A Comparison Between Mechano-Electrochemical and Biphasic Swelling Theories for Soft Hydrated Tissues

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1 A Comparison Between Mechano-Electrochemical and Biphasic Swelling Theories for Soft Hydrated Tissues W. Wilson C. C. van Donkelaar 1 J. M. Huyghe Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, P.O. Box 513, 5600 MB, The Netherlands Biological tissues like intervertebral discs and articular cartilage primarily consist of interstitial fluid, collagen fibrils and negatively charged proteoglycans. Due to the fixed charges of the proteoglycans, the total ion concentration inside the tissue is higher than in the surrounding synovial fluid (cation concentration is higher and the anion concentration is lower). This excess of ion particles leads to an osmotic pressure difference, which causes swelling of the tissue. In the last decade several mechano-electrochemical models, which include this mechanism, have been developed. As these models are complex and computationally expensive, it is only possible to analyze geometrically relatively small problems. Furthermore, there is still no commercial finite element tool that includes such a mechano-electrochemical theory. Lanir (Biorheology, 24, pp , 1987) hypothesized that electrolyte flux in articular cartilage can be neglected in mechanical studies. Lanir s hypothesis implies that the swelling behavior of cartilage is only determined by deformation of the solid and by fluid flow. Hence, the response could be described by adding a deformation-dependent pressure term to the standard biphasic equations. Based on this theory we developed a biphasic swelling model. The goal of the study was to test Lanir s hypothesis for a range of material properties. We compared the deformation behavior predicted by the biphasic swelling model and a full mechano-electrochemical model for confined compression and 1D swelling. It was shown that, depending on the material properties, the biphasic swelling model behaves largely the same as the mechano-electrochemical model, with regard to stresses and strains in the tissue following either mechanical or chemical perturbations. Hence, the biphasic swelling model could be an alternative for the more complex mechano-electrochemical model, in those cases where the ion flux itself is not the subject of the study. We propose thumbrules to estimate the correlation between the two models for specific problems. DOI: / Keywords: Cartilage, Swelling, Triphasic, Quadriphasic, Finite Element Method Introduction Biological tissues like intervertebral discs and articular cartilage primarily consist of interstitial fluid, collagen fibrils and negatively charged proteoglycans. Due to the fixed charges of the proteoglycans, the cation concentration inside the tissue is higher than in the surrounding synovial fluid. This excess of ion particles leads to an osmotic pressure difference, which causes swelling of the tissue 1. Mow et al. 2 formulated a biphasic mixture theory for articular cartilage where the collagen proteoglycan matrix is described as an incompressible porous-permeable solid matrix, and the interstitial fluid as incompressible. Lai et al. 3 derived a small deformation mechano-electrochemical extension of this biphasic theory. The three phases included in this theory were an incompressible solid, an incompressible fluid and a monovalent ionic phase. This theory was generalized to finite deformations by Huyghe and Janssen 4. Gu et al. 5 generalized the theory for multiple electrolyte systems. 1 Corresponding author: C. C. van Donkelaar, Dept. Biomedical Engineering, WH 4.118, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Telephone: ; fax: ; c.c.v.donkelaar@tue.nl Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division December 2, 2003; revision received August 4, Associate Editor Gerard A. Ateshian. In order to apply the mechano-electrochemical theory to large biological systems, or applied biomechanical problems, the mechano-electrochemical constitutive equations must be implemented into a finite element model FEM. Simon et al. 6 proposed a poroelastic finite element formulation that included transport and swelling effects in soft tissue. Frijns et al. 7 implemented the mechano-electrochemical model in a finite element model and verified it experimentally relative to onedimensional transient behavior of annulus fibrosis tissue. Levenston et al. 8 developed a variational theory for quasistatic analyses of coupled electrokinetic and fluid flow in porous media with intrinsically incompressible constituents, and tested this theory for several axisymmetric problems. Sun et al. 9 developed a FEM formulation including the coupling between mechanical, electrical and chemical events in articular cartilage. The model of Frijns et al. 7 has recently been validated by comparing its results to the analytical solution of van Meerveld et al. 10. Van Loon et al. 11 extended this model to three dimensions. The results of this work is that we are able to solve in 3D the effects of mechanical and chemical stimulation on the transient behavior of swelling materials. However, as these models are complex and computationally expensive, it is only possible to solve geometrically relatively small problems. Furthermore, there is still no commercial finite element tool that includes the mechano-electrochemical theory. Lanir 12 hypothesized that the chemical potential responds to changes in the local ionic environment with a time constant which 158 Õ Vol. 127, FEBRUARY 2005 Copyright 2005 by ASME Transactions of the ASME

