Linear Biphasic Theory Research Articles

Compressive stress relaxation behavior of articular cartilage and its effects on fluid pressure and solid displacement due to non-Newtonian flow

In this study, we investigate the effects of the power-law index and permeability parameter on the deformation of soft tissue (articular cartilage) which is bathed in the non-Newtonian fluid under stress-relaxation in compression. Ramp displacement is imposed on the surface of hydrated soft tissue. Deformation of the tissue and the fluid pressure is examined for the fast and slow rate of compression. We have employed a linear biphasic mixture theory to develop a mathematical model for compressive stress-relaxation behavior of articular cartilage for non-Newtonian flow. Numerical results indicate that shear-thinning fluids induce less solid deformation and exhibit more fluid pressure as compared to shear-thickening fluids for fast and slow rate of compression. The results also show that linear permeability induces more deformation as compared to strain-dependent nonlinear permeability due to viscoelastic nature of articular cartilage.

Experimental characterization of biphasic materials using rate-controlled Hertzian indentation

This paper describes a new method, based on Hertzian biphasic theory (HBT), to characterize properties of biphasic materials with reduced time demands, increased surface sensitivity, and reduced computational demands compared to the current gold standards. Indentation experiments were conducted at a single location on a representative osteochondral plug to demonstrate and validate the HBT method against two gold standards, linear biphasic theory (LBT) and tension–compression nonlinear biphasic theory (TCN). The (1) aggregate moduli, (2) permeability and (3) tensile moduli from HBT, LBT, and TCN were (1) HA=0.47, 0.47, and 0.40MPa, (2) k=0.0026, 0.0014 and 0.0016mm4/Ns, and (3) Et=8.7, 0.46, and 10.3MPa, respectively. The results support the HBT method and encourage its use, especially in light of its practical advantages.

Mathematical modeling of linear viscoelastic impact: Application to drop impact testing of articular cartilage

In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading. Some experimentally observed results have been explained using a non-linear viscoelastic impact model. At the same time, there is the need of simple mathematical models, which allow comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities. The objective of this study was to investigate theoretically whether the main features of articular impact could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. Based on the short-time asymptotic solution of the contact problem for a thin biphasic layer [Ateshian GA, Lai WM, Zhu WB, Mow VC. An asymptotic solution for the contact of two biphasic cartilage layers. Journal of Biomechanics 1994;27(11):1347–1360.], an asymptotic model for blunt impact of articular cartilage is developed, which is mathematically equivalent to the Maxwell impact model. In the present paper, exact analytical solutions are obtained for the main parameters of the Kelvin–Voigt and Maxwell impact models. Perturbation analysis of the impact process according to the standard viscoelastic solid model is performed. Asymptotic solutions are obtained for the drop weight impact test. The dependence of the coefficient of restitution on the impactor parameters has been studied in detail.

The Dynamic Mechanical Environment of the Chondrocyte: A Biphasic Finite Element Model of Cell-Matrix Interactions Under Cyclic Compressive Loading

