Solid Matrix Stress Research Articles

Failure in articular cartilage: Finite element predictions of stress, strain, and pressure under micro-indentation induced fracture

Articular cartilage is found at the distal end of long bones and is responsible for assisting in joint articulation. While articular cartilage has remarkable resistance to failure, once initially damaged, degeneration is nearly irreversible. Thus, understanding damage initiation is important. There are a few proposed mechanisms for articular cartilage failure initiation: (A) a single collagen fibril stress-based regime; (B) a rate-dependent regime captured by brittle failure at slow displacement rates (SDR) and ductile failure at fast displacement rates (FDR); and (C) a rate-dependent regime where failure is governed by pressurization fragmentation at SDR and governed by strain at FDR. The objective of this study was to use finite element (FE) models to provide evidence to support or refute these proposed failure mechanisms. Models were developed of microfracture experiments that investigated osmolarity (hypo-osmolar, normal osmolarity, and hyper-osmolar) and displacement rate (FDR and SDR) effects. Cartilage was modeled with a neo-Hookean ground matrix, strain-dependent permeability, nonlinear fibril reinforcement with viscoelastic fibril terms, and Donnan equilibrium swelling. Total stress, solid matrix stress, Lagrange strain, and fluid pressure were determined under the indenter tip at the moment of microfracture. Results indicated significant rate dependence across multiple outputs, which does not support (A) a single failure regime. Larger solid and fluid pressures at FDR than SDR did not support (C) a rate-dependent regime split by pressurization at SDR and strain at FDR. Consistent solid shear stresses at SDR and consistent third principal solid stresses at FDR support (B) the ductile-brittle failure regime. These findings help to shed light on the underlying mechanisms of articular cartilage failure, which have implications for the development of osteoarthritis.

Modeling biphasic hydrogels under spherical indentation: Application to soft tissues

The mechanical characterization of hydrated soft materials, such as tissues and hydrogels, plays an important role in the growing bioengineering and medical fields. Indentation techniques have been successfully employed to determine the mechanical properties of biomaterials using elastic, viscoelastic or poroviscoelastic theories partly because of their inherent simplicity and partly because the material configuration for in vivo or in vitro mechanical testing does not permit other experimental methods. In this study, we examine the mechanical behavior of gelatin hydrogels through two types of indentation tests, namely indentation-load relaxation and quasistatic indentation. The material properties extracted from the rheometer-stress relaxation experiment are used in a biphasic poroviscoelastic (BPVE) material model to simulate the indentation tests using finite element analysis (FEA). Parametric studies using FEA of the indentation-load relaxation experiment are used to estimate the hydraulic permeability and the largest time constant accurately. Force response predicted from FEA of indentation experiments using BPE and BPVE models demonstrate the significant role played by the flow-independent solid matrix relaxation behavior of soft polymeric media emphasizing the need to use a BPVE model. Investigation of the spatiotemporal variations in the pore pressure and solid matrix stress within the hydrogel and near the indenter surface provides clues to the differences seen in the estimated relaxation modulus from various tests (rheometer and indentation). From the studies on indentation testing of gelatin hydrogels, the relaxed shear modulus is an intrinsic material property of soft-tissues and hydrogels that is found to be the most appropriate material property for comparison with those obtained using other experiments.

Biphasic Analysis of Cartilage Stresses in the Patellofemoral Joint.

The objective of this study was to examine the state of stress within the solid matrix of articular cartilage in the patellofemoral joint, using anatomically faithful biphasic models of the articular layers, with the joint subjected to physiologic muscle force magnitudes. Finite element models of five joints were created from human cadaver knees. Biphasic sliding contact analyses were performed using FEBio software to analyze the response of the joint from 30 to 60 degrees of knee flexion. Results demonstrated that the collagen matrix always sustains tensile stresses, despite the fact that the articular layers are loaded in compression. The principal direction of maximum solid stresses was consistent with the known orientation of collagen fibrils in cartilage. The magnitudes of these tensile stresses under muscle forces representative of activities of daily living were well below tensile failure stresses reported in the prior literature. Results also hinted that solid matrix stresses were higher in the patellar versus femoral superficial zone. These anatomically correct finite element models predicted outcomes consistent with our understanding of structure-function relationships in articular cartilage, while also producing solid matrix stress estimates not observable from experiments alone, yet highly relevant to our understanding of tissue degeneration.