2 is an order of magnitude shorter than the permeability and mechanical time constants. Hence, the effect of time on ion concentration, and thereby on the osmotic pressure, can be neglected. If this assumption is correct, then the ion concentration can be assumed in local equilibrium, depending only on the external salt concentration and the fixed charge density, which is a function of the deformation. Note that the deformation is a function of time. If Lanir s hypothesis is correct and assuming a constant external ion concentration, the swelling behavior of cartilage would only be determined by the deformation of the solid and by fluid flow. Hence, the swelling behavior of soft hydrated tissues might be described by adding a deformation-dependent pressure term to the standard biphasic model 12. This would make modeling of swelling behavior of soft hydrated tissues easier and computationally less expensive. However, the hypothesis that the ion flow is infinitely fast compared to the fluid flow, is not correct for biological tissues in general. The goal of the study was to test whether Lanir s hypothesis could be used to determine the deformation of soft hydrated tissues as a result of mechanical and chemical loading for a range of material properties. Therefore we developed a biphasic swelling finite element model, based on the hypothesis of Lanir 12, and compared its swelling and compression behavior with the results of the full mechano-electrochemical finite element model of Frijns et al. 7. Based on the results we derived equations, which can be used to determine the correlation between the biphasic swelling and full mechano-electrochemical model, and thereby can be used to determine when the swelling of a soft hydrated tissue can be described by using a biphasic swelling model, or when a full mechano-electrochemical model is preferred. Method Following Lanir 12, assuming that the tissue consists of an incompressible solid hydrated with an incompressible fluid, the total stress in the tissue is given by pi s, (1) where p is the hydrostatic fluid pressure, s the effective stress and I the unit tensor. Due to the fixed charges, the cation concentration inside the tissue is higher than in the surrounding synovial fluid. This excess of ion particles within the matrix creates a pressure referred to as Donnan osmotic pressure, which is a driving force for fluid flow. If we include the osmotic pressure in Eq. 1, the total tissue stress becomes f I s, (2) where f is the water chemical potential and the osmotic pressure gradient, which is given by ext. (3) The internal and external osmotic pressures ( ext ) are given by 4 RTc c, (4) with the osmotic coefficient, R the gas constant, T the absolute temperature and c and c the concentrations of mobile cations and anions. Since the osmotic components are assumed to equilibrate instantaneously with the external bath, only the ion concentrations in equilibrium have to be determined. These equilibrium ion concentrations are given by 4 c F c 2 F 4 ext 2 c int c 2 2 ext, (5) 2 where c ext is the external salt concentration, c F the fixed charge density and the activity coefficients. When assuming that ext and int are equal 13,14 and by substituting Eq. 5 and 4 into Eq. 3 the osmotic pressure gradient in equilibrium becomes in RTc 2 F 4c 2 ext 2 ext RTc ext. (6) When assuming that the external salt concentration and temperature stay constant, the only nonconstant in this equation is the fixed charge density (c F ). This fixed charge density can be expressed as a function of the tissue deformation, as c F c F,0 n f,0, (7) n f,0 1J where n f,0 is the initial fluid fraction, c F,0 the initial fixed charge density and J is the determinant of the deformation tensor F. Since the osmotic pressure in equilibrium is only a function of the deformation tensor F, it can easily be implemented in a commercial finite element package. For this study we added osmotic swelling behavior to the standard biphasic model in ABAQUS using the subroutine UMAT. To include the osmotic pressures we implemented the total solid stress ( s ) as the actual solid stress ( s ) minus the osmotic pressure s s I (8) with as given in Eq. 6. Hence, Eq. 2 becomes f I s (9) Note that the pore pressure p in the biphasic model is replaced by the electrochemical potential f. In Appendix A the total set of equations of both the regular biphasic and biphasic swelling theory is given. For the behavior of the actual solid matrix a compressible neo-hookean model was used 11,15, s 1 2 K J 1 JI G J F"FT J 2/3 I. (10) The bulk modulus K and shear modulus G are defined as E K (11) 312 G E 21, (12) where E is the Young s modulus and the Poisson s ratio. To test the hypothesis that global deformation of the material is correctly determined under mechanical and chemical loading when the ion concentration is always assumed in equilibrium, we compared our biphasic swelling model to the mechanoelectrochemical model 7. To be able to compare both models the behavior of the solid matrix of the model of Frijns et al. 