Compressive properties of mouse articular cartilage determined in a novel micro-indentation test method and biphasic finite element model

The mechanical properties of articular cartilage serve as important measures of tissue function or degeneration, and are known to change significantly with osteoarthritis. Interest in small animal and mouse models of osteoarthritis has increased as studies reveal the importance of genetic background in determining predisposition to osteoarthritis. While indentation testing provides a method of determining cartilage mechanical properties in situ, it has been of limited value in studying mouse joints due to the relatively small size of the joint and thickness of the cartilage layer. In this study, we developed a micro-indentation testing system to determine the compressive and biphasic mechanical properties of cartilage in the small joints of the mouse. A nonlinear optimization program employing a genetic algorithm for parameter estimation, combined with a biphasic finite element model of the micro-indentation test, was developed to obtain the biphasic, compressive material properties of articular cartilage. The creep response and material properties of lateral tibial plateau cartilage were obtained for wild-type mouse knee joints, by the micro-indentation testing and optimization algorithm. The newly developed genetic algorithm was found to be efficient and accurate when used with the finite element simulations for nonlinear optimization to the experimental creep data. The biphasic mechanical properties of mouse cartilage in compression (average values: Young's modulus, 2.0 MPa; Poisson's ratio, 0.20; and hydraulic permeability, 1.1 x 10(-16) m4/N-s) were found to be of similar orders of magnitude as previous findings for other animal cartilages, including human, bovine, rat, and rabbit and demonstrate the utility of the new test methods. This study provides the first available data for biphasic compressive properties in mouse cartilage and suggests a promising method for detecting altered cartilage mechanics in small animal models of osteoarthritis.     

A finite element analysis of the indentation stress-relaxation response of linear biphasic articular cartilage.

The indentation problem of a thin layer of hydrated soft tissue such as cartilage or meniscus by a circular plane-ended indenter is investigated. The tissue is represented by a biphasic continuum model consisting of a solid phase (collagen and proteoglycan) and a fluid phase (interstitial water). A finite element formulation of the linear biphasic continuum equations is used to solve an axisymmetric approximation of the indentation problem. We consider stress-relaxation problems for which analytic solution is intractable; where the indenter is impermeable (solid) and/or when the interface between the indenter and tissue is perfectly adhesive. Thicknesses corresponding to a thin and thick specimen are considered to examine the effects of tissue thickness. The different flow, pressure, stress and strain fields which are predicted within the tissue, over time periods typically used in the mechanical testing of soft tissues, will be presented. Results are compared with the case of a porous free-draining indenter with a perfectly lubricated tissue-indenter interface, for which an analytic solution is available, to show the effects of friction at the tissue-indenter interface, and the effects of an impermeable indenter. While these effects are present for both thin and thick tissues, they are shown to be more significant for the thin tissue. We also examine the effects of the stiffness of the subchondral bone on the response of the soft tissue and demonstrate that the subchondral bone substrate can be modeled as a rigid, impermeable boundary. The effects of a curved tissue-subchondral bone interface, and the early time response are also studied. For physiologically reasonable levels of curvature, we will show that the curved tissue-subchrondal bone interface has negligible influence on the tissue response away from the interface. In addition, the short-time stress-relaxation responses of the tissue (e.g., at times less than 1s) demonstrate the essential role of the fluid phase in supporting the load applied to the tissue, and by extrapolation to shorter times characteristics of normal joint motion, suggest the essential role of a biphasic model in representing soft tissue behavior in joint response.     

Estimation of the Young's modulus of articular cartilage using an arthroscopic indentation instrument and ultrasonic measurement of tissue thickness

