem of elasticity theories Hollow cones under concentrad loads as an example. Zeitschrift Fur Angewandte Mathematik Und Mechanik 14, Schiesser, W.E., In: The Numerical Method of Lines. Academic Press, San Diego. Sokoloff, L., Elasticity of aging cartilage. Federation Proceedings 25, Spilker, R.L., Suh, J.K., Mow, V.C., A finite element analysis of the indentation stress-relaxation response of linear biphasic articular cartilage. Journal of Biomechanical Engineering 114, Sweigart, M.A., Athanasiou, K.A., Biomechanical characteristics of the normal medial and lateral porcine knee menisci. Proceedings of the Institution of Mechanical Engineers H 219, Yao, H., Gu, W.Y., Convection and diffusion in charged hydrated soft tissues: a mixture theory approach. Biomechanics and Modeling in Mechanobiology 6,