Wei Yong Gu

A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-electrolytes: Passive Transport and Swelling Behaviors

A new mixture theory was developed to model the mechano-electrochemical behaviors of charged-hydrated soft tissues containing multi-electrolytes. The mixture is composed of n + 2 constituents (1 charged solid phase, 1 noncharged solvent phase, and n ion species). Results from this theory show that three types of force are involved in the transport of ions and solvent through such materials: (1) a mechanochemical force (including hydraulic and osmotic pressures); (2) an electrochemical force; and (3) an electrical force. Our results also show that three types of material coefficients are required to characterize the transport rates of these ions and solvent: (1) a hydraulic permeability; (2) mechano-electrochemical coupling coefficients; and (3) an ionic conductance matrix. Specifically, we derived the fundamental governing relationships between these forces and material coefficients to describe such mechano-electrochemical transduction effects as streaming potential, streaming current, diffusion (membrane) potential, electro-osmosis, and anomalous (negative) osmosis. As an example, we showed that the well-known formula for the resting cell membrane potential (Hodgkin and Huxley, 1952a, b) could be derived using our new n + 2 mixture model (a generalized triphasic theory). In general, the n + 2 mixture theory is consistent with and subsumes all previous theories pertaining to specific aspects of charged-hydrated tissues. In addition, our results provided the stress, strain, and fluid velocity fields within a tissue of finite thickness during a one-dimensional steady diffusion process. Numerical results were provided for the exchange of Na+ and Ca++ through the tissue. These numerical results support our hypothesis that tissue fixed charge density (CF) plays a significant role in modulating kinetics of ions and solvent transport through charged-hydrated soft tissues.

New insight into deformation-dependent hydraulic permeability of gels and cartilage, and dynamic behavior of agarose gels in confined compression

Equilibrium, creep, and dynamic behaviors of agarose gels (2.0-14.8%) in confined compression were investigated in this study. The hydraulic permeabilities of gels were determined by curve-fitting creep data to the biphasic model (J. Biomech. Eng. 102 (1980) 73) and found to be similar in value to those published in the literature (AIChE J. 42 (1996) 1220). A new relationship between intrinsic permeability and volume fraction of water was found for agarose gel, capable of predicting deformation-dependent permeabilities of bovine articular cartilage and 2% agarose gel published in literature. This relationship is accurate for gels and cartilage over a wide range of permeabilities (four orders of magnitude variation). The dynamic stiffness of the gels increases with gel concentration and loading frequency (0.01-1.0Hz). The increase in dynamic stiffness with loading frequency is less pronounced for gels with higher concentrations. The results of this study provide a new insight into deformation-dependent permeability behavior of agarose gel and cartilage, and are important for understanding biological responses of cells to interstitial fluid flow in gel or in cartilage under dynamic mechanical loading.

A mixed finite element formulation of triphasic mechano‐electrochemical theory for charged, hydrated biological soft tissues

An equivalent new expression of the triphasic mechano-electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement us, modified electrochemical/chemical potentials ϵw, ϵ+and ϵ− (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton–Raphson iterative procedure is employed to handle the non-linear terms. The resulting first-order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One-dimensional (1-D) linear isoparametric element is developed. The final algebraic equations form a non-symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1-D triphasic stress relaxation problem in the confined compression configuration and a 1-D triphasic free swelling problem. The formulation accuracy and convergence for 1-D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods. Copyright

The anisotropic hydraulic permeability of human lumbar anulus fibrosus. Influence of age, degeneration, direction, and water content