We evaluated whether the use of cartilage thickness measurement would improve the ability of the arthroscopic indentation technique to estimate the intrinsic stiffness of articular cartilage. First, cartilage thickness and ultrasound reflection from the surface of bovine humeral head were registered in situ using a high-frequency ultrasound probe. Subsequently, cartilage was indented in situ at the sites of the ultrasound measurements using arthroscopic instruments with plane-ended and spherical-ended indenters. Finally, full-thickness cartilage disks (n=30) were extracted from the indented sites (thickness=799-1654microm) and the equilibrium Young's modulus was determined with a material testing device in unconfined compression geometry. We applied analytical and numerical indentation models for the theoretical correction of experimental indentation measurements. An aspect-ratio (the ratio of indenter radius to cartilage thickness) correction improved the correlation of the indenter force with the equilibrium Young's modulus from r(2)=0.488 to r(2)=0.642-0.648 (n=30) for the plane-ended indenter (diameter=1.000mm, height=0.300mm) and from r(2)=0.654 to r(2)=0.684-0.692 (n=30) for the spherical-ended indenter (diameter=0.500mm, height=0.100mm), depending on the indentation model used for the correction. The linear correlation between the ultrasound reflection and the Young's modulus was r(2)=0.400 (n=30). These results suggest that with large indenters, knowledge of the cartilage thickness improves the reliability of the indentation measurements, especially in pathological situations where cartilage thickness may be significantly lower than normal. Ultrasound measurements also provide diagnostically important information about cartilage thickness as well as knowledge of the integrity of the superficial zone of cartilage.     

Biphasic finite element modeling of hydrated soft tissue contact using an augmented Lagrangian method

A study of biphasic soft tissues contact is fundamental to understanding the biomechanical behavior of human diarthrodial joints. To date, biphasic-biphasic contact has been developed for idealized geometries and not been accessible for more general geometries. In this paper a finite element formulation is developed for contact of biphasic tissues. The augmented Lagrangian method is used to enforce the continuity of contact traction and fluid pressure across the contact interface, and the resulting method is implemented in the commercial software COMSOL Multiphysics. The accuracy of the implementation is verified using 2D axisymmetric problems, including indentation with a flat-ended indenter, indentation with spherical-ended indenter, and contact of glenohumeral cartilage layers. The biphasic finite element contact formulation and its implementation are shown to be robust and able to handle physiologically relevant problems.     

Biphasic indentation of articular cartilage--I. Theoretical analysis

A mathematical solution has been obtained for the indentation creep and stress-relaxation behavior of articular cartilage where the tissue is modeled as a layer of linear KLM biphasic material of thickness h bonded to an impervious, rigid bony substrate. The circular (radius = a), plane-ended indenter is assumed to be rigid, porous, free-draining, and frictionless. Double Laplace and Hankel transform techniques were used to solve the partial differential equations. The transformed equations and boundary conditions yielded an integral equation of the Fredholm type which was analyzed asymptotically and solved numerically. Our asymptotic analyses showed that the linear KLM biphasic material behaves like an incompressible (v = 0.5) single-phase elastic solid at t = 0+; the instantaneous response of the material is governed by the shear modulus (mu s) of the solid matrix. The linear KLM biphasic material behaves like a compressible elastic solid with material properties defined by those of the solid matrix, i.e. (lambda s, mu s) or (mu s, v s) as t----infinity. The transient viscoelastic creep and stress-relaxation behavior, 0 less than t less than infinity, of this material is controlled by the frictional drag (which is inversely proportional to the permeability k) associated with the flow of the interstitial fluid through the porous-permeable solid matrix. For given values of the Poisson's ratio of the solid matrix v s and the aspect ratio a/h, where a is the radius of the indenter and h is the thickness of the layer, the creep behavior with respect to the dimensionless time H Akt/a2 is completely controlled by the load parameter P/2 mu sa2 and the stress relaxation behavior is completely controlled by the rate of compression parameter R0 = kH A/V0h where H A = lambda s + 2 mu s and the equilibrium strain u0/h. This mathematical solution may now be used to describe an indentation experiment on articular cartilage to determine the intrinsic material properties of the tissue, i.e. permeability k, and the elastic coefficients of the solid phase (lambda s, mu s) or (mu s, v s).     

Micro-indentation hardness studies on human bones.

Bones from different portions of human skeleton are polished and cut to suitable sizes and then subjected to micro-indentation at various loads using Vicker's diamond pyramidal indenter. The Vicker's hardness number is found to differ slightly from portion to portion of the skeleton. Interferometric studies of the indentation reveal that the applied stress of indentation causes a 'pile-up' of material near the corners of the pyramidal indents.     