Cyclic mechanical loading of articular cartilage results in a complex biomechanical environment at the scale of the chondrocytes that strongly affects cellular metabolic activity. Under dynamic loading conditions, the quantitative relationships between macroscopic loading characteristics and solid and fluid mechanical variables in the local cellular environment are not well understood. In this study, an axisymmetric multiscale model of linear biphasic cell-matrix interactions in articular cartilage was developed to investigate the cellular microenvironment in an explant subjected to cyclic confined compressive loading. The model was based on the displacement-velocity-pressure (u-v-p) mixed-penalty weighted residual formulation of linear biphasic theory that was implemented in the COMSOL MULTIPHYSICS software package. The microscale cartilage environment was represented as a three-zone biphasic region consisting of a spherical chondrocyte with encapsulating pericellular matrix (PCM) that was embedded in a cylindrical extracellular matrix (ECM) subjected to cyclic confined compressive loading boundary conditions. Biphasic material properties for the chondrocyte and the PCM were chosen based on previous in vitro micropipette aspiration studies of cells or chondrons isolated from normal or osteoarthritic cartilage. Simulations performed at four loading frequencies in the range 0.01-1.0 Hz supported the hypothesized dual role of the PCM as both a protective layer for the cell and a mechanical transducer of strain. Time varying biphasic variables at the cellular scale were strongly dependent on relative magnitudes of the loading period, and the characteristic gel diffusion times for the ECM, the PCM, and the chondrocyte. The multiscale simulations also indicated that axial strain was significantly amplified in the range 0.01-1.0 Hz, with a decrease in amplification factor and frequency insensitivity at the higher frequencies. Simulations of matrix degradation due to osteoarthritis indicated that strain amplification factors were more significantly altered when loss of matrix stiffness was exclusive to the PCM. The findings of this study demonstrate the complex dependence of dynamic mechanics in the local cellular environment of cartilage on macroscopic loading features and material properties of the ECM and the chondron.

A Lagrange Multiplier Mixed Finite Element Formulation for Three-Dimensional Contact of Biphasic Tissues

A three-dimensional (3D) contact finite element formulation has been developed for biological soft tissue-to-tissue contact analysis. The linear biphasic theory of Mow, Holmes, and Lai (1984, J. Biomech., 17(5), pp. 377-394) based on continuum mixture theory, is adopted to describe the hydrated soft tissue as a continuum of solid and fluid phases. Four contact continuity conditions derived for biphasic mixtures by Hou et al. (1989, ASME J. Biomech. Eng., 111(1), pp. 78-87) are introduced on the assumed contact surface, and a weighted residual method has been used to derive a mixed velocity-pressure finite element contact formulation. The Lagrange multiplier method is used to enforce two of the four contact continuity conditions, while the other two conditions are introduced directly into the weighted residual statement. Alternate formulations are possible, which differ in the choice of continuity conditions that are enforced with Lagrange multipliers. Primary attention is focused on a formulation that enforces the normal solid traction and relative fluid flow continuity conditions on the contact surface using Lagrange multipliers. An alternate approach, in which the multipliers enforce normal solid traction and pressure continuity conditions, is also discussed. The contact nonlinearity is treated with an iterative algorithm, where the assumed area is either extended or reduced based on the validity of the solution relative to contact conditions. The resulting first-order system of equations is solved in time using the generalized finite difference scheme. The formulation is validated by a series of increasingly complex canonical problems, including the confined and unconfined compression, the Hertz contact problem, and two biphasic indentation tests. As a clinical demonstration of the capability of the contact analysis, the gleno-humeral joint contact of human shoulders is analyzed using an idealized 3D geometry. In the joint, both glenoid and humeral head cartilage experience maximum tensile and compressive stresses are at the cartilage-bone interface, away from the center of the contact area.

Effects of Degeneration on the Biphasic Material Properties of Human Nucleus Pulposus in Confined Compression

The biphasic compressive material properties of normal and degenerate human nucleus pulposus tissue were measured in confined compression. The objective of this study was to determine the effects of degeneration and age on the mechanical properties of human nucleus pulposus. The nucleus pulposus exhibits swelling behavior in proportion to proteoglycan content. In shear, the nucleus exhibits both fluid-like and solid-like properties, suggesting a biphasic nature. To date, biphasic compressive properties of human nucleus pulpous have not been reported. Human nucleus pulposus samples were tested in confined compression. Isometric swelling stress and effective aggregate modulus were measured. Linear biphasic theory was used to determine the permeability of the tissue. Mechanical behavior was correlated with proteoglycan and water content. Degeneration produced significant decreases in swelling stress (Psw = 0.138 +/- 0.029 MPa nondegenerate, Psw = 0.037 +/- 0.038 MPa degenerate) and effective aggregate modulus (H(A)(eff) = 1.01 +/- 0.43 MPa nondegenerate, H(A)(eff) = 0.44 +/- 0.19 MPa degenerate). Both properties were inversely correlated with proteoglycan content. Permeability increased with degeneration (ka = 0.9 +/- 0.43 x 10(-15) m4/N-s nondegenerate, ka = 1.4 +/- 0.58 x 10(-15) m4/N-s degenerate). Swelling is the primary load-bearing mechanism in both nondegenerate and degenerate nucleus pulposus. Knowledge of the biphasic material properties of the nucleus pulposus will aid the development of new treatment strategies for disc degeneration aimed at restoring mechanical function of the intervertebral disc.