Multiscale computational model of fluid flow and matrix deformation in decellularized liver

Currently little is known about the biomechanical environment in decellularized tissue. The goal of this research is to quantify the mechanical microenvironment in decellularized liver, for varying organ-scale perfusion conditions, using a combined experimental/computational approach. Needle-guided ultra-miniature pressure sensors were inserted into liver tissue to measure parenchymal fluid pressure ex-situ in portal vein-perfused native (n=5) and decellularized (n=7) ferret liver, for flow rates from 3–12mL/min. Pressures were also recorded at the inlet near the portal vein cannula to estimate total vascular resistance of the specimens. Experimental results were fit to a multiscale computational model to simulate perfusion conditions inside native versus decellularized livers for four experimental flow rates. The multiscale model consists of two parts: an organ-scale electrical analog model of liver hemodynamics and a tissue-scale model that predicts pore fluid pressure, pore fluid velocity, and solid matrix stress and deformation throughout the 3D hepatic lobule. Distinct models were created for native versus decellularized liver. Results show that vascular resistance decreases by 82% as a result of decellularization. The hydraulic conductivity of the decellularized liver lobule, a measure of tissue permeability, was 5.6 times that of native liver. For the four flow rates studied, mean fluid pressures in the decellularized lobule were 0.6–2.4mmHg, mean fluid velocities were 211–767μm/s, and average solid matrix principal strains were 1.7–6.1%. In the future this modeling platform can be used to guide the optimization of perfusion seeding and conditioning strategies for decellularized scaffolds in liver bioengineering.

Modeling of transient flow through a viscoelastic preparative chromatography packing

The common method for purification of macromolecular bioproducts is preparative packed-bed chromatography using polymer-based, compressible, viscoelastic resins. Because of a downstream processing bottleneck, the chromatography equipment is often operated at its hydrodynamic limit. In this case, the resins may exhibit a complex behavior which results in compression-relaxation hystereses. Up to now, no modeling approach of transient flow through a chromatography packing has been made considering the viscoelasticity of the resins. The aim of the present work was to develop a novel model and compare model calculations with experimental data of two agarose-based resins. Fluid flow and bed permeability were modeled by Darcy's law and the Kozeny-Carman equation, respectively. Fluid flow was coupled to solid matrix stress via an axial force balance and a continuity equation of a deformable packing. Viscoelasticity was considered according to a Kelvin-Voigt material. The coupled equations were solved with a finite difference scheme using a deformable mesh. The model boundary conditions were preset transient pressure drop functions which resemble simulated load/elution/equilibration cycles. Calculations using a homogeneous model (assuming constant variables along the column height) gave a fair agreement with experimental data with regard to predicted flow rate, bed height, and compression-relaxation hysteresis for symmetric as well as asymmetric pressure drop functions. Calculations using an inhomogeneous model gave profiles of the bed porosity as a function of the bed height. In addition, the influence of medium wall support and intraparticle porosity was illustrated. The inhomogeneous model provides insights that so far are not easily experimentally accessible.

Importance of Collagen Orientation and Depth-Dependent Fixed Charge Densities of Cartilage on Mechanical Behavior of Chondrocytes

The collagen network and proteoglycan matrix of articular cartilage are thought to play an important role in controlling the stresses and strains in and around chondrocytes, in regulating the biosynthesis of the solid matrix, and consequently in maintaining the health of diarthrodial joints. Understanding the detailed effects of the mechanical environment of chondrocytes on cell behavior is therefore essential for the study of the development, adaptation, and degeneration of articular cartilage. Recent progress in macroscopic models has improved our understanding of depth-dependent properties of cartilage. However, none of the previous works considered the effect of realistic collagen orientation or depth-dependent negative charges in microscopic models of chondrocyte mechanics. The aim of this study was to investigate the effects of the collagen network and fixed charge densities of cartilage on the mechanical environment of the chondrocytes in a depth-dependent manner. We developed an anisotropic, inhomogeneous, microstructural fibril-reinforced finite element model of articular cartilage for application in unconfined compression. The model consisted of the extracellular matrix and chondrocytes located in the superficial, middle, and deep zones. Chondrocytes were surrounded by a pericellular matrix and were assumed spherical prior to tissue swelling and load application. Material properties of the chondrocytes, pericellular matrix, and extracellular matrix were obtained from the literature. The loading protocol included a free swelling step followed by a stress-relaxation step. Results from traditional isotropic and transversely isotropic biphasic models were used for comparison with predictions from the current model. In the superficial zone, cell shapes changed from rounded to elliptic after free swelling. The stresses and strains as well as fluid flow in cells were greatly affected by the modulus of the collagen network. The fixed charge density of the chondrocytes, pericellular matrix, and extracellular matrix primarily affected the aspect ratios (height/width) and the solid matrix stresses of cells. The mechanical responses of the cells were strongly location and time dependent. The current model highlights that the collagen orientation and the depth-dependent negative fixed charge densities of articular cartilage have a great effect in modulating the mechanical environment in the vicinity of chondrocytes, and it provides an important improvement over earlier models in describing the possible pathways from loading of articular cartilage to the mechanical and biological responses of chondrocytes.