Indentation testing of human articular cartilage: effects of probe tip geometry and indentation depth on intra-tissue strain

Experimental determination of intra-tissue deformation during clinically applicable rapid indentation testing would be useful for understanding indentation biomechanics and for designing safe indentation probes and protocols. The objectives of this study were to perform two-dimensional (2-D) indentation tests, using indenters and protocols that are analogous to those in clinically oriented probes, of normal adult-human articular cartilage in order to determine: (1) intra-tissue strain maps and regions of high strain magnitude, and (2) the effects on strain of indenter geometry (rectangular prismatic and cylindrical) and indentation depth (40-190 microm). Epifluorescence microscopy of samples undergoing indentation and subsequent video image correlation analysis allowed determination of strain maps. Regions of peak strain were near the "edges" of indenter contact with the cartilage surface, and the strain magnitude in these regions ranged from approximately 0.05 to approximately 0.30 in compression and shear, a range with known biological consequences. With increasing indentation displacement, strain magnitudes generally increased in all regions of the tissue. Compared to indentation using a rectangular prismatic tip, indentation with a cylindrical tip resulted in slightly higher peak strain magnitudes while influencing a smaller region of cartilage. These results may be used to refine clinical indenters and indentation protocols.     

An augmented Lagrangian finite element formulation for 3D contact of biphasic tissues

Biphasic contact analysis is essential to obtain a complete understanding of soft tissue biomechanics, and the importance of physiological structure on the joint biomechanics has long been recognised; however, up to date, there are no successful developments of biphasic finite element contact analysis for three-dimensional (3D) geometries of physiological joints. The aim of this study was to develop a finite element formulation for biphasic contact of 3D physiological joints. The augmented Lagrangian method was used to enforce the continuity of contact traction and fluid pressure across the contact interface. The biphasic contact method was implemented in the commercial software COMSOL Multiphysics 4.2^® (COMSOL, Inc., Burlington, MA). The accuracy of the implementation was verified using 3D biphasic contact problems, including indentation with a flat-ended indenter and contact of glenohumeral cartilage layers. The ability of the method to model multibody biphasic contact of physiological joints was proved by a 3D knee model. The 3D biphasic finite element contact method developed in this study can be used to study the biphasic behaviours of the physiological joints.     

Investigation into the biphasic properties of a hydrogel for use in a cushion form replacement joint.

A hydrogel with potential applications in the role of a cushion form replacement joint bearing surface material has been investigated. The material properties are required for further development and design studies and have not previously been quantified. Creep indentation experiments were therefore performed on samples of the hydrogel. The biphasic model developed by Mow and co-workers (Mak et al., 1987; Mow et al., 1989a) was used to curve-fit the experimental data to theoretical solutions in order to extract the three intrinsic biphasic material properties of the hydrogel (aggregate modulus, HA, Poisson's ratio, Vs, and permeability, k). Ranges of material properties were determined: aggregate modulus was calculated to be between 18.4 and 27.5 MPa, Poisson's ratio 0.0-0.307, and permeability 0.012-7.27 x 10(-17) m4/Ns. The hydrogel thus had a higher aggregate modulus than values published for natural normal articular cartilage, the Poisson's ratios were similar to articular cartilage, and finally the hydrogel was found to be less permeable than articular cartilage. The determination of these values will facilitate further numerical analysis of the stress distribution in a cushion form replacement joint.     

Comparison of the equilibrium response of articular cartilage in unconfined compression,confined compression and indentation