7 was replaced with the compressible neo-hookean model as given in Eq. 10. The models were compared under confined compression and 1D swelling conditions. We only considered an open circuit system in which there is no streaming current over the sample. For the modeling of an open circuit system the hydraulic permeability of the biphasic swelling model was adjusted Appendix B. For the modeling of the confined compression and 1D swelling tests a cylindrical disc of tissue thickness h, equilibrated in a NaCl solution with a concentration of c ext, is placed inside an impermeable confined compression chamber Fig. 1a. This confining cylinder has an impermeable piston on top and a permeable filter at the bottom. At t0 a pressure P is applied to the piston confined compression or the external salt concentration is changed to c* free swelling Fig. 1b. Attt* the external pressure is removed or the external salt concentration is returned to c ext. For every test t* was 1.5 times the relaxation time of the biphasic swelling model. For all tests the timesteps were t*/50. The relaxation time of the biphasic swelling model was determined from a Journal of Biomechanical Engineering FEBRUARY 2005, Vol. 127 Õ 159

3 Fig. 1 Schematic representation of a confined compressionõfree swelling test: a a cylindrical disc of tissue thickness h, is placed inside a confined compression chamber with an impermeable piston on top and a permeable filter at the bottom. b At tä0 either a pressure P is applied to the piston confined compression or the external salt concentration is changed to c* free swelling.attät* the external force is removed or the external salt concentration is returned to c ext. first simulation of the biphasic swelling model with a timestep of 1 second. At the surface of the permeable filter, where free fluid flow is possible, the electro-chemical potential inside and outside the tissue must be the same f f ext. (13) The difference between the biphasic swelling model and the mechano-electrochemical model was determined for a wide range of material properties. All these parameter sets are variations on a reference set of realistic material properties for articular cartilage Table 1. As variations on the reference case D and D were varied between and mm 2 /s, k between mm 4 /Ns and mm 4 /Ns, c ext between and mmol/mm 3, c F,0 between and meq/mm 3, E between 0.1 and 0.5 MPa. Besides the material properties, the external pressure P was varied between and MPa and external salt concentration c* was varied between and mmol/mm 3. First all these parameters were changed individually. After this several randomly selected combinations of these variations were used. The internal osmotic coefficient ( in ) was assumed a linear function of the initial fixed charge density. Since the internal osmotic coefficient should be equal to the external osmotic coefficient ( ext ) for a fixed charge density of zero, and is approximately 0.83 for a fixed charge density of , in was computed as Table 1 Material properties of the reference case 7,11,16 c*110 4 mmol/mm 3 P0.25 MPa c ext mmol/mm 3 R N mm/mmol K c F, meq/mm 3 r0.4 D mm 2 /s T298 K D mm 2 /s V mm 3 /mmol E0.5 Mpa V 2.33 mm 3 /mmol F C/mmol z 1 h0.5 mm z 1 k110 3 mm 4 /Ns ext n f, in c F, (14) To compare the biphasic swelling model and the mechanoelectrochemical model we compared the global strain curves. This was done for both the confined compression and 1D swelling tests. To quantify the difference between the two models the coefficient of determination values (R 2 ) between the strain curves were computed. The R 2 values were computed per test case, for the loading and unloading parts separately. From these results a simple rule was derived, which can be used to determine when a Fig. 2 Finite element mesh and boundary conditions of the 2D simulation 160 Õ Vol. 127, FEBRUARY 2005 Transactions of the ASME

4 Fig. 3 Computed strain curves for the confined compression simulations of the reference case the R 2 values between the biphasic swelling and the mechano-electrochemical curves were and for the loading and unloading part, respectively Fig. 4 Computed strain curves for the 1D swelling experiments of the reference case the R 2 values between the biphasic swelling and the mechano-electrochemical curves were and for the swelling and shrinking part, respectively biphasic swelling model can be used and when a mechano-electrochemical model should be used to describe tissue deformation. To test if the biphasic swelling model can also be used in two dimensions, we compared the response of the biphasic swelling model in a 2D situation to the response of the full mechanoelectrochemical model of van Loon et al. 11. For the 2D simulation we used a rectangular mesh 105 mm of 400 plane strain pore pressure elements Fig. 2. The nodal displacements at the bottom plane were confined in all directions. The displacements of the nodes on the left side of the model were confined in the x-direction. At the right edge free fluid flow was allowed. The nodes on the top plane were tied so that they stay on a straight horizontal line. At the start of the simulation the external salt concentration was lowered from 0.15 to 0.05 M, after which the model was left to equilibrate for seconds. In the second step a pressure of 0.1 MPa was applied at the top surface