STUDY DESIGN Experimental investigation to determine the effect of intervertebral disc degeneration on the kinetic behavior of fluid in human anulus fibrosus. OBJECTIVES To measure the hydraulic permeability coefficient of anulus fibrosus specimens in the axial, circumferential, and radial directions to determine the anisotropic permeability behavior of nondegenerate and degenerate human intervertebral discs over a range of ages. SUMMARY OF BACKGROUND DATA Fluid, a major component of normal intervertebral discs, plays a significant role in their load-supporting mechanisms. Transport of fluid through the intervertebral disc is important for cell nutrition and disc viscoelastic and swelling behaviors. The hydraulic permeability coefficient is the most important material property governing the rate of fluid transport. However, little is known about the anisotropic behavior of this kinetic property and how it is influenced by disc degeneration. METHODS Using a permeation testing apparatus developed recently, testing was performed on 306 axial, circumferential, and radial anulus fibrosus specimens from the posterolateral region of 30 human lumbar (L2-L3) discs. A new method, flow-controlled testing protocol, was developed to measure the hydraulic permeability coefficient. RESULTS The hydraulic permeability coefficient of anulus fibrosus depended significantly on the disc degenerative grade (P = 0.0001) and flow direction (P = 0.0001). For the nondegenerate group (Grade I), the hydraulic permeability was significantly anisotropic (P < 0.05), with the greatest value in the radial direction (1.924 x 10(-15) m4/Ns) and the lowest value in the circumferential direction (1.147 x 10(-15) m4/Ns). This anisotropic kinetic (flow) behavior of anulus fibrosus varied with disc degeneration. For the Grade III specimen group, there was no significant difference in hydraulic permeability coefficient among the three major directions (P = 0.37). With disc degeneration, the hydraulic permeability coefficient was decreased in the radial direction and increased in the axial and circumferential directions. The variations of hydraulic permeability coefficient from nondegenerate discs (Grade I) to mildly degenerate discs (Grade II) in each direction were significant (P < 0.05). However, the changes in permeability from Grade II to Grade III groups were not significant (P > 0.05) except in the circumferential direction (3.8% increase; P < 0.05). CONCLUSIONS The hydraulic permeability of human nondegenerate anulus fibrosus is direction-dependent (i.e., anisotropic), with the greatest permeability in the radial direction. With disc degeneration, the radial permeability of anulus fibrosus decreases, mainly because of decreased water content, and the axial and circumferential permeability coefficients increase, mainly because of structural change, leading to more isotropic permeability behavior for Grade III discs.

Transport of fluid and ions through a porous-permeable charged-hydrated tissue, and streaming potential data on normal bovine articular cartilage

Using the triphasic mechano-electrochemical theory [Lai et al., J. biomech. Engng 113, 245-258 (1991)], we analyzed the transport of water and ions through a finite-thickness layer of charged, hydrated soft tissue (e.g. articular cartilage) in a one-dimensional steady permeation experiment. For this problem, we obtained numerically the concentrations of the ions, the strain field and the fluid and ion velocities inside when the specimen is subject to an applied mechanical pressure and/or osmotic pressure across the layer. The relationships giving the dependence of streaming potential and permeability on the negative fixed charge density (FCD) of the tissue were derived analytically for the linear case, and calculated for the nonlinear case. Among the results obtained were: (1) at a fluid pressure difference of 0.1 MPa across the specimen layer, there is a 10% flow-induced compaction at the downstream boundary; (2) the flow-induced compaction causes the FCD to increase and the neutral salt concentration to decrease in the downstream direction; (3) while both ions move downstream, relative to the solvent (water), the anions (Cl-) move with the flow whereas cations (Na+) move against the flow. The difference in ion velocities depends on the FCD, and this difference attained a maximum at a physiological FCD of around 0.2 meq ml-1; (4) the apparent permeability decreases nonlinearly with FCD, and the apparent stiffness of the tissue increases with FCD; and (5) the streaming potential is not a monotonic function of the FCD but rather it has a maximum value within the physiological range of FCD for articular cartilage. Finally, experimental data on streaming potential were obtained from bovine femoral cartilage. These data support the triphasic theoretical prediction of non-monotonicity of streaming potential as a function of the FCD.

Transport of Multi-Electrolytes in Charged Hydrated Biological Soft Tissues

A mechano-electrochemical theory for charged hydrated soft tissues with multi-electrolytes was developed based on the continuum mixture theory. The momentum equations for water and ions were derived in terms of a mechanochemical force (gradient of water chemical potential), electrochemical forces (gradient of Nernst potentials) and an electrical force (gradient of electrical potential). The theory was shown to be consistent with all existing specialized theories. Using this theory, some mechano-electrokinetic properties of charged isotropic tissues were studied. The well-known Hodgkin–Huxley equation for resting cell membrane potential was derived and the phenomenon of electro-osmotic flow in charged hydrated soft tissues was investigated. Analyses show that the tissue fixed charge density plays an important role in controlling the transport of water and ions in charged hydrated soft tissues.