At mechanical equilibrium, articular cartilage is usually characterized as an isotropic elastic material with no interstitial fluid flow. In this study, the equilibrium properties (Young's modulus, aggregate modulus and Poisson's ratio) of bovine humeral, patellar and femoral cartilage specimens (n=26) were investigated using unconfined compression, confined compression, and indentation tests. Optical measurements of the Poisson's ratio of cartilage were also carried out. Mean values of the Young's modulus (assessed from the unconfined compression test) were 0.80+/-0.33, 0.57+/-0.17 and 0.31+/-0.18MPa and of the Poisson's ratio (assessed from the optical test) 0.15+/-0.06, 0.16+/-0.05 and 0.21+/-0.05 for humeral, patellar, and femoral cartilages, respectively. The indentation tests showed 30-79% (p<0.01) higher Young's modulus values than the unconfined compression tests. In indentation, values of the Young's modulus were independent of the indenter diameter only in the humeral cartilage. The mean values of the Poisson's ratio, obtained indirectly using the mathematical relation between the Young's modulus and the aggregate modulus in isotropic material, were 0.16+/-0.06, 0.21+/-0.05, and 0.26+/-0.08 for humeral, patellar, and femoral cartilages, respectively. We conclude that the values of the elastic parameters of the cartilage are dependent on the measurement technique in use. Based on the similar values of Poisson's ratios, as determined directly or indirectly, the equilibrium response of articular cartilage under unconfined and confined compression is satisfactorily described by the isotropic elastic model. However, values of the isotropic Young's modulus obtained from the in situ indentation tests are higher than those obtained from the in vitro unconfined or confined compression tests and may depend on the indenter size in use.     

Experimental and biphasic FEM determinations of the material properties and hydraulic permeability of the meniscus in tension

Tensile tests and biphasic finite element modeling were used to determine a set of transversely isotropic properties for the meniscus, including the hydraulic permeability coefficients and solid matrix properties. Stress-relaxation tests were conducted on planar samples of canine meniscus samples of different orientations, and the solid matrix properties were determined from equilibrium data. A 3-D linear biphasic and tranversely isotropic finite element model was developed to model the stress-relaxation behavior of the samples in tension, and optimization was used to determine the permeability coefficients, k1 and k2, governing fluid flow parallel and perpendicular to the collagen fibers, respectively. The collagen fibrillar orientation was observed to have an effect on the Young's moduli (E1=67.8 MPa, E2=11.1 MPa) and Poisson's ratios (v12=2.13, v21 =1.50, v23=1.02). However, a significant effect of anisotropy on permeability was not detected (k1 =0.09x10(-16) m4/Ns, k2=0.10x10(-16) m4/Ns). The low permeability values determined in this study provide insight into the extent of fluid pressurization in the meniscus and will impact modeling predictions of load support in the meniscus.     

A dual optimization method for the material parameter identification of a biphasic poroviscoelastic hydrogel: Potential application to hypercompliant soft tissues

A dual-indentation creep and stress relaxation methodology was developed and validated for the material characterization of very soft biological tissue within the framework of the biphasic poroviscoelastic (BPVE) constitutive model. Agarose hydrogel, a generic porous medium with mobile fluid, served as a mechanical tissue analogue for validation of the experimental procedure. Indentation creep and stress relaxation tests with a solid plane-ended cylindrical indenter were performed at identical sites on a gel sample with dimensions large enough with respect to indenter size in order to satisfy an infinite layer assumption. A finite element (FE) formulation coupled to a global optimization algorithm was utilized to simultaneously curve-fit the creep and stress relaxation data and extract the BPVE model parameters for the agarose gel. A numerical analysis with artificial data was conducted to validate the uniqueness of the computational procedure. The BPVE model was able to successfully cross-predict both creep and stress relaxation behavior for each pair of experiments with a single unique set of material parameters. Optimized elastic moduli were consistent with those reported in the literature for agarose gel. With the incorporation of appropriately-sized indenters to satisfy more stringent geometric constraints, this simple yet powerful indentation methodology can provide a straightforward means by which to obtain the BPVE model parameters of biological soft tissues that are difficult to manipulate (such as brain and adipose) while maintaining a realistic in situ loading environment.     

Robust strategies for automated AFM force curve analysis--I. Non-adhesive indentation of soft, inhomogeneous materials