Quantification of cartilage biomechanical and biochemical properties via T1ρ magnetic resonance imaging

The aim of this study is to develop T1rho as an MR marker of the compositional and functional condition of cartilage. Specifically, we investigate the correlation of changes in cartilage biomechanical and biochemical properties with T1rho relaxation rate in a cytokine-induced model of degeneration. Bovine cartilage explants were cultured with 30 ng/mL of interleukin-1beta to mimic the cartilage degradation of early osteoarthritis. The average rate of T1rho relaxation was calculated from T(1rho) maps acquired on a 4.7 T research scanner. Stress-relaxation biomechanical tests were conducted with a confined compression apparatus to measure uniaxial aggregate modulus (HA) and hydraulic permeability (k0) using linear biphasic theory. Proteoglycan, collagen, and water content were measured via biochemical assays. Average T(1rho) relaxation rate was strongly correlated with proteoglycan content (R2 = 0.926), HA (R2 = 0.828), and log10 k0 (R2 = 0.862). Results of this study demonstrate that T1rho MRI can detect changes in proteoglycan content and biomechanical properties of cartilage in a physiologically relevant model of cartilage degeneration. The T1rho technique can potentially be used to noninvasively and quantitatively assess the biochemical and biomechanical characteristics of articular cartilage in humans during the progression of osteoarthritis.

Biphasic material properties of lytic bone metastases.

It is necessary to prescribe the mechanical properties of tumor tissue when modeling the metastatically involved skeleton for clarifying the mechanisms of fracture. This study provides mechanical property data for lytic bone metastases. Specimens of human lytic tumor were tested under a confined compression uniaxial creep protocol and the mechanical behavior of the tumor tissue was modeled using linear biphasic theory. The tumor tissue was found to have an aggregate modulus (HA) of 3.6 +/- 1.6 kPa and a hydraulic permeability (k) of 0.59 +/- 0.46mm4 N(-1) s(-1). Tumors with a higher percentage of stromal content were found to be stiffer and more permeable than those with a more cellular composition. No significant differences in aggregate modulus or hydraulic permeability were found between lytic metastases of different types. These data are useful for the development of models to simulate the behavior of the metastatically involved skeleton using theoretical or finite-element analysis techniques and also have significance for developing effective tumor-drug-transport models. We anticipate that specification of the mechanical behavior of this tissue may help to better focus future treatment of lytic bony metastases through better assessment of fracture risk and improved drug delivery.

A mixed-penalty biphasic finite element formulation incorporating viscous fluids and material interfaces.

The fluid viscosity term of the fluid phase constitutive equation and the interface boundary conditions between biphasic, solid and fluid domains have been incorporated into a mixed-penalty finite element formulation of the linear biphasic theory for hydrated soft tissue. The finite element code can now model a single-phase viscous incompressible fluid, or a single-phase elastic solid, as limiting cases of a biphasic material. Interface boundary conditions allow the solution of problems involving combinations of biphasic, fluid and solid regions. To incorporate these conditions, the volume-weighted mixture velocity is introduced as a degree of freedom at interface nodes so that the kinematic continuity conditions are satisfied by conventional finite element assembly techniques. Results comparing our numerical method with an independent, analytic solution for the problem of Couette flow over rigid and deformable porous biphasic layers show that the finite element code accurately predicts the viscous fluid flows and deformation in the porous biphasic region. Thus, the analysis can be used to model the interface between synovial fluid and articular cartilage in diarthrodial joints. This is an important step toward modeling and understanding the mechanisms of joint lubrication and another step toward fully modeling the in vivo behavior of a diarthrodial joint.