Diffusivity of Ions in Agarose Gels and Intervertebral Disc: Effect of Porosity

The effect of tissue porosity on ion (sodium, potassium, and chloride) diffusivity in agarose gels and porcine intervertebral disc tissues was investigated using an electrical conductivity method. An empirical, constitutive model for diffusivity (D) of solutes in porous fibrous media was proposed: D/Do=exp [−α(rs/κ1/2)β] where rs is the Stokes radius of a solute, κ is the Darcy permeability of the porous medium, Do is the diffusivity in free solution, α and β are two positive parameters whose values depend on material structure. It is found that α=1.25±0.138, β=0.681±0.059 (95% confidence interval, R2=0.92, n=72) for agarose gels and α=1.29±0.171 and β=0.372±0.088 (95% confidence interval, R2=0.88, n=86) for porcine annulus fibrosus. The functional relationship between solute diffusivity and tissue deformation was derived. Comparisons of our model prediction with experimental data on diffusion coefficients of macromolecules (proteins, dextrans, polymer beads) in agarose gels in the literature were made. Our results were also compared to the data on ion diffusivity in charged gels and in cartilaginous tissues reported in the literature. There was a good agreement between our model prediction and the data in the literature. The present study provides additional information on solute diffusivity in uncharged gels and charged tissues, and is important for understanding nutritional transport in avascular cartilaginous tissues under different mechanical loading conditions.

Effects of mechanical compression on metabolism and distribution of oxygen and lactate in intervertebral disc

The objective of this study was to examine the effects of mechanical compression on metabolism and distributions of oxygen and lactate in the intervertebral disc (IVD) using a new formulation of the triphasic theory. In this study, the cellular metabolic rates of oxygen and lactate were incorporated into the newly developed formulation of the mechano-electrochemical mixture model [Huang, C.-Y., Gu, W.Y., 2007. Effect of tension-compression nonlinearity on solute transport in charged hydrated fibrosus tissues under dynamic unconfined compression. Journal of Biomechanical Engineering 129, 423-429]. The model was used to numerically analyze metabolism and transport of oxygen and lactate in the IVD under static or dynamic compression. The theoretical analyses demonstrated that compressive loading could affect transport and metabolism of nutrients. Dynamic compression increased oxygen concentration, reduced lactate accumulation, and promoted oxygen consumption and lactate production (i.e., energy conversion) within the IVD. Such effects of dynamic loading were dependent on strain level and loading frequency, and more pronounced in the IVD with less permeable endplate. In contrast, static compression exhibited inverse effects on transport and metabolism of oxygen and lactate. The theoretical predictions in this study are in good agreement with those in the literature. This study established a new theoretical model for analyzing cellular metabolism of nutrients in hydrated, fibrous soft tissues under mechanical compression.

Effects of Swelling Pressure and Hydraulic Permeability on Dynamic Compressive Behavior of Lumbar Annulus Fibrosus

AbstractThe objective of this study was to investigate the effects of swelling pressure and hydraulic permeability on the dynamic behavior of intervertebral disk tissue in confined compression. Normal (served as a control) and trypsin-treated, axial annulus fibrosus (AF) specimens from the porcine lumbar disks were tested and their swelling strain, swelling pressure, equilibrium compressive modulus (HA, dynamic modulus, and hydraulic permeability (k) were determined at 30% and 40% swelling strain levels. The proteoglycan depletion due to trypsin treatment resulted in significantly lower values of the free swelling strain, swelling pressure, equilibrium modulus, dynamic modulus, and higher value of hydraulic permeability for trypsin-treated group, comparing to those for the control group. At the 30% swelling strain level, the equilibrium moduli were 51.84±14.53 kPa (n=8) for the control group and 15.11±5.67 kPa (n=8) for the trypsin-treated group; and the hydraulic permeabilities were 4.50E-15±1.60E-15 m4/Ns and 8.43E-15±4.29E-15 m4/Ns for control and trypsin-treated groups, respectively. No statistically significant difference in wet tissue density or dry tissue density was found between control and trypsin-treated groups. There was a significant correlation between swelling pressure and compressive (aggregate) modulus (R2=0.824, m=22). The decrease in measured dynamic modulus for trypsin-treated group was attributed to the reduced swelling pressure (or modulus HA and increased hydraulic permeability (k) due to PG depletion.

A triphasic analysis of negative osmotic flows through charged hydrated soft tissues

Osmotic flow and ion transport in a one-dimensional steady diffusion process through charged hydrated soft tissues such as articular cartilage were analysed using the triphasic theory (Lai et al., 1991, J. biomech. Engng 113, 245-258). It was found that solvent would flow from the high NaCl concentration side to the low concentration side (i.e. negative osmosis) when the fixed charge density within the tissue (or membrane) separating the two electrolyte (NaCl) solutions was lower than a critical value. The condition for negative osmosis was derived based on a linear version of the triphasic theory. Distributions of ion concentration and strain field within the tissue were calculated numerically. Quantitative results of osmotic flow rates (ordinary and negative osmosis), ion flux and electric potential across the tissue during this diffusion process suggest that the negative osmosis phenomenon is due to the friction between ions and water since they could flow through the tissues at different rates and different directions.