The atomic force microscope (AFM) has found wide applicability as a nanoindentation tool to measure local elastic properties of soft materials. An automated approach to the processing of AFM indentation data, namely, the extraction of Young's modulus, is essential to realizing the high-throughput potential of the instrument as an elasticity probe for typical soft materials that exhibit inhomogeneity at microscopic scales. This paper focuses on Hertzian analysis techniques, which are applicable to linear elastic indentation. We compiled a series of synergistic strategies into an algorithm that overcomes many of the complications that have previously impeded efforts to automate the fitting of contact mechanics models to indentation data. AFM raster data sets containing up to 1024 individual force-displacement curves and macroscopic compression data were obtained from testing polyvinyl alcohol gels of known composition. Local elastic properties of tissue-engineered cartilage were also measured by the AFM. All AFM data sets were processed using customized software based on the algorithm, and the extracted values of Young's modulus were compared to those obtained by macroscopic testing. Accuracy of the technique was verified by the good agreement between values of Young's modulus obtained by AFM and by direct compression of the synthetic gels. Validation of robustness was achieved by successfully fitting the vastly different types of force curves generated from the indentation of tissue-engineered cartilage. For AFM indentation data that are amenable to Hertzian analysis, the method presented here minimizes subjectivity in preprocessing and allows for improved consistency and minimized user intervention. Automated, large-scale analysis of indentation data holds tremendous potential in bioengineering applications, such as high-resolution elasticity mapping of natural and artificial tissues.     

A cross-validation of the biphasic poroviscoelastic model of articular cartilage in unconfined compression, indentation, and confined compression

The biphasic poroviscoelastic (BPVE) model was curve fit to the simultaneous relaxation of reaction force and lateral displacement exhibited by articular cartilage in unconfined compression (n=18). Model predictions were also made for the relaxation observed in reaction force during indentation with a porous plane-ended metal indenter (n=4), indentation with a nonporous plane ended metal indenter (n=4), and during confined compression (n=4). Each prediction was made using material parameters resulting from curve fits of the unconfined compression response of the same tissue. The BPVE model was able to account for both the reaction force and the lateral displacement during unconfined compression very well. Furthermore, model predictions for both indentation and confined compression also followed the experimental data well. These results provide substantial evidence for the efficacy of the biphasic poroviscoelastic model for articular cartilage, as no successful cross-validation of a model simulation has been demonstrated using other mathematical models.     

Extended Latanoprost Release from Commercial Contact Lenses: In Vitro Studies Using Corneal Models

In this study, we compared, for the first time, the release of a 432 kDa prostaglandin analogue drug, Latanoprost, from commercially available contact lenses using in vitro models with corneal epithelial cells. Conventional polyHEMA-based and silicone hydrogel soft contact lenses were soaked in drug solution ( solution in phosphate buffered saline). The drug release from the contact lens material and its diffusion through three in vitro models was studied. The three in vitro models consisted of a polyethylene terephthalate (PET) membrane without corneal epithelial cells, a PET membrane with a monolayer of human corneal epithelial cells (HCEC), and a PET membrane with stratified HCEC. In the cell-based in vitro corneal epithelium models, a zero order release was obtained with the silicone hydrogel materials (linear for the duration of the experiment) whereby, after 48 hours, between 4 to 6 of latanoprost (an amount well within the range of the prescribed daily dose for glaucoma patients) was released. In the absence of cells, a significantly lower amount of drug, between 0.3 to 0.5 , was released, (). The difference observed in release from the hydrogel lens materials in the presence and absence of cells emphasizes the importance of using an in vitro corneal model that is more representative of the physiological conditions in the eye to more adequately characterize ophthalmic drug delivery materials. Our results demonstrate how in vitro models with corneal epithelial cells may allow better prediction of in vivo release. It also highlights the potential of drug-soaked silicone hydrogel contact lens materials for drug delivery purposes.     

Acanthamoeba spp.: Efficacy of Bioclen FR One Step®, a povidone-iodine based system for the disinfection of contact lenses

Pathogenic strains of Acanthamoeba are causative agents of a serious sight-threatening infection of the eye known as Acanthamoeba keratitis. The prevalence of this infection has risen in the past 20 years, mainly due to the increase in number of contact lens wearers. Bioclen FR One Step® (Ophtecs Corporation) is the only available povidone-iodine based system for the disinfection of silicone hydrogel lenses and soft contact lenses on the market. Bioclen FR has been proven to be highly effective against bacteria and fungi that can cause problems for contact lens users. In this study, Bioclen FR One Step® was tested against three clinical Acanthamoeba isolates from contact lens cases. The results demonstrated that the tested Acanthamoeba clinical strains were sensitive to Bioclen FR One Step®.     