in 50 seconds, after which the model was left to equilibrate for seconds. The same material properties were used as in the reference set Table 1. To prevent numerical problems with the mechano-electrochemical model, the Youngs modulus was increased to 1 MPa in both models. As there in no analytical relationship between the hydraulic permeability of the biphasic swelling and full mechano-electrochemical model for 2D and 3D, the hydraulic permeability of the biphasic swelling model was determined by fitting the model to the results of the full mechanoelectrochemical model. The fitting procedure was performed iteratively using a multidimensional unconstrained nonlinear minimization procedure available in Matlab Version 5.3 The MathWorks Inc., To compare the biphasic swelling and full mechano-electrochemical, the displacement of the top-right node was compared. Results To illustrate the difference in deformation between an ordinary biphasic model, the biphasic swelling model and the mechanoelectrochemical model following aforementioned loading regimes, the calculated global strains for the reference case are plotted in Figs. 3 and 4, respectively. The equilibrium responses of the biphasic swelling model and the mechano-electrochemical model were equal for both the confined compression test and the 1D swelling tests, while the equilibrium strain for the regular biphasic model was lower in confined compression. For the swelling test obviously no straining occurred for a regular biphasic model. In all cases loading and unloading parts of the confined compression tests and the swelling and shrinking parts of the 1D swelling tests the errors were maximal just after changing the loading conditions. There was only a little difference between the maximal error the strain difference between biphasic swelling and the mechano-electrochemical model divided by the maximal strain of mechano-electrochemical model in the swelling and shrinking part of the 1D swelling test. The maximal error in the relaxation part of the confined compression test was half that of the loading part The local strains were computed at the moment of the largest global difference between the models during the loading part. It was found that the error in the local strain goes up to 39% Fig. 5 for confined compression and 30% for 1D swelling Fig. 6. These maximal errors were found at the location of the lowest strains, which were along the no flow boundary. For comparing the biphasic swelling model and the mechanoelectrochemical model the coefficients of correlation (R 2 ) for the Fig. 5 Computed strain curves for the confined compression simulations of the reference case as a function of height at t Ä144 s Journal of Biomechanical Engineering FEBRUARY 2005, Vol. 127 Õ 161

5 Fig. 6 Computed strain curves for the 1D swelling simulations of the reference case as a function of height at tä149 s strain curves were determined. This was done for the loading and unloading part separately. Using all resulting R 2 values, two dimensionless parameters were derived that predict the R 2 between the biphasic swelling model and the true mechanoelectrochemical model, from known and easily predicted quantities. For confined compression this dimensionless variable is given by: f c log D D J (15) E 2 k 2 and for the 1D swelling experiments f s log D D E 2 k J c F,0 c 0. (16) In these equations (D D )/(E 2 k 2 ) is the ratio between the diffusional and consolidation time van Loon et al. 11,(c F,0 /c 0 )is the ratio between the initial fixed charge density and the initial external salt concentration, and J is the volume change. Fig. 8 The correlation between the biphasic swelling model and the mechano-electrochemical model during 1D swelling R 2 Ä In Figs. 6 and 7 the coefficients of determination are plotted as a function of the variable f c and f s, respectively. The R 2 values for the unloading curve of the confined compression tests are larger than for the loading curve, especially for low values of f c Fig. 7. This difference could not be seen in the swelling test Fig. 8. To derive simple equations that can be used to quickly estimate the correlation between the two models, curves were fitted through the data points in Figs. 7 and 8. Because of the large difference between the R 2 values of the loading and unloading part of the confined compression tests, separate curves were used for the loading and unloading part. The function found for the confined compression-loading curve was f e f c, (17) and for the confined compression relaxation curve f e f c. (18) The values of f represent the correlation (R 2 ) between the biphasic swelling and the full mechano-electrochemical model. In the 1D swelling test the coefficients of determination were very high for a large ratio between the swelling and consolidation time 100, regardless of the choice of the other material properties. Since there was only a small difference between the swelling and shrinkage part of the 1D swelling experiments, we only fitted one curve for the whole swelling experiment. The function found for the 1D swelling experiment curve was f e f s. (19) Fig. 7 The correlation between the biphasic swelling model and the mechano-electrochemical model during confined compression R 2 Ä for loading and for unloading curves In this fit the data points with values of the ratio between the swelling and consolidation time larger than 100 were not included, because they are not physiological. The maximal strains for the confined compression experiments ranged from 4 to 40% and the maximal strain for the swelling experiments ranged from 30% to 7%. For the 2D simulation the optimal permeability for the biphasic swelling model was mm 4 /Ns. For the 2D simulations differences between the models were found in the transient response during swelling Fig. 9. The equilibrium response after swelling, the transient of the loading part and the equilibrium response of the loading part were the same for both models (R ) Fig Õ Vol. 127, FEBRUARY 2005 Transactions of the ASME