Cytoindentation for Obtaining Cell Biomechanical Properties

A novel biomechanical testing methodology was developed to obtain the intrinsic material properties of an individual cell attached to a rigid substrate. With use of a newly designed cell-indentation apparatus (cytoindenter), displacement-controlled indentation tests were conducted on the surface of individual MG63 cells and the corresponding surface reaction force of each cell was measured. The cells were modeled with a linear elasticity solution of half-space indentation and the linear biphasic theory on the assumption that the viscoelastic behavior of each cell was due to the interaction between the solid cytoskeletal matrix and the cytoplasmic fluid. To obtain the intrinsic material properties (aggregate modulus, Poisson's ratio, and permeability), the data for experimental surface reaction force and deformation were curve-fitted with use of solutions predicted with a linear biphasic finite element code in conjunction with optimization routines. The MG63 osteoblast-like cells had a compressive aggregate modulus of 2.05 ± 0.89 kPa, which is two to three orders of magnitude smaller than that of articular cartilage, six to seven orders smaller than that of compact bone, and quite similar to that of leukocytes. The peremeability was 1.18 ± 0.65 (×10-10) m4/N-s, which is four to six orders of magnitude larger than that of cartilage. The Poisson's ratio was 0.37 ± 0.03. The intrinsic material properties of the individual cell in this study can be useful in precisely quantifying mechanical stimuli acting on cells. This information is also needed for theories attempting to establish mechanotransductional relationships.

Interstitial fluid pressurization during confined compression cyclical loading of articular cartilage.

The objective of this study was to experimentally verify the well-accepted but untested hypothesis that cartilage interstitial fluid pressurizes variously under the action of an applied cyclical stress in confined compression over a range of loading frequencies, contributing significantly to the cartilage dynamic stiffness. Eighteen bovine cartilage cylindrical samples were tested under load control using a porous indenter in a confined compression chamber fitted with a microchip pressure transducer at its bottom. Over a static stress of 130 kPa, a cyclical stress of amplitude 33 kPa was applied with the indenter at frequencies ranging from 0.0001 to 0.1 Hz. The cartilage interstitial fluid pressure and deformation were measured simultaneously as a function of time. The displacement response at the lowest tested frequency was curvefitted in the time domain to determine the linear biphasic material properties, H(A) = 0.70+/-0.10 MPa and k0=2.4x10(-16)+/-0.64x10(-16) m4/N s. These properties were employed in the biphasic theory to predict the interstitial fluid pressure response and compare it to experiment, resulting in nonlinear coefficients of determination ranging from r2 = 0.89+/-0.15 to 0.96+/-0.03 depending on frequency. It was found for the samples of this study that above a characteristic frequency of 0.00044 Hz, the magnitude and phase of fluid pressurization matched the applied stress, reducing the tissue strain at the impermeable bottom surface to nearly zero. The findings of this study verify the hypothesis that cartilage dynamic stiffness derives primarily from flow-dependent viscoelasticity as predicted by the linear biphasic theory; they demonstrate experimentally the significance of interstitial fluid pressurization as the fundamental mechanism of cartilage load support over a wide range of frequencies.

Cytoindentation for obtaining cell biomechanical properties.