Analysis of articular cartilage as a composite using nonlinear membrane elements for collagen fibrils

To develop a composite fibre-reinforced model of the cartilage, membrane shell elements were introduced to represent collagen fibrils reinforcing the isotropic porous solid matrix filled with fluid. Nonlinear stress–strain curve of pure collagen fibres and collagen volume fraction were explicitly presented in the formulation of these membrane elements. In this composite model, in accordance with tissue structure, the matrix and fibril membrane network experienced dissimilar stresses despite identical strains in the fibre directions. Different unconfined compression and indentation case studies were performed to determine the distinct role of membrane collagen fibrils in nonlinear poroelastic mechanics of articular cartilage. The importance of nonlinear fibril membrane elements in the tissue relaxation response as well as in temporal and spatial variations of pore pressure and solid matrix stresses was demonstrated. By individual adjustments of the collagen volume fraction and collagen mechanical properties, the model allows for the simulation of alterations in the fibril network structure of the tissue towards modelling damage processes or repair attempts. The current model, which is based on a physiological description of the tissue structure, is promising in improvement of our understanding of the cartilage pathomechanics.

Simultaneous Measurement of Interstitial Fluid Pressure and Load in Rat Skin After Strain Application In Vitro

Skin provides the flexible, protective covering of the body. It consists of a network of fibrous proteins embedded in a viscoelastic gel. Theoretical models of soft tissue demonstrate that behavior of such systems is strongly influenced by the relationship between the interstitial fluid pressure (P(if)) and solid matrix stress. A microtensiometer for loading skin uniaxially in vitro was, therefore, developed and used in conjunction with the established servo-null micropipette technique to measure P(if). Dorsal rat skin specimens were preloaded to 100 mN, where P(if) was 2.3 +/- 1.3 mmHg (mean +/- SE, n = 12) above ambient, and then strained by 4%. Load instantaneously increased and the subsequent decay was described by the function, F(t) = F(1)[1-C(f)Ln(t)]. F(1), related to the instantaneous elasticity, was 272 +/- 42 mN (n = 12) while, C(f) was 0.0894 +/- 0.0026 [Ln(s)](-1) (n = 12). A similar function P(t) = P(s)(1) x [1-C(ps)Ln(t)], where P(s)(1) = 27 +/- 5 mmHg and C(ps) = 0.1274 +/- 0.0097 [Ln(s)](-1) (n = 12) fitted the decay of P(if) after 20 s with a residual > or = 0.82, though, P(if) fell more rapidly over the initial 10 s. P(if) and stress can be measured simultaneously with the apparatus, though more precise determination of the depth at which pressure is measured is required for quantitative comparison of the magnitude of these two parameters.

The acetabular labrum seal: a poroelastic finite element model

Objective. The aim of the study is to investigate the labrum’s ability to seal a pressurised layer of synovial fluid within the joint, and to study the influence of this sealing mechanism on cartilage deformation, interstitial fluid pressure and collagen solid matrix stresses. Background. Cartilage degeneration has been observed in conjunction with labrum pathology. However, little is known about the function of the labrum. Experimental observations have been reported which are consistent with a sealing function of the labrum. Methods. The model was an axisymmetric geometric approximation of the acetabular and femoral cartilage layers and the surrounding labrum. A poroelastic formulation was used to account for the solid and fluid components of these hydrated tissues. A sensitivity analysis of the labrum material properties was carried out. Results. With a compressive load of 1200 N applied across the joint model, the labrum could seal a layer of pressurised fluid between the femur and acetabulum, thus preventing contact of the articulating surfaces. With this sealing effect, loads were transferred across the joint predominantly by uniform pressurisation of the interstitial fluid of the cartilage layers. In the absence of this sealing, strains within the solid matrix of the cartilage layers were higher (e.g. 20% vs. 3%). Conclusions. The labrum can seal against fluid expression from the joint space. This sealing function protects the cartilage layers of the hip. Relevance Current treatments for labrum damage and early arthrosis may compromise the sealing function of the labrum. With continued study of the function and importance of the labrum, new surgical repair strategies can be developed to maintain the overall function of the hip joint.