6 Fig. 9 Computed displacement curves for the 2D simulation. u x and u y are the horizontal and vertical displacement of the top-right node of the model, respectively. Discussion Based on the hypothesis that electrolyte flux can be neglected in mechanical and diffusion studies of charged materials 12 we have developed a biphasic swelling model. This biphasic swelling model is a simplification of the full mechano-electrochemical model 7,11. It was shown that depending on the material properties, the overall deformational behavior of both models was similar. Hence, the biphasic swelling model can be a good alternative for the complex mechano-electrochemical model for studying tissue deformation. This makes the modeling of the swelling behavior of soft hydrated tissues much easier, and enables swelling simulation in commercial finite element packages. Furthermore, the simulations are much more stable and require less computation time. Hence, problems that cannot yet be analyzed with the full mechano-electrochemical model can be approached with the present model. It was found that the maximal errors in the local strains were larger than in the global strains. Since the local strains were computed at the moment of the largest global difference between the models during the loading part, the results in Figs. 5 and 6 represent the worst case scenario. Although the maximal error in local strain in this situation went up to 39%, the biphasic swelling model was still much better than the regular biphasic model. The aim of both models is to describe the behavior of soft hydrated tissues. This requires the fitting of unknown material parameters to experimental data for both models. We have recently shown that for our recently developed fibril-reinforced poroviscoelastic model all unknown material parameters could be determined this way 17. As there is no analytical relationship between the hydraulic permeability of the biphasic swelling and the full mechano-electrochemical model for two and three dimensions, the hydraulic permeability of the biphasic swelling model was determined by fitting the model to the results of the full mechano-electrochemical model, using the same procedure as previously used 17. The optimal permeability found for the biphasic swelling model was 11.96% lower then the hydraulic permeability of the full mechano-electrochemical model. The large difference between the models seen during swelling in two dimensions was caused by negative osmosis that occurred in the full mechanoelectrochemical model. Obviously, such effects are not included in the biphasic swelling model as it assumes immediate ion equilibrium. Hence, the biphasic swelling model should be used with care in interpreting the transient swelling. For this reason the swelling part was excluded while fitting the hydraulic permeability of the biphasic swelling model. It should be noted that negative osmosis has never been found experimentally. To be able to estimate the correlation between the biphasic swelling model and a full electromechanical model, two dimensionless parameters ( f c and f s ) together with three simple exponential equations were introduced. With these three equations the correlation between the biphasic swelling model and full mechano-electrochemical model can be estimated. All variables in these equations are known material parameters, except for the volume change J, which can be estimated from initial test simulations. It should also be noted that if the model is applied to a particular material such as articular cartilage, the unknown parameters are fitted to experimental data, independent of the full mechanoelectrochemical model. Obviously, if both models are fitted to experimental data the difference between the models will be smaller than the differences that are shown in this paper. It was found that the R 2 for the swelling experiments was always high for a high ratio between the ion diffusion and consolidation time. This is an obvious consequence of the assumption of zero ion diffusion time of the biphasic swelling model. It was also found that the R 2 for the relaxation of the confined compression test was higher than for the loading curve. Whether the biphasic swelling model can be a good alternative for the full mechano-electrochemical model is highly dependent on what one is interested in. It has been shown that the biphasic model can be used to describe the deformation of the soft hydrated tissues following mechanical and chemical loading. But since the ion flux in the biphasic swelling model is assumed infinitely fast, this model can obviously not be used to