A novel biomechanical testing methodology was developed to obtain the intrinsic material properties of an individual cell attached to a rigid substrate. With use of a newly designed cell-indentation apparatus (cytoindenter), displacement-controlled indentation tests were conducted on the surface of individual MG63 cells and the corresponding surface reaction force of each cell was measured. The cells were modeled with a linear elasticity solution of half-space indentation and the linear biphasic theory on the assumption that the viscoelastic behavior of each cell was due to the interaction between the solid cytoskeletal matrix and the cytoplasmic fluid. To obtain the intrinsic material properties (aggregate modulus, Poisson's ratio, and permeability), the data for experimental surface reaction force and deformation were curve-fitted with use of solutions predicted with a linear biphasic finite element code in conjunction with optimization routines. The MG63 osteoblast-like cells had a compressive aggregate modulus of 2.05+/-0.89 kPa, which is two to three orders of magnitude smaller than that of articular cartilage, six to seven orders smaller than that of compact bone, and quite similar to that of leukocytes. The permeability was 1.18+/-0.65 (x10(-10)) m4/N-s, which is four to six orders of magnitude larger than that of cartilage. The Poisson's ratio was 0.37+/-0.03. The intrinsic material properties of the individual cell in this study can be useful in precisely quantifying mechanical stimuli acting on cells. This information is also needed for theories attempting to establish mechanotransductional relationships.

Mechanics of interstitial-lymphatic fluid transport: theoretical foundation and experimental validation

Interstitial fluid movement is intrinsically linked to lymphatic drainage. However, their relationship is poorly understood, and associated pathologies are mostly untreatable. In this work we test the hypothesis that bulk tissue fluid movement can be evaluated in situ and described by a linear biphasic theory which integrates the regulatory function of the lymphatics with the mechanical stresses of the tissue. To accomplish this, we develop a novel experimental and theoretical model using the skin of the mouse tail. We then use the model to demonstrate how interstitial–lymphatic fluid movement depends on a balance between the elasticity, hydraulic conductivity, and lymphatic conductance as well as to demonstrate how chronic swelling (edema) alters the equipoise between tissue fluid balance parameters. Specifically, tissue fluid equilibrium is perturbed with a continuous interstitial infusion of saline into the tip of the tail. The resulting gradients in tissue stress are measured in terms of interstitial fluid pressure using a servo-null system. These measurements are then fit to the theory to provide in vivo estimates of the tissue hydraulic conductivity, elastic modulus, and overall resistance to lymphatic drainage. Additional experiments are performed on edematous tails to show that although chronic swelling causes an increase in the hydraulic conductivity, its greatly increased distensibility (due to matrix remodeling) dampens the driving forces for fluid movement and leads to fluid stagnation. This model is useful for examining potential treatments for edema and lymphatic disorders as well as substances which may alter tissue fluid balance and/or lymphatic drainage.

Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization at an impermeable contact interface in confined compression

Interstitial fluid pressurization has long been hypothesized to play a fundamental role in the load support mechanism and frictional response of articular cartilage. However, to date, few experimental studies have been performed to verify this hypothesis from direct measurements. The first objective of this study was to investigate experimentally the hypothesis that cartilage interstitial fluid pressurization does support the great majority of the applied load, in the testing configurations of confined compression creep and stress relaxation. The second objective was to investigate the hypothesis that the experimentally observed interstitial fluid pressurization could also be predicted using the linear biphasic theory of Mow et al. (J. Biomech. Engng ASME, 102, 73–84, 1980). Fourteen bovine cartilage samples were tested in a confined compression chamber fitted with a microchip piezoresistive transducer to measure interstitial fluid pressure, while simultaneously measuring (during stress relaxation) or prescribing (during creep) the total stress. It was found that interstitial fluid pressure supported more than 90% of the total stress for durations as long as 725±248 s during stress relaxation (mean±S.D., n=7), and 404±229 s during creep ( n=7). When comparing experimental measurements of the time varying interstitial fluid pressure against predictions from the linear biphasic theory, nonlinear coefficients of determination r 2=0.871±0.086 (stress relaxation) and r 2=0.941±0.061 (creep) were found. The results of this study provide some of the most direct evidence to date that interstitial fluid pressurization plays a fundamental role in cartilage mechanics; they also indicate that the mechanism of fluid load support in cartilage can be properly predicted from theory.