Computation of solid matrix stresses in cartilage of incongruous joints using a biphasic contact finite element formulation

Computation of solid matrix stresses in cartilage of incongruous joints using a biphasic contact finite element formulation

The effects of a subchondral bone perforation on the load support mechanism within articular cartilage

Impact loading of diarthrodial joints of skeletally mature animals can crack the zone of calcified cartilage and subchondral bone (ZCC-SB), often without immediate disruption to the joint surface. A common sequelae to transarticular impact is the development of osteoarthritis (OA) in the traumatized joint. An understanding of how and why damage to cartilage occurs and progresses in deep regions of cartilage would therefore increase our understanding of the etiology of this type of traumatic OA. In this study, a 2D linear biphasic finite element analysis was performed to determine articular cartilage deformations, and interstitial fluid flow and pressurization, under both uniform and non-uniform hydrodynamic pressure distributions. The crack in the ZCC-SB was modelled by a “moderate-size” perforation at which there is no load support or fluid pressurization (i.e. a free surface under a free-draining condition). The effects of such a defect on the cartilage solid matrix stresses and strains, and interstitial fluid flows and pressures have been calculated. Results show that in normal cartilage, consistent with our previous contact analysis (G.A. Ateshian, W.M. Lai, W.B. Zhu and V.C. Mow, A biphasic model for contact in diarthrodial joints, Adv. Bioeng., ASME, 22 (1992) 191–194), interstitial fluid pressurization plays a dominant role in providing support when cartilage is loaded. Hence, normally, the solid matrix of cartilage is shielded from the high stresses of joint loading. In cartilage with a perforation in the ZCC-SB, however, the solid matrix stresses and strains are significantly increased as a consequence of a diminished fluid pressurization in the region of the defect. Thus, to maintain local equilibrium, there is a transfer of internal load support from interstitial fluid pressure to solid matrix stress defeating the normal stress-shielding effect. This stress transfer suggests a mechanism for progressive solid matrix damage, thus increasing the likelihood of cartilage failure following a single episode of impact.

The biphasic poroviscoelastic behavior of articular cartilage: Role of the surface zone in governing the compressive behavior

Surface fibrillation of articular cartilage is an early sign of degenerative changes in the development of osteoarthritis. To assess the influence of the surface zone on the viscoelastic properties of cartilage under compressive loading, wer prepared osteochondral plugs from skeletally mature steers, with and without the surface zone of articular cartilage, for study in the confined compression creep experiment. The relative contributions of two viscoelastic mechanisms, i.e. a flow-independent mechanism [Hayes and Bodine, J. Biomechanics 11, 407–419 (1978)], and a flow-dependent mechanism [Mow et al. J. biomech. Engng 102, 73–84 (1980)], to the compressive creep response of these two types of specimens were determined using the biphasic poroviscoelastic theory proposed by Mak. [ J. Biomechanics 20, 703–714 (1986)]. From the experimental results and the biphasic poroviscoelastic theory, we found that frictional drag associated with interstitial fluid flow and fluid pressurization are the dominant mechanisms of load support in the intact specimens, i.e. the flow-dependent mechanisms alone were sufficient to describe normal articular cartilage compressive creep behavior. For specimens with the surface removed, we found an increased creep rate which was derived from an increased tissue permeability, as well as significant changes in the flow-independent parameters of the viscoelastic solid matrix. From these tissue properties and the biphasic poroviscoelastic theory, we determined that the flow-dependent mechanisms of load support, i.e. frictional drag and fluid pressurization, were greatly diminished in cartilage without the articular surface. Calculations based upon these material parameters show that for specimens with the surface zone removed, the cartilage solid matrix became more highly loaded during the early stages of creep. This suggests that an important function of the articular surface is to provide for a low fluid permeability, and thereby serve to restrict fluid exudation and increase interstitial fluid pressurization. Thus, it is likely that with increasing severity of damage to the articular surface, load support in cartilage under compression shifts from the flow-dependent modes of fluid drag and pressurization to increased solid matrix stress. This suggests that it is important to maintain the integrity of the articular surface in preserving normal compressive behavior of the tissue and normal load carriage in the joint.