investigate ion fluxes in soft hydrated tissues. For these kinds of investigations a full mechano-electrochemical model should be used. Although the dimensionless variables in Eqs. 15 and 16 should obviously include the ratio between the diffusional and consolidation time and the ratio between the fixed charge density and the external salt concentration, the actual forms of these equations were determined by trial and error. In this study we used a compressible neo-hookean model for the description of the solid matrix. Since the same model for the solid matrix was used for the mechano-electrochemical and biphasic-swelling model, the choice of this constitutive model is not expected to influence the results as presented in this paper. Even though for some sets of material parameters the maximal error between the two models was more than 10%, the biphasic swelling model behaves more realistic than a regular biphasic model see Figs. 2, 3 which does not incorporate swelling at all. Hence, if one is interested in swelling materials and does not have access to a full mechano-electrochemical model, the biphasic swelling model is a valuable alternative. In summary, it was shown that depending on the material properties the biphasic swelling model behaves largely the same as the full mechano-electrochemical model. Hence, the biphasic swelling model can be a good alternative for the more complex mechanoelectrochemical model. Simple equations were derived which can be used to quickly estimate the correlation between the two models. Acknowledgment This research was financially supported by the Dutch Organization for Scientific Research ZonMw. We thank Silvia Wognum for the full mechano-electrochemical simulations in two dimensions. Nomenclature B frictional coefficients D diffusivity of ion E Young s modulus Journal of Biomechanical Engineering FEBRUARY 2005, Vol. 127 Õ 163

7 F deformation tensor F Faraday s constant G shear modulus I second order unit tensor J relative volume change detf K bulk modulus L e electrical conductivity L ep electro-osmotic permeability L p closed-circuit hydraulic permeability of the mechanoelectrochemical model L conductances P external pressure R gas constant T absolute temperature V partial molar volumes of ion c ext external NaCl concentration fixed charge density c F c concentrations of mobile ion f Determinant of correlation between the biphasic swelling model and the mechano-electrochemical model f c f s Dimensionless variable Dimensionless variable h sample height i electric flux j volume flux k hydraulic permeability n f,0 initial fluid fraction n z ext int volume fraction of component p hydrostatic fluid pressure r hindrance factor u displacement valence of mobile ion osmotic coefficient of the external solution internal osmotic coefficient activity coefficient of ion osmotic pressure gradient 0 electrochemical potential in the reference state f electrochemical potential Poisson s ratio electrical potential internal osmotic pressure ext s external osmotic pressure solid stress Appendix A: Difference Between Biphasic Swelling and Regular Biphasic Theory A typical biphasic theory implementation in a commercial finite element code like ABAQUS Hibbit, Karlson & Sorensen, Inc., Pawtucket, RI, USA consists of the following equations: mechanical balance: e p0, stress strain law of the solid: e e, mass balance: u q 0, (A1) (A2) (A3) Darcy s law: q k p, (A4) with boundary conditions: u0 and p0 (A5) along the interface with an external bath. Here c is the effective stress, p is the fluid pressure, q the fluid flux, k the permeability and u the displacements. In the biphasic swelling theory the following equations account for Donnan osmotic swelling: Darcy s law: q k f k p, (A6) boundary conditions: u0 and f 0. (A7) Here f is the electrochemical potential. The stress strain law and Darcy s law are the same as in the regular biphasic theory. Appendix B: Determination of the Open Circuit Hydraulic Permeability The volume flux j and electric flux i are 18 jl p pl ep (B1) and il ep pl e, (B2) where p is the pressure gradient, is the potential gradient, L p is the closed-circuit hydraulic permeability of the mechanoelectrochemical model (k closed ), L ep is the electro-osmotic permeability and L e is the electrical conductivity. In an open circuit there is no streaming current (i0). Hence, can be expressed as L ep p. (B3) L e By substituting Eq. B3 in Eq. B1 we get j L p L 2 ep L e p, (B4) where L p L 2 ep /L e is the hydraulic permeability of the biphasic model. The electrical conductivity L e and electro-osmotic permeability L ep are given by 18 and L e F 2 L ep F where L is given by f,,i f,,i z z L f,,i f,,i V z L, f,,i n n B1 (B5) (B7) L f,,i V V. (B6) The components of B are given by B RTnf c, D (B7a) B RTnf c, D (B7b) B f B f 1rB, (B7c) B f B f 1rB, (B7d) B B 0, (B7e) B ff n f 2 k 1rBf 1rB f. (B7f) The volume fractions of the ions are given by References n n f V c, n n f V c. (B8a) (B8b) 1 Urban, J. P. G., Maroudas, A., Bayliss, M. T., and Dillon, J., 1979, Swelling Pressures of Proteoglycans at the Concentrations Found in Cartilagenous Tissues, Biorheology, 16, pp Mow, V. C., Kuei, S. C., Lai, W. M., and Armstrong, C. G., 1980, Biphasic 164 Õ Vol. 127, FEBRUARY 2005 Transactions of the ASME

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