Biomechanical Modeling of Repair Articular Cartilage: Effects of Passive Motion on Osteochondral Defects in Monkey Knee Joints

A new biomechanical methodology was developed, based on the linear biphasic finite element formulation and nonlinear optimization techniques under conditions of creep indentation, to determine the in situ intrinsic material properties of healing neocartilage in osteochondral defects. This biomechanical methodology was then used in conjunction with morphological and histological analyses to quantify neochondrogenesis in large (3.2 mm in diameter) osteochondral defects in the femoral condyle and patella of eight cynomolgus monkeys. Postoperatively, the animals were managed with either intermittent passive motion or cast immobilization, followed by 6 weeks of ad libitum cage activity. Cast-immobilization patellar neocartilage had a higher aggregate modulus than intermittent passive motion tissue (p < 0.05) and intermittent passive motion repair tissue in the medial condyle appeared to be slightly more permeable than cast-immobilization neocartilage (p < 0.10). Normal articular cartilage was four times stiffer (p < 0.0001) and had a larger Poisson's ratio (p < 0.01) than repair (intermittent passive motion and cast-immobilization combined) neocartilage. Histologic studies did not demonstrate differences in the volume or type of repair tissue formed in osteochondral defects treated with intermittent passive motion or cast immobilization. Our biomechanical methods, based on the linear biphasic theory for articular cartilage, allow for the first-time simultaneous determination of all three intrinsic material properties of neocartilage in situ on the joint. We conclude that for large, full-thickness defects at 8 weeks postoperatively, cast immobilization induced stiffer and less permeable cartilage than intermittent passive motion, but the cast-immobilization cartilage was less stiff and more permeable than normal cartilage. Our results suggest that for large, ungrafted defects, intermittent passive motion may not be an effective postoperative management modality for cartilage healing enhancement.

Application of linear biphasic theory to finite element analysis of head impulse loading

A finite element model of the human head by linear biphasic theory is developed to study the dynamic response of the human head to impact. Intracranial tissues are modelled as a binary mixture, i.e. the fluid and solid phases. To validate the biphasic finite element formulation, the result of the numerical analysis of a one-dimensional wave propagation problem is compared with that of analytic solution. The permeabilities of the subarachnoid space and brain which may reproduce the same coup and contre-coup CSF (cerebral spinal fluid) pressures from the monophasic model are searched in the specified range of skull permeability. Then the intracranial pressure distributions from the biphasic model for the frontal impact are compared with those from the monophasic model. In general, the biphasic model produces a more injurious intracranial pressure distribution than the monophasic model. The pressure distribution from the biphasic model shows a little higher contre-coup pressure in the frontal lobe than in the occipital region. This finding is in agreement with those clinical findings that contre-coup injuries are more frequently found in the frontal lobe. Another numerical simulation is conducted to characterize the effect of the volume ratios between two phases in the skull and subarachnoid space. From the results, it can be seen that the variation of the volume ratio in the subarachnoid space affects the intracranial pressure distribution of the lateral part while the variation in the skull does not.

Nonhomogeneous permeability of canine anulus fibrosus.

This report examines the permeability coefficient and aggregate modulus of slices of anulus cut from canine lumbar intervertebral discs. To examine the influence of radial position on the properties of these materials, including outer samples with intact anulus edge. The outer edge of anulus fibrosus shows radial bulge during axial compression of motion segments. The radial bulge increases monotonically when the axial compression is sustained for several hours, until a plateau is reached. Triphasic modeling of axial compression shows that this time course of radial bulge can not be obtained using a uniform permeability coefficient according to values in the literature. Confined consolidation experiments (controlled load) were designed to measure the time course of uniaxial deformation of samples of anulus that were 4 mm in diameter and 1 mm tall. The rotation symmetry axis of the samples was defined in the radial direction of the disc. The radial permeability coefficient and the aggregate modulus were determined using the consolidation data and the linear biphasic theory. The permeability coefficient was lower at the periphery than in deeper layers of the anulus. Outer samples with outer surfaces that were 0.0-0.5 mm from the anulus edge had an average permeability coefficient of (1.02 +/- 0.57) x 10(-16) m4/Ns (n = 24). Inner samples that were 2.0-2.5 mm from the anulus edge had an average permeability coefficient of (2.81 +/- 0.98) x 10(-16) m4/Ns (n = 13). The aggregate modulus HA of outer samples was significantly higher (HA = 1.56 +/- 0.34 MPa) than that of inner samples (HA = 1.31 +/- 0.47 MPa). The fact that the outer anulus is less permeable than the inner anulus may explain why radial bulge of anulus fibrosus increases monotonically in time to an equilibrium value during sustained axial compression of a motion segment.