An augmented lagrangian method for sliding contact of soft tissue

Despite the importance of sliding contact in diarthrodial joints, only a limited number of studies have addressed this type of problem, with the result that the mechanical behavior of articular cartilage in daily life remains poorly understood. In this paper, a finite element formulation is developed for the sliding contact of biphasic soft tissues. The augmented Lagrangian method is used to enforce the continuity of contact traction and fluid pressure across the contact interface. The resulting method is implemented in the commercial software COMSOL Multiphysics. The accuracy of the new implementation is verified using an example problem of sliding contact between a rigid, impermeable indenter and a cartilage layer for which analytical solutions have been obtained. The new implementation's capability to handle a complex loading regime is verified by modeling plowing tests of the temporomandibular joint (TMJ) disc.     

Robust strategies for automated AFM force curve analysis-II: adhesion-influenced indentation of soft, elastic materials

In the first of this two-part discourse on the extraction of elastic properties from atomic force microscopy (AFM) data, a scheme for automating the analysis of force-distance curves was introduced and experimentally validated for the Hertzian (i.e., linearly elastic and noninteractive probe-sample pairs) indentation of soft, inhomogeneous materials. In the presence of probe-sample adhesive interactions, which are common especially during retraction of the rigid tip from soft materials, the Hertzian models are no longer adequate. A number of theories (e.g., Johnson-Kendall-Roberts and Derjaguin-Muller-Toporov), covering the full range of sample compliance relative to adhesive force and tip radius, are available for analysis of such data. We incorporated Pietrement and Troyon's approximation (2000, "General Equations Describing Elastic Indentation Depth and Normal Contact Stiffness Versus Load," J. Colloid Interface Sci., 226(1), pp. 166-171) of the Maugis-Dugdale model into the automated procedure. The scheme developed for the processing of Hertzian data was extended to allow for adhesive contact by applying the Pietrement-Troyon equation. Retraction force-displacement data from the indentation of polyvinyl alcohol gels were processed using the customized software. Many of the retraction curves exhibited strong adhesive interactions that were absent in extension. We compared the values of Young's modulus extracted from the retraction data to the values obtained from the extension data and from macroscopic uniaxial compression tests. Application of adhesive contact models and the automated scheme to the retraction curves yielded average values of Young's modulus close to those obtained with Hertzian models for the extension curves. The Pietrement-Troyon equation provided a good fit to the data as indicated by small values of the mean-square error. The Maugis-Dugdale theory is capable of accurately modeling adhesive contact between a rigid spherical indenter and a soft, elastic sample. Pietrement and Troyon's empirical equation greatly simplifies the theory and renders it compatible with the general automation strategies that we developed for Hertzian analysis. Our comprehensive algorithm for automated extraction of Young's moduli from AFM indentation data has been expanded to recognize the presence of either adhesive or Hertzian behavior and apply the appropriate contact model.     

An analysis of the unconfined compression of articular cartilage

Analytical solutions have been obtained for the internal deformation and fluid-flow fields and the externally observable creep, stress relaxation, and constant strain-rate behaviors which occur during the unconfined compression of a cylindrical specimen of a fluid-filled, porous, elastic solid, such as articular cartilage, between smooth, impermeable plates. Instantaneously, the "biphasic" continuum deforms without change in volume and behaves like an incompressible elastic solid of the same shear modulus. Radial fluid flow then allows the internal fluid pressure to equilibrate with the external environment. The equilibrium response is controlled by the Young's modulus and Poisson's ratio of the solid matrix.     

Effect of Staphylococcus epidermidis on Hydrogel Contact Lens Retention on the Rabbit Eye

A slime-producing isolate of Staphylococcus epidermidis attached to FDA Group II hydrogel contact lenses persisted on rabbit eyes for up to 14 days, but except for minor redness of the eye no other effect was observed. Eye flora of eight representative New Zealand White rabbits included four different species of Staphylococcus including S. epidermidis and one species of Micrococcus, none of which produced overtly obvious biofilms. The slime-producing strain of S. epidermidis adhered more effectively to lenses than a non-slime-producing strain, and lenses challenged with the slime-producing strain remained on the rabbit eye for longer time periods than those with a non-slime-producing strain. Bacteria associated with the contact lens may affect the retention of the lens on the rabbit cornea during experimental studies.