Dynamic behavior of a biphasic cartilage model under cyclic compressive loading

It is well known that dynamic mechanical loads can stimulate the biosynthetic activity of articular cartilage. Studying the mechanical environment of chondrocytes under dynamic loading conditions can help to explain this mechano-biological phenomenon in articular cartilage. In this study, the linear biphasic theory was used to examine the dynamic mechanical behavior of articular cartilage under a cyclic compressive force. We first studied the dynamic confined compression of a cartilage disk as a simplified one-dimensional model and then investigated the role of the relatively impermeable subchondral bone structure on the dynamic behavior of the cartilage extracellular matrix (ECM). Under an assumption of articular cartilage as a biphasic composite structure of a porous elastic solid matrix and interstitial fluid, the porous ECM of the articular cartilage was repeatedly compressed and expanded during the loading-unloading phases of the cyclic compressive force. One interesting finding of this study was the oscillating positive (supra-ambient)-negative (sub-ambient) hydrostatic pressure within the cartilage ECM under cyclic compressive loading. The pattern of the dynamic behavior of the cartilage ECM strongly depended on the loading frequency and the primary diffusion characteristic time, τ d. This finding is consistent with those of previous studies (Guilak et al., 1990 Adv Biomech. ASME, 225–228; Mow et al., 1990, Biomechanics of Diarthrodial Joints, pp. 215–260.

Compressive mechanical properties of the human anulus fibrosus and their relationship to biochemical composition.

To enhance understanding of the biomechanical role of the intervertebral disc, the compressive properties and biochemical composition of nondegenerate samples of anulus fibrosus were determined as a function of radial position, region, and level. Because of the large swelling propensity of this tissue, a method was developed to test excised specimens while maintaining their in situ geometry and hydration. Using an analysis based on linear biphasic theory, the compressive modulus, hydraulic permeability, and isometric swelling pressure of the anulus fibrosus were determined and correlated with the tissue composition. The findings indicate that the anulus fibrosus is inhomogeneous, with regional and radial variations in both material properties and biochemical composition. The results of this study suggest that both structural and compositional factors may determine the mechanical behavior.

A mixed‐penalty finite element formulation of the linear biphasic theory for soft tissues

AbstractHydrated soft tissues of the human musculoskeletal system can be represented by a continuum theory of mixtures involving intrinsically incompressible solid and incompressible inviscid fluid phases. This paper describes the development of a mixed‐penalty formulation for this biphasic system and the application of the formulation to the development of an axisymmetric, six‐node, triangular finite element. In this formulation, the continuity equation of the mixture is replaced by a penalty form of this equation which is introduced along with the momentum equation and mechanical boundary condition for each phase into a weighted residual form. The resulting weak form is expressed in terms of the solid phase displacements (and velocities), fluid phase velocities and pressure. After interpolation, the pressure unknowns can be eliminated at the element level, and a first order coupled system of equations is obtained for the motion of the solid and fluid phases. The formulation is applied to a six‐node isoparametric element with a linear pressure field. The element performance is compared with that of the direct penalty form of the six‐node biphasic element in which the pressure is eliminated in the governing equations prior to construction of the weak form, and selective reduced integration is used on the penalty term. The mixed‐penalty formulation is found to be superior in terms of tendency to lock and sensitivity to mesh distortion. A number of example problems for which analytic solutions exist are used to validate the performance of the element.